Identifying rich clubs in spatiotemporal interaction networks

TL;DR

The paper addresses the challenge of identifying rich club effects in spatially-weighted temporal networks, where traditional TRC methods ignore edge weights. It introduces the spatially-weighted temporal rich club (WTRC) and a unified framework to separate topological, weighted, and temporal components, including a normalization against null models. The core metric is $M(k, \Delta)_{norm} = \frac{\max(\overline{C}_{k, \Delta})}{\max(\overline{C}_{k, \Delta,rand})}$, and a topological temporal counterpart (TTRC) is defined by setting weights to 1 for comparability. Through two case studies—Wisconsin congressional districts and nationwide county mobility during COVID-19—the authors show that WTRC uncovers significant, temporally persistent weighted rich clubs that static WRC or unweighted TRC miss, with clear implications for redistricting, transportation, and epidemiology; they also provide a public codebase for replication and extension.

Abstract

Spatial networks are widely used in various fields to represent and analyze interactions or relationships between locations or spatially distributed entities.There is a network science concept known as the 'rich club' phenomenon, which describes the tendency of 'rich' nodes to form densely interconnected sub-networks. Although there are established methods to quantify topological, weighted, and temporal rich clubs individually, there is limited research on measuring the rich club effect in spatially-weighted temporal networks, which could be particularly useful for studying dynamic spatial interaction networks. To address this gap, we introduce the spatially-weighted temporal rich club (WTRC), a metric that quantifies the strength and consistency of connections between rich nodes in a spatiotemporal network. Additionally, we present a unified rich club framework that distinguishes the WTRC effect from other rich club effects, providing a way to measure topological, weighted, and temporal rich club effects together. Through two case studies of human mobility networks at different spatial scales, we demonstrate how the WTRC is able to identify significant weighted temporal rich club effects, whereas the unweighted equivalent in the same network either fails to detect a rich club effect or inaccurately estimates its significance. In each case study, we explore the spatial layout and temporal variations revealed by the WTRC analysis, showcasing its particular value in studying spatiotemporal interaction networks. This research offers new insights into the study of spatiotemporal networks, with critical implications for applications such as transportation, redistricting, and epidemiology.

Figures (14)

• Figure 1: Calculations for all rich club types. For simplicity, only the numerator portion of the rich club calculation--derived from the graph being analyzed--is shown. The denominator portion follows the same calculation as the numerator but applied to a randomized version of the graph.
• Figure 2: Methods for network randomization. Edge switching changes the placement of edges in the graph while maintaining every node's degree. Weight decorrelation randomizes weight allocation while maintaining graph topology. Sequence shuffling randomly shuffles the order of temporal snapshots in a temporal network, while maintaining the weight allocation and topology within each snapshot.
• Figure 3: Steps for calculating the WTRC coefficient in a dynamic spatial interaction network. $W(G)$ represents the graph size of the rich club sub-network at a given time step. Only the numerator of Equation \ref{['eq:normalized_M']} is shown, for clarity. The denominator is calculated in the same way, but on an appropriately randomized network.
• Figure 4: An example of the temporal snapshots that make up a weighted spatiotemporal network. Light purple arcs represent flows between regions, with arc thickness indicating the volume of those flows. Rich club flows are those flows that originate in a rich club region (dark purple) and end in a rich club region (dark purple).
• Figure 5: WTRC scan results for (a) Congressional District 2 and (b) Congressional District 3 in the State of Wisconsin. The rich-club census tracts associated with the highlighted WTRC coefficient in each Congressional District (1.6 and 2.0 respectively) are shown in dark color on the map.