PreprintPDF Available
Preprints and early-stage research may not have been peer reviewed yet.

Abstract and Figures

A wealth of evidence shows that real world networks are endowed with the small-world property i.e., that the maximal distance between any two of their nodes scales logarithmically rather than linearly with their size. In addition, most social networks are organized so that no individual is more than six connections apart from any other, an empirical regularity known as the six degrees of separation. Why social networks have this ultra-small world organization, whereby the graph's diameter is independent of the network size over several orders of magnitude, is still unknown. Here we show that the 'six degrees of separation' are the property featured by the equilibrium state of any network where individuals weigh between their aspiration to improve their centrality and the costs incurred in forming and maintaining connections. Thus, our results show how simple evolutionary rules of the kind traditionally associated with human cooperation and altruism can also account for the emergence of one of the most intriguing attributes of social networks.
Content may be subject to copyright.
Why are there six degrees of separation in a social network?
I. Samoylenko,1, 2, D. Aleja,3, E. Primo,3K. Alfaro-Bittner,3E. Vasilyeva,1, 4 K. Kovalenko,1D. Musatov,1, 5 A.
M. Raigorodskii,1, 6 R. Criado,3D. Romance,3D. Papo,7M. Perc,8, 9, 10, 11 B. Barzel,12, 13, 14 and S. Boccaletti1, 3, 15, 16
1Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region, 141701, Russia.
2National Research University Higher School of Economics, 6 Usacheva str., Moscow 119048, Russia.
3Universidad Rey Juan Carlos, Calle Tulip´
an s/n, 28933 M´
ostoles, Madrid, Spain.
4P.N. Lebedev Physical Institute of the Russian Academy of Sciences, 53 Leninsky prosp., 119991 Moscow, Russia.
5Russian Academy of National Economy and Public Administration, pr. Vernadskogo, 84, Moscow, 119606, Russia.
6Moscow State University, Leninskie Gory, 1, Moscow, 119991, Russia.
7Department of Neuroscience and Rehabilitation, University of Ferrara, Ferrara, Italy.
8Faculty of Natural Sciences and Mathematics, University of Maribor, Koroˇ
ska cesta 160, 2000 Maribor, Slovenia.
9Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 404332, Taiwan.
10Complexity Science Hub Vienna, Josefst¨
adterstraße 39, 1080 Vienna, Austria.
11Alma Mater Europaea, Slovenska ulica 17, 2000 Maribor, Slovenia.
12Department of mathematics, Bar-Ilan University, Ramat-Gan, Israel.
13The Gonda Multidisciplinary Brain Research Center, Bar-Ilan University, Ramat-Gan, Israel.
14Network Science Institute, Northeastern University, Boston, MA., US.
15CNR - Institute of Complex Systems, Via Madonna del Piano 10, I-50019 Sesto Fiorentino, Italy.
16Complex Systems Lab, Department of Physics, Indian Institute of Technology, Indore - Simrol, Indore 453552, India.
(Dated: November 18, 2022)
A wealth of evidence shows that real world networks are endowed with the small-world property i.e., that
the maximal distance between any two of their nodes scales logarithmically rather than linearly with their size.
In addition, most social networks are organized so that no individual is more than six connections apart from
any other, an empirical regularity known as the six degrees of separation. Why social networks have this ultra-
small world organization, whereby the graph’s diameter is independent of the network size over several orders
of magnitude, is still unknown. Here we show that the ‘six degrees of separation’ are the property featured
by the equilibrium state of any network where individuals weigh between their aspiration to improve their
centrality and the costs incurred in forming and maintaining connections. Thus, our results show how simple
evolutionary rules of the kind traditionally associated with human cooperation and altruism can also account for
the emergence of one of the most intriguing attributes of social networks.
INTRODUCTION
In the short story ”Chains” (1929), the Hungarian writer
Frigyes Karinthy described a game where a group of people
was discussing how the members of the human society were
closer together than ever before. To prove this point, one
participant proposes that any person out of the entire Earth
population (around 1.8 billion at that time) could be reached
using nothing except each personal network of acquaintances,
betting that the resulting chain would be of no more than five
individuals. The story coined the expression ‘six degrees of
separation’ to reflect the idea that all people of the world are
six or fewer social connections apart from each other. The
concept influenced a great deal of early thought on social
networks, and was later generalized to that of ‘small world’
networks, where the maximal social distance (the diameter of
the network) scales logarithmically, rather than linearly, with
the size of the population [1].
After early studies on the structure of social networks
by Michael Gurevich [2] and Manfred Kochen [3], Stanley
Milgram performed his 1967 famous set of experiments on
social distancing [4, 5] where, with a limited sample of
a thousand individuals, it was shown that people in the
United States are indeed connected by a small number of
acquaintances. Later on, Duncan Watts recreated Milgram’s
experiments with Internet email users [6] by tracking 24,163
chains aimed at 18 targets from 13 countries and confirmed
that the average number of steps in the chains was around
six. Furthermore, many experiments conducted at a planetary
scale on various social networks verified the ubiquitous
character of this feature: i) a 2007 study by Jure Leskovec
and Eric Horvitz (with a data set of 30 billion conversations
among 240 million Microsoft Messenger users) revealed the
average path length to be 6 [7, 8], ii) the average degree of
separation between two randomly selected Twitter users was
found to be 3.435 [9], and iii) the Facebook’s network in 2011
(721 million users with 69 billion friendship links) displayed
an average distance between nodes of 4.74 [10].
Despite such abundant and consistent evidence, a clear
explanation of the mechanisms through which social networks
organize into ultra-small world states (where the diameter
does not depend on the system size over several orders of
magnitude), is still missing. Several significant questions also
remain unanswered. Namely, why does such a collective
property emerge? What are its fundamental mechanisms?
Why is the common shortest path length between units of
a social network six, rather than five or seven or any other
number, implying an average distance which is also not far
from six?
In this Article we rigourously show that, when a simple
compensation rule between the cost incurred by nodes in
arXiv:2211.09463v1 [physics.soc-ph] 17 Nov 2022
2
maintaining connections and the benefit accrued by the chosen
links is governing the evolution of a network, the asymptotic
equilibrium state (a Nash equilibrium where no further actions
would produce more benefit than cost [11]), features a
diameter which does not depend on the system’s size, and
is equal to 6. In other words, we theorematically prove that
any network where nodes strive to increase their centrality by
forming connections if and only if their cost is smaller than the
payoff tends to evolve into an ultra-small world state endowed
with the ’six degree of separation’ property, irrespective of its
initial structure.
RESULTS
A game theoretical model for network evolution
Consider the general case which is schematically depicted
in Fig. 1, where the Nnodes of a network Vare rational
agents of a game. At each step mof the game, each agent
vVselects (independently of the choices made by the other
agents at the same step) a potential neighborhood Nv(m)
made of kv(m)other nodes of V. The agent then decides
whether it is more profitable to form connections with the
nodes in Nv(m)or to remain connected with the nodes in
Nv(m1). The decision is based on a balance between the
payoff and the cost functions associated with the change of
neighborhood.
As for the cost function, we assume that node vpays a
unitary cost c > 0to maintain a connection with each node u
belonging to its neighborhood (and that node ucannot refuse
the connection paid for by v). Moreover, to be as generic as
possible, we either assume the unitary cost to be a constant, or
to depend on the network size as c=c(N).
As for the benefit function, if agents are rational, it is
logical to assume that their goal is to increase their importance
within the network. This can be naturally framed in terms of
betweenness centrality [12], which indeed provides a measure
of the influence exerted by a node on the information flow
within a network. This is defined as follows. First of all, if
vand sare two nodes of a connected network, the distance
l(v, s)is taken to be the number of edges forming the shortest
path between them. Then, the betweenness centrality (or
degree of mediation) BC(v)is taken to be Ps6=v6=t
σst(v)
σst ,
where s, t Vare all possible pairs of different vertices that
do not match with v,σst(v)is the number of shortest paths
between the vertices sand tpassing through the vertex v, and
σst is the total number of shortest paths between the vertices
sand t.
