Modeling Perpetrators' Fate-to-Fate Contagion in Public Mass Shootings In The United States Using Bivariate Hawkes Processes

TL;DR

This work models fate-to-fate contagion in U.S. public mass shootings using a two-type bivariate Hawkes process, distinguishing between events where the perpetrator dies at the scene and those where the perpetrator survives. The study finds the strongest cross-effect from 'live' to 'die at the scene' with a cross-excitation of about $0.343$ and a contagion timescale near $20$ days, while the reverse direction is not statistically significant; self-excitation also exists but is comparatively weaker. Analyses of pre- and post-2000 periods reveal a shift toward stronger cross-excitation from live to die after 2000, along with changes in contagion timescales and self-excitation strengths, likely reflecting changes in media dynamics and public discourse in the digital era. The results offer quantitative insight into how media visibility and narrative persistence may shape near-term patterns of public mass shootings, with implications for monitoring and prevention strategies. Overall, the paper demonstrates that incorporating the fate of perpetrators as a contagion channel reveals asymmetric and temporally evolving dynamics not captured by single-type models.

Abstract

This study examines how the fate of a perpetrator in a public mass shooting influences the fate of subsequent perpetrators. Using data from 1966 to 2024, we classify incidents according to whether the perpetrator died at the scene or survived the attack. Using a bivariate Hawkes process, we quantify the cross-excitation effect, which is the triggering effect that each event type exerts on the other, i.e., "die at the scene"$\rightarrow$ "live" and "live"$\rightarrow$ "die at the scene", as well as the self-excitation effects, i.e., "die at the scene"$\rightarrow$ "die at the scene" and "live"$\rightarrow$ "live". Our results show that the strongest spillover was from "live" incidents to "die at the scene", where we estimate that 0.34 (0.09, 0.80) of "die at the scene" incidents are triggered by a prior event in which the offender survived the attack. This pathway also exhibits the longest estimated contagion timescale: approximately 20 days. In contrast, the reverse influence, that is, "die at the scene"$\rightarrow$"live", is not statistically significant, with the lower bound of its 95% confidence interval nearly equal to zero. We also find that "die at the scene" events can only cause their own type, where 0.139 (0.01, 0.52) of such incidents are caused by previous "die at the scene" events, with the shortest contagion timescale of roughly 20 hours.

Figures (7)

• Figure 1: An example of a simple temporal point process. The black bullets represent the occurrences of an event, e.g., 911 calls.
• Figure 2: An illustration of a marked temporal point process. The red bullets correspond to the occurrences of the events of type 1, while the black ones account for the events of type 2. An example could be gang violence online (black bullets) and offline (red bullets).
• Figure 3: Example of a univariate Hawkes process conditional intensity using an exponential kernel. The parameters used in this simulation are, $\alpha = 0.1$, $\beta = 0.5$, $\mu = 0.6$ and $T = 300$.
• Figure 4: Example of a bivariate Hawkes process conditional intensity. The parameters used are, $\alpha = 0.50.10.20.6$, $\beta = 0.30.20.81$, and $\mu = (0.1, 0.2)$ with $T = 300$ (in days).
• Figure 5: Bivariate Hawkes process coupled conditional intensity for two event types: shooter dies at the scene (type 1) and shooter lives (type 2). The period covered is from August 1966 to September 2024, a total of 21,219 days.