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Connectivity of finite anisotropic random graphs and directed graphs

Published online by Cambridge University Press:  24 October 2008

Joel E. Cohen
Joel E. Cohen
Affiliation:
Rockefeller University, New York, NY 10021, U.S.A.
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Abstract

For graphs on a finite set of vertices with arbitrary probabilities of independently occurring edges, the reliability is defined as the probability that the graph is connected, and the redundancy as the expected number of spanning trees of the graph. Analogous measures of connectivity are defined for random finite directed graphs with arbitrary probabilities of independently occurring directed edges. Recursive formulas for computing the reliability are known. Determinantal formulas, based on matrix-tree theorems, for computing the redundancy are given here. Among random graphs with a given sum of edge probabilities, the more evenly the probabilities are distributed over potential edges, the larger the redundancy. This inequality, proved using the theory of majorization, in combination with examples shows unexpectedly that conflicts between reliability and redundancy can arise in the design of communication networks modelled by such random graphs. The significance of these calculations for the command and control of nuclear forces is sketched.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 99 , Issue 2 , March 1986 , pp. 315 - 330
Copyright
Copyright © Cambridge Philosophical Society 1986

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