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Spacetimes with semantics II (supplement)∗
On the scaling of functional spaces,
from smart cities to cloud computing
Mark Burgess
February 19, 2016
Abstract
The study of spacetime, and its role in understanding functional systems has received little attention
in information science. Recent work, on the origin of universal scaling in cities and biological systems,
provides an intriguing insight into the functional use of space, and its measurable effects. Cities are
large information systems, with many similarities to other technological infrastructures, so the results
shed new light indirectly on the scaling the expected behaviour of smart pervasive infrastructures and
the communities that make use of them.
Using promise theory, I derive and extend the scaling laws for cities to expose what may be extrapo-
lated to technological systems. From the promise model, I propose an explanation for some anomalous
exponents in the original work, and discuss what changes may be expected due to technological ad-
vancement.
Contents
1 Introduction 2
2 Bettencourt’s model of scaling in cities summarized 3
2.1 Scalingphenomenology................................... 3
2.2 Definitions.......................................... 4
2.3 Outlineoftheapproach................................... 4
2.4 Explicit and implicit assumptions behind a mean field model . . . . . . . . . . . . . . . 5
2.5 The fraction of volume that is sparse infrastructure network . . . . . . . . . . . . . . . . 7
2.6 Inflating the community: sustainable city volume at steady state . . . . . . . . . . . . . 8
2.7 Scaling of spatial infrastructure and output yields . . . . . . . . . . . . . . . . . . . . . 9
2.8 Remarks about the calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.9 Does innovation characterize superlinearity? . . . . . . . . . . . . . . . . . . . . . . . . 12
2.10 Gunther’s ‘Universal Scaling Law’ and spacetime involvement in scaling . . . . . . . . . 13
∗These notes are a continuation my series on semantic spacetimes. This document is inspired by the studies of coarse-grained
universal scaling in cities [1–3], and a comparison with models developed over the past decade or two on information systems,
e.g. [4, 5]. As IT systems grow in scale, is natural to expect a bridge between the behaviours of cities, software networks, and other
functionally ‘smart’ spaces, and one hopes for a better understanding of pervasive information technology in social contexts. Work
on this, from the low level viewpoint, has already begin in [6, 7]. I want to show how these views relate.
1
3 A promise theory derivation of the city model, and beyond 16
3.1 Brief recap’ of promise theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2 Agents and super-agents, long and short range interactions . . . . . . . . . . . . . . . . 17
3.3 Promise networks: functional interactions . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.4 Forces that condense a cluster . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.5 Promise networks that percolate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.6 Scaling of roles and interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.6.1 Linear consumption by independent agents . . . . . . . . . . . . . . . . . . . . 23
3.6.2 Sublinear economy of scale, and spacetime involvement . . . . . . . . . . . . . 23
3.6.3 Deriving Metcalfe’s law from promise networks: the importance of conditional
dependency..................................... 24
3.7 Conditional dependency as an explanation for multiple superlinearity classes . . . . . . . 25
3.8 Service discovery of dependencies in a network . . . . . . . . . . . . . . . . . . . . . . 28
4 Remarks on technological infrastructure and collaborative networks 31
4.1 Online communities versus infrastructure clusters . . . . . . . . . . . . . . . . . . . . . 31
4.2 The productivity of agents in human-machine systems . . . . . . . . . . . . . . . . . . . 33
4.3 Separation of long and short range interactions (modules) . . . . . . . . . . . . . . . . . 34
4.4 Semantic separation versus dynamical separation (silos) . . . . . . . . . . . . . . . . . . 34
4.5 The mobility of agents in human-machine communities . . . . . . . . . . . . . . . . . . 35
4.6 Scaling of voluntary cooperation versus imposition . . . . . . . . . . . . . . . . . . . . 35
4.7 The rate of change in the system and utilization . . . . . . . . . . . . . . . . . . . . . . 35
5 Summary 36
A The value of promises 40
B Defining the edge of a city or community by membership promises 41
C The phase of a community: mobility of agents, and interaction catalysts 42
D Effective power law scaling from Amdahl’s and Gunther’s law 43
1 Introduction
Two recent works have shed light on the role of infrastructure in the scaling of functional systems. The
observation of universal, scale free behaviours, both in observed data, and in a ‘mean field’ description of
cities [1,2], and the observation, in biology, of metabolic scaling laws for organisms, following a century
of observation [8]. Both of these were explained from the behaviours of their internal networks.
Cities are an example of collaborative community networks, where humans and technology mix
within a semantically (i.e. functionally) rich space, equipped with infrastructure. One may ask how
cities differ from apparently similar communities across different scales, including tribes, collectives,
companies, software installations, and even countries. Understanding the dominant processes that make
these shared environments smart, creative, and productive, is a worthy investment, given how 21st century
life relies on them so much for its success and survival.
In this note, I review and build on Bettencourt’s model of cities [1], and discuss its implications for
pervasive information technology (IT). By building on the lessons of cities, I hope to to foster a better
understanding of a broader range of functional systems, especially in information technology, while trying
to preserve the simplicity of Bettencourt’s arguments. I begin by summarizing an interpretation that lays
2
the foundations for a more microscopic formulation of the model, using promise theory [9]. The latter
may be used to relate outcomes to intentions and mechanisms, in a way that respects the idea of scaling.
2 Bettencourt’s model of scaling in cities summarized
This section is a review (and trivial generalization) of the city scaling arguments, and data used by Bet-
tencourt and collaborators at the Santa Fe Institute (SFI) [1–3], in a form suitable for comparison with
other work. In section 3, I propose a deeper justification for the model, with some embellishments. Some
interpretations may be my own.
2.1 Scaling phenomenology
Measurable attributes, of finite dynamical systems, typically scale in proportion to some measure of their
size. Size may refer to the number of agents N(persons) in the system, or per unit volume Vof the
system, and Nand Vmay or may not be related. This is a point of view that is a basic tool of analysis in
physical systems. For cities, Nis used as the scaling variable. Across an ensemble of many systems of
different size, the measurements one obtains may scale in three broad ways (see figure 1):
(a) (b) (c)
Figure 1: Scaling of a square quantity, relative to the circular system size (a) sublinear, (b) linear, (c) superlinear. If
an cost is superlinear, to applies a braking force on city size.
1. Sublinear scaling of quantities qof the infrastructure machinery q∝Nβ <1. This indicates
economies of scale, because, as the system size grows, the cost becomes relatively cheaper. In
cities, it is observed to apply to the transport networks that animate the system (arterial systems,
roads, cables, petrol stations, etc).
2. Linear scaling, simply proportional to the number q∝Nβ=1. In cities, this seems to apply to
direct consumption of goods and resources per capita (jobs, houses, water consumption, etc).
3. Superlinear yields of produce or ‘output’, where q∝Nβ >1, which is driven by interactivity be-
tween the parts of the system and its consequences (wages, rents, patents, crime, disease, GDP). If
a process rate is superlinear, then the corresponding time for the process to run will be sublinear,
and vice versa.
It is of particular interest when these patterns seem to apply across such a broad range of scales. This
suggests some emergent universality, whose origin and mechanism is worth understanding.
As part of a protracted project to uncover the behaviours of cities or urban metropolitan districts,
Bettencourt has proposed a mean field model to explain observed scaling behaviour of certain economic
measures, making only elementary assumptions about the processes within [1]. The model predicts the
main features of the data, by assuming a dynamical universality, but seems to fall short of describing
the full range of observed scaling exponents β. Empirical data from [2] revealed sublinear, linear, and
3
superlinear scaling behaviour in the variety of accessible data. The data came initially from mostly
American cities (see table 1), and have since been demonstrated in European cities in [10]. All cities,
thus far, have a more or less comparable level of technological development, and thus fit plausibly into a
statistical ensemble.
SUBLINEAR Linear SUPERLINEAR
Fuel sales Housing New patents
Fuelling stations Employment Inventors
Length of cables Power consumption R&D employment
Road surface Water consumption Other creative employment
Disease (AIDS)
Crime
Table 1: Examples of city outputs with sub- and superlinear scaling per capita, reported in [2].
Cities are not just dynamical systems, they also exhibit complex semantics, or purposeful, intentional,
patterned behaviours. In the physics of inanimate systems, the markers of semantics are comprised of
only a few simple labels; charges, force laws, and interaction graphs. These are constant over time, and
accepted as universal ‘physical law’. However, in human functional systems, the range of interactions,
and their assumed meanings, is much richer, and may depend of time and context. Coarse graining, by
creating a mean field model, is a standard physical approach which eliminates the types, labels, and other
semantics of networks, and exposes the universality of scaling phenomena. However, this simplicity is a
trade-off: semantics are also what sustains the arrangement and composition of functional processes, at a
deeper level, and could lead to actionable predictions.
In these notes, I show how semantics provide some additional structure and constraints on dynamics,
and how both short and long range interactions may be distinguished through functional dependency. This
leads to a possible explanation for the discrepancy between data and predictions of superlinear scaling
exponents in [1,2] using a slightly more detailed model than [1], based on promise theory [9].
2.2 Definitions
Consider a city, in the coarsest approximation, as a bounded homogenous mass, sustained by external
supplies and internally generated wealth. This is analogous to a model of gravity and pressure versus
volume, whose equilibrium defines an average city size. Bettencourt likens the arrangement to a star,
which gravitates together from the benefits of city infrastructure, and expands through the accretion of
new inhabitants.
The city is populated by Ninhabitants, which I will call ‘agents’, which and may include machines,
animals, etc., or any proxies for human intent that lead to economic output. The data on cities, used
in the comparison, are based on human population numbers, so Nwill refer to the people, in the first
approximation. The agents are distributed within a volume V, in Ddimensions. Cities are more or less
two dimensional (D= 2), in spite of high rise regions, because the more or less 1 dimensional networks
connecting them lie mainly in a plane.
2.3 Outline of the approach
•Let the number of agents in the network, or city community, be:
N=NI+N0(1)
4
where N0is a partial dead-weight population that is not interacting with the city infrastructure
(as user or provider)1, and the functional networks generating the yields are agents from the NI
population. Bettencourt does not distinguish between NIand N0, but it is useful to track this
distinction throughout the scaling argument.
A dependence on Ncan mean two different things, in the formulae: an average value of a static
population, across an ensemble of different cities, or a dynamically growing value of Nwithin a
single city, over the course of its evolution. The interpretation of the results is sensitive to this
distinction, requiring some care. If universality were completely true, they would be the same.
•A city is a volume Vof agents, accreted from a wider region, with finite compressibility, by virtue
of some average space requirements per person, and a pressure a (non-detailed) balance condi-
tion, which matches the output shared amongst interactions to the resources that can be fed to the
connected agents.
•An estimate is made of the minimum fraction of the volume to be infrastructure VIthat could
connect the city’s agents together into a virtual mesh.
•The city infrastructure is assumed to be ‘sparsely’ utilized, at equilibrium, in the sense that the
technology on which ensemble cities are based can absorb fluctuations in space and time, without
signifiant contention or expansion2, else it would choke to a standstill. This is a technical (statisti-
cal) property, which is necessary to achieve economies of scale, through spacetime multiplexing3.
A city may appear to be busy, even crowded at certain moments, but ‘sparse’ means that it could
technically get a lot more crowded, on average, without collapsing from congestion.
•Various functional output yields Y, of the city may be calculated in a form suitable for ensemble
comparison, to determine their scaling with N. Some outputs stem from individual agents, and
some are from the interactions between them.
2.4 Explicit and implicit assumptions behind a mean field model
To make a mean field explanation plausible, over a range of scales and circumstances, some assumptions
are required. The mean field model should not be confused with a detailed model of physical processes.
It is an effective representation that emphasizes universal aspects of behaviour. Expanding on what is
stated in [1]:
•The ability to form an ensemble over multiple cities, assumes that when similar outputs are promised,
they are made by the same kinds of agents, with the same basic capabilities [13]. If one city has a
technology that makes the same output twice as productive, this will not necessarily respect the en-
semble’s universality assumption, leading to anomalies. The latter assumption suggests that there
might be difficulties comparing third world cities, primitive societies, or fully automated production
facilities with more homogenized average candidates.
•The distribution of agents within a city cannot be predicted or described by a mean field model.
I’ll return to this issue in section 3. The agents involved in a specific output ‘yield’ Ymight be
concentrated into regions, however, there may also be multiple regions with the same role, adding
1The dead weight could be green spaces, but more likely slum dwellings such as those that dominate Baltimore. Hot-beds for
crime and shadow economy. This merely interferes with city output as far as the world is concerned.
2In Feynman’s words, there is ‘plenty of room at the bottom’. Fluctuations in network utilization generally exhibit long-tailed
behaviour [11,12], so we must be far away from the point of collapse, in spite of superficial appearances like traffic jams.
3Arrival processes like Poisson and L´
evy distributed events have the property that a convolution of multiple flows is form
invariant.
5
up to the total partial volume VYused in the expressions. All are lumped into a single equivalent
volume VYfor the purpose of total comparison (see figure 2)..
