Graphs over time: densification laws, shrinking diameters and possible explanations
Proceedings of the eleventh ACM SIGKDD international conference on Knowledge discovery in data mining
You Are Who You Talk To: Detecting Roles in Usenet Newsgroups
HICSS '06 Proceedings of the 39th Annual Hawaii International Conference on System Sciences - Volume 03
Knowledge sharing and yahoo answers: everyone knows something
Proceedings of the 17th international conference on World Wide Web
On the minimal density of triangles in graphs
Combinatorics, Probability and Computing
Networks: An Introduction
Algorithms, networks, and social phenomena
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part II
Decompositions of triangle-dense graphs
Proceedings of the 5th conference on Innovations in theoretical computer science
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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.