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The 'social brain hypothesis' for the evolution of large brains in primates has led to evidence for the coevolution of neocortical size and social group sizes, suggesting that there is a cognitive constraint on group size that depends, in some way, on the volume of neural material available for processing and synthesizing information on social relationships. More recently, work on both human and non-human primates has suggested that social groups are often hierarchically structured. We combine data on human grouping patterns in a comprehensive and systematic study. Using fractal analysis, we identify, with high statistical confidence, a discrete hierarchy of group sizes with a preferred scaling ratio close to three: rather than a single or a continuous spectrum of group sizes, humans spontaneously form groups of preferred sizes organized in a geometrical series approximating 3-5, 9-15, 30-45, etc. Such discrete scale invariance could be related to that identified in signatures of herding behaviour in financial markets and might reflect a hierarchical processing of social nearness by human brains.
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doi: 10.1098/rspb.2004.2970
, 439-444272 2005 Proc. R. Soc. B
W.-X. Zhou, D. Sornette, R. A. Hill and R. I. M. Dunbar
Discrete hierarchical organization of social group sizes
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Proc. R. Soc. B (2005) 272, 439–444
doi:10.1098/rspb.2004.2970
Published online 17 February 2005
Discrete hierarchical organization of social group
sizes
W.-X. Zhou
1,2
, D. Sornette
2,3,4
, R. A. Hill
5
and R. I. M. Dunbar
6
1
State Key Laboratory of Chemical Reaction Engineering, East China University of Science and Technology, Shanghai 200237,
China
2,3
Institute of Geophysics and Planetary Physics, and Department of Earth and Space Sciences, University of California,
Los Angeles, CA 90095, USA
4
Laboratoire de Physique de la Matie
`
re Condense
´
e, CNRS UMR 6622 and Universite
´
de Nice-Sophia Antipolis,
06108 Nice Cedex 2, France
5
Evolutionary Anthropology Research Group, Department of Anthropology, University of Durham, 43 Old Elvet,
Durham DH1 3HN, UK
6
British Academy Centenary Project, School of Biological Sciences, University of Liverpool, Crown Street,
Liverpool L69 7ZB, UK
The ‘social brain hypothesis’ for the evolution of large brains in primates has led to evidence for the coevolu-
tion of neocortical size and social group sizes, suggesting that there is a cognitive constraint on group size
that depends, in some way, on the volume of neural material available for processing and synthesizing infor-
mation on social relationships. More recently, work on both human and non-human primates has suggested
that social groups are often hierarchically structured. We combine data on human grouping patterns in a
comprehensive and systematic study. Using fractal analysis, we identify, with high statistical confidence, a
discrete hierarchy of group sizes with a preferred scaling ratio close to three: rather than a single or a continu-
ous spectrum of group sizes, humans spontaneously form groups of preferred sizes organized in a geometri-
cal series approximating 3–5, 9–15, 30–45, etc. Such discrete scale invariance could be related to that
identified in signatures of herding behaviour in financial markets and might reflect a hierarchical processing
of social nearness by human brains.
Keywords: social brain hypothesis; social group size; log-periodicity; fractal analysis
1. INTRODUCTION
Attempts to understand the grouping patterns of humans
have a long history in both sociology (Coleman 1964) and
social anthropology (Kottak 1991; Scupin 1992). While
these approaches have been largely sociological in focus,
attempts to understand grouping patterns in non-human
primates have had a largely ecological focus (see Dunbar
1988). However, there has been recent interest in the
extent to which group size and grouping patterns in
primates might be constrained by cognitive factors
(Dunbar 1992, 1998). The latter interests arise out of what
has become known as the ‘social brain hypothesis’.
The social brain hypothesis (Byrne & Whiten 1988;
Barton & Dunbar 1997) argues that the evolution of
primate brains was driven by the need to coordinate and
manage increasingly large social groups. Since the stability
of these groupings is based on intimate knowledge of other
individuals and the ability to use this knowledge to manage
social relationships effectively, the computational capacity
of the brain (presumed to be broadly a function of its size)
is assumed to impose a species-specific limit on group size.
Attempts to increase group size beyond this threshold must
inevitably result in reduced social stability and, ultimately,
group fission. Dunbar (1992, 1998; Joffe & Dunbar 1997;
also Sawaguchi & Kudo 1990) showed that there is a log-
linear relationship between social group size and relative
neocortex volume in primates, and argued that this
relationship reflected the computational capacity that any
given species could bring to bear on its social relationships.
Extrapolating these findings to humans led to the predic-
tion that humans had a cognitive limit of approximately
150 on the average number of individuals with whom
coherent personal relationships could be maintained (Dun-
bar 1993). Evidence to support this prediction has come
from a number of ethnographic and sociological sources
(Dunbar 1993). The fact that these relationships are not
simply a matter of memory for individuals but, rather, of
integrating and managing information about the constantly
changing relationships between individuals within a group,
is indicated by the fact that relative neocortex size corre-
lates with a number of core aspects of social behaviour and
socialization in primates (Byrne 1995; Pawlowski et al.
1998; Joffe 1997; Lewis 2000; Byrne & Corp 2004).
It has, however, always been recognized that both
human and non-human primate groups are internally
highly structured (e.g. Dunbar 1988). Further analyses
(Kudo & Dunbar 2001) have indicated that at least one
level of structuring (the grooming clique) also correlates
with neocortex size. While the significance of these tiered
groupings is not always apparent, there is strong prima facie
evidence to suggest that human social groups (like those of
other primates) consist of a series of sub-groupings
Author for correspondence (rimd@liverpool.ac.uk).
