Abstract
We investigate a randomly evolving process of subgraphs in an underlying host graph using the spectral theory of semigroups related to the Tsetlin library and hyperplane arrangements. Starting with some initial subgraph, at each iteration, we apply a randomly selected edit to the current subgraph. Such edits vary in nature from simple edits consisting of adding or deleting an edge, or compound edits which can affect several edges at once. This evolving process generates a random walk on the set of all possible subgraphs of the host graph. We show that the eigenvalues of this random walk can be naturally indexed by subsets of edges of the host graph. We also provide, in the case of simple edits, a closed-form formula for the eigenvectors of the transition probability matrix and a sharp bound for the rate of convergence of this random walk. We consider extensions to the case of compound edits; examples of this model include the previously studied Moran forest model and a dynamic random intersection graph model. Evolving graphs arise in a variety of fields ranging from deep learning and graph neural networks to epidemic modeling and social networks. Our random evolving process serves as a general stochastic model for sampling random subgraphs from a given graph.