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  • 1. Lambiotte, Renaud
    et al.
    Salnikov, Vsevolod
    Rosvall, Martin
    Umeå University, Faculty of Science and Technology, Department of Physics.
    Effect of memory on the dynamics of random walks on networks2015In: Journal of Complex Networks, ISSN 2051-1310, E-ISSN 2051-1329, Vol. 3, no 2, p. 177-188Article in journal (Refereed)
    Abstract [en]

    Pathways of diffusion observed in real-world systems often require stochastic processes going beyond first-order Markov models, as implicitly assumed in network theory. In this work, we focus on second-order Markov models, and derive an analytical expression for the effect of memory on the spectral gap and thus, equivalently, on the characteristic time needed for the stochastic process to asymptotically reach equilibrium. Perturbation analysis shows that standard first-order Markov models can either overestimate or underestimate the diffusion rate of flows across the modular structure of a system captured by a second-order Markov network. We test the theoretical predictions on a toy example and on numerical data, and discuss their implications for network theory, in particular in the case of temporal or multiplex networks.

  • 2.
    Smiljanic, Jelena
    et al.
    Umeå University, Faculty of Science and Technology, Department of Physics. Scientific Computing Laboratory, Center for the Study of Complex Systems, Institute of Physics Belgrade, University of Belgrade, Pregrevica 118, Belgrade, Serbia.
    Blöcker, Christopher
    Umeå University, Faculty of Science and Technology, Department of Physics.
    Edler, Daniel
    Umeå University, Faculty of Science and Technology, Department of Physics. Gothenburg Global Biodiversity Centre, Box 461, Gothenburg, Sweden; Department of Biological and Environmental Sciences, University of Gothenburg, Carl Skottsbergs Gata 22B, Gothenburg, Sweden.
    Rosvall, Martin
    Umeå University, Faculty of Science and Technology, Department of Physics.
    Mapping flows on weighted and directed networks with incomplete observations2021In: Journal of Complex Networks, ISSN 2051-1310, E-ISSN 2051-1329, Vol. 9, no 6, article id cnab044Article in journal (Refereed)
    Abstract [en]

    Detecting significant community structure in networks with incomplete observations is challenging because the evidence for specific solutions fades away with missing data. For example, recent research shows that flow-based community detection methods can highlight spurious communities in sparse undirected and unweighted networks with missing links. Current Bayesian approaches developed to overcome this problem do not work for incomplete observations in weighted and directed networks that describe network flows. To overcome this gap, we extend the idea behind the Bayesian estimate of the map equation for unweighted and undirected networks to enable more robust community detection in weighted and directed networks. We derive an empirical Bayes estimate of the transitions rates that can incorporate metadata information and show how an efficient implementation in the community-detection method Infomap provides more reliable communities even with a significant fraction of data missing.

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