Dissimilarity measures for generalized Lotka-Volterra systems on networks

Alejandro P. Riascos; Maryam Chaib De Mares; Nicol\'as A. M\'arquez

REVIEW 4 major objections 4 minor 28 references

Dissimilarity measures for generalized Lotka-Volterra systems on networks

T0 review · 4 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read The paper introduces a dissimilarity measure that compares two generalized Lotka-Volterra dynamical processes sharing initial conditions but differing in interaction parameters, network topology, or governing equations, and claims it can de

desk verdict A clean but conventional trajectory-distance measure for gLV systems, with a few genuinely nice demonstrations and one overblown stability claim that rests on a single network topology. read the letter →

arxiv 2511.12701 v2 pith:JBQJC5VF submitted 2025-11-16 cond-mat.stat-mech physics.bio-ph

Nicolás A. Márquez , Maryam Chaib De Mares , Alejandro P. Riascos This is my paper · ORCID

classification cond-mat.stat-mechphysics.bio-ph

keywords generalizedLotka–Volterradissimilaritymeasuresnetworkdynamicsecologicalstabilitymodularnetworksnonlinearpredator–preystructuralsensitivity

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper aims to establish a general, quantitative way to compare two generalized Lotka-Volterra (gLV) dynamical processes that start from the same initial conditions but differ in interaction strengths, network structure, or even the mathematical form of their dynamics. The proposed dissimilarity measure, D(t), tracks the normalized sum of absolute abundance differences over time, with a maximum D_max and a long-time limit D_inf that together capture both transient and stationary behavior. Using this measure, the authors show that structural differences invisible at the adjacency-matrix level can still produce markedly divergent dynamics in small directed networks, and that in modular networks the fraction of negative interactions acts as a control parameter for a sharp transition from stable to unstable behavior. A sympathetic reader would care because this offers a practical tool for robustness analysis and instability prediction in ecology, microbiome science, and other fields that model complex systems with nonlinear interactions.

What carries the argument

The central object is the dissimilarity D(t) = (1/S0) Σ_i |x_i(t) − x*_i(t)|, comparing a reference process x_i(t) and a modified process x*_i(t) that share initial conditions and total initial abundance S0, evolving under gLV dynamics with possibly different interaction matrices Λ vs Λ*, different network topology, or different governing equations. Its maximum over time, D_max, condenses the comparison into a single scalar, and its long-time limit D_inf indicates whether the two processes converge to the same steady state. The sign-flip protocol — taking a reference interaction matrix Λ and constructing Λ* with the same absolute weights but each sign kept positive with probability p — is th

What would settle it

Repeat the sign-flip protocol on a different network geometry (e.g., two cliques, a scale-free modular graph, or a larger number of cliques) or with weights drawn from an exponential or Gaussian distribution, and check whether ⟨D_max⟩ still rises sharply near 1-p≈0.2 and diverges for 1-p>0.7. If the thresholds shift or vanish, the claimed transition is not generic.

Watch

Extended reading notes

Core claim

The paper claims that a scalar, time-dependent dissimilarity D(t) — defined as the normalized sum of absolute abundance differences between two gLV processes that share initial conditions — can serve as a universal comparator for generalized Lotka-Volterra systems. Defined for two-species predator–prey models (Eq. 4) and for arbitrary N-species networks (Eq. 13), with a maximum D_max (Eq. 14) and long-time limit D_inf, the measure captures both transient divergence and stationary differences. The authors demonstrate that in directed three-node graphs, D(t) distinguishes dynamics that are indistinguishable at the adjacency-matrix level; in a 12-node modular network, increasing the fraction of

Load-bearing premise

The claim that a threshold fraction of negative interactions (above ~20%) drives instability and divergence (above ~70%) is based on simulations of a single 12-node clique-ring network with uniformly drawn weights; if these thresholds depend on network size, topology, or weight distribution, the broad conclusion that the fraction and distribution of negative interactions control the stability transition would be unsupported.

