Simple random walk on distance-regular graphs

A survey is presented of known results concerning simple random walk on the class of distance-regular graphs. One of the highlights is that electric resistance and hitting times between points can be explicitly calculated and given strong bounds for, which leads in turn to bounds on cover times, mixing times, etc. Also discussed are harmonic functions, moments of hitting and cover times, the Green's function, and the cutoff phenomenon. The main goal of the paper is to present these graphs as a natural setting in which to study simple random walk, and to stimulate further research in the field

Distance-regular graphs

This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page

Graphs and networks theory

This chapter discusses graphs and networks theory

Absorbing Random Walks Interpolating Between Centrality Measures on Complex Networks

Centrality, which quantifies the "importance" of individual nodes, is among the most essential concepts in modern network theory. As there are many ways in which a node can be important, many different centrality measures are in use. Here, we concentrate on versions of the common betweenness and it closeness centralities. The former measures the fraction of paths between pairs of nodes that go through a given node, while the latter measures an average inverse distance between a particular node and all other nodes. Both centralities only consider shortest paths (i.e., geodesics) between pairs of nodes. Here we develop a method, based on absorbing Markov chains, that enables us to continuously interpolate both of these centrality measures away from the geodesic limit and toward a limit where no restriction is placed on the length of the paths the walkers can explore. At this second limit, the interpolated betweenness and closeness centralities reduce, respectively, to the well-known it current betweenness and resistance closeness (information) centralities. The method is tested numerically on four real networks, revealing complex changes in node centrality rankings with respect to the value of the interpolation parameter. Non-monotonic betweenness behaviors are found to characterize nodes that lie close to inter-community boundaries in the studied networks

Planar Maps, Random Walks and Circle Packing

This open access book focuses on the interplay between random walks on planar maps and Koebe’s circle packing theorem. Further topics covered include electric networks, the He–Schramm theorem on infinite circle packings, uniform spanning trees of planar maps, local limits of finite planar maps and the almost sure recurrence of simple random walks on these limits. One of its main goals is to present a self-contained proof that the uniform infinite planar triangulation (UIPT) is almost surely recurrent. Full proofs of all statements are provided. A planar map is a graph that can be drawn in the plane without crossing edges, together with a specification of the cyclic ordering of the edges incident to each vertex. One widely applicable method of drawing planar graphs is given by Koebe’s circle packing theorem (1936). Various geometric properties of these drawings, such as existence of accumulation points and bounds on the radii, encode important probabilistic information, such as the recurrence/transience of simple random walks and connectivity of the uniform spanning forest. This deep connection is especially fruitful to the study of random planar maps. The book is aimed at researchers and graduate students in mathematics and is suitable for a single-semester course; only a basic knowledge of graduate level probability theory is assumed

Localization for Linearly Edge Reinforced Random Walks

We prove that the linearly edge reinforced random walk (LRRW) on any graph with bounded degrees is recurrent for sufficiently small initial weights. In contrast, we show that for non-amenable graphs the LRRW is transient for sufficiently large initial weights, thereby establishing a phase transition for the LRRW on non-amenable graphs. While we rely on the description of the LRRW as a mixture of Markov chains, the proof does not use the magic formula. We also derive analogous results for the vertex reinforced jump process.Comment: 30 page

Using electric network theory to model the spread of oak processionary moth, Thaumetopoea processionea, in urban woodland patches

Context: Habitat fragmentation is increasing as a result of anthropogenic activities, especially in urban areas. Dispersal through fragmented habitats is key for species to spread, persist in metapopulations and shift range in response to climate change. However, high habitat connectivity may also hasten the spread of invasive species. Objective: To develop a model of spread in fragmented landscapes and apply it to the spread of an invasive insect in urban woodland. Methods: We applied a patch-based model, based on electric network theory, to model the current and predicted future spread of oak processionary moth (OPM: Thaumetopoea processionea) from its source in west London. We compared the pattern of ‘effective distance’ from the source (i.e. the patch ‘voltage’ in the model) with the observed spread of the moth from 2006 to 2012. Results: We showed that ‘effective distance’ fitted current spread of OPM. Patches varied considerably in their ‘current’ and ‘power’ (metrics from the model), which is an indication of their importance in the future spread of OPM. Conclusions: Patches identified as ‘important’ are potential ‘pinch points’ and regions of high ‘flow’, where resources for detection and management will be most cost-effectively deployed. However, data on OPM dispersal and the distribution of oak trees limited the strength of our conclusions, so should be priorities for further data collection. This application of electric network theory can be used to inform landscape-scale conservation initiatives both to reduce the spread of invasives and to facilitate large-scale species’ range shifts in response to climate change