BC(v)quantifies how relevant the intermediary role played
by vin the graph is. However, one immediately realizes that
the contribution in BC(v)of the shortest paths in which vis
the unique intermediary between sand tis equal to that of
paths in which vis just one of a long chain of intermediaries.
To account for such a difference, one may adopt a generic
weighted version of the betweenness centrality, W B C(v),
actual friends potential new friends
?
How much would it cost
me connecting with the
new friends?
How much benet
would I get?
a
b
if benet #cost if benet > cost
FIG. 1: The game theoretical framework. The structure of a social
network evolves following simple rules of a game. Panel a: At each
step of the game, the individuals forming part of the network (like
the red woman in the picture) have to decide whether to stay with the
neighborhood formed by their actual friends, or to change to another
neighborhood formed by potential new friends. The current and new
neighborhoods may overlap (in our picture, the blue man and the
yellow woman are members of both sets). The decision is based on a
careful evaluation of the cost incurred and of the benefit gained with
the change. Panel b: The decision is merely utilitarian. If the benefit
is not overcoming the cost, then individuals maintain their current
neighborhood (left picture). If, on the contrary, the payoff exceeds
the cost, then individuals relinquish their current neighborhood and
move to the new one (right picture). The structure of the network
then evolves until converging to its Nash equilibrium (if it exists) i.e.,
to the configuration where no changes of neighborhood are allowed,
as no individual has anything to gain in abandoning acquaintances.
which is defined as
W BC (v) = X
s6=v6=t
σst(v)
σst ·f(l(s, t)),(1)
where fis a strictly decreasing function of its argument (as
longer paths must contribute less). One can think of Eq. (1) as
follows: each pair s, t of vertices creates some utility, which
is then distributed equally among all shortest paths from sto
t, and then each intermediary vertex in each path obtains a
fraction equal to f(l(s,t))
σst .
With these simple rules in mind, the Nagents play the
game. When the game converges to a Nash equilibrium
(a configuration where no agent has anything to gain
by changing its own neighborhood, as all of them have
already attained their optimal adjacency), we can demonstrate
rigorously that the obtained structure is endowed with the six
degrees of separation attribute.
3
B
A
CD
i
ii
iii
1st neighbors
1st + 2nd
neighbors
1st + 2nd + 3rd
neighbors
2-independent
set
1-independent
set
FIG. 2: l-independence of nodes. Sketch of a generic graph, with
node A at the center. The first, second and third neighbors of node A
are respectively located within the yellow, pink, and gray region. The
l-independent set of a graph is the set of nodes such that the distance
between any two of them is larger than l. The black nodes (A, B, C
and D) form the 2-independent set of the graph, as all of them are at a
distance larger than 2 from each other. The black nodes together with
the ones depicted in light blue form, instead, the 1-independent set.
Note that the light blue nodes do not participate in the 2-independent
set. Finally, the red nodes belong neither to the 1-independent set nor
to the 2-independent set.
2-independent sets and the emergence of ultra-small world
states
Before we demonstrate our main results, we first need to
introduce the concept of 2-independence of network’s nodes.
In traditional graph theory, a 1-independent set (or internally
stable set, or anti-clique) Sis a set of vertices such that any
pair of them isn’t connected by a graph’s edge. This is to say
that each edge in the graph has at most one endpoint in S. As
a consequence, any two vertices of Sare at a distance which
is strictly larger than one.
One can now generalize the latter definition, and designate
as a l-independent set Slthe set of network’s nodes such that
the distance between any pair of its members is larger than
l[13]. It follows that nodes belonging to Sldo not necessarily
belong to Sl+1 (see Fig. 2 for an illustrative sketch of the
comparison between a 2-independent set and a classical 1-
independent set).
Why are 2-independent sets important in our framework?
This can be understood by looking at Fig. 3. In panel a, the
three vertices 1,2,7 are originally part of a 1-independent set.
Now, if vertex 7 forms the yellow edges (7,1) and (7,2), it
is removed from the set but it does not change the distance
between nodes 1 and 2. It only contributes to the multiplicity
of shortest paths between nodes 1 and 2. As the number of
alternative shortest paths may be very large in large sized
networks, the minimum possible benefit obtained from gluing
a 1-independent set (as node 7 would do by forming edges
with nodes 1 and 2) may be very small with the growth of
12
3
4
5
6
ab
7
12
3
4
5
6
7
A
B
C
D
FIG. 3: Independence of nodes and Nash equilibrium. Panel
a: When only black links are considered, vertices 1,2,7 form a 1-
independent set. For consistency with Fig. 2, nodes 1 and 2 are
colored in light blue. As vertex 7 forms the yellow edges (7,1)
and (7,2) it is removed from the 1-independent set (this change is
depicted by coloring the lowest part of the node in yellow), but
the two new connections do not remove nodes 1 and 2 from the 1-
independent set, since they only contribute to the multiplicity of the
shortest paths between 1 and 2. Panel b: When only black links are
considered, vertices 1,2,7 form a 2-independent set. As the yellow
connections are formed, vertex 7 reduces the distance between nodes
1 and 2 from at least 3 down to 2. As a consequence, nodes 1 and
2 can only be part of a 1-independent set. For this reason, the upper
half of nodes 1 and 2 is depicted in black, indicating that these nodes
initially belonged to the 2-independent set, and the lower half in light
blue, indicating that by receiving the connections from node 7, they
become members of a 1-independent set. Node 7 is half colored in
black (as it initially belonged to the 2-independent set), and half in
yellow (as the two new connections remove it from all independent
sets).
the network’s size. From the latter point it follows that the
presence of independent sets of large size may be compatible
with the Nash equilibrium.
A totally different situation occurs when we consider 2-
independent sets, as in panel b of Fig. 3. Indeed, when
forming the yellow connections with nodes 1 and 2, vertex
7 is actually reducing their distance from at least 3 down
to 2. Therefore, regardless of which other edge exists in
the network involving vertices 1 and 2, vertex 7 receives a
minimum benefit equal to f(2). This is equally valid for any
other vertex of the 2-independent set which would form edges
with all other members of the set: it would receive at least the
same benefit from each pair of nodes in the set. Therefore, the
minimal benefit obtained from gluing a 2-independence set of
size xis x1
2f(2), which may be rather substantial. For this
reason, sizeable 2-independent sets cannot exist in the Nash
equilibrium.
The process of gluing large size 2-independence sets
is precisely what regulates the spontaneous emergence of
the six degrees of separation. Namely, it can be proved
theorematically that such a process determines that
i) at the Nash equilibrium the graph necessarily contains
a vertex vof very large degree, and
ii) node vis at the center of the network and displays the
remarkable property of being at a distance of no more
than 3 from any other node of the graph.
4
The latter implies that the shortest path between any pair of
nodes i, j in the graph will be smaller or equal than 6, as there
will be maximum 3 edges forming the shortest path from i
to vand maximum three edges also to form the shortest path
from vto j. Therefore, the diameter Dof the network will be
exactly 6.
The proofs of the Theorems and Lemmas involved are
available in the Supplementary Information Appendix (SI).
An illustrative case
For the sake of a better illustration , let us now focus on the
case described as follows.
i) The agents start the game when they are already
connected by means of a pristine graph where, in
addition, there exists at least one node with sufficiently
high degree.
ii) Each agent vadopts as benefit function
W BC (v) = X
s6=v6=t
σst(v)
σst ·1
l(s, t)α,(2)
with αbeing a strictly positive parameter. Comparing
with Eq. (1), this means that the weighting factor
is f(l(s, t)) = 1
l(s,t)α, and that Eq. (2) coincides,
for α= 1, with the classical weighted betweenness
centrality [12].
iii) Agents sequentially add new connections to their
neighborhood if and only if there is a positive
balance between the extra-utility brought by the new
connections and the extra-cost.