Received 29 April 2004
Accepted 29 September 2004
439
#
2005 The Royal Society
on May 14, 2011rspb.royalsocietypublishing.orgDownloaded from
arranged in a hierarchically inclusive sequence (Hill &
Dunbar 2003).
In this sequence, the core social grouping is the support
clique, defined as the set of individuals from whom the
respondent would seek personal advice or help in times of
severe emotional and financial distress; its mean size is
typically 3–5 individuals (Dunbar & Spoor 1995). Above
this may be discerned a grouping of 12–20 individuals
(often referred to as a sympathy group) that characteristi-
cally consists of all the individuals with whom one has
special ties; these individuals are typically contacted at least
once per month (Dunbar & Spoor 1995; Hill & Dunbar
2003). The ethnographic data on hunter-gatherer societies
(summarized in Dunbar 1993) point to a grouping of
30–50 individuals as the typical size of overnight camps
(sometimes referred to as bands); these groupings are often
unstable, but their membership is always drawn from the
same set of individuals, who typically number ca. 150 indi-
viduals. This last grouping is often identified in small-scale
traditional societies as the clan or regional group. Beyond
these, at least two larger-scale groupings have been
identified in the ethnographic literature: the megaband of
ca. 500 individuals and the tribe (a linguistic unit,
commonly of 1000–2000 individuals) (Dunbar 1993).
In this paper, we provide the first systematic analysis of
human grouping patterns, using data collated from the
literature. Using spectral analysis, we show that there is a
consistent pattern in the size of these groupings and, more
importantly, that successive groupings in the hierarchy
have a constant ratio.
2. MATERIAL AND METHODS
There is no universally accepted procedure for analysing human
social groups, and all methods attempting to identify group sizes
suffer from at least some sources of bias (small sample size, large
inter-individual variability or differences in the criteria used to
include individuals). Our strategy is to include all reasonable data
and attempt to extract useful signals above the noise level by a
careful analysis of the global dataset. We therefore searched the
sociological and other literatures for quantitative data on social
group and social network sizes in humans. For these purposes, we
sought studies that provided quantitative data on the size of indivi-
duals’ social networks, irrespective of how the social network itself
was defined.
Most such studies focus on a particular kind of network (among
those defined above in x 1). In addition to the data listed in
Dunbar (1993), Dunbar & Spoor (1995) and Kudo & Dunbar
(2001), we add the following data. The USA 1998 General Social
Survey reports a mean size of 3.3 for support cliques in the USA
(Marsden 2003). The mean sizes of sympathy groups are reported
by Buys (1992) to be 14.0 in Egypt, 15.1 in Malaysia, 13.5 in
Mexico, 13.8 in South Africa and 10.2 in the USA (Latkin et al.
1995). In separate samples in The Netherlands, they were repor-
ted to be 15.0 in 1995 (Kef 1997; Kef et al. 2000), 15.0 in 1992,
14.3 in 1992–1993, 14.8 in 1995–1996 and 14.2 in 1998–1999
(van Tilburg & van Groenou 2002), finally, Adams et al. (2002)
reported them to be 14.4 in Mali (West Africa). Although a num-
ber of these studies have been carried out in the same country, we
have considered each study to be an independent sample since
they involve different datasets; nevertheless, averaging
The Netherlands samples and treating them as a single data point
does not alter the conclusions drawn.
Only one study sought to estimate the size of successive social
groupings for individual subjects (Hill & Dunbar 2003). These
data were obtained from an analysis of Christmas card
distribution lists, in which 42 UK-domiciled subjects logged the
identities of all individuals in the households to which cards were
sent and their relationships to these individuals. Participants were
asked both to list everyone in the household to which they were
sending a card and to state the quality of their relationship with
each individual (using two metrics: how often they contacted the
individual, and the emotional intensity of the relationship scored
on a 0–10 Likert-type scale: for details, see Hill & Dunbar
(2003)). Because this study uniquely provides data on the differ-
ent grouping levels of which any one individual is a member, we
treat these data separately from the census data obtained from the
literature search.
3. RESULTS
We begin by analysing the data on groupings reported in
the social networks literature. (The Christmas card distri-
bution data will be dealt with separately: see below.)
Figure 1 plots the sizes of the different grouping levels
identified in the various studies.
We begin with a qualitative analysis of the data in figure
1, using the groupings that have conventionally been
defined (see x 1). First, we denote S
1
as the mean support
clique size, S
2
the mean sympathy group size, S
3
the mean
band size, S
4
the mean community group size, and S
5
and
S
6
the mean sizes of mega-bands and large tribes, respect-
ively. Averaging across these grouping levels, the data give
mean values of S
0
¼ 1 (individual or ego), S
1
¼ 4:6,
S
2
¼ 14:3, S
3
¼ 42:6, S
4
¼ 132:5, S
5
¼ 566:6 and S
6
¼
1728. To determine the possible existence of a
discrete hierarchy, we construct the series of ratios S
i
/S
i1
10
0
10
1
10
2
10
3
10
4
0
5
10
15
20
25
30
network sizes
references
Figure 1. Presentation of our dataset of 61 group sizes. The
ordinate is an arbitrary ordering of data sources and the
abscissa gives the group sizes reported in each source. The
symbols refer to the classification used in each of the studies:
circles (support cliques), triangles (sympathy groups),
diamonds (bands), stars (cognitive groups), and squares
(small and large tribes). This classification is not used in our
systematic analysis summarized in the other figures, to avoid
any bias.
440 W.-X. Zhou and others Social group size organization
Proc. R. Soc. B (2005)
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of successive mean sizes:
S
i
=S
i 1
¼ 4:58, 3:12, 2:98, 3:11, 4:28, 3:05,
for i ¼ 1, ...,6: ð3:1Þ
This suggests that humans form groups according to a
discrete hierarchy with a preferred scaling ratio between 3
and 4 (the mean of S
i
/S
i1
is 3.52).
