The Effect of Faults on Network Expansion
  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 alpha, a large connected component with basically the same expansion as the original network exists for up to a constant times alpha n faults. This result is tight in the sense that every graph G of size n and uniform expansion alpha(.), i.e. G has an expansion of alpha(n) and every subgraph G' of size m of G has an expansion of O(alpha(m)), can be broken into sublinear components with omega(alpha(n) n) faults.   For random faults we observe that the situation is significantly different, because in this case the expansion of a graph only gives a very weak bound on its resilience to random faults. More specifically, there are networks of uniform expansion O(sqrt{n}) that are resilient against a constant fault probability but there are also networks of uniform expansion Omega(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. 
Amitabha Bagchi, Ankur Bhargava, Amitabh Chaudhary, David Eppstein,   and Christian Scheideler