arxiv: 2604.26939 · v1 · submitted 2026-04-29 · 🧮 math.PR · cs.SI· q-bio.PE

Recognition: unknown

Degree-dependent and distance-dependent contact rates interpolate between explosive, exponential and polynomial epidemic growth

Zylan Benjert , J\'ulia Komj\'athy , Johannes Lengler , John Lapinskas , Ulysse Schaller

Authors on Pith no claims yet

Pith reviewed 2026-05-07 08:47 UTC · model grok-4.3

classification 🧮 math.PR cs.SIq-bio.PE

keywords epidemic growthcontact networksdegree-dependent transmissiondistance-dependent transmissionfirst passage percolationsubexponential growthsuperspreadersnetwork geometry

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The pith

Degree and distance dependent contact rates can shift epidemic growth from explosive to polynomial on the same network.

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

The paper seeks to explain why epidemic spreading can show super-exponential, exponential, or polynomial growth using one underlying contact network. It does this by letting transmission rates depend on the number of contacts each person has and on the physical distance between people. Simulations on real-world networks and theoretical proofs show that even weak versions of these dependencies can slow the overall growth rate substantially. This matters because it offers a way to understand changing growth patterns across pandemic waves without assuming the network itself changes.

Core claim

In a network model with geometry, transmission rates that depend on degree and distance allow the spreading process to interpolate between explosive, exponential, and polynomial growth phases. The growth rate is controlled by the combination of the network geometry, the prevalence of weak long-range ties, and the presence of superspreaders. This is demonstrated through data-driven simulations and proven rigorously using a spatial multiscale argument in long-range heterogeneous first passage percolation.

What carries the argument

Degree-dependent and distance-dependent contact rates within a spatial network model.

If this is right

Consecutive waves of the same pandemic can exhibit different growth rates even when the basic spreading rules stay similar.
The overall speed of spread depends on geometry, weak ties, and superspreaders together.
Models that ignore these rate dependencies will overestimate growth in geometrically structured networks.
Small changes in how contacts scale with distance or degree can produce large shifts in epidemic curves.

Where Pith is reading between the lines

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

Interventions that reduce long-distance contacts or limit interactions with high-degree individuals could be directly incorporated into growth forecasts using this model.
The same dependencies might explain varying spread speeds in non-disease contexts like the diffusion of information across social networks with spatial structure.
Further work could test how sensitive the growth type is to the exact functional form of the degree and distance dependencies.

Load-bearing premise

Real contact networks have an underlying geometry where distance between individuals affects transmission rates in a sublinear way.

What would settle it

Simulations on geometric networks where removing the degree and distance dependencies does not change the observed growth rates from polynomial or subexponential back to exponential.

Figures

Figures reproduced from arXiv: 2604.26939 by Johannes Lengler, John Lapinskas, J\'ulia Komj\'athy, Ulysse Schaller, Zylan Benjert.

**Figure 1.** Figure 1: The Gowalla network. Visualizations of the network restricted to (a) the US and (b) Europe, as well view at source ↗

**Figure 2.** Figure 2: Illustration of the Geometric Inhomogeneous Random Graph model for three different settings of view at source ↗

**Figure 3.** Figure 3: Heatmaps of the epidemic spread: four different universality classes. Bottom row: Gowalla network view at source ↗

**Figure 4.** Figure 4: The four different phases for the epidemic curve view at source ↗

**Figure 5.** Figure 5: Phase diagrams of epidemic growth for the contact-dependent SI. On all diagrams, parameter view at source ↗

**Figure 6.** Figure 6: Illustration of the 3-edge paths constructed during the first (a) and second (b) iterations of the view at source ↗

**Figure 7.** Figure 7: Enlarged version of Figure 5a, with explicit formulas for the boundaries. view at source ↗

**Figure 8.** Figure 8: Enlarged version of Figure 5b, with explicit formulas for the boundaries. view at source ↗

**Figure 9.** Figure 9: Enlarged version of Figure 5c, with explicit formulas for the boundaries. view at source ↗

**Figure 10.** Figure 10: Enlarged version of Figure 5d, with explicit formulas for the boundaries. view at source ↗

**Figure 11.** Figure 11: Four epidemic curves I(t) as a function of t on the random reference network obtained from the Gowalla dataset, with the same parameters as on view at source ↗

