We study the problem of recovering a planted matching in randomly weighted complete bipartite graphs K n,n . For some unknown perfect matching M * , the weight of an edge is drawn from one distribution P if e ∈ M * and another distribution Q if e ∈ M * . Our goal is to infer M * , exactly or approximately, from the edge weights. In this paper we take P = exp(λ) and Q = exp(1/n), in which case the maximum-likelihood estimator of M * is the minimum-weight matching M min . We obtain precise results on the overlap between M * and M min , i.e., the fraction of edges they have in common. For λ ≥ 4 we have almost perfect recovery, with overlap 1 − o(1) with high probability. For λ < 4 the expected overlap is an explicit function α(λ) < 1: we compute it by generalizing Aldous' celebrated proof of the ζ(2) conjecture for the un-planted model, using local weak convergence to relate K n,n to a type of weighted infinite tree, and then deriving a system of differential equations from a message-passing algorithm on this tree.

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The threshold model is widely used to study the propagation of opinions and technologies in social networks. In this model individuals adopt the new behavior based on how many neighbors have already chosen it. We study cascades under the threshold model on sparse random graphs with community structure to see whether the existence of communities affects the number of individuals who finally adopt the new behavior. Specifically, we consider the permanent adoption model where nodes that have adopted the new behavior cannot change their state. When seeding a small number of agents with the new behavior, the community structure has little effect on the final proportion of people that adopt it, i.e., the contagion threshold is the same as if there were just one community. On the other hand, seeding a fraction of population with the new behavior has a significant impact on the cascade with the optimal seeding strategy depending on how strongly the communities are connected. In particular, when the communities are strongly connected, seeding in one community outperforms the symmetric seeding strategy that seeds equally in all communities.