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Quantifying Influence and Information Transfer in a Modified Vicsek Model with Non-reciprocal Interactions

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A modified Vicsek model with non-reciprocal interactions shows that a model-internal influence measure and normalized transfer entropy identify the same transition points across three distinct phase transitions, points that classical…

desk verdict A genuinely new influence-based Vicsek model with solid pairwise analytics, but the collective-level TE comparison rests on an underspecified aggregation step that needs fixing before the central claim is testable. read the letter →

arxiv 2506.20888 v4 pith:NMYLNA3N submitted 2025-06-25 cond-mat.stat-mech nlin.AO

Jiahuan Pang , Wendong Wang This is my paper · ORCID
classification cond-mat.stat-mechnlin.AO
keywords Vicsekmodelnon-reciprocalinteractionstransferentropyinfluencephasetransitionspartialinformationdecompositioncollectivemotionnoiseeffects
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 separates two ideas that are often treated as the same in collective-motion research: influence, which it defines inside the model as one particle's weighted reaction to another's heading, and information transfer, measured from observed headings with transfer entropy. The authors build a modified Vicsek model with non-reciprocal interactions between two subpopulations, influencers and followers, and derive analytic formulas for pairwise influence. They find that transfer entropy grows quasi-linearly with influence when noise is fixed and that noise on the influencer boosts information transfer while noise on the follower suppresses it. At the collective level, an influencer-neighbor-averaged influence and normalized transfer entropy form two-branched, inverted-V relations over the control parameters and peak at the same points in all three phase transitions, points that differ from the susceptibility peaks and reflect a shift in the relative importance of influencers' presents versus followers' presents on followers' futures. The same model, used as a testbed for partial information decomposition, singles out two methods that attribute transfer entropy mostly to unique rather than synergistic information.

What carries the argument

The argument is carried by an explicit, model-calculable definition of influence. For a particle $i$, the pairwise instantaneous influence of $j$ on $i$ is $A_{j\to i}(t) = |w_{j\to i}|\, F(\theta_j(t)-\theta_i(t))\, s_{ij}(t) / \sum_k |w_{k\to i}| s_{ik}(t)$ when $w_{j\to i}\ge 0$, and the same expression with $F(\theta_j(t)+\pi-\theta_i(t))$ when $w_{j\to i}<0$, where $F$ wraps angles into $(-\pi,\pi]$ and $s_{ij}$ is the neighborhood indicator; particle orientations update as $\theta_i(t+\Delta t)=F(\theta_i(t)+A_i(t)+\beta_i(t))$ with uniform noise of strength $\eta$. From this definition the authors derive closed-form expressions for the time-averaged absolute influence in two-particle systems (Eqs. (23) and (24)) and, for the collective, the influence measure $\langle |\langle A\rangle_{R_I}|\rangle_{N_F,T}$ obtained by averaging pairwise influences over influencer-neighbors before averaging over followers and time. The complementary quantity is normalized transfer entropy, $\mathrm{TE}/I_{\rm total}$, computed between the aggregated angle of a follower's influencer-neighbors and the follower's future angle; the aggregation of all influencer-neighbors into one variable is a mean-field-style reduction that makes the many-body information-theoretic computation feasible. The central role of these two quantities is that they both show an inverted-V peak at the same parameter values, linking phase transitions to a shift in relative information importance.

What would settle it

Redo the collective information-transfer analysis without collapsing each follower's influencer-neighbors into a single aggregated angle, for instance by estimating multivariate transfer entropy or by averaging per-neighbor pairwise transfer entropies, and check whether normalized transfer entropy still peaks at $\eta=1.05\pi$ and $\eta=0.5\pi$; if the peak moves to the susceptibility transition points ($1.2\pi$ and $0.9\pi$), the claimed information-based transition marker depends on the mean-field-style aggregation rather than on the system's intrinsic dynamics.