**Figure 12.** Figure 12: Typical infection paths in the four different phases for the epidemic run on Europe part of the view at source ↗

**Figure 13.** Figure 13: This plot shows the edge-length distributions for the synthetic Gowalla network. Left: The view at source ↗

read the original abstract

It is a fundamental question in epidemiology to estimate, model and predict the growth rate of a pandemic. Analogously, analysing the diffusion of innovation, (fake) news, memes, and rumours is of key importance in the social sciences. The resulting epidemic growth curves can be classified according to their growth rates. These have been found to range from exponential to both faster super-exponential curves and slower subexponential or polynomial curves. Previous research has lacked a unified explanatory framework capable of accommodating super-exponential, (stretched) exponential, and polynomial growth patterns within the same contact network. In this paper we propose a simple agent-based network model that can capture all these phases. We provide such a framework by modelling how transmission rates depend on spatial distance and on individuals' numbers of contacts. By comparing the growth rate of spreading processes with or without degree-dependent and/or distance-dependent contact rates through data-driven and synthetic simulations on real and modelled networks with underlying geometry, we find evidence that even a 'sublinear presence' of these causes may cause a significant slow down of the growth rate on the same underlying network. We find that the growth rate is governed by a combination of three factors: geometry, the prevalence of weak ties, and superspreaders. We confirm our results with rigorous proofs in a theoretical model, using a spatial multiscale-argument in long-range heterogeneous first passage percolation. Our results give a plausible explanation of why the consecutive waves of a single pandemic can differ in their growth even if their spreading mechanisms are similar.

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 paper shows how adding degree and distance dependence to contact rates on a fixed geometric network can shift epidemic growth from explosive to exponential to polynomial, with simulations and a multiscale percolation argument.

read the letter

The main point is a single network model that produces all three growth regimes by making transmission rates depend on node degree and spatial distance. They keep the underlying graph fixed and only change those rate functions, which is a clean way to isolate the effect. Simulations on real and synthetic networks with geometry back the claim that even weak versions of these dependencies can slow the spread noticeably. The theory uses a spatial multiscale argument in long-range heterogeneous first-passage percolation to confirm the regimes formally. That combination of controlled numerics and percolation analysis is what is actually new here relative to earlier work on individual growth types. The paper does a decent job tying the slowdown to three factors: geometry, weak ties, and superspreaders. The controlled setup makes the results easier to interpret than many fitting exercises. The soft spots are limited but real. The abstract rates the soundness only at 4 because the full proofs and simulation details are not visible yet, so it is possible the multiscale argument has gaps when the long-range edges are very heterogeneous. The central assumption that real contact networks have usable underlying geometry and that rates can be modeled as simple sublinear functions of degree and distance is stated clearly but could be sensitive to confounding factors not captured in the model. No obvious circularity or unsupported leaps appear from what is given. This work is aimed at people in mathematical epidemiology and social diffusion modeling who need mechanistic accounts for why consecutive waves or different processes show different growth on similar networks. A reader already working with percolation or network geometry would get the most out of it. I would send it to peer review because the mix of simulation evidence and rigorous theory is substantial enough to justify referee time, even if revisions are needed on the proof details.

Referee Report

2 major / 3 minor

Summary. The manuscript proposes an agent-based network epidemic model in which transmission rates depend on node degree and spatial distance. By comparing spreading processes on fixed geometric networks with and without these dependencies, the authors claim that even sublinear forms of degree- and distance-dependence can produce a transition from explosive/super-exponential to exponential to polynomial growth. The claim is supported by data-driven and synthetic simulations on real and modeled networks together with rigorous proofs that employ a spatial multiscale argument in long-range heterogeneous first-passage percolation. The growth rate is attributed to the interplay of geometry, weak ties, and superspreaders, offering an explanation for differing growth rates across consecutive waves of the same pandemic.

