[Paper Review] The various facets of random walk entropy

Z. Burda, Jarosław Duda+2 more|arXiv (Cornell University)|Apr 21, 2010

Statistical Mechanics and Entropy5 references3 citations

TL;DR

This paper introduces and compares two distinct random walk models—generic random walk (GRW) and maximal entropy random walk (MERW)—on graphs, where MERW maximizes global path entropy by making all paths of fixed length and endpoints equally probable. The key result is that MERW exhibits strong localization in weakly diluted lattices, trapping particles in the largest defect-free regions due to a classical Lifshitz phenomenon, while GRW spreads uniformly, highlighting a fundamental difference in stationary distributions and dynamics under disorder.

ABSTRACT

We review various features of the statistics of random paths on graphs. The relationship between path statistics and Quantum Mechanics (QM) leads to two canonical ways of defining random walk on a graph, which have different statistics and hence different entropies. Generic random walk (GRW) is in correspondence with the field-theoretical formalism, whereas maximal entropy random walk (MERW), introduced by us in a recent work, is motivated by the Feynman path-integral formulation of QM. GRW maximizes entropy locally (neighbors are chosen with equal probabilities), in contrast to MERW which does so globally (all paths of given length and endpoints are equally probable). The stationary distribution for MERW is given by the ground state of a quantum-mechanical problem where nodes whose degree is smaller than average act as repulsive impurities. We investigate static and dynamical properties GRW and MERW in a variety of examples in one and two dimensions. The most spectacular difference arises in the case of weakly diluted lattices, where a particle performing MERW gets eventually trapped in the largest nearly spherical region which is free of impurities. We put forward a quantitative explanation of this localization effect in terms of a classical Lifshitz phenomenon.

Motivation & Objective

• To clarify the statistical and entropic differences between generic random walk (GRW) and maximal entropy random walk (MERW) on graphs.
• To establish that MERW, based on the Feynman path integral, maximizes global path entropy, unlike GRW which maximizes local entropy.
• To investigate how disorder (e.g., diluted lattices) affects the stationary distributions and dynamics of GRW and MERW.
• To provide a quantitative explanation for MERW localization using the classical Lifshitz phenomenon in disordered systems.
• To demonstrate that MERW stationary distributions are governed by the Perron-Frobenius eigenvector of the adjacency matrix, linked to a quantum-mechanical tight-binding model with repulsive impurities.

Proposed method

• Define GRW as a Markov process where each step is chosen uniformly among neighbors, maximizing local entropy production.
• Define MERW as a Markov process where all paths of fixed length and endpoints are equally probable, maximizing global entropy.
• Use the adjacency matrix $ A $ to count paths and derive the largest eigenvalue and corresponding Perron-Frobenius eigenvector $ \psi_1 $, which determines the MERW stationary distribution $ \pi_a = \psi_{1a}^2 $.
• Model the effect of node degree heterogeneity in non-regular graphs as a repulsive potential $ V_a = k_{\text{max}} - k_a $ in a quantum-mechanical tight-binding Hamiltonian.
• Apply Lifshitz theory to estimate the size $ R_1 $ of the largest nearly spherical defect-free region in diluted lattices, predicting $ R_1 \sim (\ln L / |\ln p|)^{1/d} $.
• Compare GRW and MERW dynamics via numerical simulations on $ 40 \times 40 $ lattices with varying defect concentrations $ q $, tracking the evolution of probability distributions over time.

Experimental results

Research questions

• RQ1 How do GRW and MERW differ in their path statistics and stationary distributions on irregular or disordered graphs?
• RQ2 Why does MERW localize in the largest defect-free region of a weakly diluted lattice, while GRW does not?
• RQ3 What is the quantitative connection between MERW localization and the classical Lifshitz phenomenon in disordered systems?
• RQ4 How do the dynamical behaviors of GRW and MERW differ during the transient phase before reaching stationarity?
• RQ5 What role does the Perron-Frobenius eigenvector play in determining the stationary distribution of MERW on arbitrary graphs?

Key findings

• MERW stationary distribution is proportional to the square of the Perron-Frobenius eigenvector component $ \psi_{1a}^2 $, which corresponds to the ground state of a quantum-mechanical Hamiltonian with repulsive impurities at low-degree nodes.
• In weakly diluted 2D lattices, MERW localizes the particle in the largest nearly spherical defect-free region, with estimated radii $ R_1 = 34.3 $, 10.8, 4.78, and 3.34 for defect concentrations $ q = 0.001, 0.01, 0.05, 0.1 $, respectively.
• For $ d \geq 2 $, the size of the largest Lifshitz region scales as $ R_1 \sim (\ln L / |\ln p|)^{1/d} $, and the energy of the ground state scales as $ E_1 \sim (|\ln p| / \ln L)^{2/d} $, indicating logarithmic growth with system size.
• The crossover between localized and extended regimes in MERW occurs when $ qL^d \sim \ln L $, meaning even a small number of impurities can drive localization in large systems.
• Dynamical simulations show GRW spreads uniformly over the lattice, while MERW explores a sequence of metastable, increasingly larger defect-free regions before settling into the true ground state, indicating two time scales: fast local relaxation and slow tunneling between regions.
• The stationary distribution of MERW on a large $ d $-dimensional lattice with finite disorder concentration $ q $ is localized in a region of volume $ \Omega_1 \sim R_1^d \sim \ln L / q $, growing logarithmically with system size $ L $.