To avoid any biases that might be present in
the published census data, we next undertake a more
systematic analysis that uses all the available data rather
than just their means. The sample in figure 1 has 61 group-
ing clusters (including the ego) with estimates of mean size
s
i
available for i ¼ 1,2, ...,61 clusters. We consider this
sample to be a realization of a distribution whose sample
estimation can be written as:
fsðÞ¼
X
61
i ¼1
d s s
i
ðÞ, ð3:2Þ
where d is Dirac’s delta function. Figure 2 shows the prob-
ability density function f(s) obtained by applying a Gaus-
sian kernel estimation approach (Silverman 1986).
Our challenge is to extract a possible periodicity in this
function in the ln(s) variable, if any. If the grouping clusters
form a series of harmonics, the harmonics will have a con-
stant ratio, and we would expect a periodic oscillation of
f(s) expressed in the variable ln(s) (known as its ‘log-period-
icity’; Sornette 1998).
Standard spectral analysis applied to f(s) is dominated
by the trend seen in figure 2, with a peak at a very low log-
frequency corresponding to the whole range of the group
sizes. We thus turn to generalized q-analysis or (H, q )-
analysis (Zhou & Sornette 2002a), which has been shown
to be very sensitive and efficient for such tasks. The q-
analysis is a natural tool to describe discrete scale invar-
iance (DSI) in fractals and multifractals (Erzan 1997;
Erzan & Eckmann 1997). The (H, q )-analysis consists in
constructing the (H, q )-derivative
D
H
q
fs
ðÞ
¼
fsðÞfqsðÞ
1 qðÞs½
H
: ð3:3Þ
Introducing an exponent H different from 1 allows us to
detrend f(s) in an adaptive way (that is, detrend it with dif-
ferent values of [(1 q )s]
H
at different s values). Note that
the limit H ¼ 1 and q ! 1 retrieves the standard definition
of the derivative of f. A value of q strictly less than 1 makes it
possible to enhance possible discrete scale structures in the
data. To keep a good resolution, we work with
0:65 6 q 6 0:95, because smaller values of q require more
data for small values of s. To put more weight on the small
group sizes (which are probably more reliable since they are
obtained by conducting general surveys in larger represen-
tative populations), we use 0:5 6 H 6 0:9. A typical (H,
q)-derivative with H ¼ 0:5 and q ¼ 0:8 is illustrated in a
semi-log plot in figure 3.
We then use a Lomb periodogram analysis (Press et al.
1996) to extract the log-periodicity in f(s). Figure 4
presents the normalized Lomb periodograms of D
H
q
fsðÞfor
different pairs of (H, q ) with 0:5 6 H 6 0:9 and
0:65 6 q 6 0:95. This figure illustrates the robustness of
our result. For the specific values H ¼ 0:5 and q ¼ 0:8
shown in figure 4, the highest peak is at x
1
¼ 5:40 with
height P
N
¼ 8:67. The preferred scaling ratio is thus
k ¼ exp 2p
=
x
1
ðÞ3:2. The confidence level is 0.993
under the null hypothesis of white noise (Press et al. 1996).
If the underlying noise decorating the log-periodic struc-
ture is correlated with a Hurst index of 0.6, the confidence
level decreases to 0.99; if the Hurst index is 0.7 (which cor-
responds to an unreasonably large noise correlation), the
confidence level falls to 0.85 (Zhou & Sornette 2002b).
The Lomb periodograms also exhibit a second peak at
x
2
¼ 9:80 with height P
N
¼ 5:48. This can be interpreted
as the second harmonic component x
2
2x
1
of the funda-
mental component at x
1
¼ 5:40. The amplitude ratio of
the fundamental and the harmonic is 1.26. The coexistence
of the two peaks at x
1
and x
2
2x
1
strengthens the stat-
istical significance of a log-periodic structure. To see this,
we constructed 10
4
synthetic sets of 61 values uniformly
distributed in the variable ln(s) within the interval [0,
ln(2000)]. By construction, these 10
4
sets, which are
exactly of the same size as our data and span the same
interval, do not have log-periodicity and thus have no
characteristic sizes. We then applied the same procedure as
10
–1
10
0
10
1
10
2
10
3
10
4
0
0.1
0.2
0.3
0.4
0.5
0.6
s
f (s)
Figure 2. Probability density function f(s) of size s estimated
with a Gaussian kernel estimator in the variable ln(s) with a
bandwidth h ¼ 0:14. Varying h by 100% does not change f(s)
significantly.
10
0
10
1
10
2
10
3
10
4
–0.20
–0.15
–0.10
–0.05
0
0.05
0.10
0.15
0.20
s
D
q
f (s)
H
Figure 3. Typical (H, q)-derivative D
H
q
f(s) of the probability
density f(s) as a function of size s with H ¼ 0:5 and q ¼ 0:8.
Social group size organization W.-X. Zhou and others 441
Proc. R. Soc. B (2005)
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for the real dataset to these synthetic datasets and obtained
10
4
corresponding Lomb periodograms. Finally, we per-
formed the following tests on these Lomb periodograms:
find the highest Lomb peak (x, P
N
). If P
N
> 8:5, check if
there is at least another peak at 2x
^
1 with its P
N
larger
than 5.5. A total of 238 sets among the 10
4
passed the test,
suggesting a probability that our signal results from chance
equal to 0.024. The probability that there are at least two
peaks (one in 4:9 < x < 5:9 with P
N
> 8:5 and the other in
9:5 < x < 11:5 with P
N
> 5:5) is found equal to 77/10
4
,
giving another estimation of 0.992 for the statistical confi-
dence of our results.
