Estimating the Percolation Centrality of Large Networks through Pseudo-dimension Theory

In this work we investigate the problem of estimating the percolation centrality of every vertex in a graph. This centrality measure quantifies the importance of each vertex in a graph going through a contagious process. It is an open problem whether the percolation centrality can be computed in $\mathcal{O}(n^{3-c})$ time, for any constant $c>0$. In this paper we present a $\mathcal{O}(m \log^2 n)$ randomized approximation algorithm for the percolation centrality for every vertex of $G$, generalizing techniques developed by Riondato, Upfal e Kornaropoulos (this complexity is reduced to $\mathcal{O}((m+n) \log n)$ for unweighted graphs). The estimation obtained by the algorithm is within $\epsilon$ of the exact value with probability $1-\delta$, for {\it fixed} constants $0 < \epsilon,\delta \leq 1$. In fact, we show in our experimental analysis that in the case of real world complex networks, the output produced by our algorithm is significantly closer to the exact values than its guarantee in terms of theoretical worst case analysis.