Related papers: Resistance Distance and Linearized Optimal Transpo…
In this paper we address the problem of understanding the success of algorithms that organize patches according to graph-based metrics. Algorithms that analyze patches extracted from images or time series have led to state-of-the art…
We have developed a steady state theory of complex transport networks used to model the flow of commodity, information, viruses, opinions, or traffic. Our approach is based on the use of the Markov chains defined on the graph…
The computation of resistance distance is pivotal in a wide range of graph analysis applications, including graph clustering, link prediction, and graph neural networks. Despite its foundational importance, efficient algorithms for…
Random walks over directed graphs are used to model activities in many domains, such as social networks, influence propagation, and Bayesian graphical models. They are often used to compute the importance or centrality of individual nodes…
Random walks on graphs are a fundamental concept in graph theory and play a crucial role in solving a wide range of theoretical and applied problems in discrete math, probability, theoretical computer science, network science, and machine…
Optimal transportation provides a means of lifting distances between points on a geometric domain to distances between signals over the domain, expressed as probability distributions. On a graph, transportation problems can be used to…
This paper is inspired by the problem of understanding in a mathematical sense the Liouville quantum gravity on surfaces. Here we show how to define a stationary random metric on self-similar spaces which are the limit of nice finite…
There have lately been several suggestions for parametrized distances on a graph that generalize the shortest path distance and the commute time or resistance distance. The need for developing such distances has risen from the observation…
We study directed, weighted graphs $G=(V,E)$ and consider the (not necessarily symmetric) averaging operator $$ (\mathcal{L}u)(i) = -\sum_{j \sim_{} i}{p_{ij} (u(j) - u(i))},$$ where $p_{ij}$ are normalized edge weights. Given a vertex $i…
We introduce a statistical mechanics formalism for the study of constrained graph evolution as a Markovian stochastic process, in analogy with that available for spin systems, deriving its basic properties and highlighting the role of the…
We formulate a new model for transport in stochastic media with long-range spatial correlations where exponential attenuation (controlling the propagation part of the transport) becomes power law. Direct transmission over optical distance…
We develop a generalized resistance geometry based on Kron reduction and effective resistance for directed graphs, paralleling classical undirected graph theory. For strongly connected directed graphs, we prove a Fiedler--Bapat identity…
Long-distance characteristics of small-world networks have been studied by means of self-avoiding walks (SAW's). We consider networks generated by rewiring links in one- and two-dimensional regular lattices. The number of SAW's $u_n$ was…
Following the recent work of Sznitman (arXiv:0805.4516), we investigate the microscopic picture induced by a random walk trajectory on a cylinder of the form G_N x Z, where G_N is a large finite connected weighted graph, and relate it to…
We analyze the distribution of the distance between two nodes, sampled uniformly at random, in digraphs generated via the directed configuration model, in the supercritical regime. Under the assumption that the covariance between the…
We establish and generalise several bounds for various random walk quantities including the mixing time and the maximum hitting time. Unlike previous analyses, our derivations are based on rather intuitive notions of local expansion…
The distance matrix $\mathcal{D}(G)$ of a graph $G$ is the matrix containing the pairwise distances between vertices. The transmission of a vertex $v_i$ in $G$ is the sum of the distances from $v_i$ to all other vertices and $T(G)$ is the…
Renormalization of complex networks requires principled criteria for assessing whether a coarse-graining preserves dynamical content. We prove that discrete harmonic morphisms -- surjective maps preserving harmonic functions -- provide the…
We propose random walks on suitably defined graphs as a framework for finescale modeling of particle motion in an obstructed environment where the particle may have interactions with the obstructions and the mean path length of the particle…
We consider transport in a fluid flow of arbitrary complexity but with a dominant flow direction. This is the situation encountered, for example, when analyzing the dynamics of sufficiently small particles immersed in a turbulent fluid and…