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Networks with Complex Weights: Green Function and Power Series

Author

Listed:
  • 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)

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.

Suggested Citation

  • Anna Muranova & Wolfgang Woess, 2022. "Networks with Complex Weights: Green Function and Power Series," Mathematics, MDPI, vol. 10(5), pages 1-20, March.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:5:p:820-:d:764124
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