Another metric consists in quantifying the area below the
significant peaks found in the Lomb periodogram of our
data and comparing them with those in the synthetic sets.
We count the area of the main peak of the Lomb period-
ogram at x and add to it the areas of its harmonics whose
local maxima fall in the intervals [(k ð1=5ÞÞx,
ðk þð1=5ÞÞx] for k ¼ 2,3, ..., around all its harmonics.
The area associated with a peak is defined as the region
around a local maximum delimited by the two closest local
minima bracketing it. The fraction of synthetic sets which
give an area thus defined larger than the value found for the
real data is 6–7%, depending on the specific values H and q
used in the analysis.
We applied the same analysis to individual social
networks based upon the exchange of Christmas cards
(Hill & Dunbar 2003). This study indicated that contem-
porary social networks might be differentiated based on the
frequency of contact between individuals, but that both
‘passive’ and ‘active’ factors may determine contact
frequency. Controlling for the passive factors (distance
apart, and whether the contact was overseas or a work
colleague) allowed the hierarchical network structure to be
examined based on the residual (active) contact frequency.
Starting from the residual contact frequencies, we
constructed their (H, q )-derivative with respect to the num-
ber of people contacted for each individual, obtained the
Lomb spectrum of the (H, q )-derivative and then averaged
them over the 42 individuals in the sample (figure 5). The
very strong peak at x ¼ 5:2 is consistent with the previous
results with a preferred scaling ratio from the expression k
¼ exp 2p=x
1
ðÞ3:3 (Sornette 1998) for the smaller
grouping levels in this study (i.e. group sizes below 150).
In summary, all these tests suggest that the evidence in
support of our hypothesis is significantly unlikely to result
from chance, but rather reflects the fact that human group
sizes are naturally structured into a discrete hierarchy with
a preferred scaling ratio close to 3.
4. DISCUSSION
Collating a variety of measures collected under a wide
range of conditions and in different countries, we have
documented a coherent set of characteristic group sizes
organized according to a geometric series with a preferred
scaling ratio close to three. The fact that the signature of
this scaling ratio comes through so strongly despite the fact
that the data derive from a variety of different small- and
large-scale societies suggests that it is very much a universal
feature. Were it the case that scaling ratios differed between
societies, pooling data would have tended to obscure any
relationships that might have been present.
Indeed, it turns out that similar hierarchies can be found
in other types of human organizations, although the
consistency of the patterning has not previously attracted
comment. Of these, the military probably provides the best
examples. In the land armies of many countries, one
typically finds sections (or squads) of ca. 10–15 soldiers,
platoons (of three sections, ca. 35), companies (3–4
platoons, ca. 120–150), battalions (usually 3–4 companies
plus support units, ca. 550–800), regiments (or brigades)
(usually three battalions, plus support; 2500þ), divisions
(usually three regiments) and corps (2–3 divisions). This
gives a series with a multiplying factor from one level to the
next close to three. Could it be that the army’s structures
have evolved to mimic the natural hierarchical groupings of
everyday social structures, thereby optimizing the cognitive
processing of within-group interactions?
0 5 10 15 20 25 30 35 40
0
2
4
6
8
10
12
ω
P
N
( )
ω
Figure 4. Normalized Lomb periodograms P
N
(x)asa
function of angular log-frequency x of the (H, q )-derivative
D
H
q
f(s) for different pairs of (H, q ) with 0:5 6 H 6 0:9 and
0:65 6 q 6 0:95.
0 10 20 30 40 50
0.5
10
1.5
20
2.5
30
3.5
40
4.5
ω
P
N
( )
ω
Figure 5. Average Lomb periodogram P
N
(x) of the (H, q )-
derivative D
H
q
f(s) with respect to the number of receivers of
the residual contact frequency for each individual in the
Christmas card experiment, as a function of the angular log-
frequency x of the (H, q )-derivative, over the 42 individuals
and different pairs of (H, q ) with 1 6 H 6 1 and
0:80 6 q 6 0:95.
442 W.-X. Zhou and others Social group size organization
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Stock market behaviour provides another example of the
same kind of phenomenon (and one that we happen to
have investigated). The existence of a discrete hierarchy of
group sizes may provide a key ingredient in rationalizing
the reported existence of DSI in financial time series in so-
called ‘bubble’ regimes characterized by strong herding
behaviours between investors (Sornette 2003). Johansen et
al. (1999, 2000) have proposed a model to explain the
observed DSI in stock market prices as resulting from a dis-
crete hierarchy in the interactions between investors.
Recent analyses of DSI in market regimes with a strong
herding component have also identified the presence of a
strong harmonic at 2x, similar to the findings reported here
( Johansen & Sornette 1999; Sornette & Zhou 2002).
Strong herding behaviour occurs only when groups of
investors coordinate their buy and sell orders; the coordi-
nated buy and sell orders that occur during a strong herd-
ing market phase thus expresses, better than at any other
time, the natural inner structure of the community of tra-
ders. By contrast, herding is absent when investors disagree
on what will be the next market move; as a result, the aggre-
gate market orders do not express the inner hierarchical
structure of the community.
The fact that DSI is found only during stock market
regimes associated with a strong herding behaviour sug-
gests that it may reflect the fact that a discrete hierarchy of
naturally occurring group sizes characterizes human inter-
actions whether they be hunter-gatherers or traders. The
findings reported here suggest that this discrete hierarchy
may have its origins in the fundamental organization of any
social structure and be deeply rooted within the cognitive
processing abilities of human brains.
