Matteo Sfragara: Competing growth on the configuration model via first-passage percolation and long-range jumps

Abstract: In this talk we study two-type competing first-passage percolation on random graphs generated by the configuration model with power-law degree distributions of exponent \(\tau\). Starting from two uniformly chosen vertices, the infections spread through the graph with random edge passage times, and each vertex becomes permanently occupied by the first type that reaches it.

We begin by introducing the classical nearest-neighbor model and reviewing known results across different degree regimes, highlighting how the structure of the graph affects the outcome of the competition. In the infinite-mean regime when \(\tau \in (1,2)\), the dynamics are dominated by high-degree vertices, leading to an extreme “winner-takes-all” phenomenon: with high probability, as the graph size n tends to infinity, one type occupies the entire graph except for the starting vertex of the other type, although both types still have a positive probability of winning.

In this regime, we then extend the model by incorporating long-range jumps, allowing each infected vertex to infect uniformly chosen vertices at a rate \(\gamma_n\). This global spreading mechanism competes with the local edge-based dynamics and significantly alters the behavior of the system. In particular, it induces a phase transition: below a critical threshold, the “winner-takes-all” phenomenon persists, while above it, long-range jumps enable macroscopic coexistence between the two types.

These results provide insight into how graph heterogeneity and transmission mechanisms interact, with applications ranging from epidemic spread to competing information or malware propagation in complex networks.