Two evolving social network models

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

Duke Scholars

Author Richard Timothy Durrett Mathematics