Two evolving social network models

Sam R. Magura and Vitchyr He Pong, North Carolina School of Science and Math, Durham, NC
David Sivakoff and Rick Durrett, Duke U.

Abstract. In our first model, individuals have opinions in [0,1]d. Connections are broken at rate proportional to their length L and a randomly chosen end point x connects to an individual chosen at random. If version (i) the new edge is always accepted. In version (ii) a new connection of length L' is accepted if L'≤L and with probability L/L' if it is longer. Our second model is a dynamic version of preferential attachment. Edges are chosen at random for deletion, then one endpoint (again chosen at random) connects to vertex z with probability proportional to f(d(z)) where f(k) = θ (k+1) + (1-θ) (ν +1), and ν is the average degree. In words, this is a mixture of degree-proportional and at random rewiring. The common feature of these models is that they have stationary distributions that satisfy the detailed balance condition and are given by explicit formulas. In addition, the first model is closely related to long range percolation, and the second to the configuration model of random graphs. As a result, we can obtain explicit results about the degree distribution, connectivity and diameter for both models.

Preprint


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