Expander properties in random regular graphs with edge faults

Abstract

Let H be an undirected graph. A random graph of type-H is obtained by selecting edges of H independently and with probability p. We can thus represent a communication network H in which the links fail independently and with probability f=1−p. A fundamental type of H is the clique of n nodes (leading to the well-known random graph G _n,p ). Another fundamental type of H is a random member of the set G ^d _n of all regular graphs of degree d (leading to a new type of random graphs, of the class G ^d _n,p ). Note that G _n,p =G ⁿ⁻¹ _n,p . The G ^d _n,p model was introduced in ([11]).

Information about the remaining (with high probability) structure of type-H random graphs is of interest to applications in reliable network computing. For example, it is well known that any member of G ^d _n is almost surely an efficient certifiable expander. Expanders are widely used in Computer Science. We have shown in ([11]) that G ^d _n,p has a giant component of small diameter even when d=O(1). We wish to determine the minimum value of p for which the giant component of G ^d _n,p remains a certifiable expander with high probability. In this paper we show that the second eigenvalue of the adjacency matrix of the giant component of G ^d _n,p is concentrated in an interval of small width around its mean, and that its mean is O((dp)^3/4), provided that dp>256. Thus, the giant component of a random member of G ^d _n,p remains, with high probability, a certifiable efficient expander, despite the link faults, provided that dp>256.

Work supported in part by the ESPRIT III Basic Research Programme of the EC under contract No. 9072 (Project GEPPCOM) and contract No. 7141 (Project ALCOM II)