Editorial extensions

If this is right

Dynamical comparison supersedes purely structural network comparison: two graphs with identical link counts and identical sum of matrix differences can still exhibit very different dissimilarity, as shown for three-node graphs.
The fraction of negative interactions in a modular network acts as a control parameter: below ~20% sign-flip probability the reference and modified dynamics stay close, above it they separate sharply, and for more than ~70% the modified process diverges.
Localized structural changes confined to a single community are buffered: even complete sign inversion inside one clique does not destabilize the whole 12-node system, whereas the same fraction of global sign flips does.
The measure works across functional forms: comparing a reference gLV equation to a modified one with interaction term x_j^γ detects an asymmetry around γ=1, with larger dissimilarity for γ>1 than γ<1.
The framework offers a practical tool for microbiome and ecology studies, where inferred interaction matrices are uncertain and alternative functional forms are plausible.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the dissimilarity is normalized by initial abundances and uses identical initial conditions, it isolates the effect of the interaction structure; one could invert the perspective and use it to quantify sensitivity to initial-condition uncertainty by sampling initial states instead of perturbing Λ.
The sharp threshold near 1-p≈0.2 and divergence near 1-p>0.7 in a modular network may be a finite-size signature of a broader phase transition; testing other N, other modular connectivities, and other weight distributions could reveal whether these values scale or shift.
The localized-vs-global contrast suggests a design principle for interventions in microbial communities: treatments aimed at one community may be safer than global network changes, assuming the buffer is not an artifact of the specific clique-ring topology.
Extending D(t) to stochastic gLV dynamics (with demographic or environmental noise) could produce a robustness index that flags systems on the verge of instability earlier than deterministic trajectories.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should read this if you care about comparing dynamical processes on networks. The core measure is a normalized L1 distance between two gLV trajectories with the same initial conditions, and it is well-defined and easy to compute. The two-species analysis is clear, and the exhaustive N=3 graph comparison in Fig. 3 is the best part: it genuinely shows that pairwise dynamical similarity does not track simple structural counts. For instance, G2 is closer to G5 than G1 is, even though both have the same number of edges and identical aggregate entry differences. That is a real, reproducible observation and it makes the case for dynamical comparison.

The gamma asymmetry in Section 3.5 is also interesting—the dissimilarity is much larger for gamma > 1 than for gamma < 1, even at the same absolute deviation from the reference. That is not obvious a priori and suggests the measure can compare across functional forms.

Now the soft spots, in proportion. The biggest problem is the stability claim. The abstract says the fraction of negative interactions 'control the transition from stable to unstable dynamics,' and the conclusion calls this a way to 'predict instabilities.' What the paper actually shows is one 12-node network (three cliques of four in a ring), one weight distribution U(0,1], 200 realizations per p, and no scaling with N or topology. That is a numerical observation about a single ensemble, not a general result. The stress-test note is right: in random Lotka-Volterra systems, thresholds depend on N and interaction strength, so the generality claim is unsupported.

Also, D_max is not a predictive diagnostic in the way the abstract implies. It is computed after simulating both systems, so when the modified system diverges, the L1 distance diverges by construction. The sharp rise at 1-p ≈ 0.2 could be an early-warning candidate, but the paper does not test whether it predicts instability before it happens. That needs to be softened.

The abstract's 'invisible at the adjacency-matrix level' is an overstatement. The three-node example compares graphs with equal edge counts, not identical adjacency matrices—so the difference is visible if you look at the full matrix. The point is still valid (edge count is not enough), but the wording oversells.

Minor issues: no code or data provided, integration details are thin, and there are small notation inconsistencies. None of that is fatal.

Bottom line: the framework is useful and the N=3 and gamma results deserve airtime, but the paper overclaims on stability and generality. I would send it to peer review, not desk reject, but with a clear request to either add a multi-topology scaling analysis or limit the claims. A serious referee could make this into a solid methodological note.

For my own work, I would not cite it in the next year—the measure is too simple to need this reference, and the stability claims are not yet reliable. But I would bring it to a reading group if we were discussing network-comparison methods.