•A key assumption is that infrastructure filaments take up only a small or ‘sparse’ fraction of volume
of the city VIV, but can cope with all of its load requirements. This sparse use of resources,
with the city volume, will be important to justifying the superlinear scaling.
•In this rendition of the model, I add one trivial extension to [1]: the infrastructure network connects
a fraction of NI≤Nof the total population Ntogether, leaving some residual number N0of
city dwellers unconnected, or non-participatory. This allows us to track the possible impact of a
non-productive mass.
•The infrastructure network is not a mesh network itself, but it delivers sufficient capacity and in-
terconnectivity to allow a virtual mesh of coverage, i.e. any agent can reach any other agent with
equal average cost.
•The maximum output of the city community is assumed to follow Metcalfe’s law, which estimates
the productivity of a network proportionally to the maximum number of links that can be made [14].
This has been criticized theoretically (for a sample see [15, 16]). However, recently this conjecture
has received empirical support from social media studies [17]4. I will derive this result, and its
limitations, from promise theory in section 3.6.3, and also show that this cannot be the only measure
of productivity to reproduce the data.
•Let us measure both value and cost in the dimensions of ‘money’ [M]. In [1], the author uses power
(energy per unit time) as the proposed currency; however, money might be easier to grasp for many
readers.
A
A
A
B
B
C
A
B
C
Figure 2: The yield agency model is a pudding model, in which different yields may be spread about inside the city
bounds, but they may all be lumped into a single equivalent volume for the purpose of scaling the ingredients.
Regarding the economic output of the city, the model is quite simple.
•The city’s activities yield outputs, labelled by different Y.
•These come from groups of agents that interact and collaborate in an unspecified way. The group
occupies an aggregate volumes VY, for each Y(see figure 2). These patches are virtually contigu-
ous even when they are physically distributed, and so they may overlap in space.
4Metcalfe originally assumed that value creation would be proportional to N2while costs grew proportional to N. The study
[17] indicates that both grow quadratically with network vertex count, though these ideas are still disputed [15,16].
6
•By assuming that VYis a fraction of the N2
Iinteraction mesh, one implicitly allows the members
of a yield producer to be located anywhere within the equal-cost network.
The economic output due to a process Ymay be written approximately as:
Y=gY
vcoop
Y
VI
N2
I,(2)
where gYhas units of money [M], and NIis assumed dimensionless. In many cases we can assume
that NI'N, so that the entire city is active (no dead weight or free riders), but there is no need to
make this identification yet.
2.5 The fraction of volume that is sparse infrastructure network
In the volume of the city, network links are counted using a continuum approximation in terms of volumes,
and fractions of volumes. This is an unfamiliar step in computer science, but it plays an important
role in deriving the scaling laws, and this case can offer valuable lessons. The minimum size of the
infrastructure network can be estimated by squeezing the total sparse volume into a narrow, approximately
one dimensional pipeline, with a small cross section. This is only plausible of the network utilization is
really sparse, since then the total interaction can be compressed into the lower dimensional network, by
multiplexing. The average distance between agents inside the city (in Ddimensions) is
d=V
N1
D
.(3)
An infrastructure network has the topology of a graph, in the mathematical sense, but it may also have
sufficient structure to pervade space5. It is embedded in a real world volume, and needs to reach the
agents distributed within. If the agents cluster around the network, the system will remain largely one
dimensional; however, if the network penetrates the space homogeneously (either by wiring or by the
movement of inhabitants who use it), then (again, in the spirit of generality) the effect of this ‘space
filling’, or fractal invasion6, may be captured by an effective (Hausdorff) dimension H < D, so that we
may write the order of magnitude estimate for the infrastructure volume:
VI≥gIV
NH
D
LD−HNI,(4)
where gI<1is a dimensionless constant that indicates the fraction of nodes in NIspanned by the
particular infrastructure being considered. Lis some fixed scale with the dimensions of length [L], so
that
[V] = [L]D.(5)
and LDVIV. In other words, the volume is the effective average linear volume swept out by a
fixed cross section LD−H, as it feeds into the NInodes connected by the infrastructure7. This has the
5This was an important argument in deriving the biological scaling laws [8].
6The model cannot formally distinguish between the intricacy of the infrastructure itself and the movement of agents around it,
but it makes sense to assume that it is the motion of people and mobile agents that is complex, rather than the system of roads and
wires of the city. I’m grateful to Luis Bettencourt for this comment.
7It is well known that the scaling of ad hoc communications networks, where agents are distributed randomly is like √N. This
is easily understood from the spatial geometry: mobile phones occupy some approximately two dimensional area, so the diameter is
of order N1
2; alternatively, they have average separation d'V/√N, so the distance across the group is of the order N d '√N.
The linearity of the process gets mixed up with the geometry of the embedding space.
7
schematic form of NIqueues that are serialized paths of dimension V1/D. For H= 0 the nodes are
unconnected, for H= 1 roads are serial or linear, and for D > H > 1, the roads or channels have an
effective fractal ‘thickness’, from a coarse-grain perspective (see fig. 3). It turns out that we only need to
look at H= 1, as it is serialization rather than physical dimension that is important.
The assumption that we can squash a volume Vinto a smaller linearized volume VIis the key process,
or universal mechanism for comparison between cities. It expresses what we mean by a ‘sparsely utilized’
infrastructure, i.e. the gas of inhabitants is somewhat compressible. One thinks first of physical channels,
such as roads, cables, and transportation costs etc; however, any serial stream of work could constitute
work process from a number of agents within the volume V. Thus we write the infrastructure volume as
d
d
L
V
VI
Figure 3: The volume of the infrastructure sparse network is negligible compared to the ball of the city.
approximately (rewriting (4)):
VI≥gIVH
DLD−HNIN
−H
D,(6)
representing NIserial queues of supporting services, being fed from a Ddimensional region. The V1/D
scaling represents a serialization of the work from across the homogeneous region.
2.6 Inflating the community: sustainable city volume at steady state
Although the city population and volume are presumed to accrete, as people come, attracted by the
promises of the city, a given population has to be sustainable. The total population and volume of the
city must satisfy an equation of state, that explains the average balance of these payments. The scaling of
these payments is the same, but in reverse, so the balance ends up only as a sign to the coefficients. From
the structure, there must be three main parts:
•Agent sustenance: the existence constraint on agent survival, individually, represents a separate
concern that does not play directly into the scaling of the city (every agent for itself). It is repre-
sented, implicitly, through the assumption of constant N. This has two aspects: the attractive force
that brings people to the city, and the resources that feed and supply the balance of payments. Few
cities are self-sufficient within their bounds. These aspects remain formally unexplained in [1], and
thus do not play a detailed role in the data described.
•Work output sustenance: output yields Y, produced by the various agents of the city, whether
individuals or factories, are assumed to make the city cash flow positive, together with the input
of external resources, making that the city is economically viable. This feedback is not modelled
directly, only assumed by the positive signs of the coupling coefficients along links, and the as-
sumption of steady state. Thus, cities might be profitable, or borrowing money to reach this steady
state. There is no way to capture these factors in the mean field approximation.
8
•City volume sustenance: what we can model is the supply of resources must balance the outputs
of the city.
Running cost ≤Resource supply in −Transport cost.(7)
The running and transport costs are assumed to be low, else the agents would not be able to sustain
their existence. The costs are in the links and in the linearized supply through the infrastructure.
The feeding of resources (perhaps from outside the city) through the linearized infrastructure (RHS)
powers interactions within the partial city volumes, over many activities, (LHS), and effectively
inflates the volume of the city by placing a scale Vover which resources must be transported. This
yields a simple (non-detailed) balance condition:
Cost of interaction links ≤Supply through infrastructure
gY
vY
VN2
I≤cYVH
DN, (8)
where we assume that work cost is proportional to the distance travelled (analogous to W=F·dl),
and cYis the cost per unit length of path transport along the infrastructure (dimensions [M][L]−H).
The positive coupling gYincludes the balance of payments to keep the city viable. Rearranging
this inequality gives the implied constraint on the sustainable volume:
V≥aN2
I
ND
D+H
(9)
where a= (gYvmaintain
Y/CY)D
D+H.
2.7 Scaling of spatial infrastructure and output yields
Substituting the steady state volume of the city into the expression for the infrastructure volume, the
model now predicts the three kinds of scaling from the introduction:
1. Sublinear. The scaling of infrastructure itself in terms of the volume, we get:
VI=gIaH/D N2
I
N−2HD+H2
D(D+H)
(10)
setting D= 2 and H= 1 into (6),
VI'N2
I
N5
6
.(11)
For pervasive N'NI, this yields
VI'N5
6,(12)
giving the sublinear scaling observed by [1]. When the infrastructure cost is basically absent, this
scales like ‘every man for himself’, like an ideal gas of non-interacting agents.
2. Linear. For individual agent consumption, the scaling is trivially linear, by assumption, both inputs
(consumption C) and outputs O.
C−=e−N(13)
C+=e+N(14)
where the dimensions [e]=[M]are of money.
9
3. Superlinear. The positive economic yield of a process in the city, due to interactions may be
written as a fraction of the possible N2
Ioutput that can be channelled through the volume VI, with
a different constant of proportionality for each output:
Y+
Y=GY
N2
I
VI
,(15)
where GYis assumed positive, absorbing the costs of interaction, and from (6) we have an expres-
sion for the infrastructure volume, up to invariant unknowns, which are simply constants that do
not depend on the state variables Nor V. So, substituting (10) into (15),
Y+
Y=gIGYLH−DV−H
DNINH
D,(16)
and substituting for volume in (9), since it also depends on NI:
Y+
Y=gIGYLH−Da−HD
D(D+H)N2HD+H2
D(H+D)N
D2−HD
D(D+H)
I,(17)
If we substitute D= 2, and H= 1, this scales as
Y+
Y'N7
6
I1 + N0
NI5
6
.(18)
And if we further assume that the infrastructure network is pervasive, so that N'NI, then
Y+
Y'N7
6
I'N7
6.(19)
This reproduces the superliner scaling identified theoretically in [1], and this result matches about
half of the superlinear scaling data quite well. Other data show significantly higher values for
the scaling. If the infrastructure network is insignificantly small NIN0, then there would be
binomial corrections to a 1/N scaling. The ‘dead population’ reduces the power law slightly (in
binomial corrections), so we could expect processes, for which most of the city cannot contribute,
to scale below the 7/6=1.16th power, thus this cannot explain the higher scaling exponent. An
enhanced explanation is proposed in section 3.7.
2.8 Remarks about the calculation
In spite of a smattering of city related narrative, this calculation is based on a very simple and universal
argument about resources exchanged between a Ddimensional volume and a one dimensional supply
network. There must be sufficient spacetime volume associated with ensemble-standard infrastructure to
accommodate growth in NI, or an increase in density of the users, without saturating it. Multiplexing in
space and time, is they key to this.
•Measures, relating to the output of self-sufficient agents, scale linearly, as one would expect. If we
double the number of suppliers, there is twice as much availability.
•Shared resources may exhibit economies of scale if they become relatively fewer per person, as
cities grow, without impairing output. These economies of scale seem to be quite well captured by
the model, using a value of β= 5/6'0.8to match data, which suggests that the result makes
sense both for Ninterpreted as an ensemble average and as a growth parameter over time.
10
•Finally, there are superlinear outputs, whose behaviour is more subtle. A production output Ymay
be a fraction, not of N, but of N2because the maximum output can depend on the interactions
between agents, and the agents working alone. Output may or may not depend on the volume
of the city, i.e. perhaps only the number of agents or currents in the pudding, and perhaps the
space they occupy in the course of their interactions; this depends on the kinematics of the detailed
processes.
For every scaling benefit, there may also be scaling costs, with the opposite sign: contention for shared
resources, spiralling costs of equilibration, etc. Some general comments about the mean field approxima-
tion:
•People have different jobs, capabilities, and habits. In the mean field representation, such detail
is not represented explicitly, but this does not imply that they don’t exist. Indeed, they must be
present to account for different flavours of output. However, this is not a one-to-one association.
Formally, the outputs are fractions of the total amount of possible produced work, averaged over
all specializations, mechanisms, and functions. To understand why this is plausible, we need to
understand that outputs are the result of combinations of jobs, in collaboration (see section 3.5).
They are assumed equally likely on average across cities of the ensemble. The diversity of a city
may increase with size8, but this does not matter as long as we assume that all jobs are the same.
The data suggest that this does matter (see section 3.7).
•A city must have ‘potential barriers’, or entry costs for construction and community building, that
depend on the level of technology available. This is also not represented in the model. There might
be significant debt associated with building, which is invisible in this picture. These are transient
responses, invisible in steady state behaviour.
