Networks with complex weights: Green function and power series

TL;DR

This paper extends the analysis of network kernels and Green functions to networks with complex weights, establishing conditions under which properties like transience and recurrence are preserved, thus broadening the applicability of Markov chain methods.

Contribution

It introduces Green functions for complex-weighted networks and demonstrates how classical Markov chain concepts extend via analytic continuation.

Findings

Green functions for complex weights are defined and analyzed.

Comparison results relate complex-weighted kernels to positive-weighted cases.

Transience and recurrence notions extend to infinite complex-weighted networks.

Abstract

We introduce a Green function and analogues of other related kernels for finite and infinite networks whose edge weights are complex-valued admittances with positive real part. We provide comparison results with the same kernels associated with corresponding reversible Markov chains, i.e., where the edge weights are positive. Under suitable conditions, these lead to comparison of series of matrix powers which express those kernels. We show that the notions of transience and recurrence extend by analytic continuation to the complex-weighted case even when the network is infinite. Thus, a variety of methods known for Markov chains extend to that setting.