Referee Report

4 major / 4 minor

Summary. The paper introduces a dissimilarity measure for two generalized Lotka-Volterra (gLV) processes that share initial conditions but differ in interaction parameters, network structure, or governing equations. The measure D(t) is defined in Eq. (13) as the normalized L1 distance between the two trajectories, with summary statistics D_max (Eq. 14) and the long-time limit D_inf. The authors apply the framework to two-species predator-prey systems, to all non-isomorphic connected directed graphs with N=3, and to a 12-node modular network with randomly sign-flipped interactions. They conclude that the fraction and distribution of negative interactions control the transition from stable to unstable dynamics and that dissimilarity measures can predict instabilities. The core definition is simple and well posed, and the N=3 census is instructive, but several broad interpretive claims go beyond the evidence presented.

Significance. If the full claims were supported, the framework would supply a practical, systematic method for comparing gLV systems across parameter, topology, and functional-form changes, with potential applications in ecology and microbiome research. The core measure is mathematically clean and parameter-free, and the three-node graph census (10^4 realizations per pair) convincingly shows that simple aggregate structural statistics such as edge count do not order dynamical similarity. The extension to different nonlinearities in Section 3.5 is a useful generalization. However, the alleged stability-transition thresholds are derived from one network topology and from a post-hoc divergence measure; the advertised predictive power is not demonstrated. The strengths are real but the conclusions need to be scaled back or supported by additional experiments.

major comments (4)

[Abstract; §4; Eq. (14)] The abstract and conclusions claim that D_max 'predicts instabilities' and serves as an early-warning diagnostic. As defined, D_max is computed after both the reference and modified systems have been integrated; when the modified system diverges, the L1 distance in Eq. (13) grows without bound, so D_max necessarily diverges. The sharp rise at 1-p≈0.2 in Fig. 5(e) is therefore a post-hoc description of the stability boundary, not a prediction. To support 'predict instabilities,' the authors would need an out-of-sample or pre-divergence test, e.g., examining whether D(t) exhibits a characteristic signature before divergence occurs. As written, the predictive claim is unsupported.
[§3.3–3.4; Figs. 5(e), 6(e)] The transition near 1-p≈0.2 and the localized-vs-global perturbation contrast are demonstrated for a single network configuration: N=12, three cliques of four connected by a ring, with Λ_ij iid U(0,1] and 200 realizations per p. No variation in N, community structure, degree sequence, weight distribution, or self-regulation strength is reported. Since random-matrix stability thresholds for Lotka-Volterra systems are known to depend on N, connectivity, and interaction variance (May 1972; Allesina & Tang 2012, both cited in §2.3), the broad conclusion that 'the fraction and distribution of negative interactions control the transition from stable to unstable dynamics' is not supported by the evidence. A scaling analysis or a sweep over these parameters is required.
[§3.3; construction of Λ and Λ*] In Sections 3.3 and 3.4 the reference interaction matrix Λ always has all positive entries drawn from (0,1], and the modified matrix Λ* is obtained by sign-flipping each entry with probability 1-p. Thus 1-p is not the fraction of negative interactions in the ecosystem; it is the probability that a previously positive interaction becomes negative. The reference state contains no negative interactions, so the experiments do not address how mixtures of positive and negative interactions in both systems affect stability. This conflates perturbation size with interaction-type composition and should be stated explicitly and discussed.
[§3.2; Fig. 3 discussion] The text states that the differences between graphs G1 and G2 are 'invisible at the adjacency-matrix level.' This is an overstatement: G1 and G2 are non-isomorphic, so their adjacency matrices differ, and standard adjacency-based metrics such as Hamming distance distinguish them. What actually coincides are aggregate quantities: the number of edges and the sum of matrix differences with G5. The legitimate point is that these simple aggregate statistics do not order the systems by dynamical similarity. The abstract's phrase 'differences that are invisible at the adjacency-matrix level' should be revised accordingly.

minor comments (4)

[Fig. 7 caption] The caption labels the panels as '(a) p=0.8, (b) p=0.7, (c) p=0.6, (b) p=0.5'; the last label should be (d).
[Eq. (13)] The double bars ||·|| are used for the absolute value of a scalar quantity. Using single vertical bars would avoid confusion with a norm.
[§3.3; numerical methods] No details are given for the numerical integration (solver, step size, divergence criterion, simulation horizon, or handling of unbounded growth). The location of the threshold in Fig. 5(e) may depend on these choices; please report them.
[§3.5; Eq. (15)] The text says the reference process follows Eq. (12) and the modified process follows Eq. (15) with the same Λ. It would be helpful to state explicitly that the same rescaling and initial-condition ensemble are used for both, to make the comparison fully parallel.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: D(t) and D_max are explicitly defined as normalized L1 trajectory distances, and the reported thresholds are numerical outputs of the stated simulations, not fitted or imported predictions.

