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Exploring Edge Probability Graph Models Beyond Edge Independency: Concepts, Analyses, and Algorithms

Desirable random graph models (RGMs) should (i) generate realistic structures such as high clustering (i.e., high subgraph densities), (ii) generate variable (i.e., not overly similar) graphs, and (iii) remain tractable to compute and control graph statistics. A common class of RGMs (e.g., Erd\H{o}s-R'{e}nyi and stochastic Kronecker) outputs edge probabilities, and we need to realize (i.e., sample from) the edge probabilities to generate graphs. Typically, each edge's existence is assumed to be determined independently for simplicity and tractability. However, with edge independency, RGMs theoretically cannot produce high subgraph densities and high output variability simultaneously. In this work, we explore realization beyond edge independence that can produce more realistic structures while maintaining high traceability and variability. Theoretically, we propose an edge-dependent realization framework called binding that provably preserves output variability, and derive closed-form tractability results on subgraph (e.g., triangle) densities in generated graphs. Practically, we propose algorithms for graph generation with binding and parameter fitting of binding. Our empirical results demonstrate that binding exhibits high tractability and generates realistic graphs with high clustering, significantly improving upon existing RGMs assuming edge independency.

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