Frontiers in Computer Education

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

community

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.

1 BACKGROUND

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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