Journal of Statistical Physics

, Volume 167, Issue 5, pp 1233–1243 | Cite as

Information Dimension of Stochastic Processes on Networks: Relating Entropy Production to Spectral Properties

  • Oliver Mülken
  • Sarah Heinzelmann
  • Maxim Dolgushev


We consider discrete stochastic processes, modeled by classical master equations, on networks. The temporal growth of the lack of information about the system is captured by its non-equilibrium entropy, defined via the transition probabilities between different nodes of the network. We derive a relation between the entropy and the spectrum of the master equation’s transfer matrix. Our findings indicate that the temporal growth of the entropy is proportional to the logarithm of time if the spectral density shows scaling. In analogy to chaos theory, the proportionality factor is called (stochastic) information dimension and gives a global characterization of the dynamics on the network. These general results are corroborated by examples of regular and of fractal networks.


Networks Fractals Entropy Stochastic thermodynamics 



We thank Alex Blumen for fruitful discussion and valuable comments. M.D. acknowledges the support through Grant No. GRK 1642/1 of the Deutsche Forschungsgemeinschaft.


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© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Physikalisches InstitutUniversität FreiburgFreiburgGermany
  2. 2.Institut Charles SadronUniversité de Strasbourg and CNRSStrasbourg CedexFrance

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