On the influence maximization problem and the percolation phase transition

We analyze the problem of network influence maximization in the uniform independent cascade model: Given a network with N nodes and a probability p for a node to contaminate a neighbor, find a set of k initially contaminated nodes that maximizes the expected number of eventually contaminated nodes. This problem is of interest theoretically and for many applications in social networks. Unfortunately, it is a NP-hard problem. Using Percolation Theory, we show that in practice the problem is hard only in a vanishing neighborhood of a critical value p=p_c. For p>p_c there exists a “Giant Cluster” of order N, that is easily found in finite time. For p<p_c the clusters are finite, and one can find one of them in finite time (independent of N). Thus, for most social networks in the real world the solution time does not scale with the size of the network.