title: Thermodynamic characterization of networks using graph polynomials.
abstract: In this paper, we present a method for characterizing the evolution of time-varying complex networks by adopting a thermodynamic representation of network structure computed from a polynomial (or algebraic) characterization of graph structure. Commencing from a representation of graph structure based on a characteristic polynomial computed from the normalized Laplacian matrix, we show how the polynomial is linked to the Boltzmann partition function of a network. This allows us to compute a number of thermodynamic quantities for the network, including the average energy and entropy. Assuming that the system does not change volume, we can also compute the temperature, defined as the rate of change of entropy with energy. All three thermodynamic variables can be approximated using low-order Taylor series that can be computed using the traces of powers of the Laplacian matrix, avoiding explicit computation of the normalized Laplacian spectrum. These polynomial approximations allow a smoothed representation of the evolution of networks to be constructed in the thermodynamic space spanned by entropy, energy, and temperature. We show how these thermodynamic variables can be computed in terms of simple network characteristics, e.g., the total number of nodes and node degree statistics for nodes connected by edges. We apply the resulting thermodynamic characterization to real-world time-varying networks representing complex systems in the financial and biological domains. The study demonstrates that the method provides an efficient tool for detecting abrupt changes and characterizing different stages in network evolution.
authors.affiliation: Department of Computer Science, University of York, York, YO10 5GH, United Kingdom.
authors.forename: Cheng
authors.initials: C
authors.lastname: Ye
authors.affiliation: Institute of Physics at São Carlos, University of São Paulo, PO Box 369, São Carlos, São Paulo, 13560-970, Brazil.
authors.forename: César H
authors.initials: CH
authors.lastname: Comin
authors.affiliation: Institute of Physics at São Carlos, University of São Paulo, PO Box 369, São Carlos, São Paulo, 13560-970, Brazil.
authors.forename: Thomas K Dm
authors.initials: TK
authors.lastname: Peron
authors.affiliation: Institute of Physics at São Carlos, University of São Paulo, PO Box 369, São Carlos, São Paulo, 13560-970, Brazil.
authors.forename: Filipi N
authors.initials: FN
authors.lastname: Silva
authors.affiliation: Institute of Mathematical and Computer Sciences, University of São Paulo, PO Box 668, São Carlos, São Paulo, 13560-970, Brazil.
authors.forename: Francisco A
authors.initials: FA
authors.lastname: Rodrigues
authors.affiliation: Institute of Physics at São Carlos, University of São Paulo, PO Box 369, São Carlos, São Paulo, 13560-970, Brazil.
authors.forename: Luciano da F
authors.initials: Lda F
authors.lastname: Costa
authors.affiliation: Department of Environmental Sciences, Informatics and Statistics, Ca' Foscari University of Venice, Dorsoduro 3246-30123 Venezia, Italy.
authors.forename: Andrea
authors.initials: A
authors.lastname: Torsello
authors.affiliation: Department of Computer Science, University of York, York, YO10 5GH, United Kingdom.
authors.forename: Edwin R
authors.initials: ER
authors.lastname: Hancock