Models for random graphs with variable strength edges

Two models for random graph formation are introduced that use a set of vertices with an associated set of vectors. A random process determines whether edges will be formed or clusters of connected vertices destroyed: edge-formation between vertices with similar vectors is preferred, and cluster destruction is controlled by picking an edge at random, with the probability of destruction being greater if the edge connects two vertices with dissimilar vectors. Differential equations for edge-strength and cluster-size distributions are derived and presented, and solutions to these equations are compared with numerical simulations of the models. The models are shown to have robust power-law cluster-size distributions for all parametric variations of the model, with an exponent of -52. The edge-strengths are shown to have an approximately Gaussian distribution, which does not vary with cluster size.