Renormalization group theory for percolation in time-varying networks

• Published in: arXiv.org
• Main Authors: Karschau, Jens, Zimmerling, Marco, Friedrich, Benjamin M
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 28.05.2018
• Subjects: Computer Science - Networking and Internet Architecture; Cyber-physical systems; Group theory; Markov analysis; Markov processes; Nodes; Percolation theory; Physics - Cellular Automata and Lattice Gases; Physics - Statistical Mechanics; Protocol (computers); Switching theory; Wireless communications; Wireless networks
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1708.05704

Motivated by multi-hop communication in unreliable wireless networks, we present a percolation theory for time-varying networks. We develop a renormalization group theory for a prototypical network on a regular grid, where individual links switch stochastically between active and inactive states. The question whether a given source node can communicate with a destination node along paths of active links is equivalent to a percolation problem. Our theory maps the temporal existence of multi-hop paths on an effective two-state Markov process. We show analytically how this Markov process converges towards a memory-less Bernoulli process as the hop distance between source and destination node increases. Our work extends classical percolation theory to the dynamic case and elucidates temporal correlations of message losses. Quantification of temporal correlations has implications for the design of wireless communication and control protocols, e.g. in cyber-physical systems such as self-organized swarms of drones or smart traffic networks.