•The technology at the time of building ought to affect the density of the city too: as transport im-
proves and gets cheaper, it enables greater distances to be covered in the same amount of time, i.e.
lower density. Conversely, it enables more efficient transport of provisions to sustain a high popu-
lation density. One might not expect old cities like Rome or London to be immediately comparable
with Brasilia or Shenzhen9. Similar questions could apply to the pace of life. Does this depend
only on the size of the city, or also on cultural norms? What if we compared Dakar with Seoul, for
example? The scaling depends only on relative density however.
•To be comparable in an ensemble average, one could expect that cities would need to be comparable
in productive technology, composition and wastage. A machine could do many times the work of
a manual laborer, for instance, so it would not be fair to compare a city with mechanization to a
city of horses and carts. The data in [2] were taken from mainly American cities, which have a
relatively uniform level of technological infrastructure, and arise from similar epochs. There is no
particular reason why they would be comparable to the slums of Mumbai. But this remains to be
discovered.
•By distinguishing between Nand NI, it is possible to see how about the productive capacity of the
city could be altered by the presence of unproductive regions. From the expansion of (18), it would
seem that the higher the proportion of the population that cannot contribute to a process, the lower
the scaling power of outputs might become.
It is known, for instance, that a typical ‘80/20 rule’ (power law behaviour) is almost universal
for networks, i.e. 80 percent of the yield typically comes from 20 percent of the agents [11, 18],
8This could be checked by comparing the size of yellow pages directory for a community to its white pages directory.
9Just 30 years old, ‘Shenzhen speed’ is the stuff of legend in China.
11
and the effect of certain hubs on transmission is crucial to understanding effective partitioning of
regions [19]. This suggests that a graph theoretic explanation can add to the simple mean field
picture.
Coarse graining extracts only the most generic features of a system, in the limit of large N, i.e. at
the coarsest scales. It is unable to probe the separation of such scales, by weak local interactions, or
distinguish long and short range interactions. At smaller range, the semantics of interactions often become
vital to the functioning of a system. In physics, ‘semantics’ are simple and mostly fixed by ‘physical law’,
e.g. they manifest as different ‘charges’, ‘forces’ or allowed interaction types. In a human-technology
system, however, there are many more flavours of interactive behaviour that may be distinguished, and
their number and patterns might even change over time.
The urban scaling predictions reviewed here have been partially matched to data, with Non the
order of 103to 107[1]; this offers strong evidence that underlying semantics of comparable cities are
unimportant at these scales. However, the argument above underestimates some of the data significantly,
especially where the data exhibit a high level of uncertainty, estimated in the margins for β. This suggests
that there is something not captured adequately by the preceding argument.
There are two possibilities: either inhomogeneities across the ensembles are significant, or the outputs
do not all follow a single model, and we are looking for either an addition or a refinement. I shall try
to shed some light on these issues, in the next section, by deriving the model from a more microscopic
model, using promise theory.
2.9 Does innovation characterize superlinearity?
Some aspects of the data, analyzed in [2], are suggestive of a fit with the single universality model in [1],
but not all. The authors attributed the superlinear scaling quantities to creativity or innovation [1, 2],
though the link between this explanation and the calculation of the exponent is incomplete. Below are
approximate representations of few samples of the superlinear scaling data from [2] (see reference for
details of the numbers):
MEA SU RE AP PROX. AVER AGE βSOURCE
Wages 1.12 ±0.02 USA
GDP 1.13 −1.26 ±0.1EU, Germany, China
Patents 1.27 ±0.02 USA
Private R&D employment 1.34 ±0.05 USA
R&D employment 1.26 ±0.1china
R&D establishments 1.19 ±0.03 USA
AIDS cases 1.23 ±0.05 USA
As pointed out by the authors, many of the numbers have a high level of uncertainty, due to the difficulty
of fitting data from disparate sources10. Even with generous margins, the single predicted value of β=
7/6=1.16 does not plausibly agree with all of these measures, however, and the deviation seems to
not be attributable to a normal variation. The data in the table below, by contrast, were collected in
an independent study in the UK that attempted to verify the superlinearity hypothesis in only a single
measure: patents (see reference [13] for details):
MEA SU RE AP PROX. AVER AGE βSOURCE
Patents 1.13 ±0.1UK all cities
Patents 0.99 ±0.1UK small cities
10My hat goes off to the authors for making this effort though!
12
The authors of this study point out the difficulty in defining what constitutes a valid city for the purpose
of ensemble comparison. Those values are significantly lower than those in the first (SFI) table. While
the SFI data for patents do not agree with Bettencourt’s model, the first set of UK data agree much better
(1.13 is close to 1.16), but the second restricted set do not (0.99 instead of 1.16). These discrepancies
warrant an explanation.
From the perspective of promise theory, which I will apply in more detail in the next section, one
thing stands out about the measures in the first table: the outputs are of two qualitatively different types:
some measures are ‘promises’ and some are ‘agents’.
•Patents, wages, GDP, disease, are related to the promises of an output, i.e. produce that relate to a
production process.
•Jobs related to research are not outputs but occupations. They represent the state of agents. They
are not (directly) the result of a process11.
If we consider the version model of [1], there are two freedoms that remain to alter (17). One is to imagine
the existence of a large fractional N0population (e.g. by invoking the 80/20 rule for the distribution of
outputs), relative to the population NI, involved in making each particular promise. However, this would
have the effect of reducing the value, so, while it might potentially explain the UK data, it could not
explain the SFI data. The second is to assume H > 1, which suggests a significant self-similarity in the
infrastructure that relates to patent production. However, this does not seem credible as the infrastructure
that enables patents is principally researchers, which do not think in paths that fill space. Something about
the universality assumptions need to be reconsidered. A possible resolution is provided in section 3.7.
One clue to these behaviours could be that the selections represent highly specialized sub-populations,
rather than homogeneous fractions. Recall that the way superlinearity arises is that an efficiency of scale
effectively gets better at large N, leading to an effective amplification with the size of the population.
Wages and GDP to involve the largest fraction of the population of a city community. All the other
measures are based on highly specialized populations to which few contribute. Another point is that
patents are not merely the result of an economic process, but are sometimes weapons used politically
and strategically as part of non-collaborative economic warfare between companies. This suggests that
semantics would play a role in their explanation, and potentially skew some data (cities with companies
like Apple and Samsung, well known for software patent fights12 might appear differently).
In section 3.7, a generic possible explanation is proposed, based on the generalized semantics of
scaling agents into clusters (superagents). If one distinguishes agents from their promises, then staged
efficiencies, arising from functional dependency, can be compounded or reduced, altering the βvalue
hierarchically.
Infrastructure ×1.16
−−−−→ Superagents ×1.16
−−−−→ Produce/Output (20)
To derive this plausibly, we need to formulate a simple promise theory of communities.
2.10 Gunther’s ‘Universal Scaling Law’ and spacetime involvement in scaling
Before looking at a deeper model, I want to comment on another scaling model, from the world of
information technology (IT). Superlinear scaling has also been observed in high performance cluster
computing [20], albeit for a different reason. Most IT models are essentially one dimensional in nature,
describing serial processes, divided into parallel one-dimensional threads. The amplification of output in
this kind of flow network is based on queueing results, which may approximated quite well by a simple
11They could conceivably be related to a training process, but then should be be counted as population or output?
12Software patents usually have low production costs, as they are often trivial and frivolous inventions.
13
current flow models, when workloads patterns are sparse and evenly insensitive to mixing (e.g. Poisson,
or L´
evy processes) [21].
Amdahl’s law, and Gunther’s generalized ‘universal’ scaling law [20,22,23], describe the ‘speedup’,
or amplification of output, in a basically serial process, by the addition of workers. Jobs may be speeded
up, but are ultimately limited by the serial coordination of work. Suppose we have NLlocal, parallel
worker threads; then this takes the general parametric form:
S(NL) = NL
1 + α(NL−1) + βNL(NL−1) (21)
The output rate for a collective of NLagents is S(NL)times that of a single agent (see appendix D). The
two terms in the denominator represent two kinds of network process, at close range. The linear term is
a bottleneck term, where all the agents contend over a limited shared resource. As long as α > 0, scaling
will be sublinear. The quadratic term is the cost of making the mesh of agents agree about something at
each stage of the process. This is a cooperation, or ‘homogenization of information’ (coherency) term,
which puts a sharp brake on scaling. For example, in database or cache replication, information has
to be replicated consistently, which is expensive. This term reduces the speed up to the point where
performance can actually grow worse with increasing N(see figure 4).
α = 0, β=0 α > 0, β = 0 α > 0, β > 0
(a) (b) (c)
Figure 4: The Gunther universal scaling law for its control parameters. (a) linear scaling, (b) cost of
sharing resources and diminishing returns from contention, (c) Negative returns from incoherency and
the cost of equilibration.
The expression (21) cannot exhibit a superlinear result for positive coefficients. Nonetheless, super-
linear scaling has been observed in computing clusters [20]. How can these facts be resolved? Gunther
argues that superlinear scaling is not possible without violating work conservation; but, by artificially
making α < 0, as a parametric fit, it is possible to simulate higher dimensional effects in this one dimen-
sional projection. This is a one dimensional view of a process that is actually two or three dimensional.
Datacentre clusters, with dense interconnect networks approach mesh-levels that fill the effective volume
of the datacentre during their internal processes13. As this trend continues, we can expect datacentres to
behave a lot like the model of cities discussed.
The appearance of superlinear scaling came as a shock in IT, because there is no way to understand
α < 0from a microscopic model. However, it can be understood as a renormalization effect, i.e. an
effective parametric representation of the projected output. Qualitatively, the serial limit on scaling could
13This is called East-West traffic in the IT industry.
14
be cheated as follows, in a sparsely utilized system. When the average state of the system is highly un-
derutilized to begin with, then some of the efficency can be regained by close packing of the utilization14.
If dense packing can absorb the increased size of a system (like a city or computational process), then
that economy of scale can enable greater output for a relatively smaller infrastructure (renormalized neg-
ative growth). Could this lead to a superlinear speedup? The phenomenon of packing is related to queue
parallelization, where for example G/G/N is provably more efficient than G/G/1×N[4, 24]. This is
because wasted idle time can be eliminated by arranging by an efficient packing of work. This cannot
persist for ever, as eventually the sparse utilization of the limiting work agents must fill up to capacity.
As it does so, the amount of contention (αfor shared bottlenecks, and βfor equilibration of state [25])
must increase rapidly. The response time Rof a queue (proportional to its average length divided by the
dispatch rate µtakes the form:
R'E(n)
µ=1
µ−λ(22)
This queue is only stable when λµ, indicating sparse usage. The scaling indicated in the city results
indicates cities that have not peaked in their infrastructure utilization yet. Superlinear scaling sounds like
a good thing but it is unstable, as the queueing projection illustrates.
The rational queueing expression, in Gunther’s Universal Scaling Law (21), can never explain frac-
tional scaling exponents seen in cities, but it can demonstrate some projected superlinearity with α < 0.
To feed superlinearity, we need something more than parallel serial processes where the work is done by
Npoint sources. Only if the work is done by N2interactions can a partial efficiency make the expo-
nent greater than unity. To get this, we need to feed higher dimensional volumes into lower dimensional
volumes. This is what is going on in the mean field theory of [1].
Interaction output =Maximum output(N2)×Fraction of infrastructure involved(N)(23)
From a graph theoretical perspective, this is a change in average connectivity of the infrastructure network
(i.e. the average degree of nodes k[4]). If the fraction is a fraction of a volume rather than a line
or a number, there are dimensional exponents involved, which represent the contact efficiency by close
packing the city population. With more dimensions, a larger surface area can be used for interaction. If
we assume that the infrastructure network is sufficiently dense that it reaches almost everyone, then this
continuum approximation is reasonable.
density of infrastructure users =NI
VI
=nINδ(D).(24)
where δ(D)=1/D(D+ 1), for H= 1. Recalling that this volume is really a continuum approximation
of a network of nodes, this translates into an average node degree utilization (or locally used connectivity)
within the infrastructure channel15
k(N) = NI
VI
=nINδ(D).(25)
Assuming the infrastructure is pervasive so N'NI, the equivalent serialized infrastructure volume, for
a single process, is something like:
VI=V
Nδ
×NI'N1−δ×cross section,(26)
=capture volume per agent ×span of agents ×cross section,
14This is the motivation for packetization (atomization) of networks, and context switching in time sharing operating systems.
15Note that this is not a real connectivity, which has to do with the number of nodes, but a kind of close-packing of the sparse
interactions that occur between the nodes into the infrastructure stream.
15
Using this volume, instead of the total volume of the system (city, community, etc), recognizes two things.
First that cost of the infrastructure is much less than that of the entire system; and, second, that it is the
serialization of the sparse resource over a standard cross section that we want to use for comparable work
output. This is like fitting the sparse output volume of the city into an idealized serial stream of fixed
width to see how its length scales with the number of inhabitants16. The efficiency comes from being able
to use more of an unexpected fixed cost, sparsely utilized resource along with other economies of scale.