full rationale

The paper's central measure D(t) is defined openly in Eqs. (4) and (13) as a normalized L1 distance between reference and modified gLV trajectories, with D_max defined in Eq. (14) as its maximum. All results—two-species T_50% curves, three-node graph comparisons, clique-network sign-flip curves, and the gamma-asymmetry—come from direct numerical integration of the stated equations. No parameter is fitted to data and then relabeled as a prediction; no uniqueness theorem is imported from the authors' prior work; and no ansatz is smuggled in via citation. The only explicit 'by construction' statement is the gamma=1 baseline in Fig. 7, where D_max=0 because the two processes are identical; the paper correctly labels this and does not use it as evidence. The self-citations (Riascos & Padilla 2023; Riascos 2024; Polo-González et al. 2025) are contextual literature mentions and are not load-bearing for the derivations. The abstract's word 'predict' is loose—D_max is computed after simulating both systems, and the instability thresholds are read off the same simulations—but this is a framing/overclaim issue, not circularity: the sharp rise near 1-p≈0.2 and the localized-vs-global contrast are nontrivial simulation findings that are not forced by the definition of D. Thus no step reduces by construction to its inputs.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the specific simulation choices (clique topology, uniform weight distribution, fixed-total initial conditions) and on the faithful numerical integration of the gLV ODEs. No parameters are fitted to data; the thresholds (1-p≈0.2, 0.7) are emergent numerical observations. The approach extends the authors' prior dissimilarity measures (Riascos 2024; Riascos and Padilla 2023) to gLV dynamics.

free parameters (3)

Edge-weight distribution for network simulations
Nonzero Λ_ij drawn independently from Uniform(0,1] for the clique network (Sec. 3.3, Fig. 5). The dissimilarity/stability thresholds are demonstrated only for this distribution.
Network topology (three cliques of 4 nodes in a ring)
The only modular topology tested (Sec. 3.3-3.4, Fig. 5-6). The localized-vs-distributed perturbation contrast may depend on this specific structure.
Initial-condition ensemble (fixed total S_0, random split)
Ensemble averages over initial conditions with S_0 = 1,2,3,4,5 (Sec. 3.2-3.3). The ordering of dissimilarity between graph pairs could depend on this sampling rule.

assumptions (4)

domain assumption The gLV model (Eq. 12) with common growth rate r_i = 1 and intraspecific competition Λ_ii = 1 is a faithful representation of the ecological dynamics under comparison.
Used throughout; the simplification is stated in Sec. 2.3 ('It is assumed that r_i = r ... Λ_ii = 1'). If growth-rate heterogeneity matters, the dissimilarity framework's conclusions could change.
domain assumption Numerical integration (4th-order Runge-Kutta) accurately captures the dynamics, including divergence for large negative-interaction fractions.
The instability for 1-p>0.7 is inferred from divergent numerical solutions (Sec. 3.3); step-size control and divergence detection are not specified.
standard math Standard theorems on existence/uniqueness of solutions for the polynomial gLV ODEs hold on the relevant domain.
Relied upon implicitly for the well-definedness of the trajectories used in Eqs. (4),(13).
domain assumption May (1972) and Allesina-Tang (2012) stability criteria cited as background apply to the sign-flip regime transitions discussed.
The paper invokes these results in Sec. 2.3 to frame the stability discussion, though the numerical findings are not derived from them.