When dealing with discrete hierarchies, it may be impor-
tant to distinguish between the specific group sizes and
their successive ratios. It may be that the absolute values of
the group sizes are less important than the ratios between
successive group sizes. If the ratio of group sizes is inter-
preted as a fractal dimension (specifically, the ratio is
related to the imaginary part of a fractal dimension: see
Sornette (1998) and references therein), this would imply
that, depending on the social context, the minimum
‘nucleation’ size (in the range 3–5 in previous examples)
may vary, but the ratio (close to three) might be universal.
The fundamental question, then, is to determine the origin
of this discrete hierarchy. At present, there is no obvious
reason why a ratio of three should be important.
Equally, however, we have little real understanding of
what mechanisms might limit the nucleation point to a
particular value. We do not even know, for example,
whether the constraint is a cognitive one (e.g. memory for
individual identities versus capacity to manage information
about relationships); or a time budgeting one (how much
time has to be invested in interaction with an individual to
create a bond of a particular strength, and then how many
such bonds can be fitted into a given time-scale). Nor do
we know much about how larger-scale groupings are built
up out of smaller ones. A hierarchical structure could, for
example, be built up by each individual interacting with,
say, three new individuals in an expanding network, or it
might be the result of rather discrete small subgroups held
together through a subset of individuals who act as ‘weak
links’ in the small-worlds sense—although there is some
evidence for the latter in respect of both primate social
groups (Kudo & Dunbar 2001) and at least some aspects of
human behaviour (Stiller et al. 2004). Considerable
additional work will need to be done on both these compo-
nents if we are to understand why these constraints on
human grouping patterns exist and exactly what their sig-
nificance might be.
Research by R.A.H. and R.I.M.D. was funded by the ESRC’s
Research Centre in Economic Learning and Social Evolution
(ELSE). R.I.M.D.’s research is supported by the British Acad-
emy Centenary Project and by a British Academy Research
Professorship.
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... These alters are grouped in additional external circles called mega-bands and large tribes. One of the most stunning facts about ego network circular structure is that the ratio between the size of adjacent circles appears to be a constant with a value around 3, and this holds true for ego networks of users belonging to various social environments, as shown in [24]. For a complete discussion about the properties of the ego network circles we refer the reader to [25]. ...
... In the table, we also show the average size of the obtained circles for online ego networks while, for offline networks, we report the values presented in [24], that summarise the properties of a large number of offline social networks obtained in diverse social environments. Despite the size of the circles in Facebook and Twitter ego networks appear to be very close to each other, it is worth to remind that they should not be compared directly. ...
... Looking at the scaling factors in Table 5, we can see that their values are very similar to each other and close to 3, for both Facebook and Twitter ego networks, and they are compatible with the results found offline. A scaling factor of three has been found in several offline social networks and it appears to be a fundamental property of human ego networks [24]. This result is another indication that Facebook and Twitter ego networks show a hierarchical structure remarkably similar to that found in offline environments. ...
Preprint
In the last few years, Online Social Networks (OSNs) attracted the interest of a large number of researchers, thanks to their central role in the society. Through the analysis of OSNs, many social phenomena have been studied, such as the viral diffusion of information amongst people. What is still unclear is the relation between micro-level structural properties of OSNs (i.e. the properties of the personal networks of the users, also known as ego networks) and the emergence of such phenomena. A better knowledge of this relation could be essential for the creation of services for the Future Internet, such as highly personalised advertisements fitted on users' needs and characteristics. In this paper, we contribute to bridge this gap by analysing the ego networks of a large sample of Facebook and Twitter users. Our results indicate that micro-level structural properties of OSNs are interestingly similar to those found in social networks formed offline. In particular, online ego networks show the same structure found offline, with social contacts arranged in layers with compatible size and composition. From the analysis of Twitter ego networks, we have been able to find a direct impact of tie strength and ego network circles on the diffusion of information in the network. Specifically, there is a high correlation between the frequency of direct contact between users and her friends in Twitter (a proxy for tie strength), and the frequency of retweets made by the users from tweets generated by their friends. We analysed the correlation for each ego network layer identified in Twitter, discovering their role in the diffusion of information.
... Each receiver has a table in which it associates, for any possible sender, its social closeness. Nodes estimate SC by embedding reference models of the structure of human social relationships, for example, the use of ego-network models [ZSHD05,HD03] for information diffusion in online social networks [DAC15], as discussed in Section 4.4. The primitive "IoP_send(data, {set of receivers}, SC ≥ x)" identifies IoP paths between the sender and the receivers, such that the last IoP node is "sufficiently close" to the receiver. ...
... As depicted in Figure 8, this results in a hierarchical structure of inclusive 'social circles' of alters around the ego with characteristic size and level of tie strength (the strength of the social relationship). Specifically, in the reference ego-network model [ZSHD05], there is an inner circle (called support clique) of 5 alters on average, which are considered the best friends of the ego. These alters are contacted at least once a week, and are the people from whom the ego seeks help in case of emotional distress or financial disaster. ...
... The sizes of ego network circles form a typical pattern of 5-15-50-150 alters, with a scaling ratio between adjacent circles around 3. This pattern is considered one of the distinctive features of human social networks. Evidence to support the existence of Dunbar's number and the described ego-network structure in offline social networks has come from a number of ethnographic and sociological sources [D93], [ZSHD05]. More recently, results have also been shown on the presence of Dunbar's number and the ego network structure in phone-call networks [MMLM13], [GQRZ17] and in online social networks [APCD15], [DACP15]. ...