The net result is an amplification of the output by δ:
Interaction related output Y=const
VI
N2
I'Y0N1+δ(D).(27)
The numerator is unexpectedly constant, but the infrastructure volume scales sub-linearly, the net output
appears superlinear, with these assumptions. The question is how do we know if these are the same
assumptions as the used for the measurements?
Modern datacentres and networks at scale have multiple redundant paths that make their interconnec-
tion networks space filling (e.g. Clos structures [26]). This brings higher dimensional scaling issues into
the picture. The general principle is one of close packing of utilization. When dependencies scale more
favorably than the contended processes that rely on them (relatively speaking), each process gets a larger
share of the shared resource, and is accelerated for a while, provided the total utilization remains low.
3 A promise theory derivation of the city model, and beyond
I now want to show that we can re-derive the scaling, and indeed the model of [1], from using a quasi-
atomic theory of a network, taking into account some of the more important semantics to see how univer-
sality emerges, and where its limits might lie. Promise theory strikes a balance between semantics and
dynamics, and thus between coarse graining and the chemistry of different functional agents.
3.1 Brief recap’ of promise theory
Promise theory is a formalism that describes dynamics of atomic, black-box agents, alongside their broad
functional semantics, i.e. with their intentional behaviour. The default or ground state of agents is one in
which they make no promises, and are independent or ‘autonomous’, i.e. each agent is self-sufficient and
controls its own internal resources, and each agent can make promises only about its own capabilities and
intentions17. Assuming that an agent is not deliberately deceiving, a promise may be considered the best
available local information about the likelihood of an outcome.
Promise theory can be understood in a number of ways. For the present purpose, it can be thought of
as a labelled graph theory, with some rules and constraints on interpretation. A promise from an agent S
to one of more agents Rtakes the form
Sb
−→ R(28)
where bis the body of the promise. The promise body, explains what is being promised, and has polarity
(+ to offer, and - to accept), as well as type τ(b)and constraint χ(b). All agents are considered to be
16A simple analogy is to think of a tube of toothpaste. The toothpaste comes out in a one dimensional stream of fixed width, but
we are forcing the output of a three dimensional tube through this portal, and asking: how does the amount that comes out increase
with the size of the tube if we squeeze it in the same way? By fixing the cross section, we can compare different tubes, or different
cities.
17From a physics perspective, promises look a lot like a rich array of charge flavours, for exotic forces, and the network looks a
lot like a force field.
16
indistinguishable to begin with, and these agents may work together to form aggregate ‘superagents’, by
promising cooperation [9].
In a community or city, we might imagine that the atomic agents are persons, machines, and even
software, acting as proxies for human intentions. An aggregate ‘superagent’, such as a company can
collaborate to keep new promises (see figure 5). Each aggregation into a superagent enables new promises
to exist that cannot be attributed to any of its components. To understand output yields, the key is to
ask: what kind of agent is responsible for an output? Is it a single point source agent, or a distributed
superagent, with an internal collaboration mesh? In either case, collaboration has a finite range. In
b4
b6
b5
b3
b2
b1
int
A
Aext
Πext
Πint
Figure 5: What is an agent? Agents aggregate to make superagents, with new promises possible at each scale of
agency. So what we consider to be an agent, depends on the scale at the networks under consideration.
summary:
1. Agent types are distinguished by the promises they make.
2. The way that agents make use of one anothers’ intent through promises is what we mean by agent
semantics.
3. The outcome of a promise is deduced by observation; this is called an assessment in promise theory.
An assessment by agent iabout a promise πis written αi(π).
4. A link between two agents requires a promise by both parties: one to make a service offer (+), and
the other to accept it (-).
S+b
−−→ R
R−b
−−→ S)=unidirectional transfer (29)
This kind of binding is the basis for determining the coupling strength gYfor the output yields.
Promise theory is a theory of incomplete information, and embodies controlled coarse graining over
semantic scales.
3.2 Agents and super-agents, long and short range interactions
Consider a set of agents Ai, where iruns over the population of the system (city, datacentre, etc), and
all machines and proxies for human intent. In order to collaborate, agents need to make some basic
promises [9]. Every promise may be either kept or not kept, and the average value needs to be self
sustaining. Each autonomous agent thus has a balance of payments to consider. It needs to accept fuel,
17
food, energy, money, etc. Dependencies also include raw materials which have to come from outside. If
agents can cache resources they can maintain weak coupling, else they are strongly dependent on their
environments. This applies to energy, supplies, and also inputs of information and ideas. Promises made
directly between agents are called short range.
Definition 1 (Short range interaction) A binding between adjacent agents Sand Rof the form
S+τ,χ+
−−−−→ R(30)
R−τ,χ−
−−−−→ S(31)
where τis the same in both promises, and χ+∩χ−6= 0.
A promise may be long range if it is non-local, i.e. it couples several agents together, or employs inter-
mediaries.
Definition 2 (Long range interaction) A binding between adjacent agents Sand R, through an inter-
mediate agent A
S+τ,χ+|d
−−−−−→ R(32)
R−τ,χ−
−−−−→ S(33)
I+d
−−→ S(34)
S−d
−−→ I(35)
where τis the same in both promises, and χ+∩χ−6= 0.
Note that there is no a priori notion of distance in a graph, other than the number of interactions or hops
between agents nodes. The familiar notion of distance comes about from embedding a graph in metric
space, which in turn is related to a continuum approximation.
At any scale, a promising agency that plays a functional systemic role makes promises of the following
general forms.
Ai
−f
−−→ Aext ,∀i. (36)
There must be an external agency acting as a source of this fuel f, providing
Aext
+f
−−→ Ai,∀i. (37)
Any agent Aimay depend on pre-requisite promise of dependency D, provided by another, in order to
provide service +S; according to the assisted promise law [9]:
Ai
+S|d
−−−→ Aj, Ai
−d
−−→ Ak'Ai
+S
−−→ Aj(38)
provided
Ak
+d
−−→ Ai.(39)
In this way, agents have probes, services, skills (+), and needs or receptors (-) that can unlock or cat-
alyze their functionality. The expresses its exterior promises outwardly, e.g. a door handle’s function is
recognized by its shape, just as a car and its promises are recognizable by its exterior structure. Interior
promises might be involved in making the exterior ones, but these are not generally visible at super-agent
scale. The basics of scaling semantics and agency are laid out in [7].
18
3.3 Promise networks: functional interactions
Functional networks have two aspects to their productivity, which can be described by:
•Replication of agent output: dynamics, economics.
•Combination of used services, ideation by mixing: intent, semantics, fitness for purpose.
It is the combination of these that leads to the understanding of fully functional scaling. Basic communi-
cation and supply infrastructure networks enable interactions between any pair of agencies, but specialist
functional networks are typically small and disconnected [27]. They relate specific promises or services
that are constrained by operational semantics. The scaling of such networks has been examined in [7]. It
has a few aspects:
•Agents keep promises at scale by individually promising similar capabilities in parallel, e.g. Am-
dahl’s and Gunther’s laws of scaling [20, 22, 23]. A promise that is considered kept by a single
agency scales by having multiple sub-agents form a ‘superagent’. Now Bettencourt’s insight re-
veals how output can also scale in a non-parallel fashion [1].
•When one agent depends on a promise being kept by another, in order to keep a promise, this
creates a serial dependency, introducing a queue and a handling rate scale. The agents must be able
to understand one another, with common language, else they are partitioned. Partitioning cuts off
long range interaction, and promotes long range diversity, like cultural and specialist diversity18.
Figure 6: No single type of promise binding (dark lines) leads to percolation of value in the promise graph. However,
with conditional dependency, and sufficient diversity and homogeneity, there can still be effectively close to N2links
whose value converts into a common currency.
3.4 Forces that condense a cluster
A hypothesis of promise theory is that one may define a notion of force for agents, which is attractive
when there is economic advantage, and repulsive for economic disadvantage19. The formation of super-
18The lack of a common language is effectively a channel separation, disconnecting networks into separate branches. In commu-
nication networks, channel width is sometimes shared between different partitioned agencies by using non-overlapping frequency
ranges. This is called Frequency/Wavelength Division Multiplexing (FDM). It is a form of multi-tenancy.
19It does not matter here whether we consider the force to be a Newtonian deterministic force, or a probabilistic susceptibility for
drifting closer, as in stochastic systems. We can think of a generalized energy-momentum tensor [28], for generalized agent ‘fields’.
19
agents thus comes about, for economic reasons [29], by the value of collaboration. If the promises are
unconditional, superagents will be localized. If they are conditional, the clusters are ordered and may
thus be distended or even scattered.
•Agents, which make the same kinds of promises of same polarity, tend to repel one another.
•Agents, which make complementary (binding) promises of opposite polarity tend to attract one
another.
Applied to the city problem, this suggests that the basic attraction to condense the city out of a sur-
rounding gas of agents comes from the common supply promises, which are predominantly of positive
sign, and that all agents share; for the survival infrastructure. This is held in check by the repulsion of
agents making similar promises, except where there are promises to cooperate. One may expect structures
as follows:
•Clusters of professionals bound together by cooperation promises.
•Chained transport agents, bound together by conditional promises.
•Distributed competitors, perhaps clustering around shared infrastructure hubs, e.g. malls, districts.
The embedded spacetime structure of the city should be an equilibrium configuration between the attrac-
tion to needs and desires, and the repulsion from competition with similar skills.
In promise theory, a specialized role characterizes a pattern of agents that make similar promises. By
specializing specific tasks to specific agents, each agent can be more focused in learning and adapting,
but acquires an additional cost of cooperation proportional to some positive power of the cluster size.
Superagency collaboration is a short range interaction (between interior promises) [7]. It can form long
range strong interactions, with associated cost, if it has the internal resources to support these. Limiting
interactions to short range leads to stability. The long range interactions drive superlinear effects, but also
promote ‘chaos’.
Figure 7: The geometry of superagents may fill space in different ways. Infrastructure that interconnects other
agencies is a superagent in its own right, involving linear or approximately linear cooperation between member
agents. Under preferential attachment, agents NItend to cluster around the infrastructure agency, leaving a few N0
padding out the spatial volume. The circles around the subagents may be considered infrastructure binding the agents
together.
The notions of attraction and repulsion are wired into our imaginations in terms of spatial concepts.
Even without an embedding spacetime, we can speak of agent affinities, like the interactions described
in molecular chemistry, where spacetime plays no real role. With a physical volume to embed a graph of
20
promise-keeping interactions, geometry ties range to distance, but in a virtual network (which includes
transport of messages by intermediate carrier), short range interactions can also be disseminated over a
longer effective range, by adding cost or latency (such as in telecommunications).
3.5 Promise networks that percolate
In order to justify Metcalfe’s law, there needs to be an average level of interaction that spans the complete
graph and propagates value. This doesn’t necessarily require a single promise type to dominate the entire
graph, because it is the value graph, not the promise graph that needs to link up all the nodes. However,
from the previous section, we would expect basic infrastructure to dominate. The main requirement for
this is the presence of a common (or at least interchangeable) currency between all the nodes.
Specialized exterior service promises naturally lead to small molecular clusters of component ‘atoms’
(superagents), that make specific interior promises. They seldom span large areas, because promises act
like short range interactions (which is also why superagents can be considered quasi-atomic black box
agents, see figure 6), so they do not easily bring about percolation of value in an economic zone, like
a city. The promises, which are ubiquitous, are associated with the survival of agents, and relies on the
most general kind of infrastructure in systems: power, food, air, water, etc. These are likely responsible
for the interconnection of the many smaller microcosms of value creation (small businesses in cities, and
microservices in IT) to bring about a unified community with its economies of scale.
If we let Nτbe the number of agents that consume a promise of type τ, then we expect the class of τ
related to survival to be of the order Nsurvival 'NI, in the meaning of the city model. For all other types,
Nother NI. However, we’ll see in section 3.7 that long range interactions are also needed to explain
the scaling exponents for cities. The size of the effective network is not therefore given by the adjacency
matrix of the underlying infrastructure network, but rather by the typed promise graph.
It is useful to recall the definition of a promise network (see [7]).
Definition 3 (Promise adjacency matrix) The directed graph adjacency matrix which records a link if
there is a promise of any type τ, and body bij (τ)between the labelled agents.
Π(τ)
ij =1 iff Ai
bij (τ)
−−−−→ Aj,
0)∀bij (τ)6=∅(40)
The adjacency is the effective topology of the spacetime network, as far as the agents are concerned. The
link-occupancy of this matrix, for a given promise type, is a linear sum whose value is generally much
lower than that of the total possible mesh of interactions. Thus, for any promise type τ,
NI
X
i,j=1
Π(τ)
ij =Nτ(Nτ−1) N2
I,(41)
Note that an agent can make a promise to itself too, so the upper limit could be written N2
I. The value-
percolating connectivity or degree of a node
Πij =X
τ
Π(τ)
ij ,(42)
ki'X
j
Πij .(43)
We can also write this in terms of the direct valuation of the promises, in terms of the actual matrix of
promises πij [7]:
ki'X
j
vC(πij ).(44)
21
where vCis the value of the promise as calibrated and assessed by a common central agency (see appendix
A).
