Modularity and edge sampling
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Suppose that there is an unknown underlying graph G on a large vertex set, and we can test only a proportion of the possible edges to check whether they are present in G. If G has high modularity, is the observed graph G' likely to have high modularity? We see that this is indeed the case under a mild condition, in a natural model where we test edges at random. We find that q^*(G') ≥ q^*(G)-ε with probability at least 1-ε, as long as the expected number edges in G' is large enough. Similarly, q^*(G') ≤ q^*(G)+ε with probability at least 1-ε, under the stronger condition that the expected average degree in G' is large enough. Further, under this stronger condition, finding a good partition for G' helps us to find a good partition for G.
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