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.

我们在统一独立级联模型中分析网络影响最大化的问题： $ñ$ 节点和概率 $p$ 为了使节点污染邻居，请找到一组 $ķ$最初受污染的节点，使最终受污染的节点的预期数量最大化。从理论上讲，这个问题对于社交网络中的许多应用都是有意义的。不幸的是，这是一个NP难题。使用渗流理论，我们表明，在实践中，仅在消失的临界值附近，问题很难解决$p=p_C$。为了$p>p_C$ 存在一个秩序的“巨型集群” $ñ$，这很容易在有限的时间内找到。为了$p<p_C$ 群集是有限的，并且可以在有限的时间内找到其中一个（独立于 $ñ$）。因此，对于现实世界中的大多数社交网络，解决时间不会随网络规模的增长而变化。