Abstract
The micro-canonical, canonical, and grand canonical ensembles of walks defined in finite connected undirected graphs are considered in the thermodynamic limit of infinite walk length. As infinitely long paths are extremely sensitive to structural irregularities and defects, their properties are used to describe the degree of structural imbalance, anisotropy, and navigability in finite graphs. For the first time, we introduce entropic force and pressure describing the effect of graph defects on mobility patterns associated with the very long walks in finite graphs; navigation in graphs and navigability to the nodes by the different types of ergodic walks; as well as node’s fugacity in the course of prospective network expansion or shrinking.
Highlights
The precursor of a concept of statistical ensembles and the related ergodic hypothesis formulated by Boltzmann [1,2] were met with a violently negative reaction by the great majority of scientists for clumsiness, absurd, and paradoxical consequences [3], it allowed the theoretical calculation of the equations of state for the first time
In contrast to complex network theory, we consider the statistical ensembles of walks defined on a finite connected undirected graph in the thermodynamic limit of very long walks n → ∞, which has previously never been addressed
We review three classical thermodynamic ensembles defined by Gibbs [11]—the microcanonical (Section 2), canonical (Section 4), and grand canonical (Section 8) ensembles of very long walks defined in finite connected undirected graphs—and demonstrate that the concept of ergodic ensembles might be applied to quite abstract objects of discrete mathematics
Summary
The precursor of a concept of statistical ensembles and the related ergodic hypothesis formulated by Boltzmann [1,2] were met with a violently negative reaction by the great majority of scientists for clumsiness, absurd, and paradoxical consequences [3], it allowed the theoretical calculation of the equations of state for the first time. The series of intrinsic random walks (introduced in Section 5) make up equal probabilities to all walks of a given length starting at a node providing an example of the canonical ensemble of walks defined on the finite graph. The grand canonical ensemble describes the statistics of local fluctuations of the growth rate of the numbers of long walks around the chemical potential as n → ∞ (Section 8) The distribution of these fluctuations follows Fermi–Dirac statistics and marks graph’s defects and boundary nodes hosting dramatically less very long walks than others. Plays the role of chemical potential describing the change to free energy after absorbing a new edge to a very long walk in a κ-regular graph in a micro-canonical ensemble. According to (4), the topological entropy of the graph μ can be interpreted as the effective dimension of space of the graph, dG, in a micro-canonical ensemble of very long walks
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