Spectral Bounds in Random Graphs Applied to Spreading Phenomena and Percolation

In this paper, we derive nonasymptotic theoretical bounds for the influence in random graphs that depend on the spectral radius of a particular matrix, called the Hazard matrix. We also show that these results are generic and valid for a large class of random graphs displaying correlation at a local scale, called the LPC random graphs. In particular, they lead to tight and novel bounds in percolation, epidemiology and information cascades. The main result of the paper states that the influence in the sub-critical regime for LPC random graphs is at most of the order of $O(\sqrt{n})$ where $n$ is the size of the network, and of $O(n^{2/3})$ in the critical regime, where the epidemic thresholds are driven by the size of the spectral radius of the Hazard matrix with respect to 1. As a corollary, it is also shown that such bounds hold for the size of the giant component in inhomogeneous percolation, the SIR model in epidemiology, as well as for the long-term influence of a node in the Independent Cascade Model.