Networks with Complex Weights: Green Function and Power Series

Anna Muranova and Wolfgang Woess
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Anna Muranova: Faculty of Mathematics and Computer Science, Uniwersytet Warmińsko-Mazurski, 10-710 Olsztyn, Poland
Wolfgang Woess: Institute of Discrete Mathematics, Graz University of Technology, 8010 Graz, Austria

Mathematics, 2022, vol. 10, issue 5, 1-20

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.

Keywords: weighted graph; network; Green kernel; recurrence; transience (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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