The blessing of transitivity in sparse and stochastic networks
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The interaction between transitivity and sparsity, two common features in empirical networks, implies that there are local regions of large sparse networks that are dense. We call this the blessing of transitivity and it has consequences for both modeling and inference. Extant research suggests that statistical inference for the Stochastic Blockmodel is more difficult when the edges are sparse. However, this conclusion is confounded by the fact that the asymptotic limit in all of the previous studies is not merely sparse, but also non-transitive. To retain transitivity, the blocks cannot grow faster than the expected degree. Thus, in sparse models, the blocks must remain asymptotically small. Previous algorithmic research demonstrates that small "local" clusters are more amenable to computation, visualization, and interpretation when compared to "global" graph partitions. This paper provides the first statistical results that demonstrate how these small transitive clusters are also more amenable to statistical estimation. Theorem 2 shows that a "local" clustering algorithm can, with high probability, detect a transitive stochastic block of a fixed size (e.g. 30 nodes) embedded in a large graph. The only constraint on the ambient graph is that it is large and sparse--it could be generated at random or by an adversary--suggesting a theoretical explanation for the robust empirical performance of local clustering algorithms.
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