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.

A Nested Alignment Graph Kernel Through the Dynamic Time Warping Framework

“…Moreover, we would also like to further develop novel graph kernels through the dynamic time warping framework associated with other types of (hyper)graph characteristic sequences, e.g., the cycle numbers identified by the Ihara zeta function, the time-varying entropies computed from the continuoustime or discrete-time quantum walk [9,8], and the depth-based hypergraph complexity traces [3]. Finally, we are also interested in developing novel graph kernels for timevarying financial market networks [29], using the dynamic time warping framework.…”

Analyzing graph time series using a generative model

“…For instance, Ye et. al [3] present a novel 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. This approach combines the theoretical tools developed for studying graph structure in the context of statistical mechanics of complex networks and clearly point the potentials of the current approach to study real-world time-varying networks.…”

Network Edge Entropy from Maxwell-Boltzmann Statistics

“…The Maxwell-Boltzmann distribution relates the microscopic properties of particles to the macroscopic thermodynamic properties of matter [4]. It applies to systems consisting of a fixed number of weakly interacting distinguishable particles.…”

“…The properties of this physical heat bath system are described by a partition function with the energy microstates of the network represented by a suitably chosen Hamiltonian. Usually, the Hamiltonian is computed from the adjacency or Laplacian matrix of the network, but recently, Ye et al [4], have shown how the partition function can be computed from a characteristic matrix polynomial instead.…”

“…This provides a convenient route to network characterization [3,5]. By populating the energy states with particles which are in thermal equilibrium with a heat bath, this thermalization, of the occupation statistics for the energy states can be computed using the Maxwell-Boltzmann distribution [4,5]. The properties of this physical heat bath system are described by a partition function with the energy microstates of the network represented by a suitably chosen Hamiltonian.…”