The effect of faults on network expansion

  • Authors:

  • Amitabha Bagchi;Ankur Bhargava;Amitabh Chaudhary;David Eppstein;Christian Scheideler

  • Affiliations:

  • University of California, Irvine, CA;Johns Hopkins University, Baltimore, MD;University of California, Irvine, CA;University of California, Irvine, CA;Johns Hopkins University, Baltimore, MD

  • Venue:
  • Proceedings of the sixteenth annual ACM symposium on Parallelism in algorithms and architectures
  • Year:
  • 2004

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Abstract

In this paper we study the problem of how resilient networks are to node faults. Specifically, we investigate the question of how many faults a network can sustain so that it still contains a large (i.e. linear-sized) connected component that still has approximately the same expansion as the original fault-free network. For this we apply a pruning technique which culls away parts of the faulty network which have poor expansion. This technique can be applied to both adversarial faults and to random faults. For adversarial faults we prove that for every network with expansion α, a large connected component with basically the same expansion as the original network exists for up to a constant times α • n faults. This result is tight in the sense that every graph G of size n and uniform expansion α (•),i.e. G has an expansion of α (n) and every subgraph G' of size m of G has an expansion of O (α (m)), can be broken into sublinear components with w(α (n) • n) faults.For random faults we observe that the situation is significantly different. In this case the expansion of a graph only gives a very weak bound on its resilience to random faults. Specifically, there are networks of uniform expansion O(≾n) that are resilient against a constant fault probability but there are also networks of uniform expansion Ω(1/log n) that are not resilient against a O(1/log n) fault probability. Thus, a different parameter is needed. For this we introduce the span of a graph which allows us to determine the maximum fault probability in a much better way than the expansion can. We use the span to show the first known results for the effect of random faults on the expansion of d-dimensional meshes.