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Structure of a large social network
Ga´bor Csa´nyi∗†
TCM Group, Cavendish Laboratory, University of Cambridge
Madingley Road, Cambridge CB3 0HE, United Kingdom
Bala´zs Szendro˝i‡
Department of Mathematics, Utrecht University
PO. Box 80010, NL-3508 TA Utrecht, The Netherlands
(Dated: February 2, 2008)
We study a social network consisting of over 104 individuals, with a degree distribution exhibiting
two power scaling regimes separated by a critical degree kcrit, and a power law relation between
degree and local clustering. We introduce a growing random model based on a local interaction
mechanism that reproduces all of the observed scaling features and their exponents. Our results
lend strong support to the idea that several very different networks are simultenously present in the
human social network, and these need to be taken into account for successful modeling.
PACS numbers: 89.75.Da, 89.75.Hc, 89.75.Fb, 89.65.Ef
The ubiquity of networks has long been appreciated:
complex systems in the social and physical sciences can
often be modelled on a graph of nodes connected by
edges. Recently it has also been realized that many net-
works arising in nature and society, such as neural net-
works [1], food webs [2], cellular networks [3], networks of
sexual relationships [4], collaborations between film ac-
tors [1, 5] and scientists [6, 7], power grids [1, 8], Internet
routers [9] and links between pages of the World Wide
Web [10, 11, 12] all share certain universal characteris-
tics very poorly modelled by random graphs [13]: they
are highly clustered “small worlds” [1, 14, 15] with small
average path length between nodes, and they have many
highly connected nodes with a degree distribution often
following a power law [5, 10]. The network of humans
with links given by acquaintance ties is one of the most
intriguing of such networks [14, 15, 16, 17], but its study
has been hindered by the absence of large reliable data
sets.
— The new data set The WIW project was started by
a small group of young professionals in Budapest, Hun-
gary in April 2002 on the web site www.wiw.hu with
the aim to record social acquaintance. The network is
invitation–only and new members join by an initial link
connecting to the person who invited them. New links
are recorded between members after mutual agreement.
This scheme results in members preferring to use their
real names and effectively prevents proliferation of mul-
tiple pseudonyms. Because of the relatively short age
of the network, links formed between people newly ac-
quainted through the web site have a minimal structural
effect; thus the majority of the links represent genuine
social acquaintance and the WIW develops as a growing
∗Corresponding author
†Electronic address: gabor@csanyi.net
‡Electronic address: szendroi@math.uu.nl
subgraph of the underlying social acquaintance network.
Indeed, the growth process of the WIW network is es-
sentially equivalent to the “snowball sampling” method
well known to sociologists [17], and to the crawling meth-
ods used to investigate the World Wide Web and other
computer networks. We study the WIW network using
two anonymous snapshots taken in October 2002 (with
12388 nodes and 74495 links) and January 2003 (with
17496 nodes and 127190 links).
The degree distribution of the WIW network is plotted
on Figure 1. The graph shows two power law regimes
P (k) ∼
{
k−1.0 if k < kcrit
k−2.0 if k > kcrit
The two regimes are separated by a critical degree kcrit ≈
25. The exponent γ2 ≈ −2 of the large-k power law
falls in a range that has often been observed before in
a variety of contexts [3, 4, 5, 6, 7, 8, 9, 10, 11, 12].
The value γ1 ≈ −1 of the small-k power law exponent
is much less common, observed before only in some sci-
entific collaboration networks [6] and food webs [2]. The
possibility of a double power law was discussed in [7, 18],
but the WIW network is the first data set which conclu-
sively demonstrates the existence of double power law be-
haviour. The two snapshots give essentially identical dis-
tributions. Since the network grew by about 50% during
this period, the described distribution can be regarded
as essentially stationary in time.
The two scaling regimes in the degree distribution of
the WIW graph are indicative of two distinct growth pro-
cesses: the invitation of new members, and the record-
ing of acquaintance between already registered members.
The degree distribution of the invitation tree graph is
shown on Figure 2, where a power law is observed for
large degrees with an exponent γ ≈ −3. Since this dis-
tribution is qualitatively different from the total degree
distribution, it is reasonable to conclude that there are
at least two different types of social linking at play here:
the network of friends defined by ties strong enough to

2100 101 102 103
10−4
10−3
10−2
10−1
100
k
P(
k)
−1.0
−2.0
FIG. 1: The degree distribution of the WIW network (dia-
monds), with a small-k power law P (k) ∼ k−1.0 and a large-
k power law P (k) ∼ k−2.0 separated by a critical degree
kcrit ≈ 25. The solid line gives the degree distribution of
the model of the text with edge/node ratio m = 15, q = 0.5
and size V = 2× 105, averaged over 50 graphs.
