# A Multigraph Approach to Social Network Analysis

By Shafie, Termeh | Journal of Social Structure, January 1, 2015 | Go to article overview

# A Multigraph Approach to Social Network Analysis

Shafie, Termeh, Journal of Social Structure

(ProQuest: ... denotes formulae omitted.)

1 Introduction

Network data involving relational structure representing interactions between actors are commonly represented by graphs where the actors are referred to as vertices or nodes, and the relations are referred to as edges or ties connecting pairs of actors. Research on social networks is a well established branch of study and many issues concerning social network analysis can be found in Wasserman and Faust (1994), Carrington et al. (2005), Butts (2008), Frank (2009), Kolaczyk (2009), Scott and Carrington (2011), Snijders (2011), and Robins (2013).

A common approach to social network analysis is to only consider binary relations, i.e. edges between pairs of vertices are either present or not. These simple graphs only consider one type of relation and exclude the possibility for self relations where a vertex is both the sender and receiver of an edge (also called edge loops or just shortly loops). In contrast, a complex graph is defined according to Wasserman and Faust (1994):

If a graph contains loops and/or any pairs of nodes is adjacent via more than one line the graph is complex. [p. 146]

In practice, simple graphs can be derived from complex graphs by collapsing the multiple edges into single ones and removing the loops. However, this approach discards information inherent in the original network. In order to use all available network information, we must allow for multiple relations and the possibility for loops. This leads us to the study of multigraphs which has not been treated as extensively as simple graphs in the literature.

As an example, consider a network with vertices representing different branches of an organ-isation. The edges apparent in such a network may then comprise of information, money, and personnel flows including cooperation, support, friendship and antagonism. These different edges should be considered simultaneously in order to understand the inter-organisational behaviour. Robins and Pattison (2006) emphasise analysing these different edges jointly to understand social processes in the network and its implications for an organisation's performance. In an organisational network, it is also evident that different kinds of ties may appear within the same branch creating loops. For instance, friendships may be more common between individuals within a branch, which may also indicate a higher propensity to turn to these friends for advice or support. A multigraph has the capacity to gather and represent all of this information.

A common definition of multigraphs (also called multiple networks) is graphs having several kinds of ties on the same vertex set (e.g. Robins 2013; Ranola et al. 2010; Koehly and Pattison 2005). However, by convention, loops are in many cases excluded from consideration. We emphasise the fact that social network data often comprise of loops in their natural state and include this in our definition: multigraphs are graphs where multiple edges and loops are permitted.

In this paper we present a new multigraph approach that may be used to analyse networks with multiple edges and loops of different kinds. We describe multigraph data structures with examples of their natural appearance together with a description of the possibility to obtain multigraphs using blocking, aggregation and scaling. A novel way of representing multigraphs using edge multiplicities is introduced and we quantify graph complexity by the distribution of edge multiplicities. A random multigraph model based on independent edge assignments (IEA) to sites of vertex pair is given and we derive several complexity statistics under IEA. Further, it is described how these measures can be used to analyse local and global network properties and to convey structural dependencies in social networks. Finally, a brief discussion is given on the possibilities and limitations of the presented multigraph approach, together with suggestions for future reasearch topics. …

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