Agents can keep multiple promises and multiple types of promise ‘simultaneously’ over a given
timescale T, by multiplexing their time at a rate that is much faster, i.e. tmulti 1/T to avoid the
queueing instability. On the assumption of sufficiently sparse packing:
X
τ
N
X
i,j=1
Π(τ)
ij ≤N(N−1).(45)
For the economic output of a promise network, we care more about the assessments of which promises
were kept than the number of promises that were made (see appendix A). Each agent assesses promises
individually, and they may not agree. However, to compare to city statistics, we may assume that an sta-
tistical bureau agency has been appointed by the city to calibrate these assessments αofficial(πτ)according
to a single scale. Promise-keeping is an average over time. Provided the sum time to keep a promise, for
all τ, for each agent, is much less than each time interval of the assessments, we can reduce α(π)to a
frequency ‘probability’. Another way of saying this is: provided the cost of keeping the promises is less
than the budget of each agent.
These estimates are maximal. The size of a functional cluster is not really related to any of these
graphs, because there are semantic constraints. Specific functional behaviour, in a single promise type,
is a strict limitation, which leads to very sparse subgraphs. To gauge an average measure of the total
economic impact of all functional interactions, we have to assume:
•The functions are successful in driving an economy.
•The density of implicit interactions is quite high, else a given output Ywill not be represented by
an average mesh density.
•There are some long range interactions that make the partially connected graph totally connected
on average, even if only at a low level. The survival promises probably fulfill this role.
In reality, a city or community might be partitioned into quite independent regions, leading to a modular
reducible form [30]. If one imagines the specific network, which delivers output Y, it may be some
maximally quadratic polynomial of NI, related to the structure function of the network, but it may also
be significantly less than this. What will tend to lead to percolation of interactions, which bind in a mesh,
is the existence of long range, pre-conditional promises, i.e. dependencies. Then, the sum will be some
polynomial of NI, such that
X
τ
N
X
i,j=1
Π(Y)
ij 'c1N+c2N2.(46)
If i, j run over all the individual agents within city limits, then these matrices are sparse and fragmented
for each τ. Suppose we assume that there is no dominant infrastructure, only small clusters of voluntary
cooperation, as in [27]. Then, the aggregate graph for a city output Y, would need to have sufficient
random cooperative connectivity to form a process that generates output algorithmically. The density can
be estimated, if we assume that, for promise type τ, an agent has an average valency V, then we would
need
τmax
X
τ=1
Vτ
X
ij
αCΠ(τ)
ij 'NVτmax ≤N2.(47)
22
Moreover, since the diversity of promise types is unlikely to be greater than the population it stems from,
one estimates N
V≤τmax ≤N, which does not seem realistic. Taking these estimates into account, it
seems most likely that a few types of promise relationships dominate the connectivity and value creation,
i.e. the basic ‘survival’ (food, water, sanitation, communications) promises, and the specialized outputs
contribute relatively little to the total output, except when directly dependent on the survival promises.
This is a further suggestion that the attraction to urban life stems from core interconnection infrastructure,
rather than from independent diversity.
3.6 Scaling of roles and interactions
Consider now the role of dependence in keeping promises as a determinant for scaling.
3.6.1 Linear consumption by independent agents
For promises that are requirements for survival, every agent more or less deterministically depends on
some source infrastructure agent S, the number of promises is one to one:
Number of consumed =
NI−1
X
i=1
Π(τ)
iS 'NI−1.(48)
If the source is distributed over several agents, then this is still true:
Number of consumed =
S
X
s=1
NI−1
X
i=1
Π(τ)
is 'NI−1.(49)
Thus, the number of promises needed to supply this demand is also proportional to NI:
Ninfra ≡Nτ'NI.(50)
3.6.2 Sublinear economy of scale, and spacetime involvement
When agents interact through links, on a small scale, the chemistry of their interactions may be based on
simple counting. Normally, in promise theory, we count by agent or by link. However, as the numbers
of converging links become so great that counting is impractical, there is no way to liken a process to a
simple Poisson arrival queue, and we resort to flow counting based on density arguments. Let’s now show
that this is equivalent to volume of the infrastructure VIin [1]:
Consider a number of agents Ninfra who provide infrastructure (gas stations, supermarket, etc) for a
number of clients Nclient, which in turn offer a service conditionally, based on the dependence.
πinfra :Ainfra
+infra#V
−−−−−−→ {Aclient}(51)
{Aclient}−infra
−−−→ Ainfra (52)
{Aclient}+service|infra
−−−−−−−→ Ainfra .(53)
Suppose that each infrastructure agent Ainfra can promise to service Vclients simultaneously; then, using
a simple valency argument, we have a detailed balance equation for the interactions at steady state:
α+Ninfra V ≥ α−Nclient .(54)
23
Thus for simple counting of distinguishable agents, we may estimate the number of infrastructure agents
needed to support a number of clients:
Ninfra ≥α−
α+1
V
| {z }
intrinsic
×Nclient.(55)
where α−/α+may be interpreted as the affinity for the service, or the reciprocal compressibility. This
scales linearly with the number of clients in the catchment area of the infrastructure. Moreover, there
is no way, in this detailed formulation that we can count otherwise. The only economy of scale in this
arrangement is the standard linear multiplexing result for the marshalling of Vqueues into a single queue
with Vservers, noted in section 2.10.
However, if we now ask how to count the number of clients that can be fed into a single infrastruc-
ture agent, in a spatial volume, with dimensional multiplexing, then the best estimate is to serialize the
counting, as before:
Nclient =Vcatchment
Nusers 1
D
C(D)×Nusers,(56)
where we imagine a catchment volume Vcatchment, containing any number of agents Nusers who are inter-
ested in the infrastructure service, and we serialize them along a tube of constant cross section C(D)(see
figure 8). Although these numbers only apply to a small mesoscopic volume of space, in a homogenous
infrastructure
catchment volume
Figure 8: How spacetime involvement compresses serialized agent links into an effective flow of fixed cross section.
city, this will apply to the entire city, so we are justified in taking
Nuse 'NI,(57)
which, combined with an equation of state for the volume, reproduces the earlier result
Ninfra 'V1/DN1−1/D .(58)
In this argument, it is clear that only the active agents NIplay a role in the counting, and process flow,
hence this also justifies why we can assume NI→Nin [1].
3.6.3 Deriving Metcalfe’s law from promise networks: the importance of conditional dependency
A key assumption in the scaling argument of city outputs in [1], is what is Metcalfe’s law [14–16], which
proposes that the value of a network scales like the square of the total number of nodes, i.e. value is
24
generated by the number of possible links between agents20. This has received some empirical support
in [17], but has also been criticized in [15, 16]. More importantly, interaction value generation is not the
same as output. Agents can also produce wealth without interacting, if they have all the prerequisites
(short range interactions). In section 3.7, we’ll see that long range interaction forms the basis for one
kind of value creation, but not the only one.
Promise theory predicts that links represent value in the following way. Consider then the sum of all
impartial promise valuations by third party C. If we assume that all agents assess the value of interactions
as strictly positive, then:
Mean value =X
τ
NI
X
i,j=1
vC(πτ
ij )≤chαiαjiNI(NI−1) (59)
where NI= maxτ(dim(τ)) (see appendix A about promise valuations). In spite of the quadratic appear-
ance from the result, this is a linear sum, so it acts automatically as a linear averaging measure. Also,
for any given specialization τ, the filling fraction of the promise network is likely low; thus, a key as-
sumption is that, when properly documented, agent’s specialized promises in fact depend on many others
conditionally, forming a wide reaching network of progressively weak coupling. Conditional promises
propagate the range of value interaction [9,31]. This is the ecosystem effect.
The weakness of coupling is not a problem provided the city is reasonably homogeneous in density
over the timescales of the ensemble parameters, because of the assumption of overall sparseness. If we
define an effective density for the network, which describes some probable average level ρ∈[0,1] of
‘intercourse’ between agents (any kind of sustained relationship), then it is fair to write the value of a
network of promises:
Mean value =X
τ
NI
X
i,j=1
vC(πτ
ij ) = c ρ NI(NI−1).(60)
provided the total density of promises forms an SCC of order NImembers. This value can be distorted
from the quadratic form by significant inhomogeneity. Now, for most cities, NI1and ρ, αi1, so
for strictly positive value interactions:
v'c ρN 2
I.(61)
This is Metcalfe’s law. It depends on the assumption of strictly positive value (i.e. no non-profitable
interactions), and sufficient density of promises to involve everyone in the city who belongs to the infras-
tructure. Why is this plausible, when most specialization leads to modularity? One reason is that modu-
larity is only a separation of scales, not an elimination of dependency: dependencies form an ecosystem.
Nearest neighbours might hold the greatest semantic importance to a given function, but this reductionist
viewpoint is not independently sustained without the eigenstability of the entire web [30].
3.7 Conditional dependency as an explanation for multiple superlinearity classes
We can note briefly why certain system processes (or occupations, in a city) scale differently. In a special-
ization society, singular individuals or agencies rarely have all the prerequisites to complete their work
without assistance. They need to collaborate and depend on others. Thus other agents take on the role
of effective infrastructure for one another. It is the accessibility of this dependency on one another that
throttles output, and can modulate scaling behaviour.
20Metcalfe’s law does not refer to the cost/value of physical connectivity, which (once again) can be much sparser than a mesh at
low utilization. Indeed, that is important, else the net profit approaches zero. Rather, it refers to a correlation of agents’ promised
activities, linking their behaviours and generating value by interaction.
25
The argument for superlinear scaling in cities in [1] uses interactions, with arbitrary (unknown mean
field) range that can span the city, to model economic output. However, from a promise theory perspec-
tive, it is reasonable to ask the question: should we consider the output to be a fraction of N2, representing
the maximum output due to interactions (as in Metcalf’s law), or should we consider the output to come
directly from the agents, as a fraction of N, as in Amdahl’s law. This offers an explanation for the
anomalous superlinear exponents in the data for [2]. The superlinear scaling was initially associated with
‘innovation’ activities21. However, the promise theory shows how one does not need to invoke a process
of innovation to explain the scaling. To drive the long range cohesion of the whole community network,
specialists come to depend on specialized services (e.g. patents depend on lawyers). This leads to a
number of cases:
1. Interaction scaling: as proposed in [1], for interactive value creation.
ALab
+patent
−−−−→ Aobserver (62)
ALab
±interact
−−−−−→ Aservices (63)
Patent agencies are interacting at arbitrary range with a significant fraction the total promise graph,
as a part of the ecosystem.
Y'Y0v
VIN2→N1+δ'N7
6=N1.16.(64)
The amplified value relies on the interplay between long range mixing, and short range isolation.
2. Scarce agent scaling: skilled specialist experts’ output is proportional to the number of skilled
agents, since their queue is sparse, and not filled by a wide volume of demand. However, the same
economy of scale applies to their services when they are depended on, as ‘infrastructure’, by others.
Y'Y0v
VIN→Nδ'N1
6=N0.16.(65)
3. Interaction promises with a scarce dependency: such as in the case of a service that depends on
a source of agents to fulfill a dependency. e.g. patents can only be produced by labs that depend
on the outputs of specialized R&D employees and lawyers, working in private relationships, or in
secrecy. The expression in (76) assumes a promise configuration like that of the assisted promise
law [9], with a main output based on a number of agents that provide input. The dependencies
produce raw output, and the ‘lab’ agency collates and represents the collaborative mixing, e.g.
ALab
+patent|research,legal
−−−−−−−−−−−→ Aobserver (66)
ALab
±interact
−−−−−→ Aservices (67)
ALab
−research
−−−−−→ Astaff (68)
ALab
−legal
−−−→ Alawyer (69)
Astaff
+research
−−−−−→ ALab (70)
Alawyer
+legal
−−−→ ALab (71)
More generically, with two stages in the process of promise keeping, each experiencing scaling
(see figure 9),
21In [32] an explanation based on spanning tree branching processes was postulated, but was not credible.
26
R S D
−d
+d
−Y
AND
+Y|d
Figure 9: A two stage (long range) dependency has two economies of scale, when fed by a spacetime workflow.
The probability of promises kept is multiplicative, like the logical ‘AND’ of the promises.
S+Y|d
−−−→ R(72)
R−Y
−−→ S(73)
S−d
−−→ D(74)
D+d
−−→ S(75)
the total process picks up two ‘economies of scale’: the delivery of Yconditionally AND the
delivery of the conditional dependence.
Y'Y0v
VIN2×v
Vdepend N→N1+2δ'N4
3=N1.33.(76)
where D= 2 is used for the numerical values. These values accord better with the cited data in [2],
and tie in with the story about queueing.
What characterizes this interaction is the high level of specialization required to fulfill the depen-
dencies. If the network is sparse, this is more difficult than if it is dense and diverse. This is the
specialization gamble. With specialization comes individual efficiency, but also risk of instability
by disconnection from key dependencies [33].
