The Linking Probability of Deep Spider-Web Networks

We consider crossbar switching networks with base b (that is, constructed from b × b crossbar switches), scale k (that is, with b inputs, b outputs and b links between each consecutive pair of stages) and depth l (that is, with l stages). We assume that the crossbars are interconnected according to the spider-web pattern, whereby two diverging paths reconverge only after at least k stages. We assume that each vertex is independently idle with probability q, the vacancy probability. We assume that b ≥ 2 and the vacancy probability q are fixed, and that k and l = ck tend to infinity with ratio a fixed constant c > 1. We consider the linking probability Q (the probability that there exists at least one idle path between a given idle input and a given idle output). In a previous paper it was shown that if c ≤ 2, then the linking probability Q tends to 0 if 0 < q < qc (where qc = 1/b (c−1)/c is the critical vacancy probability), and tends to (1 − ξ) (where ξ is the unique solution of the equation ( 1 − q(1 − x) b = x in the range 0 < x < 1) if qc < q < 1. In this paper we extend this result to all rational c > 1. This is done by using generating functions and complex-variable techniques to estimate the second moments of various random variables involved in the analysis of the networks.