In practice, at each step mof the game, the potential
neighborhood Nv(m)of each agent vVis equal to
Nv(m1) plus pother nodes. The pnew edges are then
added only if W BC (v)pc i.e., only if the extra weighted
betweenness centrality is larger or equal to the extra cost pc.
When no agent is able to incorporate any further edge, the
network is said to have reached its asymptotic equilibrium.
It should be remarked that such a final state cannot formally
be associated to a Nash equilibrium, because the option of
removing existing links is not contemplated in the game, and
therefore there is no certainty that agents, in their asymptotic
states, are in their optimal adjacency configuration.
The following Theorem can be proved:
i) if vis a node of the pristine graph with koriginal
connections, and
ii) if H {3,4,5, ...}is some integer number strictly
larger than 2, and
iii) if, for the considered values of cand α, the inequality
1
2α1
(H+ 2)αkc, (3)
H
neighbors
a
b
(H +1)
neighbors
(H + 2)
neighbors
v u
v u
FIG. 4: The emergence of the ultra-small world state. Panel a:
Sketch of a hypothetical network where nodes vand uare separated
by a distance H+1. The neighbors of vare then at either H(the light
blue node), or H+ 1 (the green node s), or H+ 2 (the red nodes)
edges from u. For a better visualization, paths of different lengths
are marked with the corresponding colors. Panel b: A direct (yellow)
link is added between vand u. Our study demonstrates rigorously
(see Theorem 3 of the SI) that the network configuration of panel a
is incompatible with an equilibrium state.
is satisfied,
then, in the equilibrium state of the network, the node vis
linked to all other nodes of the graph by no more than Hlinks,
implying that the diameter of the equilibrium network does
not exceed 2H.
In practice, the theorem guarantees that the asymptotic state
of a network evolving from an initial condition that satisfies
condition (3) is an ultra-small world state (and, for H= 3,
also the emergence of the 6 degrees of separation property).
The proof of the Theorem (see SI for details) is given by
contradiction i.e., by supposing that there is a node uin the
final state of the network whose distance from vis at least
H+ 1 i.e., l(u, v)H+ 1. To better illustrate the situation,
we depicted in panel a of Fig. 4 the case in which nodes vand
uare separated by a distance H+ 1. In that circumstance,
the nodes directly connected to v(the neighbors of v) may be
found at either H(the light blue node), or H+ 1 (the green
node), or H+ 2 (the red nodes) edges from u. Looking at the
figure, it is easy to understand that all network’s shortest paths
which end in uand start in either the green or the light blue
node cannot pass through v. Therefore, the only contribution
to the benefit function of vfrom shortest paths ending in u
is coming from those paths which start in the red nodes, the
neighbors of vthat are at distance H+ 2 from u.
When one, instead, includes a direct link between vand u
[the yellow link in panel b of Fig. 4], then the shortest path
between any neighbor of v(denoted by w) and ubecomes
5
wvu, since H3. Calculating then the value of
W BC (v)corresponding to the addition of such a link,
and recalling that the equilibrium requires W BC (v)to be
smaller than the cost c, one easily get to an expression which
is in explicit contradiction with condition (3) (see the SI for
full details).
A realistic case
We shall remark that our approach is valid independently
on the specific degree distribution properties of the pristine
graph. However, the maximum degree of a node in a
scale-free network generated by the preferential attachment
method [14] is known to scale as N[15, 16] and this implies
that, for these networks, condition (3) is (from a given size on)
always verified for any value of fixed cost cand any value of
α, thus making them very good candidates for initializing the
formation of ultra-small world structures.
Therefore, to illustrate power and generality of the
above Theorem, we performed a massive numerical trial by
initializing our game on networks of Nnodes generated
with the Barab´
asi-Albert (BA) algorithm [14], for α= 1
(i.e., adopting as benefit the weighted betweenness centrality),
H= 3 and c= 0.15N(to ensure a coherent scaling of the
cost with that of the maximum degree in the network). With
these stipulations, condition (3) becomes 0.3k0.15N.
As k2N[15, 16], this means that condition (3) is verified
at each value of N, and one then expects that the diameter at
equilibrium would not exceed 6.
It is important to remark here that estimating the benefit
function (2) requires the retrieval of the global structure of
network’s pathways at each step of the game. However,
such information is in general not available to the agents
of real social networks. Indeed, computing Eq. (2)
becomes prohibitively costly as the size of the network
increases, requiring (with the fastest existing algorithms)
O(NL)operations (being Lthe total number of links in the
network) [17, 18].
For this reason, it is much more realistic and much less
computationally demanding to assume that agents only use
local information. We then consider a scenario wherein at
each step mof the game, a (large degree) node vis chosen. v
incorporates an edge with another node uif
a) 0.3kc, where kis the degree of v,
b) the distance between uand vis larger than 3.
In this way, it is only required to check that the subgraph
formed by vand its first and second neighbors has zero
overlap with the subgraph formed by uand its first neighbors,
and the method is not hurting for the knowledge of the overall
shortest paths’ structure. At the same time, the adoption
of local information makes our study’s main claims even
stronger, because it proves that a global network property (the
network diameter) may emerge as a result of a game in which
25
20
15
10
5
CDD
N
102103104
25
20
15
10
5
6
0 2 4 6 8 10
x103
N
CDD
Equilibrium state, ultra-small world
R
a
n
d
o
m
l
y
a
d
d
e
d
l
i
n
k
s
,
s
m
a
l
l
-
w
o
r
l
d
I
n
i
t
i
a
l
c
o
n
d
i
t
i
o
n
,
s
m
a
l
l
-
w
o
r
l
d
FIG. 5: The emergence of the six degrees of separation. Ensemble
average hDivs. Nfor different sets of networks. Light blue
line: BA scale-free networks that are used as initial conditions for
the evolution of the game. Green line: Networks generated at the
equilibrium state of the game. Red line: Networks constructed by
randomly adding to the initial condition of each game the same
number of links needed to reach the game equilibrium. A horizontal
black dashed line is positioned at D= 6 to indicate that the
network’s structure obtained at the equilibrium features the ultra-
small world property, with the concurrent emergence of the six
degrees of separation Inset: log-lin plot of hDivs. N. The
logarithmic scaling of the light blue and red lines is clearly visible.
agents only share local information, which is what actually
happens in almost all real circumstances.
Note that, if an edge connecting uand vis added, the above
conditions implies that W BC (v)c. Indeed, if the node
usatisfies condition b), it can easily be shown (using the same
arguments that the reader finds in the SI for the proof of the
Theorem) that the maximum contribution to vof the shortest
paths between uand a neighbor of vis 1/5. Adding the new
edge, such a contribution raises to 1/2, and this means that
W BC (v)0.3k,
where kis the number of connections of v. Therefore, if
condition a)holds, then condition W B C(v)cis also
satisfied. This implies that our local method is actually more
restrictive when incorporating edges, and yet sufficient to
give evidence of the predictions of the Theorem for nodes
satisfying condition (3).
The results of our simulation trial are presented in Fig. 5.
At each value of the network size N,10,000 different
realizations of a BA scale-free network are generated. The
ensemble average hDiof the value of the network diameter
6
is plotted as a light blue line in the figure, showing a small-
world behavior (a logarithmic scaling with N, well visible in
the log-lin plot of the inset).
Each of the generated networks is then taken as initial
condition for the evolution of our game, following the
conditions a) and b) described above, until reaching the final,
equilibrium state. hDifor the reached equilibria is reported as
a green line in the figure, and it is clearly seen that an ultra-
small world state emerges with hDi= 6 (a value marked by
an horizontal dashed line).
A legitimate objection is that adding links to a graph (and
therefore increasing the graph’s density) always results in
decreasing the network’s diameter, and therefore a proper
comparison has to be offered to assess the relevance of the
obtained results. For this purpose, in all trials we took diligent
note of the total number of links added before reaching the
equilibrium. Then, we took back the initial condition of the
specific trial, and added exactly the same number of links,
but this time in a fully random way i.e., without caring
about the fulfillment of the game conditions a) and b). The
obtained values of hDiare reported as a red line in the figure.
