Phase Boundary of a Stochastic Watts-Threshold SIS Model on Random Networks

Heger Arfaoui; Slimane BenMiled; Yasmine Beji

arxiv: 2606.30069 · v1 · pith:W7X2FVLD · submitted 2026-06-29 · cs.SI

Phase Boundary of a Stochastic Watts-Threshold SIS Model on Random Networks

Yasmine Beji , Heger Arfaoui , Slimane BenMiled This is my paper

Reviewed by Pith T0 review T1 audit T2 compute T3 formal T4 kernel 2026-06-30 03:47 UTCgrok-4.3pith:W7X2FVLD record.json open to challenge →

classification cs.SI

keywords complex contagionWatts thresholdSIS modelphase boundaryrandom networksepidemic thresholdMonte Carlo simulationlogistic regression

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The pith

A six-parameter interaction model captures the phase boundary of the stochastic Watts-threshold SIS model on random networks, invariant across topologies.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper reconstructs the extinction-persistence phase boundary for a stochastic Watts-threshold SIS model of complex contagion with recovery, in the joint space of transmission rate, adoption threshold and infectious duration. It shows that the boundary is well described by a six-parameter interaction model whose form does not change between Erdos-Renyi and Barabasi-Albert networks. The adoption threshold dominates feasibility, the transition is sharp, and the work supplies a quantitative reference for the complex-contagion analogue of the classical SIS threshold. A sympathetic reader would care because the result gives concrete conditions under which reinforced adoption persists or dies out on networks.

Core claim

The extinction-persistence phase boundary of the stochastic Watts-threshold SIS model is well described by a six-parameter interaction model whose structure is invariant across Erdos-Renyi and Barabasi-Albert networks. The transition is sharp, with the 10-90% extinction-probability band spanning only Δθ ≈ 0.005-0.008, and the adoption threshold is the dominant parameter governing epidemic feasibility, with transmission rate and infectious duration playing secondary and asymmetric roles.

What carries the argument

The six-parameter interaction model for the phase boundary in the joint parameter space of transmission rate β, adoption threshold θ and infectious duration d, obtained from adaptive Delaunay-based sampling and weighted logistic regression on Monte Carlo trials.

If this is right

The adoption threshold θ dominates the feasibility of epidemic persistence or extinction.
Transmission rate β and infectious duration d play secondary and asymmetric roles in setting the boundary location.
The phase transition is sharp, with the extinction probability changing rapidly over a small interval Δθ ≈ 0.005-0.008.
The structure of the six-parameter model is the same for both Erdos-Renyi and Barabasi-Albert networks.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the invariance extends, the same functional form could be tested for predicting boundaries on other random or empirical networks.
The dominance of θ implies that small shifts in the number of required reinforcing neighbors can switch an outbreak from persistence to extinction.
The reported asymmetry between β and d suggests that control measures shortening infectious periods may have different leverage than measures reducing transmission probability.

Load-bearing premise

The adaptive Delaunay-based sampling combined with weighted logistic regression on the Monte Carlo trials produces an unbiased reconstruction of the true continuous phase boundary in the (β, θ, d) space.

What would settle it

New Monte Carlo simulations on held-out parameter combinations or a third network topology that deviate substantially from the predictions of the fitted six-parameter model would show the description does not hold.

Figures

Figures reproduced from arXiv: 2606.30069 by Heger Arfaoui, Slimane BenMiled, Yasmine Beji.

**Figure 1.** Figure 1: Illustration of the threshold adoption rule. A focal susceptible node with degree k = 7 and threshold θ = 0.4 requires at least ⌈0.4 × 7⌉ = 3 simultaneous successful transmissions from infected neighbors to become infected. In this example, only 2 of 4 infected neighbors transmit successfully; the threshold is not met and the node remains susceptible. bors that successfully transmit to it in that time step… view at source ↗

**Figure 2.** Figure 2: Adaptive Delaunay boundary refinement (ER network). Left: initial coarse grid. Center: point cloud after 7 iterations, showing enrichment near the transition region. Right: cumulative regime distribution across iterations. 3 Methods 3.1 Adaptive Delaunay Boundary Refinement The extinction-persistence boundary occupies a thin manifold in the threedimensional parameter space. Uniform grid sampling would all… view at source ↗

**Figure 3.** Figure 3: ER network extinction-probability landscape. Top row: (β, θ) plane at fixed d. Bottom row: (d, θ) plane at fixed β. Solid contour: P(ext) = 0.5; dashed: 0.1 and 0.9 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: Critical adoption threshold θc at P(ext) = 0.5. Left: θc vs. β at several fixed d (well-separated curves). Right: θc vs. d at several fixed β (bunched curves). Solid: ER; dashed: BA. The BA boundary is systematically above ER across the parameter space. hierarchy θ ≫ β > d is preserved. Two quantitative differences emerge, visible in [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

