Path Evaluation and Centralities in Weighted Graphs - An Axiomatic Approach

Abstract

We study the problem of extending the classic centrality measures to weighted graphs. Unfortunately, in the existing extensions, paths in the graph are evaluated solely based on their weights, which is a restrictive and undesirable assumption for a variety of settings. Given this, we define a notion of the path evaluation function that assesses a path between two nodes by looking not only on the sum of edge weights, but also on the number of intermediaries. Using an axiomatic approach, we propose three classes of path evaluation functions. Building upon this analysis, we present the first systematic study how classic centrality measures can be extended to weighted graphs while taking into account an arbitrary path evaluation function. As an application, we use the newly-defined measures to identify the most well-linked districts in a sample public transport network.

Graph Spectral Embedding using the Geodesic Betweeness Centrality

“…It is also possible to compute edge betweenness centrality in a way that takes into account the original weight information. For example, by looking into the path evaluation function that assesses a path between two nodes that is combining both the sum of edge weights and the number of shortest distance path as was proposed in [34]. Another possible direction is to solve a generalized Sylvester equation which would incorporate both matrices corresponding to the original weight information and the edge betweenness centrality.…”

Path Querying in Graph Databases: A Systematic Mapping Study

“…Defined as the evaluation of a path with the purpose of obtaining its cost, which can be its length (the number of edges between them) or some other property. Some of the properties that can be assessed for a path include normalization, monotonicity, relocation, homogeneity, and additivity [68].…”

Artificial intelligence for improving public transport: a mapping study

“…Conceptual/theoretical results include studies whose results are on a conceptual/theoretical level and thus, not realized in practice. They mainly focus on exploring solutions based on artificial data and modelling (Cao et al 2011;Dimanche et al 2017;Sosnowska and Skibski 2018), and the results serve as contributions to potential further development. Experiments based on artificial data and Experiments based on real data include studies that show applied experiments.…”

“…• Timetable scheduling support (Bembalkar and Game 2019;Othman and Tan 2018;Tan et al 2011;Tekin et al 2018;Xie et al 2004). • Analysing PT systems (Mayaud et al 2019;Sosnowska and Skibski 2018).…”