{"id":"b262dc16-4cfb-4a25-99de-5b63885d0c44","arxiv_id":"2506.20888","paper_version":4,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"In a modified Vicsek model with non-reciprocal interactions, pairwise influence correlates quasilinearly with transfer entropy at fixed noise, and both influence and normalized transfer entropy identify the same three phase transitions.","lead":"This paper modifies the Vicsek flocking model so that each neighbor's influence on a particle can be computed directly, then compares that influence with transfer entropy, a popular information-flow statistic. It reports testable links between the two at the pair level, and shows that a normalized information measure and the model's influence both mark the same flocking phase transitions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central claim rests on an unspecified 'single-variable' aggregation of influencer-neighbor angles in the collective TE calculation; without a precise estimator, the reported coincidence with influence is not independently testable.","rationale":"The reader's CONDITIONAL verdict is reasonable and I did not find an internal contradiction that forces rejection. The strongest claim is the collective two-branched relation between influence and normalized TE across three transitions. The most load-bearing point is that the source variable θ_neigh in the collective TE calculation is never defined as a mathematical random variable. The description in Sec. V can be read as pooling all influencer-neighbor angle samples into one marginal; that would yield a pairwise TE weighted by neighborhood multiplicity, not a transfer entropy from the aggregated influencer-neighborhood, and duplicated follower present/future samples can bias conditional MI. Without the code or a precise definition, the agreement with influence at 1.05π, 0.5π, and w=0 cannot be independently checked. The reader instead flagged the equal-update-target assumptions in Appendices C and D; those affect the analytic η/8 law and the asymptotic constants but not the empirical peak coincidence itself, and the paper explicitly acknowledges the assumption fails for large weights. Thus the TE aggregation issue is more central to the headline claim. A scalar circular-mean re-computation is a concrete test that would settle whether the reported coincidence is a robust model property or an estimator artifact. This concern strengthens the conditions for acceptance but does not change the overall conditional verdict.","tokens_in":51020,"tokens_out":10162,"duration_ms":125231,"concrete_test":"Recompute the collective normalized TE curves in Fig. 10 with a precisely defined scalar source: for each follower i and time t, take θ_neigh(t) as the circular mean of the influencer-neighbor angles, then estimate TE from that scalar source to θ_i(t+Δt) conditional on θ_i(t) using the same 8-bin estimator and also a 16-bin estimator. If the peaks at η=1.05π, η=0.5π, and w=0 persist under this well-defined aggregation, the ambiguity is benign; if they shift or disappear, the central two-branched relation is an artifact of the unspecified neighbor-pooling procedure.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The collective-level claim depends on a transfer-entropy computation whose source variable is not well defined. In Sec. V, the authors say that the θ_j(t) values of all influencer-neighbors of a follower are treated as a single variable θ_neigh(t), and that a joint distribution p(θ_neigh(t), θ_i(t), θ_i(t+Δt)) is then built and TE computed. The text never specifies how a variable-length neighborhood of angles is reduced to one random variable, or how the empirical distribution is formed. If the implementation pools every neighbor angle as an independent sample, then the computed quantity is a neighborhood-multiplicity-weighted pairwise TE, with the follower's present and future duplicated across samples for each neighbor; this is not transfer entropy from the aggregated influencer-neighborhood, and the duplicated conditioning samples can bias the conditional mutual information. The observed coincidence between normalized TE and the influence measure at η=1.05π, η=0.5π, and w=0 could then be an artifact of that pooling rather than a property of the model. The paper describes the construction as mean-field-like in spirit, but it does not give the operational definition needed to verify the central claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":51302,"tokens_out":6445,"duration_ms":80810,"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":[{"comment":"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.","section":"Sec. V, collective TE construction (paragraph beginning 'To compute the information received...')"