Measuring event concentration in empirical networks with different types of degree distributions

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).