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Extended reading notes

Core claim

The authors' central claim is that a model-calculable quantity they call influence can be derived analytically in a modified Vicsek model with non-reciprocal interactions, and that this quantity tracks the same collective transitions as normalized transfer entropy. In the two-particle system, the time-averaged absolute influence $\langle |A_{I\to F}|\rangle_T$ is a non-decreasing function of both noise $\eta$ and interaction strength $|w_{I\to F}|$; it is linear in $\eta$ with slope $1/3$ at low noise for large positive weights, saturating at $\pi/2$, while for $w_{I\to F}=-1$ it remains constant at $\pi/4$ for all noise strengths. At fixed $\eta$, transfer entropy from influencer to follower is quasi-linear in this influence, while normalized transfer entropy collapses to a Boltzmann sigmoid as influence grows for $w_{I\to F}>0$, excluding the fully chaotic $\eta=2\pi$ case. Turning off influencer and follower noise separately shows the dual role of noise: influencer noise enriches and increases information transfer, follower noise suppresses it. At the collective level, the absolute influencer-neighbor-averaged influence $\langle |\langle A\rangle_{R_I}|\rangle_{N_F,T}$ follows the linear law $\eta/8$ in the ordered state and approaches $0.058$ as $\eta\to 2\pi$ for the simulated parameters, and normalized transfer entropy computed from aggregated influencer-neighbor angles peaks at the same points as this influence: $\eta=1.05\pi$ for the aligned-disordered transition, $\eta=0.5\pi$ for the chiral-disordered transition, and $w_{I\to F}=0$ for the aligned-chiral transition. These points differ from the susceptibility peaks at $1.2\pi$ and $0.9\pi$, which the authors interpret as evidence that the transitions are marked by a change in the relative information contributed by influencers' presents versus followers' presents to followers' futures. Finally, the model is used to compare partial information decomposition methods, and the authors conclude that the $I_{ccs}$ and $I_{imi}$ decompositions, which assign most of the transfer entropy to unique information, fit the physical intuition of influence better than the $I_{min}$ method.

Load-bearing premise

The analytic results assume that, without noise, every particle and its influencers move toward one common direction at each time step, so heading differences come only from noise and many neighbor noises cancel on average; when some particles pull much harder than others, when neighborhoods are small, or when particles cluster unevenly, that assumption fails and the exact formulas (including the linear $\eta/8$ growth law and the $0.058$ asymptotic value) become approximations.

Editorial extensions

If this is right

  • Normalized transfer entropy can serve as an information-based order parameter: it locates the aligned-disordered, chiral-disordered, and aligned-chiral transitions at the same parameter values as the influence measure, showing where one subpopulation's present outweighs the other's in shaping followers' futures.
  • Because influence and transfer entropy are quasi-linearly related at fixed noise but not across noise values, transfer entropy alone cannot be equated with influence; a follower receiving more bits is not necessarily being influenced more strongly, since the source of the noise matters.
  • The dual noise effect implies that studies of leader-follower dynamics must separate noise on the leader from noise on the follower: leader noise can genuinely increase information transfer while follower noise suppresses it.
  • The analytic influence formulas (the $1/3$ slope, the $\pi/2$ and $\pi/4$ constants, and the $\eta/8$ collective law) give predictions that can be compared directly with future simulations of the modified Vicsek model and similar alignment models.
  • The partial information decomposition comparison suggests that in this system the unique-information component of transfer entropy is physically meaningful, so PID-based quantities rather than raw transfer entropy should be used when attributing directed influence in collective motion.

Reading between the lines

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

  • Beyond the paper, the same aggregation-and-normalization recipe could be applied to other non-reciprocal active matter models, for instance non-reciprocal Ising or chiral active systems, to test whether normalized transfer entropy generically marks transition points that susceptibility misses.
  • Because the collective transfer entropy calculation aggregates influencer-neighbors into one angle, the claimed transition-point agreement may weaken at low density or small interaction radius where this mean-field-style reduction is less accurate; a density and radius scan would settle this.
  • If the model's internal influence could be measured in experiments on programmable active matter, the $\eta/8$ law would offer a quantitative calibration: measured influence growing linearly with applied noise would confirm that the experimental update rule matches the assumed one-step alignment.
  • The distinction the paper draws suggests that observed leader-follower relations in natural flocks, typically inferred from transfer entropy, may mix unique, shared, and synergistic contributions; using PID-based unique information could change which individuals are identified as leaders.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper introduces a modified Vicsek model with non-reciprocal pairwise weights and defines a quantitative notion of influence A_{j→i}(t) as the normalized angular-difference term in the update rule. For a two-particle system it derives analytic expressions for the time-averaged absolute influence, compares them with transfer entropy (TE) and normalized TE, and reports that noise on influencers enhances information transfer while noise on followers suppresses it. For the many-particle system it defines four collective influence averages, selects ⟨|⟨A⟩_{R_I}|⟩_{N_F,T} as the main influence-based order parameter, and shows that this quantity and normalized TE both exhibit peaks at what the authors identify as the transition points of the aligned-disordered, chiral-disordered, and aligned-chiral transitions. The paper then evaluates seven partial information decomposition (PID) methods on XOR/OR/ADD test cases and selects I_ccn and I_imi as most suitable for the modified Vicsek model.

