Beyond Degree Distributions: Local to Global Structure of Social Contact Graphs

Abstract

The structure and dynamical properties of networked systems are often characterized by the degree distribution of the underlying graph. The degree distributions of many real world networks have often been found to be power laws, and in this approach, many properties of dynamical processes occurring on these networks, such as attack rates of epidemics, have been related to the power law exponents. A potential problem in many of these results is that the variability among the space of graphs having a specified degree distribution is not usually taken into account.

We show by explicit construction that local properties such as the degree distribution or assortativity are not sufficient to characterize the global dynamics of diffusion processes, such as epidemics. We use a simple Markov chain based on “edge flips’” to generate random graphs with given degree and assortativity properties, and study epidemic properties of such graphs. We find that the “epidemic curves” in these graphs are significantly different, and the difference increases as the Markov chain gets further from its starting point.

Our work provides a cautionary note on the use of local random graph models to infer global network and dynamical properties. Furthermore, it introduces the edge flipping methodology as a tool for investigating the effects of a broad range of hypothesized local network structures on the global dynamics of any process.