Exact solutions for models of evolving clustered networks with addition and deletion via age-resolved message passing

Abstract

Evolving networks experience vertex addition and deletion over time via processes which shape their structure and function. In many empirical networks, vertices do not arrive or depart in isolation but instead within groups or cohorts: for example, coauthorship networks grow by the addition of cliques whenever a new paper with multiple authors is published. In this paper, we propose a stochastic network evolution model based on block graphs that evolve through the addition of fully connected subgraphs ($m$-cliques) and the deletion of individual vertices. We derive a master equation governing the time evolution of the joint degree distribution and obtain an exact closed-form solution, valid for an arbitrary vertex deletion rate under uniform attachment, that describes both growing and constant-size networks. From this solution we obtain the marginal degree distributions and the clustering coefficient in closed form and the size of the giant connected component and percolation threshold governing robustness to random vertex failure using age-resolved message passing. Our results generalise existing single-vertex addition-deletion models to clique-based dynamics at arbitrary rates of turnover, and are confirmed by Monte Carlo simulation.