Preprint
Cyber-Physical convergence, the fast expansion of the Internet at its edge, and tighter interactions between human users and their personal mobile devices push towards an Internet where the human user becomes more central than ever, and where their personal devices become their proxies in the cyber world, in addition to acting as a fundamental tool to sense the physical world. The current Internet paradigm, which is infrastructure-centric, is not the right one to cope with such emerging scenario with a wider range of applications. This calls for a radically new Internet paradigm, that we name the Internet of People (IoP), where the humans and their personal devices are not seen merely as end users of applications, but become active elements of the Internet. Note that IoP is not a replacement of the current Internet infrastructure, but it exploits legacy Internet services as (reliable) primitives to achieve end-to-end connectivity on a global-scale. In this visionary paper, we first discuss the key features of the IoP paradigm along with the underlying research issues and challenges. Then we present emerging networking and computing paradigms that are anticipating IoP
... The level of SC is a property of the social link (in the sense of Figures 3d and 3e), which models the trust between the IoP nodes connected through that link. Nodes estimate the SC of their links (more precisely, the social link between their users) by exploiting reference models of the structure of human social relationships, such as the egonetwork models [ZSHD05,HD03]. As said before, the IoP graph is a multilayer graph, where different types of links may exist among the users. ...
... As depicted in Figure 6, this results in a hierarchical structure of inclusive 'social circles' of alters around the ego with characteristic size and level of tie strength (the strength of the social relationship) [LQ17]. Specifically, in the reference egonetwork model [ZSHD05], there is an inner circle (called support clique) of 5 alters on average, which are considered the best friends of the ego. These alters are contacted at least once a week, and are the people from whom the ego seeks help in case of emotional distress or financial disaster. ...
... These people represent the social relationships that the ego maintains actively, spending a non-negligible amount of time and cognitive resources interacting with them so as to prevent the corresponding social relationships decaying over time. Evidence to support the existence of Dunbar's number and the described ego-network structure in offline social networks has come from a number of ethnographic and sociological sources [D93], [ZSHD05]. More Increasing social tightness recently, results have also been shown on the presence of Dunbar's number and the ego network structure in phone-call networks [MMLM13], [GQRZ17] and in online social networks [APCD15], [DAC15]. ...
Preprint
The cyber-physical convergence, the fast expansion of the Internet at its edge, and tighter interactions between human users and their personal mobile devices push towards a data-centric Internet where the human user becomes more central than ever. We argue that this will profoundly impact primarily on the way data should be handled in the Next Generation Internet. It will require a radical change of the Internet data-management paradigm, from the current platform-centric to a human-centric model. In this paper we present a new paradigm for Internet data management that we name Internet of People (IoP) because it embeds human behavior models in its algorithms. To this end, IoP algorithms exploit quantitative models of the humans' individual and social behavior, from sociology, anthropology, psychology, economics, physics. IoP is not a replacement of the current Internet networking infrastructure, but it exploits legacy Internet services as (reliable) primitives to achieve end-to-end connectivity on a global-scale. In this opinion paper, we first discuss the key features of the IoP paradigm along with the underlying research issues and challenges. Then, we present emerging data-management paradigms that are anticipating IoP.
... Indeed, there is considerable evidence to show that, within natural social networks, individual alters can be ranked in order of declining investment by ego (e.g. Saramäki et al. 2014) and that these rankings fall into a natural series of layers with a scaling ratio of ~3 that yields breakpoints at around 5, 15, 50 and 150 alters (Zhou et al. 2005, Hamilton et al. 2007). These layers correspond to marked differences in both the frequency of contact with alters and in rated emotional closeness (Roberts et al. 2009, Sutcliffe et al. 2012, seemingly reflecting a combination of temporal and cognitive constraints that give rise to the layered structure of networks. ...
... As we discuss hereafter, we have found an additional internal layer that has not previously been identified, which we denote as layer 0. The scaling ratio between layers are very close to 3 for all the k-means configurations and all the datasets. For comparison, Table 1 also gives the characteristic sizes of each of these layers in offline egocentric personal social networks, as determined from face-to-face contacts, for which Zhou et al. (2005) found a scaling ratio of ~3.2. Note that for the Twitter dataset, the results with k=5 match the offline layer sizes much better than those for k=4 (which tends to combine two of the middle layers) for the results found by k-means. ...
... Our analyses of three different online datasets confirm the layered structure found in offline face-to-face social networks. For all the online datasets, the scaling ratio for the various layers identified by the analyses, and the respective sizes of these layers, are extremely close to those observed in offline networks (Hill and Dunbar 2003, Zhou et al. 2005, Hamilton et al. 2007). These layers have previously been identified only from samples of quite modest size (grouping levels in small scale societies, Christmas card distribution lists: all N<<1000). ...
Preprint
We use data on frequencies of bi-directional posts to define edges (or relationships) in two Facebook datasets and a Twitter dataset and use these to create ego-centric social networks. We explore the internal structure of these networks to determine whether they have the same kind of layered structure as has been found in offline face-to-face networks (which have a distinctively scaled structure with successively inclusive layers at 5, 15, 50 and 150 alters). The two Facebook datasets are best described by a four-layer structure and the Twitter dataset by a five-layer structure. The absolute sizes of these layers and the mean frequencies of contact with alters within each layer match very closely the observed values from offline networks. In addition, all three datasets reveal the existence of an innermost network layer at ~1.5 alters. Our analyses thus confirm the existence of the layered structure of ego-centric social networks with a very much larger sample (in total, >185,000 egos) than those previously used to describe them, as well as identifying the existence of an additional network layer whose existence was only hypothesised in offline social networks. In addition, our analyses indicate that online communities have very similar structural characteristics to offline face-to-face networks.
... The number of active relationships is known as the Dunbar's Number and is around 150 both in offline and online social networks (Dunbar, 1993;Dunbar et al., 2015). The second aspect is the hierarchical organization of alters into concentric social circles around the ego (Zhou et al., 2005;Hill & Dunbar, 2003). These circles are hierarchically inclusive: each circle includes all the alters of its inner circles, whereas rings refer to the portions of circles that exclude inner circles. ...