They are a throttle on the process, because their absence could stop it altogether. Hence, we are
justified in using the product ‘AND’ for combining the probably values in (76). This is not a
(a) (b)
Figure 10: Structural recursion in an ecosystem is not like a branching process of containers (a), but rather the
agents overlap with other regions of the same network to access their virtual functions. Thus their outputs are not
concealed as interior substructure, but exposed as part of the flat internetwork (b). The result is that a second order
recursion picks up a second economy of scale, in turn increasing the superlinearity of the derived output.
hierarchical system interaction, because the services are not necessarily hidden from the long range
dynamics by internal components of the superagent ‘lab’ (see figure 10 (b)). But in organizational
theory, one normally assumes that all organizations are hierarchically organized (see figure 10 (a)).
27
4. Recursive promise dependency. Let’s consider what happens when the ecosystem network is
based on a hierarchy of interaction ranges, i.e. promises are made recursively in fully protected
shells. The agency produces a service using full community infrastructure, but also has some
specialist dependency contained entirely within (see figure 10 (a)).
Acompany
+Aspecialist
+solution
−−−−−→Aclient
−−−−−−−−−−−−−−−→ Aclient (77)
would be viewed as a recursive operation on the infrastructure, and the economics of scale would
apply to both times the (different) infrastructures were used.
Ypatent =N2
I
Vpatent
(78)
Vpatent =gR&D VR&D
NR&D 1
D
NR&D (79)
VR&D =V
NI1
D
(80)
Substituting for VR&D/NR&D from the last expression into the former,
Vpatent =gR&D V
NI1
D2
NR&D (81)
Inserting this into the output expression
Ypatent 'N1+ 1
D2−1
D(D+1) 'N13
12 'N1.08.(82)
This value is almost linear, which is what we might expect on a self-contained specialization, since
the outside world would not be able to tell the difference between a single agent and a single
superagent.
The value is also smaller than case (1) above, not larger, so short range hierarchical scaling cannot
explain the anomalously large exponents measured in cities. On the other hand, there are some
smaller exponents in this range. It is interesting to examine these measures from the perspective of
the promises represented, and their range in the embedding space.
There is a simple prediction here: long range interaction via dependency seems to increase output
superlinearity, by compounding ecomomies of scale, i.e. dependency brings strong long range coupling
and activates a larger amount of the N2mesh. A similar effect can be obtained artificially in [1], by
slightly increasing the Hausdorff dimension of the infrastructure H > 1. This does would correspond to
a more pervasive generic infrastructure network, which is an opaque explanation at best. It seems unclear
how to justify it.
Short range dependency is basically invisible at larger scales. This observation might help to explain
superlinear seen in technological contexts too, through coordination [20], but we have to be careful not to
mix together effects that come about due to higher dimensionality, with other mechanisms for increasing
the utilization of dependent resources.
3.8 Service discovery of dependencies in a network
The ability for agents to discover and match with other agents, that make complementary promises, is
the basis of functional scaling, and the semantics of cooperation and innovation. It depends on either
28
physical or virtual mobility of the agents. Kinetically, agents may follow a random walk, as in ballistic
discovery. A second possibility is that cheap intermediaries perform the discovery of specialized roles22 .
We can say that two agents are either
•Physically close.
•Virtually close.
Promise theory also suggests that they may be close in two ways: dynamically close or semantically close
(such as when related meanings are similar). The former depends on the length scales of the system (e.g.
city) and its structure. The latter can be assumed approximately independent of these scales, because the
carriers are very light, cheap, or fast (or, in the case of semantic distance, purely cognitive). If the cost of
discovery can be neglected, the cost equation is different: collaboration can be cheaper, and the value of
being in close proximity for a particular specialization is reduced23.
Directories, maps, and indices [7] are the keys for agents to virtualize discovery of dependencies, and
locate one another without physical search in spacetime. Telephone directories map coordinate addresses
to names. Yellow pages map coordinates to specializations. Similar specializations are grouped. Shop-
ping malls and industrial estates act as physical directories, where clients can expect to find services in a
small volume. Directories may be discovered themselves, or formed by voluntary registration. The value
of new bindings overcomes the tendency for similar specializations to repel one another: similar agents
may be attracted implicitly (covalently) by the intermediate attraction to clients. Apart from predicting
the importance of directories for smart cities, and organizations, this also predicts that the availability of
directory information could affects the productivity of a city as a function of size. This effect might not
be clearly visible in the ensemble data, since we would need to compare cities at the same value of N.
The scaling estimates of the city are based on infrastructure where physical motion of the population is
based on the cost of traversing some fraction of the length of the city. We can repeat the output calculation
to neglect this cost, as is the case in services that do not require physical transport.
•Physical interaction (transport/mobility): people move around using transport infrastructure to
experience their environment. There is a promise for people to observe their surroundings, for
something related to subject τ, and this promise is kept fractionally ατ∈[0,1] during their walk.
Let the linear range of the agent Aibe some dimensionless fraction per unit time rTexplore of the
size of the city V1/D, where ris the speed in units of city size24. If the density of impulses per unit
length of city region Iis assumed constant relative to the transport rate (because this is the basis
of commerce, i.e. what the city is trying to optimize for people’s finite time), then the number of
impulses Iτof type τ, experienced on such a walk, may be written:
Iτ∝riTexplore V1/D Iατ(83)
where ατis the probability that the person or agent will be receptive to impulses in its environment
that are relevant to promises of type τ.
Although there is room for inhomogeneous variations in the city regions, in the transport rate r,
and the density of offerings I, this will not change the average scaling argument much, as long as
Nis large. I make the assumption here that the density of experiences Iis constant, even though
the density of people is related to the city size. This is because the size of a city is constrained by
the time rather than the distance (and we are suppressing explicit time).
22Electrons play this role in molecular chemistry, or telecommunications in the human realm.
23A dependency does not just have to be discovered, but also maintained in a persistent relationship, which accumulates cost over
time.
24If the person’s path is detailed, one could include the Hausdorff dimension of the path and use vH/D
ias the range, as Bettencourt
suggests. I’ll ignore this for now, as humans do not tend to move in fractal paths, as his data suggest.
29
The range will be some fraction of the size of the city, available by transport infrastructure V1/D.
The cost of physically fishing for ideas thus takes the form
C'cYNIV1
D,(84)
in agreement with the work model of [1]. This applies for physical city interactions, and leads to
the same output scaling expression in [1].
Y+
Y'N2D+1
D(1+D)N
D2−D
D(D+1)
I,(85)
'N7
6
I1 + N0
NI5
6
.(86)
•Virtual interaction (teleport/messaging): people are immobile and send messages to one another,
watch entertainment, browse, read, talk, etc. These activities occupy an increasing amount of the
time spent by people, not least because it can easily be interleaved with work time. The rate is no
longer related to the size of the city, nor is there any obvious boundary to what can be discovered
online (since the range of the Internet is even more diverse than a city)25. In this case, the impulses
are more likely to be related to availability of the fountain itself (e.g. ‘bandwidth’ B) multiplied
the time spent.
Iτ∝B Texplore Iατ(87)
Discovery of information is the main issue. Before search engines, there were only directories
such as white pages (by person) and yellow pages (by promise type). However delivery of what is
discovered might still involve spatial constraints, e.g. locating a new car online does not allow it
to be teleported to the buyer’s location. However, 3d printing technology might change this, for a
class of problems, soon.
Here it is not the locations that matter, but the rate at which impulses are absorbed. Once again,
this is constant. When friends, books, or movies are communicating ideas to us, this happens at a
rate that depends only on how quickly we can get hold of a stream. How users discover locations
online, or by telephone is a separate question. Directories [7], advertisements, and chance all play
a role here. The cost of fishing for ideas is thus now independent of the city size. For a community
of multiple super-agents, the analogous expression is:
C'cNIBTexplore .(88)
If we imagine a community with no other infrastructure except its telecommunications network,
and substitute (88) into the detailed balance equation:
gYvY
VN2
I≥cNIBTexplore .(89)
Following through the calculation for the yield estimate identically, we find the scaling is no longer
superlinear (D= 2, H = 1)
Y'NH
DN(1−H
D)
I'N1
2
I1 + N0
NI−1
2
.(90)
25Because telecommunications networks are global, it does not make sense to relate their cost to the size of the city (though this
depends on exactly how we model the costs), so the cost depends more on its usage than on its extent. We simply assume that it
exists and has sufficient capacity for the NIconnected residents.
30
This simple result reflects the intuition that, if we neglect the ‘universal cost’ of telecommunications
from the community accounting, then the value generated as a result of collaborative processes is
proportional only to the fraction of participants who span the diameter of the city or community.
This reproduces the well-known result for mobile ad hoc networking (MANET) [21,34].
The rate of output based on trawling of ideas and gestation in closed workgroups will be
Iwork
τ=NDcphysIphys
τ+cvirtIvirt
τ(91)
and for the entire city of NWworkplaces:
Icity =X
τ
Iwork
τ'NWIwork.(92)
The physical channel agrees with Bettencourt’s expression for mixing volume in (4), where NI=
NDNW, and the virtual channel accords with mobile ad hoc networks. It would interesting to exam-
ine communities physically remote from services to see how these predictions match reality.
4 Remarks on technological infrastructure and collaborative net-
works
Cities are just one form of smart adaptive space, for which we now have a new and fascinating insight
in the form of statistical scaling data. Remarkably little data are available for computer installations,
software development, or smart warehouses, since these reside in the private sector; nonetheless, there
may well be insights to gain from studying the relationship between theory and practice, where we can.
Some results may have sufficient dynamical similarity to other cases to infer valuable lessons. Although
there are qualitative differences between biological organisms and cities, the main features that makes
computer systems different from cities are the size and timescales involved. Datacentres are still tiny
compared to cities (in terms of active agents N). Structural changes take place on the order of seconds
rather than weeks, and the rates increase as technology advances (see figure 11). The chief lesson we can
derive from cities, which might be applied to other smart infrastructure, is the involvement of spacetime
relationships in counting, at large N.
In [6, 7], a generalized abstraction of functional spaces was developed, starting from ‘atomic’ irre-
ducible considerations. By developing the city scaling theory in this framework, one could hope to bridge
the gap between disciplines, and promote future analysis of the effect of changing costs and technology. In
information technology, dynamical infrastructure, known as cloud computing, offering co-located shared
compute, storage and caching, as well as providing facilities for community software development, shared
repository models, and finally the pervasive Internet of Things, represent both present and upcoming chal-
lenges for infrastructure modelling.
4.1 Online communities versus infrastructure clusters
We must be clear about the difference between online communities and physical networks. It has not
escaped the notice of [3] that universality of cities implies network scaling in technology-assisted com-
munities, and some data concerning online communities has been examined with interesting and large
superlinear behaviour:
31
IT INFRASTRUCTURE COMMON CIT Y
N'10 −107
Functional modules
Long and short range interactions
<10−3sNET WORK ∆tinfra >103s
10−6s−1sAGE NT S ∆tinteraction 1s−104s
Seconds TR AN SP ORT T IM E Hours
software INNOVATIO N hardware
code, services PROMISES goods, services
code, memes, habits EPIDEMIC TRANSMISSION replicants, memes, habits
Membership SE MA NT IC E DG E Membership
Latency threshold DY NAMIC EDGE Density threshold
Servers, storage TE NAN CY U NI T Homes, offices, storage
Containers, hosts, private nets PARTITIONING cubicles, rooms, buildings
Process groups, clusters, datacentres SU PE R-AG EN CI ES cubicles, rooms, buildings
H= 1 −NTRAWL PATH DIMENSION HH = 1-2
1−3EMBEDDING DIMENSION D= 2 −3
Figure 11: Comparing cities with IT infrastructure.
MEA SU RE Average βSource
DNS hosts 1.28 ±0.06 Internet
Total web pages 2.03 ±0.1Internet
Active web pages 1.68 ±0.1Internet
Contributors 1.61 ±0.1Wikipedia
External links 1.59 ±0.2Wikipedia
Internal links 1.21 ±0.02 Wikipedia
Insufficient details are provided to suggest an explanation for the numbers26; however, a promise theory
approach like the one begun here, may easily play a role in understanding the structural dependences.
Online communities are sociologically interesting, but more practical, from a engineering viewpoint,
is to understand how scaling behaviour modulates productivity in smart infrastructure, that mixes physical
and virtual mobility. Such ‘smart spaces’ enact a kind of computation to optimize their conditions, and
even exhibit some algorithmic qualities. Users interact with hosted services, which are themselves a
community of (software) agents, acting as proxies for human intentions. This client interaction behaves
like a long-range weak coupling: an autonomous ‘gas’ of visitors27, and malls and directories act as
catalysts for value generating interactions. Applications and companies, on the other hand, consume
unique and shared resources as residents of the infrastructure. They have superagent boundaries around
company and functional concerns, which limit internal processes to short-range (and typically strong
coupling) interactions, in the manner of a tenancy [7]. Some brief remarks below concern what one
might expect to study and find in software agents that are proxies for human intent, and are enhanced
with significant automation.
