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RANDOM GRAPH MODELS OF SOCIAL NETWORKS

In mathematics, a "graph", in the context of the field known as
"graph theory", is a mathematical object composed of points known
as "vertices" or "nodes" and lines connecting some (possibly
empty) subset of them, known as "edges".

A "random graph", in this context, is a graph in which properties
such as the number of nodes, edges, and connections between them
are determined in some random way.

... ... M.E. Newman et al (Columbia University, US) discuss
social networks, the authors making the following points:

1) A social network is a set of people or groups of people,
"actors" in the jargon of the field, with some pattern of
interactions or "ties" between them. Friendships among a group of
individuals, business relationships between companies, and
intermarriages between families are all examples of social
networks that have been studied in the past. Network analysis has
a long history in sociology, the literature on the topic
stretching back at least half a century to the pioneering work of
Rapoport, Harary, and others in the 1940s and 1950s. Typically,
network studies in sociology have been data-oriented, involving
empirical investigation of real-world networks, the investigation
often followed by graph theoretical analysis aimed at determining
the centrality or influence of the various actors.

2) Most recently, after a surge in interest in network structure
among mathematicians and physicists, partly as a result of
research on the Internet and on the World Wide Web, another body
of research has investigated the statistical properties of
networks and methods for modeling networks either analytically or
numerically. One important and fundamental result that has
emerged from these studies concerns the numbers of ties that
actors have to other actors, their so-called "degrees". It has
been found that in many networks, the distribution of actors'
degrees is highly skewed, with a small number of actors having an
unusually large number of ties. Simulations and analytical work
have suggested that this skewness could have an impact on the way
in which communities operate, including the way information
travels through the network and the robustness of networks to
removal of actors.

3) The authors describe some new exactly solvable models of the
structure of social networks, the models based on random graphs
with arbitrary degree distribution. The authors provide models
both for simple unipartite networks, such as acquaintance
networks, and for bipartite networks, such as affiliation
networks. The authors compare the predictions of their models to
data for a number of real-world social networks and find that in
some cases the models are in remarkable agreement with the data,
whereas in other cases the agreement is poorer, perhaps
indicating the presence of additional social structure in the
network not captured by the random graph.

Proc. Nat. Acad. Sci. 2002 99:2566

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