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Accounting for degree distributions in empirical analysis of network dynamics

Degrees (the number of links attached to a given node) play a particular and important role in empirical network analysis because of their obvious importance for expressing the position of nodes. Some recent work might suggest the conclusion that degree distributions are expressive also of structural aspects of networks and their dynamics, where the term "structure" refers, in a loose sense, to phenomena involving sets of three or more nodes (such as transitivity, hierarchy, subgroup formation, etc.). It is argued here that there is no general straightforward relation between the degree distribution on one hand and structural aspects on the other hand, as this relation depends on further characteristics of the presumed model for the network. Therefore empirical inference from observed network characteristics to the processes that could be responsible for network genesis and dynamics cannot be based only, or mainly, on the observed degree distribution.

As an elaboration and practical implementation of this point, a statistical model for the dynamics of networks (expressed as digraphs) is proposed in which the out-degree distribution is governed by parameters that are not connected to the parameters for the structural dynamics. It could be said that such an approach treats the parameters for the degree distribution as nuisance parameters and has the purpose to minimize the influence of the observed degrees on the conclusions about the structural aspects of the network dynamics. The model is a stochastic actor-oriented model, extending the model of Snijders (Sociological Methodology, 2001). It deals with the degrees in a manner resembling Tversky's Elimination by Aspects approach: when an actor (node) changes an outgoing tie, the first aspect considered is the out-degree, and the second aspect is the resulting structure. A statistical procedure for parameter estimation in this model is proposed, and some examples illustrate the extent to which inference about structural aspects can be influenced by assumptions about actor preferences/constraints for the out-degree distribution.