The Minrank of Random Graphs

Abstract

The minrank of a directed graph $G$ is the minimum rank of a matrix $M$ that can be obtained from the adjacency matrix of $G$ by switching some ones to zeros (i.e., deleting edges) and then setting all diagonal entries to one. This quantity is closely related to the fundamental information-theoretic problems of (linear) index coding (Bar-Yossef et al. ), network coding (Effros et al. ), and distributed storage (Mazumdar, ISIT, 2014). We prove tight bounds on the minrank of directed Erdős–Renyi random graphs $G(n,p)$ for all regimes of $p\in [{0,1}]$ . In particular, for any constant $p$ , we show that $\mathsf {minrk}(G) = \Theta (n/\log n)$ with high probability, where $G$ is chosen from $G(n,p)$ . This bound gives a near quadratic improvement over the previous best lower bound of $\Omega (\sqrt {n})$ (Haviv and Langberg), and partially settles an open problem raised by Lubetzky and Stav. Our lower bound matches the well-known upper bound obtained by the “clique covering” solution and settles the linear index coding problem for random knowledge graphs.