Growing graphs with addition of communities

TL;DR

This paper introduces a new model for large networks that incorporates the addition of entire communities or cliques, extending traditional preferential attachment models to better reflect real-world social and biological networks.

Contribution

It proposes a novel graph model combining preferential attachment with the addition of complete subgraphs, and derives degree distribution relations using finite-difference equations.

Findings

Model generalizes existing preferential attachment frameworks

Results match known mathematical relations in special cases

Empirical tests confirm the model's applicability to large graphs

Abstract

Paper proposes a model of large networks based on a random preferential attachment graph with addition of complete subgraphs (cliques). The proposed model refers to models of random graphs following the nonlinear preferential attachment rule and takes into account the possibility of {\guillemotleft}adding{\guillemotright} entire communities of nodes to the network. In the derivation of the relations that determine the vertex degree distribution, the technique of finite-difference equations describing stationary states of a graph is used. The obtained results are tested empirically (by generating large graphs), special cases correspond to known mathematical relations.