Piecewise-Linear Distance-Dependent Random Graph Models

In this paper, we propose a form of random graph (network) model in which the probability of an edge (link) is dependent on a real-valued function on pairs of vertices (nodes). In general, we expect this function to satisfy the triangle inequality, and hence to take the form of a distance metric, although this is not essential. It may reflect physical distances between vertices, dissimilarity measures, or other functions. We relate the distance metric d(x,y) to the probability p(x,y) of an edge existing between vertices x and y using a piecewise-linear function defined by the triple of parameters (q, a, b): p(x,y) = q, if d(x,y) <= a = 0, if d(x,y) > b = q (b - d(x,y)) / (b - a), if a < d(x,y) <= b A number of important random graph models (Erdos-Renyi graphs, geometric random graphs, and random graphs with a hard distance limit on edges) are special cases of this definition. We use maximum likelihood estimation (MLE) to derive the best parameter triple (q, a, b) for a specific graph. The paper applies this model to a number of example networks, and discusses its practical utility. Figure 1 shows an example network of this type. A simple SIR disease simulation illustrates the utility of the model for simulation purposes, in exploring a range of network topologies. [GRAPHICS] .