As expected, the red line is always located below the light
blue one, but the remarkable result is that the new network
ensemble maintains exactly the same logarithmic scaling with
N(once again well visible in the inset), which is destined to
depart more and more from the constant value characterizing
ultra-small world states and emerging at the equilibrium of
our game.
DISCUSSION AND OUTLOOK
One of the most captivating features observed in social
networks is that no individual is more than six connections
apart from any other, a regularity known as the six degrees
of separation. Such a feature is detected ubiquitously and
independently on the number Nof individuals forming the
network, pointing to the fact that the structure of these
networks radically differs from either that of regular networks
(where the diameter scales linearly with the size) and that of
classical small-world networks (where, instead, the scaling
law is logarithmic) [1].
The quest for the reasons underneath such a fundamental
deviation has, so far, focused on finding the relationship
between the scaling properties of distances in a graph and
those of the node’s degree distribution. It was indeed proved
that scale-free networks with degree distribution p(k)kγ
and 2< γ < 3(as it is observed in all real-world networks)
display a scaling of the diameter as Dln ln N[19], which
departs from the classic logarithmic scaling of small-world
networks and yet maintains an explicit dependence on the
network size N. On the other hand, scale-free networks
featuring an asymptotically invariant shortest path (called
Mandala networks [20]) may be synthesized, which however
have an associated value of γstrictly equal to 2 and therefore
do not match any case observed in real world.
Rather than being dependent on global (i.e., degree
distribution) scaling properties, our study suggests that the
fundamental mechanism behind such observed regularity may
be found, instead, in a dynamic evolution of the networks
ruled by a compensation rule between the cost incurred by
nodes in maintaining connections and the benefit accrued
by the chosen links. Namely, we demonstrated that a
simple evolutionary process through which the cost incurred
in forming connections is counterbalanced by the benefit
in terms of increased betweenness centrality is a possible
mechanism underlying the emergence of ultra-small world
networks with six degrees of separation. Our study points
out, therefore, that evolutionary rules of the kind traditionally
associated with human cooperation and altruism [2123] can
in fact account also for the emergence of this attribute of social
networks. Furthermore, we show that such a global network
feature can emerge even from situations where individuals
have access to only partial information on the overall structure
of connections, which is indeed the case in almost all social
networks.
The compensation of costs and benefits is certainly a natural
interaction strategy through which rational agents determine
their connections [2429], and therefore our study contributes
to the understanding of why the six degrees of separation
is such a ubiquitous property across vastly different social
networks. It is, moreover, reasonable to assume that a similar
evolutionary principle may also apply to the design of man-
made or technological networks [30]: take, for instance air or
sea transportation networks [3133], in which airports/ports
may increase their volume of trades and/or tourism industry
by ”being in between” the main routes of interchanges of
passengers and goods, and in doing so they are keen to incur
in the relative costs of maintaining (or even enlarging) the
number of local connections.
On the other hand, the units of biological networks are
in general not rational agents, and it is not straightforward
to argue that benefits in terms of betweenness centrality
shaped, for instance, the structure of metabolic, genetic or
brain networks along their million year long evolutionary
path [3436]. However, one cannot rule out that other
compensation mechanisms could have played a pivotal role in
this case too, with different benefit functions (e.g. resilience
to random perturbations or failures [37, 38], or local or global
efficiency [39]) recouping for the cost to form or maintain a
specific adjacency structure. In neural structures, for instance,
it is well known that the functional gains associated with link
formation must offset the associated structural costs [4042].
Note that, for neural structures, while this principle holds in
general at evolutionary and developmental time scales, it may
also take place at much shorter scales, comparable to those of
social network dynamics.
Finally, our study also sheds light on the so-called strength
of weak ties phenomenon. This concept was introduced by
Mark Granovetter who showed that the most common way
of finding a new job is through personal contacts with distant
acquaintances, and not via close friends, as one would instead
7
have expected [43, 44]. Distant acquaintances represent
links connecting different groups of people, and therefore
provide each individual with a unique way to receive useful
information about distant groups.
Formally speaking, weak ties are links connecting nodes
that were originally located at rather large distances and
they are therefore called bridges or local bridges (see
the discussion and references in Chapter 3 of Ref. [45]).
Their importance for social interaction and communication is
strongly supported by a wide range of studies [46, 47].
The formation of links connecting nodes from 2-
independent sets as the key to the emergence of the six
degrees of separation describes exactly the case of a local
bridge formation, i.e. a weak tie in Granovetter’s sense.
Therefore, our model can also be viewed as the game-
theoretical foundation for strength of weak ties.
Acknowledgments
Authors would like to thank Gonzalo Contreras-Aso and
Jorge Tredicce for many inspiring discussions. R. Criado and
M. Romance acknowledge funding from projects PGC2018-
101625-B-I00 (Spanish Ministry, AEI/FEDER, UE) and
M1993 (URJC Grant). M.P. was supported by the Slovenian
Research Agency (Grant Nos. P1-0403 and J1-2457). The
usage of the resources, technical expertise, and assistance
provided by the supercomputer facility CRESCO of ENEA
in Portici (Italy) is also acknowledged.
The two authors have contributed equally to the Manuscript
[1] Watts, DJ. & Strogatz, SH. Collective dynamics of ‘small-
world’networks. Nature 393, 440–442 (1998).
[2] Gurevitch, M. Ph.D. thesis (Massachusetts Institute of
Technology) (1961).
[3] de Sola Pool, I & Kochen, M Contacts and influence. Social
Networks 1, 5–51 (1978).
[4] Milgram, S. The small world problem. Psychology Today 2,
60–67 (1967).
[5] Travers, J. & Milgram, S. An experimental study of the small
world problem in Social Networks. (Elsevier), pp. 179–197
(1977).
[6] Dodds, PS., Muhamad, R. & Watts, DJ. An experimental study
of search in global social networks. Science 301, 827–829
(2003).
[7] Leskovec, J. & Horvitz, E. Worldwide buzz: Planetary-scale
views on an instant-messaging network, (Citeseer), Technical
report (2007).
[8] Sanderson, K. Six degrees of messaging (2008).
[9] Bakhshandeh, R., Samadi, M., Azimifar, Z. & Schaeffer,
J. Degrees of separation in social networks in International
Symposium on Combinatorial Search. Vol. 2, (2011).
[10] Backstrom, L., Boldi, P., Rosa, M., Ugander, J. & Vigna, S.
Four degrees of separation in Proceedings of the 4th Annual
ACM Web Science Conference. pp. 33–42 (2012).
[11] Nash Jr, JF. Equilibrium points in n-person games. Proceedings
of the National Academy of Sciences 36, 48–49 (1950).
[12] Freeman, LC. A set of measures of centrality based on
betweenness. Sociometry pp. 35–41 (1977).
[13] Fink, JF. & Jacobson, MS. On n-domination, n-dependence
and forbidden subgraphs in Graph theory with applications to
algorithms and computer science. pp. 301–311 (1985).
[14] Barab´
asi, AL. & Albert, R. Emergence of scaling in random
networks. Science 286, 509–512 (1999).
[15] Bollob´
as, B. & Riordan, OM. Mathematical results on scale-
free random graphs. Handbook of Graphs and Networks: from
the genome to the internet pp. 1–34 (2003).
[16] Flaxman A. & Frieze, A. T Fenner, High degree vertices
and eigenvalues in the preferential attachment graph. Internet
Mathematics 2, 1–19 (2005).
[17] Newman, ME. Scientific collaboration networks. ii. shortest
paths, weighted networks, and centrality. Physical Review E 64,
016132 (2001).
[18] Brandes, U. A faster algorithm for betweenness centrality.
Journal of Mathematical Sociology 25, 163–177 (2001).
[19] Cohen, R. & Havlin, S. Scale-free networks are ultrasmall.
