2 8 D ec 2 00 6 epl draft What is the probability of connecting two points ?

What is the probability of connecting two points ? Abstract.-The two-terminal reliability, known as the pair connectedness or connectivity function in percolation theory, may actually be expressed as a product of transfer matrices in which the probability of operation of each link and site is exactly taken into account. When link and site probabilities are p and ρ, it obeys an asymptotic power-law behavior, for which the scaling factor is the transfer matrix's eigenvalue of largest modulus. The location of the complex zeros of the two-terminal reliability polynomial exhibits structural transitions as 0 ≤ ρ ≤ 1. Introduction. – Since the original work of Moore and Shannon [1], network reliability has been a field devoted to the calculation of the connection probability between different sites of a network constituted by edges (links, bonds) and nodes (vertices, sites), each of them having a probability of operating correctly (the reliability). This field, although mainly developed in an applied background [2], is strongly related to graph theory [3, 4], combinatorics and algebraic structures [5, 6], percolation theory [7,8], as well as numerous lattice models in statistical physics [9–12]. For instance, the all-terminal reliability Rel A , i.e., the probability that all nodes are connected, is derived from the Tutte polynomial, an invariant of the associated graph, when all edges have the same reliability p (0 ≤ p ≤ 1). This polynomial appears in the partition function for various Potts models, and has been calculated for several families of graphs [9–11]; the location of its complex zeros has also been studied [10, 11, 13]. The two-terminal reliability Rel 2 (s → t), the probability that a source s and a destination t are connected, is known as the connectivity function or pair connectedness in perco-lation theory. It has been used in modeling epidemics or fire propagation [7,8]. This approach is complementary to the effort recently devoted on complex networks, in which the network resilience, i.e., its robustness against link or node failures (sometimes following deliberate attacks) has been studied for " scale-free " random graphs [14]. Exact reliability calculations are known to be very difficult [15], except for series-parallel reducible graphs for which only successive simplifications {p series = p 1 p 2 , p parallel = p 1 //p 2 = p 1 + p 2 − p 1 p 2 } are needed. Even for planar graphs with identical edge reliabilities p …