Infinite Ergodic Walks in Finite Connected Undirected Graphs

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.

Assessing Pandemic Uncertainty on Conditions of Vaccination and Self-isolation

“…The entropy function (3) allows for the following decomposition involving the conditional entropies [ 29 , 30 ]:…”

“…where the excess entropy [29, 30, 32–34] quantifies the apparent uncertainty of the diagnosis (i.e., the forthcoming state of the Markov chains shown in Fig. 1 ) that can be resolved by studying the entire history of states observed in the past; the mutual information [ 29 , 30 , 32 , 35 ] between the present and future states of the chain conditioned on its past state, measures the efficacy of forecasting the future state of the Markov chain from the present state alone; finally, the latter term known as ephemeral entropy of the present state conditional on the future and past states of the chain [ 29 , 30 , 33 ], assessing the amount of true uncertainty about the forthcoming state of the chain that can neither be inferred from the past history, nor have any repercussion for the future states of patient.…”

“…describing the transition probabilities between states of the chain, the unpredictable information component in (6) is then given by [ 29 , 30 ]…”

“…and the mutual information between the present state and the future state conditioned on the past state [ 29 , 30 ], i.e.,…”

Single-Threshold Model Resource Network and Its Double-Threshold Modifications

“…The first one is random walks and diffusion on graphs. This mathematical tool is used to simulate and investigate a wide class of theoretical and applied problems [11][12][13][14][15].…”