Physical Review Letters 90, 058701 (2003).
[20] Sampaio Filho, C.IN., Moreira, AA., Andrade, R.FS.,
Herrmann, HJ. & Andrade, JS. Mandala networks: ultra-small-
world and highly sparse graphs. Scientific Reports 5, 1–6
(2015).
[21] Nowak, MA. Five rules for the evolution of cooperation.
Science 314, 1560–1563 (2006).
[22] Rand, DG., Arbesman, S. & Christakis, NA. Dynamic social
networks promote cooperation in experiments with humans.
Proceedings of the National Academy of Sciences 108, 19193–
19198 (2011).
[23] Rand, DG. & Nowak, MA. Human cooperation. Trends in
Cognitive Sciences 17, 413–425 (2013).
[24] Egu´
ıluz, VM., Zimmermann, MG., Cela-Conde, CJ. & Miguel,
MS. Cooperation and the emergence of role differentiation
in the dynamics of social networks. American Journal of
Sociology 110, 977–1008 (2005).
[25] Perc, M. & Szolnoki, A. Coevolutionary games—a mini review.
BioSystems 99, 109–125 (2010).
[26] MO Jackson. Social and economic networks. (Princeton
University Press), (2010).
[27] Christakis, NA. & Fowler, JH. Social contagion theory:
examining dynamic social networks and human behavior.
Statistics in Medicine 32, 556–577 (2013).
[28] Nishi, A., Shirado, H., Rand, DG. & Christakis, NA. Inequality
and visibility of wealth in experimental social networks. Nature
526, 426–429 (2015).
[29] Alvarez-Rodriguez, U., Battiston, F., de Arruda, GF., Moreno,
Y., Perc, M. & Latora, V. Evolutionary dynamics of higher-
order interactions in social networks. Nature Human Behaviour
5, 586–595 (2021).
[30] Holme, P. & Saram¨
aki, J. Temporal networks. Physics Reports
519, 97–125 (2012).
[31] Guimera, R., Mossa, S., Turtschi, A. & Amaral, LN. The
worldwide air transportation network: Anomalous centrality,
community structure, and cities’ global roles. Proceedings of
the National Academy of Sciences 102, 7794–7799 (2005).
[32] Barbosa, H., Barthelemy, M., Ghoshal, G., James, C.R.,
Lenormand, M., Louail, T., Menezes, R., and Ramasco, J.J.,
Simini, F. & Tomasini, M. Human mobility: Models and
applications. Physics Reports 734, 1–74 (2018).
[33] Lei, W., Alves, LG. & Amaral, LAN. Forecasting the
evolution of fast-changing transportation networks using
machine learning. Nature Communications 13, 1–12 (2022).
[34] Jeong, H., Tombor, B., Albert, R., Oltvai, ZN. & Barab´
asi,
AL. The large-scale organization of metabolic networks. Nature
407, 651–654 (2000).
[35] Bassett, DS. & Sporns, O. Network neuroscience. Nature
Neuroscience 20, 353–364 (2017).
[36] Zwir, I., Del-Val, C., Hintsanen, Mirka and Cloninger, KM
and Romero-Zaliz, R and Mesa, A and Arnedo, J and Salas, R
and Poblete, GF and Raitoharju, E., Raitakari, O., Keltikangas-
J¨
arvinen, L., de Erausquin, GA., Tattersall, I., Lehtim ¨
aki, T.
& Cloninger, CR. Evolution of genetic networks for human
creativity. Molecular Psychiatry 27, 354–376 (2022).
[37] Albert, R., Jeong, H. & Barab´
asi, AL. Error and attack tolerance
of complex networks. Nature 406, 378–382 (2000).
[38] Cohen, R., Erez, K., Ben-Avraham, D. & Havlin, S. Resilience
of the internet to random breakdowns. Physical Review Letters
85, 4626 (2000).
[39] Morone, F. & Makse, HA. Influence maximization in complex
networks through percolation. Nature 524, 65–68 (2015).
[40] Ram´
on y Cajal, S. Histologie du syst´
eme nerveux de l’homme
et des vert´
ebr´
es, 1909, english translation as Histology of
the nervous system of man and vertebrates (N. Swanson & L.
Swanson, Trans.). (Oxford University Press), (1995).
[41] Bullmore, E. & Sporns, O. The economy of brain network
organization. Nature Reviews Neuroscience 13, 336–349
(2012).
[42] Sterling,P. & Laughlin, S. Principles of neural design. (MIT
Press), (2015).
[43] Granovetter, MS. The strength of weak ties. American Journal
of Sociology 78, 1360–1380 (1973).
[44] Granovetter, MS. Getting a job: A study of contacts and
careers. (University of Chicago Press), (2018).
[45] Easley, D. & Kleinberg, J. Networks, crowds, and markets:
Reasoning about a highly connected world. (Cambridge
University Press), (2010).
[46] Burt, RS. Structural holes: the social structure of competition.
(Harvard University Press), (1992).
[47] Burt, RS. Structural holes and good ideas. American Journal of
Sociology 110, 349–399 (2004).
1
Supplementary Information:
Why are there six degrees of separation in a social network?
I. Samoylenko, D. Aleja, E. Primo, K. Alfaro-Bittner, E. Vasilyeva, K. Kovalenko, D. Musatov, A. M.
Raigorodskii, R. Criado, M. Romance, D. Papo, M. Perc, B. Barzel, S. Boccaletti.
In this Supplementary Information (SI), the reader finds all details of the theorematic proofs which are referred to in the main
text. The SI is divided in two main sections. The first Section contains some definition and preliminaries that are of use in all
Theorem and Lemmas and describes the results of the game theoretical approach presented in the first part of our Manuscript.
The second Section contains, instead, the details of the illustrative case (presented in the second part of our Manuscript) in which
nodes can only add links to their neighborhoods.
THE GAME THEORETICAL APPROACH
In our model, the Nnodes of a network Vare agents of a game. At each step mof the game, each node vVselects
(independently on the choices that may be made by the other agents in the same step) a potential neighborhood Nv(m)made of
kv(m)other nodes of V, and decides whether it is more profitable to form connections with all the nodes in Nv(m)or to remain
connected with the nodes in Nv(m1). The decision is based on compensation between the costs incurred in the change and
the payoff (or benefit).
To be more specific, at the step mof the game, each agent v(separately and independently) compares the configuration defined
by the adjacency matrix Am1(reflecting the state of the network after all agents have made their choice of neighborhood at
the step m1) with that of the adjacency matrix Am,u which is obtained from Am1by eliminating all the kv(m1) edges
between vand the members of Nv(m1) and adding instead all the kv(m)edges between vand the members of Nv(m).
If the benefit of the new configuration Am,u overcomes the costs of forming such kv(m)edges, then the agent vadopts the
neighborhood Nv(m), otherwise it remains linked with the members of Nv(m1). The step mof the game is concluded when
all the Nagents have made their decision and a new global network arrangement is produced, reflected by the adjacency matrix
Am, which is then used (by all agents) at the step m+ 1.
Definitions and preliminaries
Definition 1 The cost of forming a connection (u, v)is taken to be equal to some c(N)>0, which can be either a constant
or a generic function of the network size.
Definition 2. The distance between two vertices is the number of edges forming the shortest path between them.
Definition 3. The benefit function is taken to be W B C(v) = Ps6=v6=t
σst(v)
σst ·f(l(s, t)) [as in Eq. (1) of the main text],
where s, t Vare all possible pairs of vertices other than v,σst(v)is the number of shortest paths between the vertices sand t
passing through the vertex v,σst is the total number of shortest paths between vertices sand t,f(x)is an arbitrary (but positive
and strictly decreasing) function of the argument x, and l(s, t)is the length of the shortest path between vertices sand t.
Definition 4. A network satisfies the ultra-small world property if its diameter (the maximal distance between any pair of
network’s nodes) is bounded by a given value which is independent on the network’s size.
