The contact process on dynamical random trees with degree dependence

The contact process is a simple model for the spread of an infection in a structured population. We investigate the case when the underlying structure evolves dynamically as a degree-dependent dynamical percolation model. Starting with a connected locally finite base graph we initially declare edges independently open with a connection probability that is allowed to depend on the degree of the adjacent vertices and closed otherwise. Edges are independently updated with a rate depending on the degrees and then are again declared open and closed with the same probabilities. We are interested in the contact process, where infections are only allowed to spread via open edges. Our aim is to analyse the impact of the update speed and the connection probability on the existence of a phase transition. For a general connected locally finite graph, our first result gives sufficient conditions for the critical value for survival to be strictly positive. Furthermore, in the setting of Bienaymé-Galton-Watson trees, we show that the process survives strongly with positive probability for any infection rate if the offspring distribution has a stretched exponential tail with an exponent depending on the connection probability and the update speed. In particular, if the offspring distribution follows a power law and the connection probability is given by a product kernel and the update speed exhibits polynomial behaviour, we provide a complete characterisation of the phase transition.

💡 Research Summary

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The paper investigates the contact process—a classic model for infection spread—on graphs whose edges evolve over time according to a degree‑dependent dynamical percolation mechanism. Starting from a connected, locally finite base graph G, each edge {x,y} is declared open with probability p(dₓ,d_y), where dₓ and d_y are the degrees of its endpoints, and closed otherwise. Independently, each edge updates at rate v(dₓ,d_y); at an update the edge is again opened with probability p(dₓ,d_y) and closed with probability 1‑p(dₓ,d_y). The background edge process Bₜ thus stays in its stationary product measure, i.e. at any fixed time each edge is open with probability p(dₓ,d_y) independently of the others.

On top of this random evolving network the standard contact process Cₜ runs: an infected vertex recovers at rate 1, and infects each neighbor across an open edge at rate λ. The authors denote the joint process (C,B) as CPDG (contact process on a dynamical graph). They study two critical infection rates: λ₁(G) – the infimum λ for which extinction has positive probability, and λ₂(G) – the infimum λ for which the infection returns infinitely often to its starting vertex (strong survival).

The main contributions are threefold.

1. General Extinction Criterion (Theorem 3.1, Corollary 3.2).
   Assuming a uniform lower bound on the update speed (v_min > 0) and the existence of a weight function W(d) satisfying two linear inequalities (3.2) and (3.3), the authors prove λ₁(G) > 0. In other words, if edges are refreshed sufficiently often and the degree‑dependent opening probabilities decay fast enough, the infection cannot survive for arbitrarily small λ. The corollary translates these abstract conditions into concrete parameter regimes. If the connection probability obeys p(n,m) ≈ κ n^{‑α} with α ≥ 1, or if p is a product or maximum kernel of the form
   \