Transfractal Stochastic Nets

A stochastic extension of the (u,v)-flower graph recursion is given for the case where u=1, which produces small-world, transfinite dimensional hierarchical graphs. Inspired by the two equivalent (for the non-stochastic case) means of propagation, two approaches are provided. Both rely on using a list of possible v values and a probability distribution, e.g. v=[v_1, v_2] and p=[p,1-p], and both converge to the same statistics for large graph order. The first, less restrictive case entails choosing a random v value for each edge at each generation; the second, producing a more symmetric result, involves choosing a random value for v at each generation to be consistent across that step. The main contribution of these constructions is the ability to tune desirable network parameters such as assortativity and the exponent of the power law degree distribution. An additional area of interest for the second method is exact eigenvalue propagation, which has been determined for the case of v=[2,3]. As part of this work, a general result was found for propagating the eigenvalues of recursive quadrangularizations of any simple graph.