Abstract
We propose a new distribution-free model of social networks. Our definitions are motivated by one of the most universal signatures of social networks, triadic closure---the property that pairs of vertices with common neighbors tend to be adjacent. Our most basic definition is that of a $c$-closed graph, where for every pair of vertices $u,v$ with at least $c$ common neighbors, $u$ and $v$ are adjacent. We study the classic problem of enumerating all maximal cliques, an important task in social network analysis. We prove that this problem is fixed-parameter tractable with respect to $c$ on $c$-closed graphs. Our results carry over to weakly $c$-closed graphs, which only require a vertex deletion ordering that avoids pairs of nonadjacent vertices with $c$ common neighbors. Numerical experiments show that well-studied social networks with thousands of vertices tend to be weakly $c$-closed for modest values of $c$.
Highlights
There has been an enormous amount of important work over the past 15 years on models for capturing the special structure of social networks
Generative models articulate a hypothesis about what “real-world” social networks look like, how they are created, and how they will evolve in the future
The plethora of models presents a quandary for the design of algorithms for social networks with rigorous guarantees: which of these models should one tailor an algorithm to? One idea is to seek algorithms that are tailored to none of them, and to instead assume only determinstic combinatorial conditions that share the spirit of the prevailing generative models
Summary
There has been an enormous amount of important work over the past 15 years on models for capturing the special structure of social networks This literature is almost entirely driven by the quest for generative (i.e., probabilistic) models. Generative models articulate a hypothesis about what “real-world” social networks look like, how they are created, and how they will evolve in the future. They are directly useful for generating synthetic data and can be used as a proxy to study the effect of random processes on a network [3, 41, 43]. To define our notion of “socialnetwork-like” graphs, we turn to one of the most agreed upon properties of social networks – triadic closure, the property that when two members of a social network have a friend in common, they are likely to be friends themselves
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