A discontinuous phase transition in the threshold-θ geq 2 contact process on random graphs
Pith reviewed 2026-05-24 23:13 UTC · model grok-4.3
The pith
The threshold-θ contact process on random graphs with degrees at least θ+2 switches discontinuously from logarithmic extinction to exponential survival.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
If the degree distribution μ is lower bounded by θ+2 and has finite kth moments for all k>0, then for p>p1 the process survives for e^Θ(n) time w.h.p. maintaining large density, while for p<1 and small initial density it dies out in O(log n) time w.h.p.; on (θ+1)-regular graphs it dies out in n^O(1) time w.h.p.
What carries the argument
The threshold-θ contact process on random graphs with prescribed degree distribution μ, whose update rule requires θ infected neighbors for infection to occur.
If this is right
- Above p1 the process started from the all-infected state maintains positive density for time exponential in n with high probability.
- Below p=1 any sufficiently small initial set of infected nodes dies out in O(log n) time with high probability.
- On random (θ+1)-regular graphs the process dies out in time n to a polynomial power with high probability.
- The same discontinuous behavior extends to Erdős-Rényi graphs under the stated degree conditions.
Where Pith is reading between the lines
- The sharp separation between survival and extinction times may appear in other threshold-based spreading models on heterogeneous networks.
- Empirical networks whose degree sequences satisfy the lower bound could be tested for similar jumps in observed persistence times.
- Interventions that reduce effective degrees below θ+2 might force rapid extinction even at moderate transmission rates.
Load-bearing premise
The degree distribution must be bounded below by θ+2 and possess finite moments of every order.
What would settle it
Simulate the process on a sequence of random graphs with the required degree distribution and measure whether the survival time jumps from O(log n) or polynomial to exp(Θ(n)) as the infection probability crosses p1.
read the original abstract
We study the discrete-time threshold-$\theta \geq 2$ contact process on random graphs of general degrees. For random graphs with a given degree distribution $\mu$, we show that if $\mu$ is lower bounded by $\theta+2$ and has finite $k$th moments for all $k>0$, then the discrete-time threshold-$\theta$ contact process on the random graph exhibits a discontinuous phase transition in the emergence of metastability, thus answering a question of Chatterjee and Durrett \cite{cd13}. To be specific, we establish that (i) for any large enough infection probability $p>p_1$, the process started from the all-infected state w.h.p. survives for $e^{\Theta(n)}$-time, maintaining a large density of infection; (ii) for any $p<1$, if the initial density is smaller than $\varepsilon(p)>0$, then it dies out in $O(\log n)$-time w.h.p.. We also explain some extensions to more general random graphs, including the Erd\H{o}s-R\'enyi graphs. Moreover, we prove that the threshold-$\theta$ contact process on a random $(\theta+1)$-regular graph dies out in time $n^{O(1)}$ w.h.p..
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript proves that the discrete-time threshold-θ (θ≥2) contact process on random graphs with degree distribution μ (lower-bounded by θ+2 and possessing all finite moments) exhibits a discontinuous phase transition. For p>p1 the process started from the all-infected state survives for e^Θ(n) time w.h.p. while keeping positive density; for p<1 and sufficiently small initial density it dies out in O(log n) time w.h.p. Extensions to Erdős-Rényi graphs are indicated, and polynomial-time extinction is shown on (θ+1)-regular random graphs. The results answer a question of Chatterjee and Durrett.
Significance. If the derivations hold, the work supplies the first rigorous demonstration of discontinuity for θ≥2, together with explicit exponential survival and logarithmic extinction timescales that are load-bearing for the metastability claim. The explicit hypotheses on μ (minimum degree θ+2 and all moments finite) are used to obtain both the long survival and rapid extinction statements, and the paper provides machine-checkable-style probabilistic arguments under these conditions.
minor comments (2)
- The constant p1 is introduced in the abstract and results statements without an explicit definition or construction; a brief indication of how p1 is obtained (e.g., via a branching-process comparison or fixed-point equation) would improve readability even if the full construction appears later.
- The phrase 'large density of infection' in the survival statement could be replaced by a concrete lower bound (e.g., δ>0 independent of n) to make the metastability claim fully quantitative at the statement level.
Simulated Author's Rebuttal
We thank the referee for their positive assessment and recommendation to accept the manuscript. The report accurately summarizes the main results on the discontinuous phase transition for the threshold-θ contact process under the stated conditions on μ.
Circularity Check
No significant circularity detected
full rationale
The paper states its main results explicitly as conditional on the degree distribution μ having support bounded below by θ+2 and all moments finite; these hypotheses are used directly to establish the survival and extinction time bounds via standard probabilistic arguments on random graphs. No equations, fitted parameters, or self-citations are invoked in a load-bearing way that reduces the claimed times or phase transition to definitions or prior self-work by construction. The derivation relies on external techniques from percolation and contact process literature (including the cited Chatterjee-Durrett question) without renaming known results or smuggling ansatzes. The central claims remain independent of the paper's own inputs.
Axiom & Free-Parameter Ledger
axioms (1)
- standard math Standard results on the configuration model and properties of random graphs with given degree distribution having finite moments.
Reference graph
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