  The exact calculation of network reliability in a probabilistic context has
been a long-standing issue of practical importance, but a difficult one, even
for planar graphs, with perfect nodes and with edges of identical reliability
p. Many approaches (determination of bounds, sums of disjoint products
algorithms, Monte Carlo evaluations, studies of the reliability polynomials,
etc.) can only provide approximations when the network's size increases. We
consider here a ladder graph of arbitrary size corresponding to real-life
network configurations, and give the exact, analytical solutions for the all-
and two-terminal reliabilities. These solutions use transfer matrices, in which
individual reliabilities of edges and nodes are taken into account. The special
case of identical edge and node reliabilities -- p and rho, respectively -- is
solved. We show that the zeros of the two-terminal reliability polynomial
exhibit structures which differ substantially for seemingly similar networks,
and we compare the sensitivity of various edges. We discuss how the present
work may be further extended to lead to a catalog of exactly solvable networks
in terms of reliability, which could be useful as elementary bricks for a new
and improved set of bounds or benchmarks in the general case.
