Reentrant spreading in a two-species coalescence-fragmentation model with susceptible-infected-recovered dynamics

The 2025 Bondi Beach mass shooting of Jews was perpetrated by individuals inspired by ISIS (Islamic State) propaganda that increasingly featured anti-Semitic hate content following the October 2023 start of the Israel-Palestine war. There is an urgent need to get ahead of future threats by understanding how and when a newly created piece of hate content will spread systemwide online. We present a two-species coalescence-fragmentation model with susceptible-infected-recovered dynamics that incorporates the following published empirical features: (1) New pieces of hate content tend to be generated and promoted by a subset of in-built communities on less regulated platforms. (2) These ''hate'' communities create links (hyperlinks) with each other and with nonhate communities across all platforms to form dynamically evolving clusters (i.e., coalescence) across which new hate content can then spread. (3) These clusters can get broken up by moderator shutdowns (i.e., fragmentation). We present numerical solutions and derive two levels of approximate mean-field theory: effective medium theory and beyond effective medium theory. Both numerical and analytic solutions reveal that systemwide spreading is governed by reentrant threshold phases: as the fraction of hate communities varies, the system can transition from spreading to no spreading and back to spreading. The derived analytic formulas give explicit insight into how these phase boundaries might be manipulated to prevent systemwide spreading. More broadly, the reentrant phase behavior warns that policies which steadily reduce the number of hate communities can initially succeed but then backfire if pushed further, suggesting that blanket requirements for platforms to simply do ''more'' are oversimplistic.