Frontiers in Computer Education Wang (ed.)
© 2015 Taylor & Francis Group, London, ISBN 978-1-138-02797-8
A friendship network based on random and triadic-closure in a fixed
D.S. Zhao
, K. Zhao
, J. Yang & J.R. Sha
College of Modern Educational Technology, Northwest University for Nationalities, Lanzhou, Gansu, China
ABSTRACT: In this paper, we proposed a friendship network model in a fixed community based on such
connection mechanisms as: random selection and triadic-closure. That means each node in the network can
strike up friendships with strangers randomly or its two-hop neighbours (friends of its friends). We simulated the
network structure with NetworkX and found it exhibits important network topological properties: short diameters,
high clustering coefficients, and various shapes of in-degree distributions. We found that clustering coefficients
and the shapes of in-degree distributions are both related to the ratio between triadic-closure probability and
random probability. Accordingly, we gave a simple method to distinguish the effect of triadic-closure on the
structure of friendship networks in the real world from a random connection.
In recent years, there has been considerable interest
in complex networks arising in social life, largely
because their contribution to human understanding
of various social structures and processes, such as
how people meet and make friends from strangers
(Van et al. 2003), search information and obtain
job opportunities (Granovetter 1973). The studies of
social networks mainly focus on two related tracks:
the static properties of network topological structures
observed from the real world data, e.g., movie actors
networks (Watts & Strogatz 1998), scientific collabo-
ration networks (Newman 2001; Barabâsi et al. 2002),
online friendship networks (Backstrom et al. 2012), the
dynamical evolving processes measured from some
network generation models, e.g., Erdös-Rényi model
(ERDdS 1959), Barabási-Albert model (Albert et al.
1999; Bianconi & Barabási 2001), copying model
(Kleinberg et al. 1999), forest fire model (Leskovec
et al. 2007).
Social networks are composed of a lot of peo-
ple or groups who are linked together according to
a variety of relationships. The variety of connection
mechanisms lead to complex network structures. In the
previous studies, we have learned that social networks
are very different from random and regular networks.
They always have properties of small shortest paths
(Backstrom et al. 2012), high clustering coefficients,
scale-free degree distributions, and so on. Some of
these features are also seen in the world wide web
(Albert et al. 1999; Adamic & Huberman 2000; Broder
et al. 2000), internet (Yook et al. 2002), and biological
These authors contributed equally to this work, and should
be regarded as co-first authors.
networks (Wagner & Fell 2001). How do these features
emerge in social networks? There may be combina-
tions of one or more mechanisms that are connecting
people together. For example, homophily the ten-
dency of individuals to associate with those who are
similar to themselves (Tarbush & Teytelboym 2013),
mutual interests (Singer et al. 2009), the maximum-
likelihood principle (Leskovec et al. 2008), the closure
of short network cycles (Kossinets & Watts 2006),
and drive people to interact with each other and social
networks evolve over times. To observe these phenom-
ena, a lot of network models have been proposed to
reproduce statistical properties in the real world.
The most celebrated model is the Barabási-Alber t
(BA) model (Albert et al. 1999), in which when a new
node joins the network, it creates a constant number
of edges, where the destination node of each edge is
chosen proportional to the destination’s degree. The
BA model reproduces the scale-free network with
power law degree distribution, and reveals such a phe-
nomenon ‘richer get richer’ in social networks. There
are also some alternative models based on the prefer-
ential attachment mechanisms (Bianconi & Barabási
2001; Albert & Barabási 2000). David in his studies
showed that ‘winners don’t take all’, and gave an alter-
native model which combined preferential attachment
and random connection (Pennock et al. 2002). Copy-
ing model (Kleinberg et al. 1999) and forest fire model
(Leskovec et al. 2007) were based on attaching new
nodes to the network by copying or burning through
existing edges.These models reproduced densification
and shrinking diameters in social networks. Jure, Lars,
Ravi and Andrew (Leskovec et al. 2008) attributed
the macroscopic network properties to the maximum-
likelihood principle and proposed a model based
on triadic-closure mechanisms. Jackson and Rogers

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