Subgraph Frequencies: Mapping the Empirical and Extremal Geography of Large Graph Collections

International World Wide Web Conference (WWW)

Abstract

A growing set of on-line applications are generating data that can be viewed as very large collections of small, dense social graphs – these range from sets of social groups, events, or collaboration projects to the vast collection of graph neighborhoods in large social networks. A natural question is how to usefully define a domain-independent `coordinate system’ for such a collection of graphs, so that the set of possible structures can be compactly represented and understood within a common space. In this work, we draw on the theory of graph homomorphisms to formulate and analyze such a representation, based on computing the frequencies of small induced subgraphs within each graph. We find that the space of subgraph frequencies is governed both by its combinatorial properties – based on extremal results that constrain all graphs – as well as by its empirical properties – manifested in the way that real social graphs appear to lie near a simple one-dimensional curve through this space.

We develop flexible frameworks for studying each of these aspects. For capturing empirical properties, we characterize a simple stochastic generative model, a single-parameter extension of Erdos-Renyi random graphs, whose stationary distribution over subgraphs closely tracks the one-dimensional concentration of the real social graph families. For the extremal properties, we develop a tractable linear program for bounding the feasible space of subgraph frequencies by harnessing a toolkit of known extremal graph theory. Together, these two complementary frameworks shed light on a fundamental question pertaining to social graphs: what properties of social graphs are `social’ properties and what properties are `graph’ properties?

We conclude with a brief demonstration of how the coordinate system we examine can also be used to perform classification tasks, distinguishing between structures arising from different types of social graphs.

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