Definition 5. A 2-independent set Sis a set of network’s nodes such that the distance between any pair of its members is
larger than 2. As a consequence, each pair of nodes u, v Sisn’t connected directly by a network edge, nor a vertex wV
exists having simultaneously connections with uand v.
It is now necessary to make a couple of preliminary observations, that are of use in all demonstrations which are part of this
SI.
A first observation is concerned with the fact that, if a node v(which actual state satisfies W BC (v)qc, with qthe number
of neighbors) selects a potential new neighborhood where all previous connections are maintained and pnew neighbors are
2
s
l(s,t)>l(s,t)
original '
'
l(s,t)=l(s,t)
'
t
v
s
t
v
u
s
t
v
AB C
u
l(s,t)= 5
ss,t = 5
ss,t (v)=1
l(s,t)= 4
ss,t = 1
ss,t (v)=1
'
l(s,t)= 5
ss,t =5+ 3
ss,t (v)=1+ 3
FIG. 6: Schematic illustration of the two possible cases examined in Lemma 0. In all panels red and light blue curves are used to mark,
respectively, the shortest paths between sand t(whose number is σs,t ) and the shortest paths between nodes sand twhich pass through node
v(whose number is σs,t(v)). Panel A) shows an hypothetical graph Vwhere nodes sand t(colored in light blue) are separated by a distance
l(s, t)=5,σs,t = 5, and σs,t(v)=1. The next two panels illustrate the two possible cases originated by adding a link between nodes vand
u. Panel B) Case l(s, t)> l0(s, t). In this situation, both the distance between sand tand the number of shortest paths decrease. In particular,
the unique (new) shortest path between sand tis the one that passes through node v. Panel C) Case l(s, t) = l0(s, t). In this case three new
shortest paths appear (colored in green). Note that all the shortest paths colored in red are equal to the ones presented in panel A.
added and if such a new configuration is not accepted, it actually implies
W BC (v)< pc, (4)
where W BC (v)is the difference between W BC 0(v)(calculated with incorporating the pnew edges) and W B C(v). Indeed,
as the new strategy is rejected, it follows that
W BC 0(v)<(q+p)c,
where qwas the number of neighbors of vbefore incorporating the pnew edges. Moreover, as W BC (v)qc then, condition (4)
comes from the fact that
W BC 0(v)<(q+p)cW BC (v) + pc.
A second, important, observation comes from the content of the following Lemma 0, which states that the contribution to the
benefit function of vgiven by the shortest path between any two nodes (say sand t) never decreases when vacquires a new link.
Figure 6A illustrates the case in which node vaccrues an utility due to being intermediary in one of the shortest paths connecting
nodes sand t. If a link is now added between vand another node u, the benefit function for vchanges as
W BC 0(v) = X
s6=v6=t
σ0
st(v)
σ0
st ·f(l0(s, t)).(5)
where now all quantities are denoted by 0. Notice that the number of pairs (s, t)in Eq. (5) may change from the ones considered
in W BC (v)because of the added link.
Lemma 0. If a new link is added between two nodes uand vthen
σs,t(v)
σs,t ·f(l(s, t)) σ0
s,t(v)
σ0
s,t ·f(l0(s, t)),(6)
where sand tare any two nodes of V\ {v}.
Proof of Lemma 0. When adding such a link, the distance between sand tcan either decrease or remain the same. Then,
these two cases have to be separately examined.
3
Case l(s, t)> l0(s, t)(illustrated in Fig. 6B). In this case, when a link is added between uand v, one obtains that
σs,t(v)
σs,t ·f(l(s, t)) f(l0(s, t)),
because σs,t(v)σs,t ,l(s, t)> l0(s, t), and fis a strictly decreasing function of its argument. Therefore, as all the
shortest paths between sand tpass through v(see again Fig. 1B), one has that σ0
s,t(v) = σ0
s,t, and condition (6) is verified.
Case l(s, t) = l0(s, t)(illustrated in Fig. 6C). In this situation, when a link between uand vis added, xnew shortest paths
between sand tmay arise, which however have to pass all through v. Therefore, one has
σ0
s,t =σs,t +xand σ0
s,t(v) = σs,t (v) + x. (7)
Thus, as σs,t(v)σs,t , one obtains that
σs,t(v) (σs,t +x)σs,t (σs,t (v) + x),
and eventually
σs,t(v)
σs,t σs,t(v) + x
σs,t +x.
Therefore, considering both l(s, t) = l0(s, t)and Eq. (7), it comes out that condition (6) is satisfied.
Main results
Here, we describe the properties of the model’s Nash equilibria (when they exist), i.e. of those settings where it is unprofitable
for any agent to unilaterally deviate from its current strategy.
First, we state a bounding condition for the cost ensuring that, when they exist, such Nash equilibria are not empty graphs,
and therefore they contain at least a connected component.
Theorem 0. If c(N)>f(2)(N2)
2, then the empty graph is a Nash equilibrium configuration. Otherwise, an empty graph will
never be obtained at the Nash equilibrium.
Proof. Consider a vertex vof an empty graph V. From each pair of other vertices s, t V , s 6=v6=t, the contribution
to W BC (v)is no more than f(2) (which would correspond to the case in which both nodes sand tare directly linked
to v). The net benefit for vof making xconnections will be then equal to x
2·f(2) xc. Obviously, the maximum is
reached at x=N1. Then, if a vertex vin an empty graph changes its strategy and forms connections with the other
N1nodes, it receives a net utility equal to N1
2·f(2) (N1)c. Such a latter quantity will be non negative for
(N1)(N2)f(2)
2(N1)ccf(2)(N2)
2, meaning that a vertex can exist which would benefit from deviating from the
strategy of not drawing any edges. If instead c > f(2)(N2)
2, then the maximum possible gain (after deducing costs) will be
negative, and this implies that the empty graph will be a Nash equilibrium.
Condition for the existence of a vertex of a large degree
The next step is to prove that, in a non empty Nash equilibrium, there exists always a vertex of large degree.
Lemma 1. Let the maximum degree of a vertex in a graph Vbe k. Then, the maximum 2-independent set will be of size at
least |V|
k2+1 .
Proof. Let’s construct iteratively a 2-independent set Sof the required size.
For this purpose, we start by considering a generic node v1V. Then, we consider the set t1including v1, all its neighbors
(vertices of Vwhich are at distance 1 from v1) and all its neighbors of neighbors (vertices of Vat distance 2 from v1). Since the
degree of a vertex in Vdoes not exceed k, so does the maximum number of neighbors of v1. Consider now the vertex uwhich
4
is neighbor of v1. Since its degree does not exceed kand it has a connection (v, u), then the maximum number of vertices that
are at distance 2 from v1and are furthermore connected to uis k1. It follows that the number of vertices belonging to t1is
bounded from above as k(k1) + k+ 1 = k2+ 1 (a situation in which v1has kneighbors and each neighbor of v1provides a
unique set of vertices located at a distance of 2 from it).
The vertex v1is added to the set Sand the subgraph V1=V\t1is considered. V1is the graph obtained from Vby removing
all members of t1, and therefore it contains vertices that are all at a distance of at least 3 from v1(in other words, any vertex
v2V1will be 2-independent with v1). The procedure can be repeated iteratively until there are no vertices left in the resulting
subgraph: at each iteration of the procedure, a new vertex can be added to S, and no more than k2+ 1 vertices are removed from
V. As a consequence, there are at least |V|
k2+1 iterations of the procedure, which means |S| |V|
k2+1 (quod erat demonstrandum!)