Significance. If the central claims hold, the work supplies a unified, network-intrinsic mechanism for the full spectrum of observed epidemic growth regimes without altering the underlying contact structure. The controlled simulation design (fixed network, varying only the functional form of the rates) and the percolation-theoretic confirmation are notable strengths. The results could inform both epidemiological forecasting and the modeling of information diffusion, provided the geometric embedding and sublinear functional forms are shown to be robust.

major comments (2)

[§3] §3 (Simulation results): The quantitative evidence for a 'significant slow down' relies on comparing growth curves with and without degree/distance dependence, but the manuscript does not report the precise functional forms (e.g., power-law exponents) or the range of strength parameters used; without these, it is unclear whether the slowdown is generic or tied to specific post-hoc choices.
[§4] §4 (Theoretical analysis): The spatial multiscale argument in long-range heterogeneous first-passage percolation is invoked to confirm the simulation findings, yet the proof sketch does not explicitly derive how the distance-dependent rate modifies the passage-time distribution across scales; a concrete bound linking the sublinear exponent to the resulting growth regime is needed to make the interpolation claim rigorous.

minor comments (3)

[Abstract] Abstract: The phrase 'sublinear presence' is introduced without definition; it should be clarified in the model section with the exact functional dependence (e.g., rate ~ d^α with α<1).
[Figures] Figure captions (throughout): Several growth-curve panels lack error bands or indication of the number of independent realizations; adding these would improve reproducibility.
[§2] §2 (Model): The embedding procedure used to assign distances on real networks is only sketched; a short paragraph detailing the geometric embedding algorithm and its sensitivity would help readers assess the distance-dependent component.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for the constructive comments, which have helped us strengthen the presentation. We address each major comment point by point below.

read point-by-point responses

Referee: [§3] §3 (Simulation results): The quantitative evidence for a 'significant slow down' relies on comparing growth curves with and without degree/distance dependence, but the manuscript does not report the precise functional forms (e.g., power-law exponents) or the range of strength parameters used; without these, it is unclear whether the slowdown is generic or tied to specific post-hoc choices.

Authors: We agree that the precise functional forms and parameter ranges must be reported explicitly to demonstrate that the slowdown is generic rather than dependent on particular choices. In the revised manuscript we have expanded Section 3 to state the exact functional forms (sublinear power-law dependence on degree and on distance), the specific exponents employed in the primary simulations, and the full ranges of strength parameters over which the transition in growth regimes was tested. We have also added supplementary figures confirming robustness across these ranges. revision: yes
Referee: [§4] §4 (Theoretical analysis): The spatial multiscale argument in long-range heterogeneous first-passage percolation is invoked to confirm the simulation findings, yet the proof sketch does not explicitly derive how the distance-dependent rate modifies the passage-time distribution across scales; a concrete bound linking the sublinear exponent to the resulting growth regime is needed to make the interpolation claim rigorous.

Authors: The referee is right that the main-text presentation is a sketch. The complete multiscale argument, including the explicit modification of the passage-time distribution by the distance-dependent rate and the concrete bounds that relate the sublinear exponent to the resulting growth regime, appears in the supplementary appendix. In the revision we have inserted a more detailed outline of these steps into Section 4, thereby making the link between the sublinear form and the interpolated growth regimes explicit in the main text. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on independent simulations and percolation proofs

full rationale

The paper introduces an agent-based model with degree- and distance-dependent transmission rates, then validates growth-rate interpolation via controlled simulations on fixed networks (varying only the rate functions) and confirms via rigorous multiscale arguments in long-range heterogeneous first-passage percolation. No equation reduces to a fitted parameter by construction, no prediction is statistically forced from the same data subset, and no load-bearing premise rests on self-citation chains or imported uniqueness theorems. The geometry assumption is scoped explicitly to the studied networks and does not smuggle ansatzes; the central claim therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The framework rests on introducing functional forms for contact rates that depend on degree and distance; these are modeling choices rather than derived quantities. The geometric embedding of the network is taken as given for distance effects.

free parameters (2)

degree-dependence strength parameter
Controls how strongly transmission rate scales with number of contacts; varied to interpolate between growth regimes.
distance-dependence strength parameter
Controls how strongly transmission rate scales with spatial distance; varied to interpolate between growth regimes.

axioms (2)

domain assumption Contact networks possess an underlying geometric structure that permits meaningful distance-dependent rates.
Invoked for both simulations on real networks and the theoretical multiscale argument.
domain assumption Transmission rates can be expressed as functions of degree and distance.
Core modeling premise that enables the interpolation between growth types.

pith-pipeline@v0.9.0 · 5600 in / 1432 out tokens · 150825 ms · 2026-05-07T08:47:47.311016+00:00 · methodology

discussion (0)

Reference graph

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