Eur. Phys. J. B 77, 239-247 (2010)
https://doi.org/10.1140/epjb/e2010-00209-0

Quantifying structure in networks^*

¹ Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany
² Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico, 87501, USA

Corresponding author: ^a olbrich@mis.mpg.de

Received: 22 December 2009
Published online: 2 July 2010

Abstract

We investigate exponential families of random graph distributions as a framework for systematic quantification of structure in networks. In this paper we restrict ourselves to undirected unlabeled graphs. For these graphs, the counts of subgraphs with no more than k links are a sufficient statistics for the exponential families of graphs with interactions between at most k links. In this framework we investigate the dependencies between several observables commonly used to quantify structure in networks, such as the degree distribution, cluster and assortativity coefficients.