Lemma 2. Let a 2-independent set of size xbe present in the network. Then, if one vertex of such a set “glues the set”
(i.e., it forms connections with all other members of the set), it receives an additional utility equal to at least x1
2·(f(2)f(6))
Proof. Consider a 2-independent set Sand a vertex vS, and let us estimate the minimum gain that such a vertex will get
by forming edges to all other vertices of S. Notice that, before forming connections in the set, the maximum contribution to
W BC (v)received from a pair of other vertices s, t Sis f(6). This is because both distances l(v, s)and l(v , t)are at least 3,
and therefore either the shortest path between sand tis of length at least 6, or it does not pass through v(and, in this latter case,
the contribution to W BC (v)is 0). This implies that even in the case in which all shortest paths between sand tare passing
through v, the maximum possible contribution to W BC (v)from these two vertices is f(6). Now, let the vertex vform the
edges (v, s)and (v, t). Then, there exists a unique path from sto tof length 2 passing through v(see the explanatory figure 2b
in the main text), and thus such a pair of vertices will contribute f(2) to W BC (v). On its turn, this implies that the minimum
increase of W BC (v)from gluing any pair of vertices s, t Sis equal to f(2) f(6), and then (summing up over all possible
pairs of vertices s, t S) the Lemma is proved.
Theorem 1. Let c(N)>0be a size-dependent cost function of forming an edge, and let kbe a positive integer number
specifying the degree of a node in the network. If a network size ˜
Nexists starting from which (i.e. for all sizes N > ˜
N) the
relationship
2
f(2) f(6)(c(N) + f(2) f(6))(k2+ 1) < N
is satisfied, then the Nash equilibrium will always contain at least a vertex of degree at least k.
Proof. Let’s demonstrate Theorem 1 by contradiction, and assume the opposite i.e., let us suppose that there is a ksuch that
for any Nthere are equilibrium states that do not contain a vertex of degree at least k.
Due to Lemma 1, this entails the existence of a 2-independent set Sof size at least N
k2+1 .
Then, due to Lemma 2, any vertex of such a 2-independent set will increase the value of the benefit function (by forming
connections with all other vertices in S) by at least N
k2+1 1
2·(f(2) f(6)). The costs of forming these links will be instead
equal to N
k2+1 1·c.
Then one can examine the utility obtained by a vertex vSfrom the formation of such connections, taking into account
that, in order to prevent vfrom connecting with every other vertex in S, the difference between the additional utility received
and the cost must be negative. One has
1
2·N
k2+1 1·N
k2+1 2·(f(2) f(6)) N
k2+1 1·c < 0,
and, therefore,
N
k2+1 2<2c
f(2)f(6) N
k2+1 <2c
f(2)f(6) + 2.
In order to be at the Nash equilibrium, it is necessary that the above condition is verified, because otherwise it would be strictly
advantageous for a vertex from the maximum 2-independent set Sto change its strategy and glue the set S. Then, one obtains:
N
k2+1 <2c
f(2)f(6) + 2 N < 2c
f(2)f(6) + 2·(k2+ 1) =N < h2
f(2)f(6) (c+f(2) f(6))i(k2+ 1)
which is in direct contradiction with the statement of Theorem 1. This implies that our initial assumption is incorrect, and
therefore that there will be necessarily a vertex of degree at least kat the equilibrium (quod erat demonstrandum!).
5
The emergence of ultra-small world states and of the six degrees of separation property
Finally, we can prove the main result of our study, related with the fact that, if they exist, the Nash equilibria are ultra-small
world states featuring the six degrees of separation property.
Theorem 2. Let the cost c(N)of forming a link in the network satisfy
c(N)<3
rf(2) f(6)
16 ·3
pN·(f(2) f(5))2f(2) + f(6).
Then, from a given network size ˜
Non (i.e. for all networks whose size Nis larger than ˜
N), the distance between two generic
vertices v, u of the network does not exceed 6 at the Nash equilibrium.
Lemma 3. If
c(N)<3
rf(2) f(6)
16 ·3
pN·(f(2) f(5))2f(2) + f(6),
then all equilibria contain at least a vertex usuch that the distance to it from any other vertex of the network does not exceed 3.
Proof. Once again, let us proceed by contradiction, i.e. by trying to prove the opposite. Let us then consider a value of the
degree ξkξ+ 1 where ξ=1
f(2)f(5) ·3
qf(2)f(6)
2·3
pN·(f(2) f(5))2f(2) + f(6).
First, one has to notice that the inequality k2+ 1 <2(k1)2is always verified, as far as k > 4. Then, one has that
2
f(2)f(6) (c+f(2) f(6))(k2+ 1) <2
f(2)f(6) (c+f(2) f(6)) ·2(k1)2=4
f(2)f(6) (c+f(2) f(6)) ·(k1)2
4
f(2)f(6) (c+f(2) f(6)) 1
f(2)f(5) ·3
qf(2)f(6)
2·3
pN·(f(2) f(5))2f(2) + f(6)2
<
<4
f(2)f(6) 3
qf(2)f(6)
16
3
pN·(f(2) f(5))2 3
qf(2)f(6)
22
·
·1
(f(2)f(5))2·3
pN·(f(2) f(5))22=N=
=2
f(2)f(6) (c+f(2) f(6))(k2+ 1) < N.
Due to Theorem 1, one has therefore to conclude that there must be a vertex of degree kat the Nash equilibrium.
Notice that, in the derivation of the above expression, we have made use of the two inequalities c+f(2)f(6) <3
qf(2)f(6)
16 ·
3
pN·(f(2) f(5))2(from the statement of Lemma 3) and f(2) + f(6) <0(from the fact that f(x)is a strictly decreasing
function of its argument).
Let us call such a vertex u. Suppose now that a vertex vVexists such that the distance between uand vis at least 4, and
let us calculate the minimum additional utility that the vertex uwould accrue by forming a connection with v.
We now denote as Suthe set of neighbors of the vertex u, and we consider a generic vertex siSu. If the shortest path from
vto sipasses through u, then the distance from vto simust be at least 5 (otherwise it would be possible to get to ufrom valong
the edges of the network with less than 3 moves). Notice that the pair si, v contributes to W BC (u)no more than f(l(si, v)), i.e.
no more than f(5) in the present case. After forming the edge (u, v ), the distance between the vertices siand vwill become 2,
which implies that the contribution of this pair to W BC (u)will become f(2), i.e. it will increase by at least (f(2) f(5)). This
argument is valid for all other vertices of S. Then, the total benefit from holding the edge (u, v)will be at least (f(2) f(5)) ·k.
At the same time, the cost of forming the edge (u, v)is equal to c. Thus, in order to be at the Nash equilibrium, it is necessary
that
(f(2) f(5)) ·k < c k < c
f(2)f(5) .
On the other hand, one has that
6
c
f(2)f(5) <1
f(2)f(5) ·3
qf(2)f(6)
16 ·3
pN·(f(2) f(5))2f(2) + f(6)<
<1
f(2)f(5) ·3
qf(2)f(6)
2·3
pN·(f(2) f(5))2f(2) + f(6)k.
Therefore, one has that k > c
f(2)f(5) , which evidently leads to a contradiction. Therefore, all equilibrium states must
necessarily contain at least a vertex uof degree at least ksuch that each vertex vVwill necessarily be at a distance of no
more than 3 from u. Lemma 3 is proved.
Proof of Theorem 2. According to Theorem 1, there is a vertex uof degree at least kin the equilibrium state. According
to Lemma 3, the distance from any other vertex of the network to udoes not exceed 3. This automatically implies that the
maximal distance between any two vertices s, t (the diameter of the network) cannot exceed 6 at the equilibrium: no more
than 3 steps are needed for passing from sto uand no more than 3 other steps are needed from uto t(quod erat demonstrandum!).
THE ILLUSTRATIVE CASE
In the second part of the main text, we reported on an illustrative model, where nodes can modify their neighborhood only by
adding new links to the already existing ones. In particular, the game consisted in the following steps.
1) The agents start the game when they are already connected by means of a pristine graph where, in addition, there exists at
least one node with sufficiently high degree.
2) Each agent adopts as benefit function
W BC (v) = X
s6=v6=t
σst(v)
σst ·1
l(s, t)α,(8)
with αbeing a strictly positive parameter. This implies that the weighting factor is f(l(s, t)) = 1
l(s,t)α, and that Eq. (8)
coincides, for α= 1, with the classical weighted betweenness centrality.
