  We consider crossbar switching networks with base $b$ (that is, constructed
from $b\times b$ crossbar switches), scale $k$ (that is, with $b^k$ inputs,
$b^k$ outputs and $b^k$ 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 $b\ge 2$ 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 $c\le 2$, then the linking probability $Q$ tends to 0 if
$0<q<q_c$ (where $q_c = 1/b^{(c-1)/c}$ is the critical vacancy probability),
and tends to $(1-\xi)^2$ (where $\xi$ is the unique solution of the equation
$(1-q (1-x))^b=x$ in the range $0<x<1$) if $q_c<q<1$. 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.
