Abstract
Measuring event concentration often involves identifying clusters of events at various scales of resolution and across different regions. In the context of a city, for example, clusters may be characterized by the proximity of events in the metric space. However, events may also occur over urban structures such as public transportation and infrastructure systems, which are naturally represented as networks. Our work provides a theoretical framework to determine whether events distributed over a set of interconnected nodes are concentrated on a particular subset. Our main analysis shows how the proposed or any other measure of event concentration on a network must explicitly take into account its degree distribution. We apply the framework to measure event concentration (i) on a street network (i.e., approximated as a regular network where events represent criminal activities); and (ii) on a social network (i.e., a power law network where events represent users who are dissatisfied after purchasing the same product).
Highlights
Consider a non-uniformly distribution of events over different regions
The proposed framework enables us to derive a summary statistic for measuring event concentration based on Voronoi diagrams
It provides an approximation for the distribution of the sizes of Voronoi cells for regular, Poisson, and power law networks in which events are distributed uniformly at random
Summary
Consider a non-uniformly distribution of events over different regions. Past efforts to explain the mechanisms through which some regions reveal a high concentration of events (i.e., form hotspots) range from agent-based [1], game theoretic [2, 3], reaction-diffusion [4], and predator-prey [5] modeling. Let Ddd denote a random variable that represents the degree of a randomly selected node in Ndv. To derive the pmf of the sizes of the Voronoi cells resulting from a uniform distribution of events, consider the following assumptions. We are able to introduce a framework that defines the pmf of the sizes of Voronoi cells when events are distributed uniformly at random. Let Dg denote a random variable that represents the degree of a randomly selected generator node. Let F(x) = P[X = x] represent the pmf of the sizes of the Voronoi cells in the case where generator nodes (events) are uniformly distributed.
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