**Figure 5.** Figure 5: Topology comparison. Left: overlaid P(ext) = 0.5 boundaries at d = 4.0 yr. Centre: boundary shift ∆θc = θ BA c − θ ER c across (β, d) space. Right: coefficient magnitude comparison on log scale. The invariance across ER and BA topologies (same coefficient ranking, same signs, same parameter hierarchy) suggests that the dominant features of the phase diagram arise from the threshold mechanism itself rather… view at source ↗

read the original abstract

Complex contagion models, in which adoption requires reinforcement from multiple neighbors, have been extensively studied in the monotone (no-recovery) setting, but the phase diagram of threshold models with SIS-like recovery on networks remains unmapped. We study a stochastic Watts-threshold SIS model on Erdos-Renyi and Barabasi-Albert networks and reconstruct its extinction-persistence phase boundary in the joint parameter space of transmission rate $\beta$, adoption threshold $\theta$, and infectious duration $d$. Using adaptive Delaunay-based sampling and weighted logistic regression on over 180,000 Monte Carlo trials, we find that: (i) the boundary is well described by a six-parameter interaction model whose structure is invariant across both topologies; (ii) the transition is sharp, with the 10-90\% extinction-probability band spanning only $\Delta\theta \approx 0.005$-$0.008$; and (iii) the adoption threshold is the dominant parameter governing epidemic feasibility, with transmission rate and infectious duration playing secondary and asymmetric roles. The characterization provides a quantitative reference for the complex-contagion analogue of the classical SIS epidemic threshold.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This gives the first simulated phase diagram for a Watts-threshold SIS model with recovery, but the adaptive sampling and fit need validation before the sharpness and invariance claims can be trusted.

read the letter

The main takeaway is that this paper supplies the first quantitative phase boundary for a stochastic Watts-threshold SIS model on ER and BA networks, using 180k Monte Carlo trials to fit a six-parameter interaction model in (β, θ, d) space.

What is new is the joint mapping itself. Monotone threshold models have phase diagrams, but adding SIS-style recovery left the extinction-persistence surface unmapped; the work fills that gap with concrete simulation output and reports that the same functional form works on both topologies, that the threshold dominates, and that the 10-90% band is narrow.

The scale of the runs and the focus on the dominant parameter are the parts that hold up. Those are useful reference points for anyone modeling complex contagion with recovery.

The soft spots sit in the reconstruction method. The abstract describes adaptive Delaunay sampling followed by weighted logistic regression but gives no error bars, no comparison to uniform grids, and no checks against known limiting cases. The stress-test point about possible bias from adaptive refinement therefore lands: if the sampling concentrates near an evolving boundary estimate, the fitted surface and its claimed invariance could shift. Without those diagnostics the sharpness numbers (Δ heta ≈ 0.005-0.008) rest on unverified simulation fidelity.

This is for network epidemiologists who need a starting quantitative reference for threshold models that include recovery. A reader already working on phase diagrams in contagion processes will extract the parameter ranges and the topology-invariance observation even if they later rerun the fits.

It deserves a serious referee. The gap is real and the data volume is substantial; the methodological gaps are the sort that review can tighten rather than fatal flaws.

Referee Report

2 major / 1 minor

Summary. The manuscript claims to reconstruct the extinction-persistence phase boundary of a stochastic Watts-threshold SIS model on Erdős-Rényi and Barabási-Albert networks in the three-dimensional parameter space (β, θ, d) using adaptive Delaunay sampling and weighted logistic regression on 180,000 Monte Carlo trials. It reports that a six-parameter interaction model fits the boundary invariantly across topologies, with sharp transitions (Δθ ≈ 0.005–0.008) dominated by the threshold θ.

Significance. Should the numerical reconstruction prove robust, the work would supply a quantitative benchmark for complex-contagion SIS dynamics on networks, filling a gap left by monotone threshold models. The volume of simulations and the cross-topology invariance constitute strengths that could make the six-parameter form a useful reference, provided the adaptive sampling does not introduce systematic bias.

major comments (2)

[Methods (adaptive sampling)] The central claim that the six-parameter model accurately describes the phase boundary rests on the adaptive Delaunay-based sampling and weighted logistic regression producing an unbiased estimate. However, no comparison to uniform sampling or stratified designs is reported to confirm that the adaptive refinement does not bias the reconstructed surface, particularly near the boundary where most trials are allocated.
[Results (simulation details)] The abstract and results report 180k trials but provide no error bars on the fitted parameters, no validation against known limits (such as the standard SIS threshold when θ=1), and no details on network sizes or stopping criteria; these omissions are load-bearing for assessing the reported sharpness and invariance of the model.

minor comments (1)

[Notation] The definition of the infectious duration d and its relation to the recovery process should be stated explicitly in the model section for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which highlight important aspects of the methods and results that require clarification. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Methods (adaptive sampling)] The central claim that the six-parameter model accurately describes the phase boundary rests on the adaptive Delaunay-based sampling and weighted logistic regression producing an unbiased estimate. However, no comparison to uniform sampling or stratified designs is reported to confirm that the adaptive refinement does not bias the reconstructed surface, particularly near the boundary where most trials are allocated.

