New transience bounds for long walks in weighted digraphs

Fix two nodes i and j in an edge-weighted diagraph and form the following sequence: Let a(n) be the maximum weight of walks from i to j of length n; if no such walk exists, a(n) = −∞. It is known that, if G is strongly connected, the sequence a(n) is always eventually periodic with linear defect, i.e., after the transient, a(n + p) = a(n) + p · λ. In fact, the ratio λ is the largest mean weight of cycles in G. We call these cycles critical. Periodicity stems from the fact that the weights of critical cycles eventually dominate the maximum weight walks.