Significance. If the claims are correct, the paper provides a concrete testbed in which a model-defined influence can be compared with information-theoretic measures, and it reports an interesting empirical correspondence between normalized TE and influence at collective phase transitions. The analytic derivations for pairwise and collective influence, with their stated assumptions, are a useful contribution, and the PID benchmark on binary test cases is a practical addition. However, the significance is tempered by two considerations: the model defines influence as the very coupling term that appears in the update rule, so a statistical association between influence and TE is partly built in; and the collective TE calculation relies on an underspecified aggregation procedure that must be clarified before the central coincidence claims can be independently verified.

major comments (4)
  1. [Sec. V, collective TE construction (paragraph beginning 'To compute the information received...')] The construction of the source variable θ_neigh(t) is not operationally defined. The text states that the θ_j(t) values of all influencer-neighbors of a follower are 'considered as a single variable' and that a joint distribution p(θ_neigh(t), θ_i(t), θ_i(t+Δt)) is then built, but it never specifies how a variable-length set of angular values is mapped to a scalar random variable, how the empirical distribution is estimated, or whether the follower's present and future are duplicated for each neighbor when forming triplets. If the implementation pools every neighbor angle as an independent sample, the computed quantity is a neighborhood-multiplicity-weighted pairwise TE with duplicated conditioning samples, not TE from an aggregated neighborhood variable, and the duplication can bias the conditional mutual information. Because the central claim that normalized TE and influence peak at the same transition points (η=1.05π, η=0.5π, and w=0) depends directly on this estimator, the paper must give a precise estimator definition, pseudocode, or code, and ideally a test of the estimator on a synthetic example where the ground truth is known.
  2. [Sec. IV and Appendix C, Eqs. (23)–(24)] The analytic formulas for the pairwise time-averaged influence assume the equal-update-target condition of Eq. (C4), namely θ_I(t-dt)+A_{F→I}(t-dt)=θ_F(t-dt)+A_{I→F}(t-dt). The authors themselves note in Appendix C that this condition cannot be guaranteed for large w_{I→F}, and they derive a correction term θ̃(t-dt) in Eq. (C21) that is not included in Eqs. (23)–(24). Since Fig. 5 compares Eqs. (23)–(24) with simulations over the full range w=1 to 100 and η up to 2π, the paper should quantify the neglected correction and state the range of weights and noise for which the closed-form expressions are valid. Without this, the apparent agreement in Fig. 5 is not fully explained.
  3. [Appendix D, Eqs. (D2), (D7), and (D9)] The analytic results for the collective influence in the ordered state, including the η/8 law and the asymptotic value 0.058, rely on three assumptions: that θ_order=θ_i+A_i for every particle at every time step, that the number of influencer-neighbors equals the number of follower-neighbors (U_I=U_F), and that the average of the influencer-neighbor noises vanishes in Eq. (D7). The last assumption is a mean-field-style approximation whose accuracy depends on the neighborhood size and density; Appendix B tests the collapse of curves with R and N but does not directly validate the noise-averaging step. Since these formulas are used to argue that the influence measure correctly locates the ordered-state regime and the disorder limit, the paper should either prove or numerically demonstrate that the assumptions hold in the parameter range used in Figs. 9(b) and 9(d), or state the resulting error bounds.
  4. [Figs. 9(b), 9(d), 10(c), 10(g), and Fig. 14] The claim that influence and normalized TE 'clearly identify' the same transition points, and that these differ from the susceptibility peaks, is based on visual inspection of peaks without reported uncertainties. Fig. 14 shows that the susceptibility peak shifts with system size, so the difference between the influence peak at η=1.05π and the susceptibility peak at η=1.2π could be a finite-size or grid-resolution effect. The authors should provide peak positions with error bars, state the η grid spacing, and, ideally, include a finite-size analysis for the influence/TE peaks comparable to the one shown for susceptibility.
minor comments (4)
  1. [Eq. (25) and Fig. 8(c)] The fitting formula contains a typo, 'TE/Toatl MI', and the reported Boltzmann sigmoid parameters a=0.564 and k=0.223 are given without uncertainties or a goodness-of-fit measure; please add these or state that the fit is illustrative.
  2. [Sec. V and Appendix E] The paper does not state explicitly whether the same 8-bin equidistant binning used in the pairwise analysis is applied to the aggregated collective triplets, nor how many samples per follower are used when constructing the joint distribution for θ_neigh(t); this information should be provided to make the collective TE computation reproducible.
  3. [Sec. III, Eq. (16) and Fig. 7] The notation η_I and η_F in Fig. 7 is introduced only in the caption and surrounding text; it would be clearer to define these as separate noise strengths in the model section and to state explicitly how they enter Eq. (16) when they are turned off.
  4. [Sec. V, final PID discussion] The argument that I_ccn and I_imi are preferred because influence should contain a unique information component is plausible, but the negative values produced by I_ccn on the OR test case (Table IV) deserve a comment, since the paper mentions this limitation only in passing.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: influence and transfer entropy are independently defined quantities, and the reported relations between them are empirical consequences of the model rather than reductions by construction.