Authors: We agree that the absence of a direct comparison leaves open the possibility of bias in the adaptive sampling procedure. In the revised manuscript we will add a supplementary analysis that repeats the boundary reconstruction on a reduced three-dimensional grid using uniform sampling (approximately 25,000 trials) and compares the resulting six-parameter fits to those obtained with the adaptive Delaunay scheme. Any statistically significant differences will be quantified and discussed; if differences are negligible, this will support the robustness of the original surface. revision: yes
Referee: [Results (simulation details)] The abstract and results report 180k trials but provide no error bars on the fitted parameters, no validation against known limits (such as the standard SIS threshold when θ=1), and no details on network sizes or stopping criteria; these omissions are load-bearing for assessing the reported sharpness and invariance of the model.

Authors: These omissions are acknowledged. The revised manuscript will include: (i) 95 % confidence intervals on all six fitted interaction-model parameters obtained from the weighted logistic regression; (ii) an explicit validation subsection in which θ is fixed at 1 and the resulting extinction-persistence boundary is compared with the known SIS threshold on both ER and BA networks; (iii) precise statements of the network sizes (N = 5000 nodes for both topologies) and stopping criteria (each realization runs for a maximum of 10 000 time steps or until extinction, with 200 independent Monte Carlo realizations per sampled point). These additions will be placed in the Methods and Results sections. revision: yes

Circularity Check

0 steps flagged

No significant circularity; empirical reconstruction from Monte Carlo data

full rationale

The paper reconstructs the phase boundary via adaptive Delaunay sampling and weighted logistic regression applied to over 180,000 fresh Monte Carlo trials on ER and BA networks. The six-parameter interaction model is presented as an empirical description of the resulting surface, not as a first-principles derivation or prediction that reduces to the fitting procedure by construction. No self-citations are invoked as load-bearing uniqueness theorems, no ansatz is smuggled, and no fitted parameter is relabeled as an independent prediction. The derivation chain is therefore self-contained against external simulation benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the assumption that the fitted six-parameter model extracted from finite Monte Carlo runs on two specific random-graph ensembles accurately represents the infinite-network phase boundary; the model parameters themselves are free parameters fitted to data.

free parameters (1)

six parameters of the interaction model
Fitted via weighted logistic regression to the simulation outcomes to describe the extinction probability surface.

axioms (2)

domain assumption Erdos-Renyi and Barabasi-Albert networks are representative topologies for the studied contagion processes
Simulations are performed exclusively on these two ensembles.
domain assumption The stochastic threshold rule with fixed infectious duration d correctly captures complex contagion with recovery
Core modeling choice stated in the abstract.

pith-pipeline@v0.9.1-grok · 5732 in / 1324 out tokens · 60856 ms · 2026-06-30T03:47:29.022241+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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Pastor-Satorras, R., Castellano, C., Van Mieghem, P., Vespignani, A.: Epidemic processes in complex networks. Rev. Mod. Phys.87, 925–979 (2015)

2015

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In: 22nd Int

Wang, Y., Chakrabarti, D., Wang, C., Faloutsos, C.: Epidemic spreading in real networks: an eigenvalue viewpoint. In: 22nd Int. Symp. on Reliable Distributed Systems, pp. 25–34. IEEE (2003)

2003

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Centola, D., Macy, M.: Complex contagions and the weakness of long ties. Am. J. Sociol.113(3), 702–734 (2007)

2007

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Granovetter, M.: Threshold models of collective behavior. Am. J. Sociol.83(6), 1420–1443 (1978)

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Watts, D.J.: A simple model of global cascades on random networks. Proc. Natl. Acad. Sci.99(9), 5766–5771 (2002)

2002

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Dodds, P.S., Watts, D.J.: Universal behavior in a generalized model of contagion. Phys. Rev. Lett.92(21), 218701 (2004)

2004

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Dodds, P.S., Watts, D.J.: A generalized model of social and biological contagion. J. Theor. Biol.232(4), 587–604 (2005)

2005

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Gleeson, J.P., Cahalane, D.J.: Seed size strongly affects cascades on random net- works. Phys. Rev. E75(5), 056103 (2007)

2007

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Gleeson, J.P.: Cascades on correlated and modular random networks. Phys. Rev. E77(4), 046117 (2008)

2008

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Science 329(5996), 1194–1197 (2010)

Centola, D.: The spread of behavior in an online social network experiment. Science 329(5996), 1194–1197 (2010)

2010

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Miller, J.C.: Complex contagions and hybrid phase transitions. J. Complex Netw. 4(2), 201–223 (2016)

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Barab´ asi, A.-L., Albert, R.: Emergence of scaling in random networks. Science 286(5439), 509–512 (1999)

1999

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Newman, M.E.J.: Spread of epidemic disease on networks. Phys. Rev. E66(1), 016128 (2002)

2002