100 101 102
10−5
10−4
10−3
10−2
10−1
100
k
P(
k)
WIW
q = 0
q = 0.5
q = 1.0
FIG. 2: The degree distribution of the invitation tree of the
WIW graph (diamonds) exhibits a large-k power law P (k) ∼
k−3. Also plotted is the invitation tree of the model graph
with V = 2 · 105 nodes, m = 15 and different parameters q.
warrant an invitation is different from the network of
acquaintances that drives the mutual recognition, once
both parties are registered.
The density of edges in the neighbourhood of a node is
measured by the local clustering coefficient. For a node
v of degree k, the local clustering coefficient C(v) is the
number of acquaintance triangles of which v is a vertex,
divided by k(k−1)/2, the number of all possible triangles.
Figure 3 plots C(k), the average of C(v) over nodes of
degree k, against the degree, showing the existence of a
power law
C(k) ∼ k−0.33.
A relationship C(k) ∼ k−α was observed before in [19,
20], but with significantly larger exponents. Such power
100 101 102
10−1
100
k
C(
k)
FIG. 3: The correlation between the local clustering coeffi-
cient C(k) and the node degree k for the WIW graph (di-
amonds), showing a power law C(k) ∼ k−0.33. The solid
line plots the same for the model graph with parameters
V = 2 · 105, m = 15 and critical q = 0.5, averaged over
50 graphs.
laws hint at the presence of hierarchical architecture [20]:
when small groups organize into increasingly larger
groups in a hierarchical manner, the local clustering de-
creases on different scales according to such a power law.
The average clustering coefficient 〈C〉 ≈ 0.2 is obtained
as the average of C(v) over all nodes. The average diam-
eter between two members of the WIW network is about
4.5. These two measures indicate the “small–world” na-
ture of the WIW network in the sense of [1].
—Time development As Figure 4 shows, the number of
nodes of the WIW network grew approximately linearly
with time. This appears to be related to the fact that the
WIW network develops as a subgraph of the underlying
social network, and thus the availability of new members
is constrained by high clustering of the existing social
links: a substantial proportion of the aquaintances of a
newly invited member will have been invited already.
On the other hand, the number of edges also grew lin-
early with time, and thus the edge/node ratio only grew
moderately during the existence of the network. This ob-
servation is in contradiction with any purely local time-
independent edge creation mechanism. If every member
of the network actively participates in edge creation in-
depently of its age in the network, the edge/node ratio
would also increase linearly with time. This linear growth
of the edge/node ratio was not observed in the network,
and hence we conclude that the edge creatioon activity
of members necessarily decreased with time.
The fact that the edge/node ratio changes little over
time is consistent with the observed stationary nature of
the degree distribution. To see this, consider a grow-
ing network with V (t) nodes at time t and a time-
independent degree distribution P (k) with
∑
k P (k) = 1

3Apr 2002 July 2002 Oct 2002 Jan 2003
0
2
4
6
8
10
12
14
x 104
Nodes x 3
Edges
FIG. 4: The time development of the number of nodes and
edges of the WIW network. Note that the number of nodes
is multiplied by 3 for better visibility.
and finite first moment
∑
k kP (k). At time t, there are
n(k, t) = V (t)P (k)
nodes of degree k, and hence the number of edges is
E(t) =
1
2
∑
k
kn(k, t) =
V (t)
2
∑
k
kP (k).
Consequently the edge/node ratio E(t)/V (t) is essen-
tially constant, and it only changes because the maximal
degree increases. This argument applies to the WIW net-
work with stationary distribution
P (k) =
{
c1
kγ1 if k < kcritc2
kγ2 if k > kcrit
with γ1 ≈ −1, γ2 ≈ −2. This distribution is on the
boundary of distributions with finite first moment: the
first moment exists for γ2 < −2 but not otherwise.
—Network modelling A random graph process based on
linear preferential attachment for the creation of new
edges was proposed in [5] to account for the observed
power laws in natural networks. Such a process leads
indeed to a graph with a power law degree distribu-
tion [5, 21]. However, this model is by definition macro-
scopic, requiring information about the entire network in
every step. This assumption is realistic for the World
Wide Web or some collaboration networks, where all
nodes are “visible” from all others. For human social net-
works however, it is reasonable to assume some degree of
locality in the interactions. Also, the original scale-free
models are not applicable to networks with high, degree
dependent clustering coefficients. These problems mo-
tivated the introduction of new models which use local
triangle creation mechanisms [15, 22, 23, 24, 25], which
increase clustering in the network. These models have
degree-dependent local clustering, and can also lead to
power law degree distributions, though no existing model
of this kind shows a double power law.
