Discrete curvature on graphs from the effective resistance*

Abstract

This article introduces a new approach to discrete curvature based on the concept of effective resistances. We propose a curvature on the nodes and links of a graph and present the evidence for their interpretation as a curvature. Notably, we find a relation to a number of well-established discrete curvatures (Ollivier, Forman, combinatorial curvature) and show evidence for convergence to continuous curvature in the case of Euclidean random graphs. Being both efficient to approximate and highly amenable to theoretical analysis, these resistance curvatures have the potential to shed new light on the theory of discrete curvature and its many applications in mathematics, network science, data science and physics.

Transitions between cognitive topographies: contributions of network structure, neuromodulation, and disease

“…This relies on defining a suitable measure of distance on a graph. Devriendt and colleagues noted that the geodesic distance (length of the shortest path between two nodes) may be a suitable candidate, but recommended using the effective resistance instead [28,29]. Like geodesic distance, the effective resistance is predicated on the length of the paths between a pair of nodes.…”

“…However, unlike the geodesic distance, effective resistance does not only consider the shortest path between two nodes, but rather it takes into account paths of all length along the graph, such that two nodes are less distant the more paths exist between them, thereby reflecting the full topology of the network. The resistance distance ω ij between nodes i and j is large when nodes i and j are not well connected in the network, such that only few, long paths connect them, resulting in a long time for a random walker to reach one node from another, whereas a small ω ij means that they are well connected through many, predominantly short paths i and j [28]. Up to a constant, the effective resistance can be computed as the "commute time": the mean time it takes a random walker to go from node i to node j and back, for all pairs of nodes i and j [21].…”

“…(d) Transition energy to a given cognitive topography (averaged over all starting states) correlates with the mean of its NeuroSynth map. (e) Mean and variance of a NeuroSynth map are coarse descriptions that do not account for neuroanatomy.As neuroanatomically-grounded predictors of transition energy, we consider the following features of each NeuroSynth map: (i) the map's spatial alignment with the regional distribution of participation coefficients, treating the structural connectome as a network; (ii) the map's spatial alignment with the regional distribution of weighted node degree, treating the structural connectome as a network; (iii) the map's spatial alignment with the regional distribution of binary node degree, treating the structural connectome as a network; (iv) a recently developed measure of variance for distributions over a network[28,29]: unlike the usual measure of variance, which assumes independent data-points and is agnostic to their spatial location, this measure takes into account the relationships between observations. Specifically, variance of a distribution over a network is low, if high values occur at nodes that are easy to reach using a diffusion process along network paths of all length.…”

Maximal algebraic connectivity for paths with fixed total resistance

“…Esto es, la distancia se reduce al agregar aristas o aumentar pesos, por lo que resulta un a métrica natural en situaciones en las que más conectividad debiera significar menor distancia. En vista de ello, esta métrica se ha estudiado en situaciones diversas como estructuras moleculares [Babić et al, 2002], diseño de rutas aéreas [Yang et al, 2019], geometría discreta [Devriendt and Lambiotte, 2022], agrupamiento de redes [Alev et al, 2018], entre otros. La extensión de la métrica de resistencia efectiva a conjuntos infinitos juega un papel importante en la construcción de laplacianos en fractales autosimilares [Kigami, 2001, Strichartz, 2006] y en gráficas infinitas [Jorgensen and Pearse, 2010].…”