3) Agents add new connections to their neighborhood if and only if there is a positive balance between the extra-utility
brought by the new connections and the extra-cost.
In practice, at each step mof the game, each agent vVconsiders a neighborhood Nv(m)which is equal to Nv(m1)
plus pother nodes.
When all agents become unable to incorporate further edges, the network is said to have reached its asymptotic equilibrium
state.
Under this conditions, we are able to prove the following Theorem:
Theorem 3. If in the initial graph there is a node whose degree khas a value satisfying the relationship
1
2α1
(H+ 2)αkc, (9)
for some integer H {3,4,5, ...}, then the equilibrium state contains at least a node vwhich is linked to all other nodes of the
graph by no more than Hlinks. This automatically implies that the diameter of the equilibrium state of the network does not
exceed 2Hand that, therefore, such an equilibrium state is an ultra-small world state (and, for H= 3, also the emergence of the
6 degrees of separation property).
Proof of Theorem 3. The proof is given again by contradiction. i.e. by supposing that there is a node uin the final state of
the network whose distance from vis at least H+ 1 i.e., H0=l(u, v)H+ 1. Then, the nodes directly connected to v(the
neighbors of v) may be found at either H01, or H0, or H0+ 1 edges from u. In the main text, we already discussed that the
only contribution to the benefit function of vfrom shortest paths ending in uand originating in a neighbor of vis coming from
those paths starting in the neighbors of vthat are at distance H0+ 1 from u. The contribution of these paths to Eq. (8) satisfies
1
lα
s,u ·σs,u(v)
σs,u 1
(H0+ 1) α1
(H+ 2) α.
7
This is because σs,u(v)σs,u , and H+ 2 H0+ 1 ls,u . Therefore, it easily follows that
W BC (v)k
(H+ 2) α+R,
where Raccounts for the contribution of all other shortest paths in the network that pass through vand that have not been
considered so far.
When a direct link between nodes vand uis added, the shortest path between any neighbor of v(denoted now generically by
w) and ubecomes wvu. Consequently,
W BC (v)1
2α1
(H+ 2)α·k,
because the contribution of the rest of the shortest paths (which do not start in wand end in u) does not decrease (compared to
R) with the addition of the new edge (see details of the demonstration in the above Lemma 0). Once again, let us now recall that
the definition of the equilibrium state foresees explicitly that the gain from adding any edge must be smaller than the cost, that
is,
c > W BC (v)1
2α1
(H+ 2)α·k,
which is in explicit contradiction with condition (9) (quod erat demonstrandum!).
ResearchGate has not been able to resolve any citations for this publication.
Article
Full-text available
Transportation networks play a critical role in human mobility and the exchange of goods, but they are also the primary vehicles for the worldwide spread of infections, and account for a significant fraction of CO2 emissions. We investigate the edge removal dynamics of two mature but fast-changing transportation networks: the Brazilian domestic bus transportation network and the U.S. domestic air transportation network. We use machine learning approaches to predict edge removal on a monthly time scale and find that models trained on data for a given month predict edge removals for the same month with high accuracy. For the air transportation network, we also find that models trained for a given month are still accurate for other months even in the presence of external shocks. We take advantage of this approach to forecast the impact of a hypothetical dramatic reduction in the scale of the U.S. air transportation network as a result of policies to reduce CO2 emissions. Our forecasting approach could be helpful in building scenarios for planning future infrastructure. Transportation networks undergo permanent changes influenced by a variety of human-induced and natural factors. The authors propose here a machine learning framework for prediction of connections removal that could be useful in building scenarios for transportation infrastructure needs.
Article
Full-text available
The genetic basis for the emergence of creativity in modern humans remains a mystery despite sequencing the genomes of chimpanzees and Neanderthals, our closest hominid relatives. Data-driven methods allowed us to uncover networks of genes distinguishing the three major systems of modern human personality and adaptability: emotional reactivity, self-control, and self-awareness. Now we have identified which of these genes are present in chimpanzees and Neanderthals. We replicated our findings in separate analyses of three high-coverage genomes of Neanderthals. We found that Neanderthals had nearly the same genes for emotional reactivity as chimpanzees, and they were intermediate between modern humans and chimpanzees in their numbers of genes for both self-control and self-awareness. 95% of the 267 genes we found only in modern humans were not protein-coding, including many long-non-coding RNAs in the self-awareness network. These genes may have arisen by positive selection for the characteristics of human well-being and behavioral modernity, including creativity, prosocial behavior, and healthy longevity. The genes that cluster in association with those found only in modern humans are over-expressed in brain regions involved in human self-awareness and creativity, including late-myelinating and phylogenetically recent regions of neocortex for autobiographical memory in frontal, parietal, and temporal regions, as well as related components of cortico-thalamo-ponto-cerebellar-cortical and cortico-striato-cortical loops. We conclude that modern humans have more than 200 unique non-protein-coding genes regulating co-expression of many more protein-coding genes in coordinated networks that underlie their capacities for self-awareness, creativity, prosocial behavior, and healthy longevity, which are not found in chimpanzees or Neanderthals.
Article
Full-text available
We live and cooperate in networks. However, links in networks only allow for pairwise interactions, thus making the framework suitable for dyadic games, but not for games that are played in larger groups. Here, we study the evolutionary dynamics of a public goods game in social systems with higher-order interactions. First, we show that the game on uniform hypergraphs corresponds to the replicator dynamics in the well-mixed limit, providing a formal theoretical foundation to study cooperation in networked groups. Second, we unveil how the presence of hubs and the coexistence of interactions in groups of different sizes affects the evolution of cooperation. Finally, we apply the proposed framework to extract the actual dependence of the synergy factor on the size of a group from real-world collaboration data in science and technology. Our work provides a way to implement informed actions to boost cooperation in social groups.
Article
Full-text available
Recent years have witnessed an explosion of extensive geolocated datasets related to human movement, enabling scientists to quantitatively study individual and collective mobility patterns, and to generate models that can capture and reproduce the spatiotemporal structures and regularities in human trajectories. The study of human mobility is especially important for applications such as estimating migratory flows, traffic forecasting, urban planning, and epidemic modeling. In this survey, we review the approaches developed to reproduce various mobility patterns, with the main focus on recent developments. This review can be used both as an introduction to the fundamental modeling principles of human mobility, and as a collection of technical methods applicable to specific mobility-related problems. The review organizes the subject by differentiating between individual and population mobility and also between short-range and long-range mobility. Throughout the text the description of the theory is intertwined with real-world applications.
Article
Social networks play an increasingly important role in today's society. Special characteristics of these networks make them challenging domains for the search community. In particular, social networks of users can be viewed as search graphs of nodes, where the cost of obtaining information about a node can be very high. This paper addresses the search problem of identifying the degree of separation between two users. New search techniques are introduced to provide optimal or near-optimal solutions. The experiments are performed using Twitter, and they show an improvement of several orders of magnitude over greedy approaches. Our optimal algorithm finds an average degree of separation of 3.43 between two random Twitter users, requiring an average of only 67 requests for information over the Internet to Twitter. A near-optimal solution of length 3.88 can be found by making an average of 13.3 requests.
Article
Despite substantial recent progress, our understanding of the principles and mechanisms underlying complex brain function and cognition remains incomplete. Network neuroscience proposes to tackle these enduring challenges. Approaching brain structure and function from an explicitly integrative perspective, network neuroscience pursues new ways to map, record, analyze and model the elements and interactions of neurobiological systems. Two parallel trends drive the approach: the availability of new empirical tools to create comprehensive maps and record dynamic patterns among molecules, neurons, brain areas and social systems; and the theoretical framework and computational tools of modern network science. The convergence of empirical and computational advances opens new frontiers of scientific inquiry, including network dynamics, manipulation and control of brain networks, and integration of network processes across spatiotemporal domains. We review emerging trends in network neuroscience and attempt to chart a path toward a better understanding of the brain as a multiscale networked system.