We now present a new model to account for the ob-
served properties of the WIW network. As mentioned
above, the WIW can be viewed as a growing subgraph
of the underlying social acquiantance graph. This sug-
gests a model obtained by a two-step process, first mod-
elling the underlying graph, and then implementing a
growth process. The lack of available data on the un-
derlying graph however prevents us from following this
programme directly. We build instead a growing graph
in one single process, choosing the local triangle mech-
anism as our basic edge creation method. This models
the social introduction of two members of the WIW net-
work by a common friend some time in the past, such
edges being gradually recorded in the WIW network it-
self. The invitation of new members is modelled by sub-
linear preferential attachment [5], motivated by experi-
mental results on scientific collaboration networks [7, 26],
where the data permits the analysis of initial edge for-
mation. We also impose the constant edge/node ratio to
be consistent with the observed stationary distribution.
Note that this constant has to be tuned from the shape
of observed distributions, and cannot be inferred from
the WIW data directly. The reason for this is that the
WIW has a disproportionate number of nodes of degree
one (Figure 1), representing people who once responded
to the invitation but never returned, which distorts the
edge/node ratio without invalidating our other conclu-
sions.
The precise description of our process is as follows.
• We begin with a small regular graph.
• New nodes arrive at a rate of one per unit time
and are attached to an earlier node chosen with a
probability distribution giving weight kq to a node
of degree k, where q ≥ 0 is a parameter.
• Internal edges are created as follows: we select two
random neighbours of a randomly chosen node v,
and if they are unconnected, we create an edge be-
tween them. Otherwise, we select two new neigh-
bours of the same node v and try again.
• A constant number of internal edges is created per
unit time, so that the edge/node ratio equals a con-
stant m after each time step.
The degree distribution of graphs generated by our
process is shown on Figure 5. We found a very robust
large k power law of exponent γ2 ≈ −2, essentially in-
dependently of the invitation mechanism. We measured
the joint probability distribution of the degrees k,k′ of
nodes connected by new internal edges, and found that
for large values, it was proportional to kk′. This directly
leads to a power law exponent of −2 via standard mean-
field arguments [7]. The small k behaviour was found to
be sensitive to the invitation mechanism; Figure 5 shows
that a second power law only appears at a critical q. The
critial value of q depends on the edge/node ratio.

4100 101 102 103 104
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
k
P(
k)
q=0
q=1.0
q=0.5
FIG. 5: The dependence of the degree distribution of our
model graph on the parameter q, with m = 15 and V = 2 ·105
(q = 0.5, 1) and V = 5 · 104 (q = 0), averaged over 50 graphs.
To test the hypothesis that the low degree power law
is indeed related to the invitation mechanism, we plot on
Figure 2 the degree distribution of the invitation tree of
the model network for various values of the parameter q.
At the critical value, we obtain a scale-free distribution
with exponent −3. Decreasing q leads to a much sharper
drop in the curve, with an exponential tail for q = 0,
whereas increasing q above the critical value leads to a
gelation-type behaviour: new nodes connect only to very
large degree nodes.
Figure 1 shows that, for appropriate choice of param-
eters, the degree distribution of our model reproduces
that of the real network extremely well. Figure 3 plots
the dependence of the local clustering cofficient C(k) as a
function of the degree k in our model network. While no
simple power law can be observed, there is a clear trend
of decreasing clustering with increasing degree, indicative
of the presence of hierarchy in the model network [20].
—Conclusion We have presented and analyzed a large
new data set of a human acquintance network with a
stable degree distribution which exhibits a new feature:
two power law regimes with different exponents. The ob-
served approximately constant edge/node ratio is a con-
sequence of the stability of the degree distribution, and
implies that the average activity of members is time de-
pendent, whereas the growth of the number of nodes is
constrained by the underlying social network. We also
introduced a model which reproduces the observed de-
gree distribution extremely well, and concluded that the
small-k power law is a related to the scale-free nature of
the invitation tree, whereas the large-k power law is a
result of the triangle mechanism of social introductions.
Our results show that human social networks are likely
to be composed of several networks with different char-
acteristics, and directly observable processes will exhibit
a mixture of features resulting from distinct underlying
mechanisms.
Acknowledgments
We thank Zsolt Va´rady and Da´niel Varga for access
to the data of the WIW network, and Risi Kondor for
helpful discussions.
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