... The typical average sizes of circles (Roberts et al., 2009;Dunbar et al., 2015) are 1.5, 5, 15, 50, and 150. The ratio between adjacent circles is around 3, and this is considered a key aspect of human ego networks (Zhou et al., 2005). It is also known that these structures do not change much over time (Saramäki et al., 2014), and in particular can also be found in popular online social networks, i.e., Facebook and Twitter (Dunbar et al., 2015). ...
Preprint
We analyze the ego-alter Twitter networks of 300 Italian MPs and 18 European leaders, and of about 14,000 generic users. We find structural properties typical of social environments, meaning that Twitter activity is controlled by constraints that are similar to those shaping conventional social relationships. However, the evolution of ego-alter ties is very dynamic, which suggests that they are not entirely used for social interaction, but for public signaling and self-promotion. From this standpoint, the behavior of EU leaders is much more evident, while Italian MPs are in between them and generic users. We find that politicians, more than generic users, create relationships as a side effect of tweeting on discussion topics, rather than by contacting specific alters.
... The third circle (affinity group) is composed by casual friends and extended family members, while the last layer (active network ) includes people with whom the individual has occasional social interactions. The ego network model has been successfully used to characterize the social relationships in both OSN platforms, like Facebook and Twitter [30,48], and offline social networks, based on surveys [49], phone call logs [29], and Bluetooth proximity [50]. However, to the best of our knowledge, their application to model social relationships in a combination of the two worlds has not yet been explored. ...
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Context modeling and recognition represent complex tasks that allow mobile and ubiquitous computing applications to adapt to the user’s situation. The real advantage of context-awareness in mobile environments mainly relies on the prompt system’s and applications’ reaction to context changes. Current solutions mainly focus on limited context information generally processed on centralized architectures, potentially exposing users’ personal data to privacy leakage, and missing personalization features. For these reasons on-device context modeling and recognition represent the current research trend in this area. Among the different information characterizing the user’s context in mobile environments, social interactions and visited locations remarkably contribute to the characterization of daily life scenarios. In this paper we propose a novel, unsupervised and lightweight approach to model the user’s social context and her locations based on ego networks directly on the user mobile device. Relying on this model, the system is able to extract high-level and semantic-rich context features from smartphone-embedded sensors data. Specifically, for the social context it exploits data related to both physical and cyber social interactions among users and their devices. As far as location context is concerned, we assume that it is more relevant to model the familiarity degree of a specific location for the user’s context than the raw location data, both in terms of GPS coordinates and proximity devices. We demonstrate the effectiveness of the proposed approach with 3 different sets of experiments by using 5 real-world datasets collected from a total of 956 personal mobile devices. Specifically, we assess the structure of the social and location ego networks, we provide a semantic evaluation of the proposed models and a complexity evaluation in terms of mobile computing performance. Finally, we demonstrate the relevance of the extracted features by showing the performance of 3 different machine learning algorithms to recognize daily-life situations, obtaining an improvement of 3% of AUROC, 9% of Precision, and 5% in terms of Recall with respect to use only features related to physical context.
... Consequently, each agent has at least five linkneighbours (i.e., social contacts) but may have more. This number is empirically validated, as we intend to represent the closest layer of intense contacts (Zhou et al. 2005;Hamilton et al. 2007;Mac Carron et al. 2016) 3 . Moreover, we have carried out sensitivity analyses and found a higher number of links to have little impact on overall simulation results. ...
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Inequality perceptions differ along racial and gendered lines. To explain these disparities, we propose an agent-based model of localised perceptions of the gender and racial wage gap in networks. We show that the combination of homophilic graph formation and estimation based on locally limited knowledge can replicate both the underestimation of the gender or racial wage gap that empirical studies find and the well-documented fact that the underprivileged perceive the wage gap to be higher on average and with less bias. Similarly, we demonstrate that the underprivileged perceive overall inequality to be higher on average. In contrast to this qualitative replication, we also show that the effect of homophilic graph formation is quantitatively too strong to account for the empirically observed effect sizes within a recent Israeli sample on perceived gender wage gaps. As a parsimonious extension, we let agents estimate using a composite signal based on local and global information. Our calibration suggests that women place much more weight on the (correct) global signal than men, in line with psychological evidence that people adversely affected by group-based inequities pay more attention to global information about the issue. Our findings suggest that (educational) interventions about the global state of gender equality are much more likely to succeed than information treatments about overall inequality and that these interventions should target the privileged.
... This parameter usually depends on the type of social interactions considered to model the social ties among the ego and the alters. While social networks in the physical world are commonly characterized by four layers [75], five circles are a common finding for OSN [76]. Since our objective is to model all the aspects of the users' social context in the mobile environment by taking into account social interactions in both physical and cyber worlds, we firstly need to find the optimal number of circles to use in our model. ...
Preprint
Full-text available
Context modeling and recognition represent complex tasks that allow mobile and ubiquitous computing applications to adapt to the user's situation. Current solutions mainly focus on limited context information generally processed on centralized architectures, potentially exposing users' personal data to privacy leakage, and missing personalization features. For these reasons on-device context modeling and recognition represent the current research trend in this area. Among the different information characterizing the user's context in mobile environments, social interactions and visited locations remarkably contribute to the characterization of daily life scenarios. In this paper we propose a novel, unsupervised and lightweight approach to model the user's social context and her locations based on ego networks directly on the user mobile device. Relying on this model, the system is able to extract high-level and semantic-rich context features from smartphone-embedded sensors data. Specifically, for the social context it exploits data related to both physical and cyber social interactions among users and their devices. As far as location context is concerned, we assume that it is more relevant to model the familiarity degree of a specific location for the user's context than the raw location data, both in terms of GPS coordinates and proximity devices. By using 5 real-world datasets, we assess the structure of the social and location ego networks, we provide a semantic evaluation of the proposed models and a complexity evaluation in terms of mobile computing performance. Finally, we demonstrate the relevance of the extracted features by showing the performance of 3 machine learning algorithms to recognize daily-life situations, obtaining an improvement of 3% of AUROC, 9% of Precision, and 5% in terms of Recall with respect to use only features related to physical context.