full rationale

The paper defines influence through Eqs. (14) and (16) as the angular term in the orientation update, while transfer entropy is the standard conditional mutual information of Eq. (7). These are different definitions, and the pairwise relations reported in Figs. 5, 6, and 8 are measured or analytically derived from the model's noise and update structure, not obtained by substituting one definition into the other. The Boltzmann sigmoid in Eq. (25) is explicitly fitted to the data with reported parameters, not advertised as a prediction, so it is not a case of a fitted input being renamed a prediction. The collective analysis computes normalized transfer entropy from simulation distributions and compares it with an independently defined neighbor-averaged influence; the common peaks at the transition points are presented as observations. The aggregation variable theta_neigh(t) in Sec. V is under-specified, which is a reproducibility and estimation concern, but the text never defines influence in terms of the transfer entropy or vice versa, so no circular reduction can be exhibited. The only evident self-citation, Ref. [62], is a pointer to future experimental work and is not load-bearing. The derivation chain is therefore self-contained with respect to the paper's stated claims.

Assumptions & free parameters 2 free parameters · 5 assumptions · 1 invented entities

The central quantitative content of the paper rests on the model definition of influence (Eq. 14), on two alignment assumptions used in the analytic derivations, and on the binning choice for estimating information-theoretic quantities. The only genuinely fitted numbers are the sigmoid parameters a and k in Eq. (25). No independent physical entity is introduced beyond the internal model variable A_{j to i}.

free parameters (2)
  • a (sigmoid offset) = 0.564
    Fitted to normalized TE vs influence data in Eq. (25) for w>0; interpreted as self-influence of the follower. No uncertainty reported.
  • k (sigmoid scale) = 0.223
    Fitted characteristic influence scale in the Boltzmann sigmoid Eq. (25); no uncertainty reported.
assumptions (5)
  • domain assumption The update rule defines influence as the neighbor-weighted angular difference, Eq. (14).
    This is the paper's model assumption; all later influence measures are averages of this quantity.
  • ad hoc to paper Two-particle alignment assumption: in the absence of noise the influencer and follower update to the same direction at every step, Eq. (C4).
    Used to derive the analytic pairwise influence formulas Eqs. (23)-(24); the authors note in Appendix C that it is not exact when w_{I to F} is large.
  • ad hoc to paper Ordered-state mean-field assumption: all particles share a common direction theta_order with theta_order = theta_i + A_i for each particle, Eq. (D2), and the average of influencer-neighbor noises vanishes.
    Basis for the eta/8 collective law; relies on equal numbers of influencer and follower neighbors (U_I = U_F).
  • domain assumption Equal number of influencer and follower neighbors (U_I = U_F) in the collective derivation.
    Stated in Appendix D when deriving Eq. (D8); approximately true for equal subpopulation sizes at high density.
  • domain assumption Equidistant discretization into 8 bins yields reliable TE/PID estimates for continuous angular variables.
    The paper selects this binning to match PID methods that require discrete probability mass functions, but does not test sensitivity to bin count.
invented entities (1)
  • Pairwise influence A_{j to i}(t)
    purpose: Quantifies the directional pull of particle j on particle i in the modified Vicsek update, allowing influence to be compared with information transfer.
    Defined by Eq. (14) as a weighted angular difference normalized by total neighbor weight; it is an internal model variable rather than an externally measurable quantity, so there is no falsifiable handle outside the model.

