Renormalization group theory for percolation in time-varying networks

📝 Abstract

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.

💡 Analysis

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.

📄 Content

1 Scientific REPOrTs | (2018) 8:8011 | DOI:10.1038/s41598-018-25363-2 www.nature.com/scientificreports Renormalization group theory for percolation in time-varying networks Jens Karschau   , Marco Zimmerling & Benjamin M. Friedrich 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 memoryless 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. Renormalization group (RG) theory elegantly addresses percolation in static networks1–3. Percolation refers to the existence of large connected components in a random graph. Specifically, for subgraphs of a regular lattice, a giant connected component emerges above a critical lattice filling fraction, thus marking a phase transition of percolation. Percolation theory has been applied to a range of phenomena, from fluid flow in porous materials to epidemic spreading4–8. In this paper, we apply RG theory to time-varying communication networks. Our work is motivated by wireless communication networks that often exhibit unreliable links. There, a key question concerns the existence of a multi-hop path of simultaneously active links, which permits sending a mes- sage from a source to a destination node via one or several intermediate relay nodes. Real-world applications of particular relevance include self-organizing swarms of flying drones9, smart traffic networks of communicating cars10, and networks of cooperating robots in production lines11. Recent flooding and multi-path routing proto- cols were shown to be more reliable than traditional single-path routing in field experiments12,13. The emergence of ever larger wireless networks that serve as critical communication infrastructures for cyber-physical applica- tions14 prompts the need for a theoretical understanding of message losses and their temporal correlations when using these protocols15. Widely used schemes to estimate the quality of a wireless link assume that message losses are uncorrelated in time16. But temporal correlations among losses render these estimates invalid, and hence may cause existing communication protocols and control algorithms to fail17,18. This question of temporal correlations of message losses falls into a recent, application-driven interest in time-varying networks19–21. Here, we introduce a minimal model of percolation in time-varying networks, which captures key features of multi-path wireless communication with unreliable links. Most real-world applications exhibit fairly regu- lar network topologies, such as swarms of drones flying in a formation9, or sensor arrays in smart production facilities. Thus, we consider the case of network nodes distributed on a regular lattice, connected by links that stochastically switch between being active or inactive with finite switching time. The case of two states per link, active and inactive, serves as illustrative example, and corresponds to, e.g. a data transmission rate of an indi- vidual link that is either above or below the threshold, which guarantees a certain quality of service. We ask for the existence of multi-hop paths consisting of simultaneously active links that connect a designated source and destination node. We assume that transmission delays are short compared to the stochastic switching time of individual links. Indeed, in low-power wireless networks, transmission times are at most a few milliseconds per link, whereas the stochastic switching times of links can be on the order of hundreds of milliseconds. Now, if one were just interested in the probability to find a multi-hop path at a single point in time, the question would reduce cfaed, TU Dresden, 01069, Dresden, Germany. Correspondence and requests for materials should be addressed to B.M.F. (email: benjamin.m.friedrich@tu-dresden.de) Received: 11 October 2017 Accepted: 13 April 2018 Published: xx xx xxxx OPEN www.nature.com/scientificreports/ 2 Scientific REPOrTs | (2018) 8:8011 | DOI:10.1038/s41598-018-25363-2 to a bond-percolation problem for a static network, where the probability of an individual link to be act

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