On the Resilience of Bipartite Networks

Abstract

Motivated by problems modeling the spread of infections in networks, in this paper we explore which bipartite graphs are most resilient to widespread infections under various parameter settings. Namely, we study bipartite networks with a requirement of a minimum degree $d$ on one side under an independent infection, independent transmission model. We completely characterize the optimal graphs in the case $d=1$, which already produces non-trivial behavior, and we give extremal results for the more general cases. We show that in the case $d=2$, surprisingly, the optimally resilient set of graphs includes a graph that is not one of the two "extremes" found in the case $d=1$. Then, we briefly examine the case where we force a connectivity requirement instead of a one-sided degree requirement and again, we find that the set of the most resilient graphs contains more than the two "extremes." We also show that determining the subgraph of an arbitrary bipartite graph most resilient to infection is NP-hard for any one-sided minimal degree $d \ge 1$.