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Pith. "Pith review of Quantifying Influence and Information Transfer in a Modified Vicsek Model with Non-reciprocal Interactions." pith.science (2026). https://pith.science/paper/NMYLNA3N

@misc{pith2026250620888,
  author       = {Pith},
  title        = {Pith review of: Quantifying Influence and Information Transfer in a Modified Vicsek Model with Non-reciprocal Interactions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NMYLNA3N}},
  note         = {Machine review of arXiv:2506.20888}
}
read the original abstract

Understanding information transfer among individuals is fundamental to revealing the collective dynamics of complex systems. Information transfer has been quantified using various information-theoretic tools and assigned the concept of influence. However, information-theoretic measures are inherently statistical, not causal, and influence in the context of causal inference implies a causal relation, so equating influence with information transfer creates conceptual confusion and interpretational challenges. Here, we introduce an influence-based Vicsek model with non-reciprocal interactions to distinguish influence from information transfer and examine their relationship. At the pairwise level, for fixed noise strengths, influence and transfer entropy exhibit quasi-linear relations; for fixed interaction weights, influence and transfer entropy exhibit nonlinear relations. At the collective level, we find that both influence and normalized transfer entropy form two-branched relations that clearly identify the transition points across three distinct phase transitions. These transition points reveal a different aspect of the collective dynamics not captured by classical order parameters: phase transitions are associated with changes in the relative importance of influencers' presents or followers' presents on followers' futures. Finally, we use our model to assess partial information decomposition methods and identify two methods most suitable for analyzing our system, one based on pointwise surprisal changes and the other on secret key agreement. Our work is a first step in distinguishing the concept of influence from information transfer in a physical model system, provides a concrete testbed for methods emerging from the growing field of information theory-based causal inference, and offers new insights into the dynamics of complex systems.

Figures

Figures reproduced from arXiv: 2506.20888 by the authors.

Figure 2
Figure 2. FIG. 2. Schematic of the phase transitions investigated [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Characterization of the three phase transitions using the classical order parameters, the time average [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. FIG. 4. The [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figures from the paper (12 more)
Figure 5
Figure 5. Figure 5: FIG. 5. Time [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Investigation of the dual effects of noise on [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Information transfer as a function of influence [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Characterization of the phase transitions with [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Characterization of the phase transitions with information transfer. [PITH_FULL_IMAGE:figures/full_fig_p014_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Decomposition of TE for the phase transitions using three representative PID methods. The simulations of [PITH_FULL_IMAGE:figures/full_fig_p016_11.png]
Figure 13
Figure 13. Figure 13: FIG. 13. Impact of model parameters [PITH_FULL_IMAGE:figures/full_fig_p019_13.png]
Figure 12
Figure 12. Figure 12: FIG. 12 [PITH_FULL_IMAGE:figures/full_fig_p019_12.png]
Figure 14
Figure 14. Figure 14: FIG. 14. Effects of the system size [PITH_FULL_IMAGE:figures/full_fig_p020_14.png]
Figure 16
Figure 16. Figure 16: Effects of densities 𝜌𝜌 with 𝑅𝑅 = 1 and 𝐿𝐿 = 10 for 𝜂𝜂 = 0.5𝜋𝜋, 1.0𝜋𝜋, and 1.5𝜋𝜋. (a) 〈𝑣𝑣𝑎𝑎〉𝑇𝑇 versus 𝜌𝜌 for different 𝜂𝜂 . For aligned phases ( 𝜂𝜂 = 0.5𝜋𝜋 ), 〈𝑣𝑣𝑎𝑎〉𝑇𝑇 increases with 𝜌𝜌; for 𝜂𝜂 = 𝜋𝜋, increasing density first decreases 〈𝑣𝑣𝑎𝑎〉𝑇𝑇 and then increases it. T…
Figure 17
Figure 17. Figure 17: FIG. 17. The probability density function of 𝑓𝑓 [PITH_FULL_IMAGE:figures/full_fig_p023_17.png]
Figure 18
Figure 18. Figure 18: FIG. 18 [PITH_FULL_IMAGE:figures/full_fig_p028_18.png]

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.