Structure of neighborhoods in a large social network
Alina Stoica
Orange Labs and Liafa
Paris, France
stoica@liafa.jussieu.fr
Christophe Prieur
Liafa
Paris, France
prieur@liafa.jussieu.fr
ABSTRACT
We present here a method for analyzing the neighborhoods
of all the vertices in a large graph. We ¯rst give an algorithm
for characterizing a simple undirected graph that relies on
enumeration of small induced subgraphs. We make a step
further in this direction by identifying not only subgraphs
but also the positions occupied by the di®erent vertices of
the graph. We are thus able to compute the roles played
by the vertices of the graph, roles found according to a new
de¯nition that we introduce. We apply this method to the
neighborhood of each vertex in a 2:7M vertices, 6M edges
mobile phone graph. We analyze how the contacts of each
person are connected to each other and the positions they
occupy in the neighborhood network. Then we compare
their quantity of communication (duration and frequency) to
their positions, ¯nding that the two are not independent. We
¯nally interpret and explain the results using social studies
on phone communications.1
Categories and Subject Descriptors
H.2.8 [Database Management]: Database Applications|
Data Mining ; G.2.2 [Discrete Mathematics]: Graph The-
ory|Graph algorithms; J.4. [Computer Applications]:
Social and Behavioral Sciences|Sociology
General Terms
Algorithms, Human Factors, Measurement, Theory
Keywords
social networks, roles, patterns, complex networks, personal
networks
1. INTRODUCTION
The study of social networks has changed a lot since the early
pioneering works of anthropologists who decided to focus
1Supplementary material is available online at
www.liafa.jussieu.fr/~stoica/neighborhoods
on relationships instead of individuals [7, 4, 3]. After the
technical framework of social network analysis was settled
in the 1970's by the combination of mathematical tools such
as graph theory, algebra and statistics [26, 8, 32, 33], the
¯eld has been again shaken with the exponential growing of
the size of relational databases coming with the development
of communication tools. The tremendous research activity
on the structure of the World-Wide Web [14, 11, 12] that
have pre-dated Google's PageRank algorithm [9] have given
birth to a new object of study, namely complex networks,
due to the common properties found to be shared not only
by the graph of the WWW [2, 34] but also by many networks
appearing in various contexts (biology, linguistics, economics
and, of course, social networks) [29, 6].
There is thus a wide gap between these kinds of studies of
the global structure of huge networks and qualitative studies
of personal networks, sometimes built from face-to-face in-
terviews (for a historical survey of this trend, see [35]), even
though more and more such studies now take as data per-
sonal networks scraped from internet's social network ser-
vices [18]. Inbetween, the classical problem of identifying
roles in a (possibly quite large) network, introduced in the
1970's as one of the main tools of social network analysis
[37], relies on the fact that some nodes have similar po-
sitions in the sense that they are linked to the same other
nodes, which is de¯ned as the so-called structural equivalence
of nodes, or the more general notion of regular equivalence,
where two nodes are equivalent if the neighbors of the two
are equivalent to each other [5].
Our work is at the intersection of these three research trends:
we study the roles of nodes in the personal networks of all
individuals of a large (thus `complex') network. In [31] (in
French), we already compared to a classical ethnographic
study what can be achieved in terms of qualitative analy-
sis with such a large-scale (2 million) collection of personal
networks.
Now to address the issue of roles, we devised a method rely-
ing on a very popular data mining problem: the search for
frequent subgraphs in a given (possibly large) graph. On
this issue, some authors considered that frequent subgraphs
are the ones that appear in a given graph (or set of graphs)
more often than a chosen threshold. Some algorithms [19,
21] extend the apriori-based candidate generation-and-test
approach [1], while others [5, 39] use a pattern-growth ap-
proach [15]. More recently, several algorithms have been
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