Stochastic Models of Network Survivability

In this paper we consider a network which is a union of several dense subnetworks (clusters or terminals) interconnected into one "large" network by means of links (edges) and nodes. The nodes represent terminals, individuals, control centers, road intersections, bridges, etc. Edges (links) are representing roads, communications channels or Internet connection between individuals (nodes). The nodes and/or edges are subject to failures which in reality may be caused by "enemy attack", natural disasters like earthquakes, floods, or disruption of communication channels. Formally, edge failure is elimination of this edge from the network, and node failure means elimination of all edges incident to this node. Assuming that node or link failures are independent events with known probability a, we approximate using Monte Carlo simulation the probability that the network has partially or completely collapsed, i.e. fell apart into a given number of isolated clusters, each containing at least one terminal. Our approach is based on defining and approximating by means of Monte Carlo the network topological invariant called network destruction spectrum (D-spectrum) (see [8]). It describes quantitatively network behavior during the process of sequential elimination of its nodes or edges. In our case, the univariate marginal D-spectrum is identical to its signature, (see [11]). The knowledge of the D-spectrum allows to describe the temporal behavior of the network in the process of its disintegration by introducing a random mechanism (like a Poisson process) governing component failure appearance. A detailed example of a network with 3 terminals, 20 nodes and 34 edges illustrates our approach.