... Tras un rápido examen, un primer hecho que salta a la vista es que el tamaño promedio de la banda local (Grupo 2) es de solo 33,1 individuos. Este valor es bastante más bajo que el promedio ideal de 50 para este nivel de organización (Gamble 1999;Hamilton et al. 2007;Zhou et al. 2005). Son los Beatty de Nevada (2.000 msnm) los que presentan el promedio más bajo con 20 individuos (Steward 1938), ubicándose justo en la base del rango estimado para el nivel de banda local, de entre 20 a 70 personas (Wobst 1974). ...
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Ego networks have proved to be a valuable tool for understanding the relationships that individuals establish with their peers, both in offline and online social networks. Particularly interesting are the cognitive constraints associated with the interactions between the ego and the members of their ego network, which limit individuals to maintain meaningful interactions with no more than 150 people, on average, and to arrange such relationships along concentric circles of decreasing engagement. In this work, we focus on the ego networks of journalists on Twitter, considering 17 different countries, and we investigate whether they feature the same characteristics observed for other relevant classes of Twitter users, like politicians and generic users. Our findings are that journalists are generally more active and interact with more people than generic users, regardless of their country. Their ego network structure is very aligned with reference models derived in anthropology and observed in general human ego networks. Remarkably, the similarity is even higher than the one of politicians and generic users ego networks. This may imply a greater cognitive involvement with Twitter for journalists than for other user categories. From a dynamic perspective, journalists have stable short-term relationships that do not change much over time. In the longer term, though, ego networks can be pretty dynamic, especially in the innermost circles. Moreover, the ego-alter ties of journalists are often information-driven, as they are mediated by hashtags both at their inception and during their lifetime. Finally, we found that relationships between journalists are assortative in popularity: journalists tend to engage with other journalists of similar popularity, in all layers but especially in their innermost ones. Instead, when journalists interact with generic users, this assortativity is only present in the innermost layers.
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This paper examines social network size in contemporary Western society based on the exchange of Christmas cards. Maximum network size averaged 153.5 individuals, with a mean network size of 124.9 for those individuals explicitly contacted; these values are remarkably close to the group size of 150 predicted for humans on the basis of the size of their neocortex. Age, household type, and the relationship to the individual influence network structure, although the proportion of kin remained relatively constant at around 21%. Frequency of contact between network members was primarily determined by two classes of variable: passive factors (distance, work colleague, overseas) and active factors (emotional closeness, genetic relatedness). Controlling for the influence of passive factors on contact rates allowed the hierarchical structure of human social groups to be delimited. These findings suggest that there may be cognitive constraints on network size.
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Data on the number of adults that an individual contacts at least once a month in a set of British populations yield estimates of network sizes that correspond closely to those of the typical "sympathy group" size in humans. Men and women do not differ in their total network size, but women have more females and more kin in their networks than men do. Kin account for a significantly higher proportion of network members than would be expected by chance. The number of kin in the network increases in proportion to the size of the family; as a result, people from large families have proportionately fewer non-kin in their networks, suggesting that there is either a time constraint or a cognitive constraint on network size. A small inner clique of the network functions as a support group from whom an individual is particularly likely to seek advice or assistance in time of need. Kin do not account for a significantly higher proportion of the support clique than they do for the wider network of regular social contacts for either men or women, but each sex exhibits a strong preference for members of their own sex.
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The scientific study of complex systems has transformed a wide range of disciplines in recent years, enabling researchers in both the natural and social sciences to model and predict phenomena as diverse as earthquakes, global warming, demographic patterns, financial crises, and the failure of materials. In this book, Didier Sornette boldly applies his varied experience in these areas to propose a simple, powerful, and general theory of how, why, and when stock markets crash.Most attempts to explain market failures seek to pinpoint triggering mechanisms that occur hours, days, or weeks before the collapse. Sornette proposes a radically different view: the underlying cause can be sought months and even years before the abrupt, catastrophic event in the build-up of cooperative speculation, which often translates into an accelerating rise of the market price, otherwise known as a "bubble." Anchoring his sophisticated, step-by-step analysis in leading-edge physical and statistical modeling techniques, he unearths remarkable insights and some predictions--among them, that the "end of the growth era" will occur around 2050.Sornette probes major historical precedents, from the decades-long "tulip mania" in the Netherlands that wilted suddenly in 1637 to the South Sea Bubble that ended with the first huge market crash in England in 1720, to the Great Crash of October 1929 and Black Monday in 1987, to cite just a few. He concludes that most explanations other than cooperative self-organization fail to account for the subtle bubbles by which the markets lay the groundwork for catastrophe.Any investor or investment professional who seeks a genuine understanding of looming financial disasters should read this book. Physicists, geologists, biologists, economists, and others will welcomeWhy Stock Markets Crashas a highly original "scientific tale," as Sornette aptly puts it, of the exciting and sometimes fearsome--but no longer quite so unfathomable--world of stock markets.
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Conventional wisdom over the past 160 years in the cognitive and neurosciences has assumed that brains evolved to process factual information about the world. Most attention has therefore been focused on such features as pattern recognition, color vision, and speech perception. By extension, it was assumed that brains evolved to deal with essentially ecological problem-solving tasks. 1.