Concentration of the Stationary Distribution on General Random Directed Graphs

We consider a random model for directed graphs whereby an arc is placed from one vertex to another with a prescribed probability which may vary from arc to arc. Using perturbation bounds as well as Chernoff inequalities, we show that the stationary distribution of a Markov process on a random graph is concentrated near that of the "expected" process under mild conditions. These conditions involve the ratio between the minimum and maximum in- and out-degrees, the ratio of the minimum and maximum entry in the stationary distribution, and the smallest singu- lar value of the transition matrix. Lastly, we give examples of applications of our results to well-known models such as PageRank and G(n, p).Comment: 14 pages, 0 figure

Forts, (fractional) zero forcing, and Cartesian products of graphs

The (disjoint) fort number and fractional zero forcing number are introduced and related to existing parameters including the (standard) zero forcing number. The fort hypergraph is introduced and hypergraph results on transversals and matchings are applied to the zero forcing number and fort number. These results are used to establish a Vizing-like lower bound for the zero forcing number of a Cartesian product of graphs for certain families of graphs, and a family of graphs achieving this lower bound is exhibited