26 [35] has made some comments on the distribution of community sizes for physical communities.
27In [36, 37], I showed how statistical behaviours in computers behave like a gas at equilibrium, with periodic boundary condi-
tions.
32
4.2 The productivity of agents in human-machine systems
Productive output, in agent communities, is driven either by output from agents working independently,
in parallel, or from the mesh of interactions between them. It is throttled by the contention for shared
resources (serialization or queueing) and the high cost of long range coherence (i.e. equilibration of state,
or mutual calibration). These are the main features captured in Gunther’s Universal Scaling [22].
Automation allows individual agents to generate much greater outputs, without the cost of coopera-
tion, so unequal automation might skew the measured performance outputs in empirical studies, making
it difficult to compare cities and other systems28. When forming a statistical ensemble of systems (cities,
organisms, or cloud infrastructure), we have to be sure to compare similar systems. A fundamentally
different technology base would undermine the universality of data for individual data points, and could
influence the interpretation of the scaling.
As pointed out in section 3.8, the impact of using messenger channels, like telecommunications rather
the physical mixing, is that the process of information and service ‘discovery’ is fundamentally different,
altering the costs by removing the imprint of physical scales from the output scaling. Caching becomes
a crucial enabler here, because it converts quadratic N2costs in to linear Noutputs from single agents
(thanks to smart behaviour).
For technology infrastructure, the analogue of discovery by wandering around a town browsing store
fronts, in the high street or shopping mall, is browsing a database, or directory (e.g. yellow pages, or
online shopping catalogue). Information spread by rumour and reputation, in a pre-telecommunications
communities, are supplemented by targeted information about recommendations (e.g. Amazon, Google,
and Facebook ads and search).
Productivity is not only increased by automation. The cost of transporting and equilibrating goods and
services remains. In information technology, larger and larger amounts of data are now stored in physical
and data warehouses. Distribution channels, for products and services are mirrored by technologies
for data replication, such as Paxos [38] and Raft [39]. These are now being adopted widely, based on
the sense that data equilibrium is important to avoid the inconsistency of ‘many worlds’ viewpoints,
when interacting with distributed systems. However, the scaling of equilibration software is necessarily
poor, since the cost scales superlinearly (typically like N2), to maintain consistent states [20]. Resource
‘latency’ or response time is a key issue, when dependencies are not local. This too can be improved by
caching. The cost of transporting data and materials from remote suppliers, either to a city is like the cost
of reaching across the planet to rent a virtual machine platform. Ultimately, no community wants to rely
on a fragile remote resource. A cheap non-scalable solution is better than an expensive long term one.
This has driven the centralization of large cloud installations, where economies of scale can be argued
locally, as transport costs and latencies are the responsibility of the clients.
Caching and replication are strategies that decouple long-range dependences, and restore agent au-
tonomy. It is cheap to cache and replicate many technology functions today, offering a kind of ‘smart
behaviour’ like a brain to keep interactions local. The mobility of small data is high, as transferring
small data is cheap. The mobility of computation has traditionally been low, as computers were large
and cumbersome machinery. But that is reversing, thanks to encapsulation methods (superagent tech-
nologies), like machine virtualization and so-called ‘containerization’ of software. Moreover, there is
computational processing capacity everywhere today, including on smartphones in our pockets, whereas
the sheer size of data being collected is growing to impractical levels for transportation. Computation is
progressing from being a shared resource, to a ubiquitous agent capability.
28Some notes to this effect were remarked in [10].
33
4.3 Separation of long and short range interactions (modules)
Limiting unnecessary mixing due to long-range or strong-coupling interactions is essential for establish-
ing modular functionality in networks. Modularity promotes specialization, and the quiet isolation for
the gestation of lengthy processes like learning and innovation. It makes economic sense, provided the
modules do not themselves become bottlenecks. Where humans are coupled strongly, the Dunbar limits
for human valency [40] intrude on scalable design, even with the help of automation.
What does this tell us about cloud computing, microservice communities, and the Internet of Things?
Suppose the economy of scale were only of the order of 85%, as in cities, this would not a huge incentive
to centralize, if local resources were available too. The economy would have to be offset against effects
like latency and long range dependence, equilibration, etc. As our interest in data from embedded sources
increases, and we equip environments with smart ecosystems [26], the idea that cloud computing will
be localized in large datacentres seems unrealistic. Latency costs suggest a greater delocalization of
workforce to eliminate long range interaction.
Managing specialization, or separation of concerns, is a human-technology issue that we are only
just starting to grapple with at scale [41]. Modularity has long been a part of system doctrine, but the
evolution of so-called ‘microservices’, or small, specialized software services, is now being motivated
empirically, by the limitation of Dunbar valency for human agents [40, 42]. Breaking a system into
many small specialist parts, each associated with a different human owner, incurs a new cost of service
interaction, but this cost may be paid cheaply with automation to alleviate a more expensive human
burden: the hard limit of what a human brain can cope with. The density of information services in
modern society is now huge, and the complexity of interactions is significantly more than humans can
manage unassisted. Technological agents can handle the greater numbers of interactions, but only if
they are sufficiently simple that a human collaboration can understand how to design the promises they
should keep. Coordination services for replicating siloed resource clusters are being developed, based
on the experiences of large industrial providers like Google [43]. These currently rely on long-range
data equilibration methods, which is serious throttle on their performance, and must become worse when
spacetime dimension plays a role.
Scaling issues pose the question: at what point should systems (cities, datacentres, communities)
break up into decentralized regions? Cloud datacentres grew up from the economies of scale that can be
achieved through specialized expertise in infrastructure management. We have already witnessed cloud
datacentres multiply like power stations, placed at strategic geographic locations, to cover distances. The
next logical step is ubiquitous infrastructure. At this point, the economic advantages of physical clustering
(indeed cities themselves) may disappear altogether, leaving only the value of centralized meeting places,
clubs, conferences, etc for human contact.
4.4 Semantic separation versus dynamical separation (silos)
Silos that encapsulate specializations are isolated semantic functions. The appearance of silos in human
organizations is often thought of as a negative phenomenon, because it represents an exclusion of outside
interests. However, it also has a necessary and positive effect, implying a separation of short range
interactions.
This is not the same as separation of dynamical scales. Voluntary partitioning to mitigate contention
has clear advantages, if resources permit. Conversely, confusing semantic separation with physical sepa-
ration may have dynamical consequences. An excellent example of this may be seen from town planning:
for a while ‘garden cities’ were a trend, designed with the aim of tidy separation of functions, separated
by green spaces. The principle backfired, however, e.g. in Brasilia [44], were different city functions
were so distributed that residents have to travel considerable distances to reach town facilities. This led
to severe traffic congestion.
34
When the cost of collaboration between partitioned agencies grows [33], business networks can de-
generate and become non-viable, as a larger part of the infrastructure becomes non-contributing. This
is a lesson for microservices. IT architects are starting to realize this now, and are opting for ubiquitous
‘hyperconverged’ architectures, i.e. a return to total package servers to reduce latency and congestion.
Sometimes regions form by themselves, from dynamical and semantic principles, with partitions based
on language, business, geographic centrality, eigenvector centrality [45], etc.
4.5 The mobility of agents in human-machine communities
In the past, sparse machine resources were immobile, and inputs were brought to the machine for pro-
cessing. It was cheaper to move inputs to machinery. Today, mobile devices with significant processing
capability are commonplace. and encompass computers and manufacturing (aka 3d printing). Particu-
larly, in information infrastructure, the ubiquity of stationary and mobile processing capacity reduces the
need for data to be sent over large distances. Conversely, the amount of data expected to come from
domestic and industrial sources will grow massively, decreasing relative mobility of data. Mobility of
processing resources is now a crucial issue, and is enabled by containment wrapper technologies that
form intentional superagent modules around specific functional roles.
The ubiquity and miniaturization of information technology suggests that the role of space dimension
may well be a temporary phenomenon, at least for local interactions. Even with ubiquitous information
infrastructure, there will still be some role for large datacentres, particularly in the realm of storage
archives, since disaster redundancy is one of the critical issues. Similarly, ‘data gravity’ proposes moving
the smaller resource to the larger resource, e.g. move computational power to large data instead of
assuming that computational power has to be at a fixed location and transporting data.
4.6 Scaling of voluntary cooperation versus imposition
Garbage collection, sewage, and drainage services are amongst the most important dependencies for a
system. They are sinks for pushed output: scaling results for these services would be interesting indeed.
One would expect them to be more susceptible to flash floods and long tail behaviours than the corre-
sponding supply networks, as they can only fail catastrophically when a threshold is reached [31]. It
would be very interesting to know how the converse of ‘push’ methods, or imposition services, scale with
city size: pull on demand, voluntary cooperation are popping up in society to replace mass broadcasting,
e.g. video on demand, replacing television broadcasts.
4.7 The rate of change in the system and utilization
The perception of time has an interesting relationship with long range order. Time is a local concept,
as both physicists and distributed systems specialists know only too well. Without long range data syn-
chronization, clocks may not tell the same time, and not just human clocks, but the pulses that drive and
clock processes29. For example, imagine an orchestra of musicians. The orchestra can play without a
conductor is everyone has enough to do all the time, but when every instrument plays a sparse role in the
whole, coordination is difficult as the agents are mostly dormant. Multiplexing allows better utilization
of a network. When all agents are busy, i.e. utilization is high, process time passes at a faster rate, relative
to its surroundings, and coordination becomes easier. Busy agents are ready to receive new inputs and
respond with services, suggesting that a highly utilized system could be more efficient, provided that it
lies below the queueing instability.
29Each change in an agent is a tick of the network clock, and each partitioned network has its own sense of time, and time only
moves forward according to that clock when something happens (this is essentially Einsteinian relativity [6]).
35
At high speed, the world is dominated by time, and is essentially one dimensional. In a world dom-
inated by space there is multi-dimensionality, and volumes can quickly become relevant in a continuum
limit, and time is suppressed by equilibration. The higher dimensionality promotes an increased average
connectivity for the promise graph, k(N)'Nδ(D), that depends on the dimension of the embedding
space D. In the complex interaction between space and time, all possibilities are on the table.
5 Summary
Universality and scaling are powerful notions in science. Having data about the scaling of functional
processes, at large and small N, offers an invaluable insight into what we can expect of technological
systems at scale, and their increasing intrusion into human society. Understanding social sciences in
terms of laws, analogous to physical law, is an area where progress has been made over the past century.
Understanding such patterns in ‘smart’ admixtures of humans and technology seems an even more rele-
vant challenge today [4]. An obvious question becomes: is there something that can be transferred from
the study of cities to other network and community systems, e.g.
•Online user communities—human communities in a virtual space.
•Humans-software communities—interacting processes in a virtual space.
•Microservice communities—collaborating agents in a virtual space, interacting with humans.
•Cloud computing—a shared infrastructure community of contending processes.
Datacentres and software systems share a lot of similarities with cities, but there are differences too.
Information infrastructure is largely one dimensional, in most locations. In datacentres infrastructure is
only just in the process of becoming two and even three dimensional. It is also quite inhomogeneous,
and one needs to know when and where dynamical similarities might be exploited to argue dynamical
similarity [26]. As Nbecomes large, issues of space and time become much more entwined with the
more common one-dimensional algorithmic behaviours generally studied in computer science.
Universality reveals emergent laws, on broad scales. However, a fuller understand of systems, whether
human cities, smart cities, computers, or any other human structure, is only achieved by describing both
dynamics and semantics at micro- and mesoscopic scales. Just as we cannot understand medicine without
understanding the functional roles of structures inside organisms, so the functional organs in a city are
key to what it does. The universal scaling arguments for urban areas, in [1, 2], are exciting discoveries,
as they point to the involvement of spacetime in functional systems, which is still poorly understood in
information and computer science.
Datacentres are still tiny compared to cities, but the two are also growing together thanks to the
pervasive spread of the Internet. By applying promise theory to expose some short range interactions, we
find more possible exponents that match quite well with the anomalous results in citebettencourt1 (see
section 3.7). The immediate applications of this approach leads to some basic observations:
•Value creation in communities comes from a mesh of promises whose outcomes funnelled, filtered
and powered by serialized processes for supply and harvesting.
•The sparse probabilistic utilization of shared infrastructure allows agents to interleave their efforts
and achieve economies of scale. This keeps a fluctuating city in an approximately stable, steady
state over short timescales, and admits limited long term growth.
•Specialization of tasks into modulaar services allows systems to focus their time and capabilities
without the cost of context switching. The strategy of specialization also brings fragility: if the
36
cost of reconnecting the specialists grows too large, the community can fail to keep promises for
essential functions [33].
•Increasing autonomy of a system population, due to the availability of personal assistants, and
localized capabilities (smartphones, 3d printers, etc) will undermine the need for transport, and the
involvement of city scales in many processes.
