Multiplicative Path Weights in Networks: From Katz Connectivity to Mean First-Passage Times

Signal dispersion, stress propagation in interbank systems, and transmission probabilities in epidemic and reliability models are all captured by studying paths on directed graphs where the path weight is defined as the product of its edge weights. Connectivity in such networks is naturally quantified by aggregating these multiplicative path weights across nodes. This paper provides a comprehensive treatise on connectivity and ranking measures based on multiplicative path weights. We demonstrate how standard all-path connectivity, such as the standard Katz centrality, can be extended beyond its classical spectral threshold. In addition, we derive closed-form expressions for connectivity based on first-hit paths and introduce a new version of Katz centrality. Furthermore, for random edge weights, we derive resolvent-based expressions for the first and second moments, alongside variance bounds, under a path-indexed coupling framework. Finite irreducible Markov chains are a subclass of the models studied, and we show that the scaled matrix of first-hit path probabilities is the probability-generating function for first-passage and return times, and that differentiation yields the classical first-passage times. Numerical examples illustrate the application of our results in a variety of settings.