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Cite this review

Pith. "Pith review of Dissimilarity measures for generalized Lotka-Volterra systems on networks." pith.science (2026). https://pith.science/paper/JBQJC5VF

@misc{pith2026251112701,
  author       = {Pith},
  title        = {Pith review of: Dissimilarity measures for generalized Lotka-Volterra systems on networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JBQJC5VF}},
  note         = {Machine review of arXiv:2511.12701}
}

read the original abstract

In this paper, we introduce a general framework to quantify dissimilarities between generalized Lotka-Volterra dynamical processes, ranging from classical predator-prey systems to multispecies communities interacting on networks. The proposed measures capture both transient and stationary dynamics, allowing systematic comparisons across systems with varying interaction parameters, network weights, or topologies. Our analysis shows that even subtle structural changes can lead to markedly distinct outcomes: in two-species systems, interaction strength and initial conditions strongly affect divergence, while in small directed networks, differences that are invisible at the adjacency-matrix level produce divergent dynamics. In modular networks, the fraction and distribution of negative interactions control the transition from stable to unstable dynamics, with localized perturbations within cliques yielding different global outcomes than distributed ones. Beyond structural variations, the framework also applies when modified processes follow distinct nonlinear equations, demonstrating its versatility. Taken together, these results highlight that dynamical dissimilarity measures provide a powerful tool to analyze robustness, detect structural sensitivity, and predict instabilities in nonlinear systems. More broadly, this approach supports the comparative analysis of biological systems, where complex interaction networks and nonlinear dynamics are central to stability and resilience.

Figures

Figures reproduced from arXiv: 2511.12701 by the authors.

**Figure 1.** Dissimilarity in the dynamics of prey and predator populations using a reference system with 𝛼 = 1. (a) Functions 𝑢(𝜏) (original process) and 𝑢 ⋆ (𝜏) (modified process with 𝛼 ⋆ = 1.2) associated to preys as a function of 𝜏, with 𝑢0 = 𝑢 ⋆ 0 = 2, panel (b) shows the same result for predators 𝑣(𝜏) and 𝑣 ⋆(𝜏). (c) Dissimilarity metric (LV)(𝜏) in Eq. (4) for the two systems in (a)–(b), {(𝑢, 𝑣), (𝑢 ⋆, 𝑣⋆)}, as a function… view at source ↗

**Figure 2.** Comparison of two gLV dynamical processes in a graph with three nodes. (a) Reference case: a complete graph with uniform interactions, Λ𝑖𝑗 = 1. (b) Modified graph: a single interaction is altered with Λ ⋆ 𝑎𝑏 = 1 − 𝛽 for the link 𝑎 → 𝑏, where 𝛽 ≥ 0; all other entries Λ⋆ 𝑖𝑗 remain as in the reference. (c)-(e) Temporal evolution of the dissimilarity (LV)(𝑡) between the dynamics on the reference graph (a) and the modif… view at source ↗

**Figure 3.** Comparison of gLV dynamical processes for all non-isomorphic connected directed graphs of size 𝑁 = 3. The thick continuous lines show the ensemble average ⟨ (LV)(𝑡) ⟩ as a function of time 𝑡, computed from 104 realizations with random initial conditions fulfilling 0 = 1. Results are arranged in panels where row 𝜇 and column 𝜈 correspond to the comparison between the dynamics defined by the graph 𝜇 and the graph … view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Statistical analysis of ∞ for the comparison of gLV dynamical processes on graphs of size 𝑁 = 3. The panels show the probability density 𝜌(𝑥) of the rescaled variable 𝑥 = 0∞, obtained from comparisons between the dynamics of pairs of graphs 1 , 2 ,…, 5 . Results …

**Figure 5.** Figure 5: Network with three cliques with a total of 𝑁 = 12 nodes. The connection weights Λ𝑖𝑗 are chosen randomly with a uniform distribution taking values between 0 and 1 as indicated in panel (a). In this structure, the sign is randomly changed, where 𝑝 is the probability of m…

**Figure 6.** Figure 6: Dissimilarity ⟨max{𝚲, 𝚲⋆}⟩ as a function of the probability 1−𝑝 of sign-flip in the edges that are part of a clique. (a)-(d) present the structures analyzed. The sign is randomly changed in the links that are part of the clique formed by the nodes 𝑖 = 1, 2, 3, 4, wher…

**Figure 7.** Figure 7: compares two processes for different values of 𝛾 in random weighted networks with three cliques and size 𝑁 = 12. For the case presented in [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]

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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

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