•Superlinear scaling results from a dependence on exterior specialist agencies. When remote depen-
dencies are involved, staged economies of scale can accumulate bringing superlinear effects, for
each remote dependence to the ensemble. If these external dependencies could be redistributed to a
purely autonomous attribute of each agent (e.g. when phone boxes are replaced by mobile phones),
then scaling would at best be linear, and the artefact of superlinearity would disappear.
•The ability to discover promised information, to mix it and select new combinations, is the basis
of innovation and collaboration. A highly discoverable ecosystem is ‘smart’ in the sense that it can
adapt and invent. Any space can, in principle, be made smart in this way, if its agencies make the
necessary infrastructure promises.
There are two main lessons to take from these notes, that may be surprising to technologists: (i) the
behaviour of a system can involve more than one spatial dimension, and (ii) we can measure and describe
just how smart and productive functional spaces actually are across a range of scales from local to global.
Acknowledgments
Many thanks to Luis Bettencourt for patiently reading and refining my understanding of his model. I’m
grateful to Noah Brier for an invitation to Transition in late September 2015, where I met Geoffrey West
and learned about scaling in cities. This work could not have happened but for that chance meeting in a
remote city. Also thanks to John D. Cook for carefully reading a draft.
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39
A The value of promises
The value of links in a network depends on the promises they make. The value of a promise is a form of
assessment [9] that any agent can make independently. We write an assessment of whether a promise was
kept
αi(Ai
b
−→ Ak)∈[0,1] (93)
to mean the assessment by agent Aithat the promise from Ajto Akwas kept. A valuation is an estimate
of what a promise is worth to an agent. This may of may not depend on the assessment of to what extent
the promise is kept or not. Every agent assesses on its own calibrated scale. If we want a common
currency valuation for all parties, this has to be calibrated by a single agent according to its scale.
The interpretation of value is also an individual judgement that relies on trust, and may be based on
accounting of the assessments over time (reputation) [46].
My reputation ∝X
you you −b
−−→ me(94)
In words, my reputation is proportional to all the number of ‘you’ who (publicly) promise to accept my
promised service. Even unilateral promises may have some value:
VALUATI ON B Y XABO UT P ROM IS E REA SO N FO R VALUE TO X
Me me +b
−−→ you An reputation building investment
Me you +b
−−→ me A service that might help me
You me +b
−−→ you A service that might help you
You you −b
−−→ me You need the service now
Cooperative relationships are usually based on conditional assistance [9], and take the form of a condi-
tional equilibrium:
S+S|M
−−−−→ R(95)
R+M|S
−−−−→ S(96)
S−M
−−→ R(97)
R−S
−−→ S. (98)
in words, Spromises Ra service, if it receives payment M; and Rpromises to pay Mif it receives
service S. In a network without trust, this is a deadlock. But if any agent trusts the other enough to go
first, it is a cyclic generator of a long term relationship. Such relationships imply lasting value, as known
from game theory (for a review see [26]). Both agents also promise that they will take (-) what the other
is offering unconditionally. This is a signal of trust. Valuations are not necessarily rational to anyone but
the agent that makes them, and are unrelated to cost.
The economic value is that something is exchanged, which requires a binding of both + and - promises.
S+S
−−→ R(99)
R−S
−−→ S(100)
40
Both agents recognize the value of the other party, so the value exchanged is proportional their assess-
ments that the promises were kept:
vS∝αS(S+S
−−→ R)αS(R−S
−−→ S)(101)
vR∝αR(S+S
−−→ R)αR(R−S
−−→ S).(102)
In a community where such transfers are made often and between arbitrary pairs of agents, standards
of valuation are equilibrated, and may be exchange in league with a calibration agency (e.g. a bank or
government). Thus, in a well-connected community, with a spanning infrastructure, we may posit that
the value of a one-way transfer is simply
vC(ΠS
ij ) = cSαiαj,(103)
where cSis the currency value of a perfect service relationship S, and αiis an impartial assessment of
the probability with which Aiwill keep its promise to give or receive S.
B Defining the edge of a city or community by membership promises
So far, we’ve skirted around the issue of what constitutes the city limit, in the definition of a city. The
size of the city plays a role in the measurements, and the model in [1] treats the city as an economically
inflated balloon, with an edge, so we must understand what constitutes the scope of the city or semantic
space. In fact the city is more like a club with membership than a balloon with an edge.
Consider how the network we call a city is reached from beyond. How does it communicate with the
outside world? Cities have roads and other infrastructure links in and out of the city (James Blish novels
notwithstanding). Should these links be treated as if they were part of the same infrastructure mesh as
the city itself? If so, where does the city start and end? The use of dynamical scaling arguments for
everything else suggests that there might be a dynamical ‘network’ answer to this question of boundary
conditions, but this is not the case.
Consider a simple thought experiment, in which we start with two separate cities and join them to-
gether by increasing the density of connections (see figure 12). As soon as a city is connected to an
Figure 12: A simple thought experiment: when do two cities merge into a single one? As we add more an more
link capacity, when does the scaling change from N2
1+N2
2to (N1+N2)2? Providing the links are not saturated, it
takes only a single link. So where what does the edge mean?
outside network, this extends the internal network. As long as the ‘external’ network has the capacity
41
to support the aggregate level of traffic from the population NI, then it is no different from the internal
infrastructure network, and we have simply extended the boundary of the city.
One proposal might be to look for the presence of a discontinuity in the network capacity, like a
threshold event horizon, at which the response time for a network interaction changed:
Rexternal Rinternal (104)
But this doesn’t quite make sense either. Some processes are fast and some are slow, even in an urban
centre (a letter posted may take longer to arrive than an office worker takes to process and even reply to
it, while a person can take the train across town in half that time)30. An ecosystem has a broad mixture
of timescales. Only by trying to separating weakly coupled agents can we compare relative timescales
for modular components. Moreover, the entirely local gestation or production time is usually the limiting
time factor in the economic processes considered here, not the transport or delivery times.
While physically plausible, this this is not how cities are defined. They are defined semantically,
by labelling of community membership. The simple explanation [6, 9, 29] is that a city is defined to be
that collection of agents that mutually promise to be members of the city, and that are accepted as such
by the city authorities. In practice, the population must register as residents, and they receive promises
of services (including tax collection). One assumes that the transient population of any city is a small
correction to this.
The autonomy of any observer counts in making the judgement of community boundary. Each agent
can (and will) judge independently whether it considers itself a part of a region or not. This begs the
question how collected data define communities, and they really have an edge or not. Can the scaling
laws be made to fit any interconnected network? The structural considerations in section 3.7 suggest that,
with the right understanding of functional structure, the universalities can be applied properly. These
issues have been highlighted in the empirical studies in [10, 13], where it is found that the scaling is
distorted by picking the ‘wrong boundary’ in urban regions.
As we try to apply the ideas to similar networks, such as IT infrastructure and online communities,
the role of the physical city as an entity becomes unimportant. It is rather the community that resides and
interacts within it that plays the major role of mixing. The details of the infrastructure play a role, but the
universality lies in the notion of a community.
C The phase of a community: mobility of agents, and interaction
catalysts
Agents exist a priori in an unbound state, effectively a gas phase, mixing with their changing surround-
ings. By promising constrained cooperation, they can voluntarily become part of a solid phase, interacting
only with fixed neighbours. Mobility of promising agents, in the gas phase, remains important in all func-
tional systems.
There are two kinds of networks in a city:
•Supply or delivery networks which are one-way flows from source to destination. These are mainly
branching processes, but may also have simple redundancy.
•Collaboration networks with two-way interactions between communicating agents. The agents can
be people, machines, companies, etc.
30An impartial approach, based on actual network topology, would be to use the ‘Archipelago method’, to defining regions of
network eigenvector centrality. Using a hill climbing to define natural regions that are seeded on very central nodes leads to well
defined regions [45]. The question is whether these have any semantic significance. There are two ways to do it: either based on the
shared infrastructure network, or on the virtual business networks that describe the outputs Y, by defining an effective adjacency
matrix based on promise bindings.
42
Collaborative networks are based in interactions. Chemistry demonstrates that such networks can be
realized in two ways: either by using fast messengers (electrons) between slow fixed molecules (like
covalence), or by moving faster molecules between slower interaction regions (like ionic), and we may
arrange the same methods in larger systems.
•People at fixed locations can use telephone or Internet to send messages (solid state agents).
•People travel between locations, carrying messages with them (agents as a gas).
These methods may be treated as different kinds of network in the model. Promise theory is a chemistry
for generalized semantics bindings.
The model proposed by [1] is close to the appearance of a kinetic theory, but the city is not a gas
with random motions in this model. It’s phase is not defined, because the physical realization of the
network is not defined. The movement of the population could responsible for forming links between
agents, i.e. transport via the infrastructure networks (like taxis and subway); or intermediate messenger
technologies could be responsible (such as post and Internet). The phase could play a role in the scaling
of the infrastructure network in general, constraining its degrees of freedom and range. A solid phase
limits the effective dimensionality D.
Modern cities comprise a fixed infrastructure in a mainly ‘solid state’, while the agents are sometimes
bound in a solid state, and sometimes free as a gas. People move around the city in the subway like
water, but increasingly they use messages, like ‘covalently’ bonded work-molecules rather than transport
pipes. Now that distance is less costly, the ease and speed of networking is reflected in the lower density
of modern cities, versus older cities where high density enabled ease of meeting, in a kind of primordial
soup of mixed intercourse.
Cities are not the only functional networks, of course. Any community could be completely mobile,
with no fixed address, such as online communities, coming together only in meeting places that play the
role of catalysts. This makes the mixing of skills and promises more efficient31. Catalysts for bringing
agents together, like social meeting places play an increasingly important role. Open source software is
one of the important outputs of the modern world, which happens completely outside cities32. In biology,
the infrastructure networks are in a fluid state, with functional agents (cells) transported suspension,
and their promises advertised by compatability molecules and receptors on their surfaces, like exterior
superagent promises [7], exactly analogous to small businesses. IT networks are mainly solid, or quasi-
solid, even when using mobile devices, as the messages move between the agents much faster than the
agents move themselves so the motion of devices is negligible for many purposes.
D Effective power law scaling from Amdahl’s and Gunther’s law
The Amdahl and Gunther scaling relations are workload scalings, not in the usual form of universal
scaling relations. Let’s consider how we might derive an approximate power law scaling relation from
these. If we want to know how the time fraction (speedup) scales for as a function of the number of
processors N, then we can compare Nwith γN . Then, we can write, for γ > 1
T(γN )
T(N)=σ+π
γN
σ+π
N
(105)
= 1 + 1
γ−1π
σn
1 + π
σn
.(106)
31Curiously, it is also believed in biology that the cooking of food is what made humans an efficient species, supplying energy to
fuel our large brains.
32Github and other version control repository virtual code libraries function as catalytic meeting places.
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Let δ=D
D+H, where D= 1 and H=π
σn , then, approximating as a binomial expansion,
T(γN )
T(N)= 1 −γ−1
γδ. (107)
'1−γ−1
γδ
. . . (108)
'γ−δ.(109)
Thus we have an approximate power law fit for large Ncompared to π/σ, and we can write
T(N)'T0N−δ.(110)
i.e. there is a marginal relative economy of scale for small N, which decays to an essentially scale
invariant constant result. If we allow, like Gunther, for the presence of equilibration, or mesh coherence
effects, then we could use the form
T(N) = σ+π
γN +κN, (111)
where κrepresents linear time take to poll each of the worker agents. This is the case where replication
and consistency are required. With this extra term, we have
T(γN )
T(N)=σ+π
γN +κγN
σ+π
N+κN (112)
= 1 + 1
γ−1π
σ+ (γ−1) κ
σN2
1 + π
σ+κ
σN2.(113)
Now the behaviour doesn’t separate cleanly, and there are two regimes of approximate power law scaling,
with something more messy in between. Using the same procedure as before, we get an anomalous term
for κ6= 0:
T(γN )
T(N)'1−γ−1
γδ
+κ(γ−1)N2
π+σN +κN 2(114)
'γ−δ(κσ, Nsmall)(115)
'γ(κ > 0, N large)(116)
(117)
So for large N, with κ > 0, we have simply
T=T0N, (118)
i.e. the scaling cost becomes linearly worse with increasing size.
When we compare these results to a spacetime scaling, it will become apparent that this takes the
approximate form of a scaling law in a one-dimensional spacetime D= 1, with Hausdorff dimension
H=π/σn < 1. This indicates that a serial workflow, with some parallelism, is essentially a one
dimensional problem, with some fractal complexity in its trajectory due to parallelism33. Interestingly, as
the parallelism increases, the duration of the fractal dimensionality shrinks to nothing. Thus the large N
limit for serial processing tends to squeeze the degrees of freedom in the system.
33Alternatively, if we think about the problem graph theoretically, we can also say that it behaves like a D=Ndimensional
space, and a trajectory with Hausdorff dimension H=π/sigma. In a graph, the node degree k=Nis the effective dimension
of spacetime at the point [6].
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