},{"comment":"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.","section":"Sec. IV and Appendix C, Eqs. (23)–(24)"},{"comment":"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.","section":"Appendix D, Eqs. (D2), (D7), and (D9)"},{"comment":"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.","section":"Figs. 9(b), 9(d), 10(c), 10(g), and Fig. 14"}],"minor_comments":[{"comment":"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.","section":"Eq. (25) and Fig. 8(c)"},{"comment":"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.","section":"Sec. V and Appendix E"},{"comment":"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.","section":"Sec. III, Eq. (16) and Fig. 7"},{"comment":"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.","section":"Sec. V, final PID discussion"}],"recommendation":"major_revision","confidential_remarks":"The central empirical claim of the paper rests on the collective transfer-entropy computation, and the current manuscript does not specify the estimator at the level needed to verify it. If the implementation pools neighbor angles as independent samples, the reported coincidence between normalized TE and influence could be largely an artifact of the pooling scheme rather than a property of the model. I recommend that the editor request the authors to provide the exact estimator, pseudocode, or code, and to rerun the collective analysis with a clearly defined aggregated variable. The pairwise and analytic results are less problematic and should be preserved."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read the Pang-Wang preprint. The useful core: they define a pairwise influence as the normalized weighted angular difference term in a non-reciprocal Vicsek update, derive analytic expressions for its time average in the two-particle case (Eqs. 23-24), and show it has clean behavior across the aligned, chiral, and disordered phases. The dual effect of noise—influencer noise raises TE from influencer to follower, follower noise suppresses it—is a real, clearly demonstrated finding. That part is worth a serious look.\n\nThe paper is less solid at the collective level, and the problem is not the thermodynamics. The text in Sec. V says the θ_j(t) of all influencer-neighbors are 'considered as a single variable θ_neigh(t)' and a joint distribution is built, but it never specifies how variable-length neighborhoods are reduced to one random variable, or how the empirical distribution is assembled. If the implementation pools every neighbor angle as an independent sample, the quantity being computed is not transfer entropy from an aggregated influencer-neighborhood; it is a multiplicity-weighted average of pairwise TEs, with duplicated conditioning samples that can bias the conditional mutual information. Without the operational definition, the reported coincidence between normalized TE and the influence measure at the three transition points is not independently testable. This is the load-bearing gap, and it is fixable with a precise estimator description and, ideally, code.\n\nTwo smaller issues: the sigmoid fit in Fig. 8(c) excludes η=2π post hoc, which is disclosed but still weakens the 'Boltzmann relation' claim; and the transition points are single-peak locations from one parameter set, with no error bars or finite-size analysis on the influence/TE side (the susceptibility peaks do have some N-dependence discussion in Appendix B). The analytic derivations in Appendices C and D openly assume a common update target; those assumptions are stated, but they are not tested for the large-weight or finite-neighborhood cases, so the eta/8 law and the 0.058 asymptotic should be treated as approximate until checked.\n\nWho should read this: people building testbeds for information-theoretic causality measures and those working on non-reciprocal flocking models will get value from the model and the pairwise results. The collective claim needs another revision cycle. I'd send it to a referee, but with a strong request to clarify the TE estimator and to share the data/code.\n\nRecommendation: engage with it, but as a conditional accept—the collective claim is not yet reproducible.","headline":"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.","tokens_in":51738,"tokens_out":3380,"would_cite":false,"duration_ms":35414,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["Vicsek model","non-reciprocal interactions","transfer entropy","influence","phase transitions","partial information decomposition","collective motion","noise effects"],"falsifier":"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.","tokens_in":50803,"feed_emoji":"🐦","tokens_out":9583,"duration_ms":104853,"temperature":0.7,"pith_summary":"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.","feed_headline":"Influence and transfer entropy mark three flocking transitions","feed_subtitle":"In a modified Vicsek model, normalized transfer entropy and influence peak at the same three points, revealing what order parameters miss.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the original Vicsek model update rule and the mean-speed order parameter that the modified model is built on.","marker":"[29]"},{"why":"Defines transfer entropy, the paper's central information-theoretic measure of directed information transfer.","marker":"[10]"},{"why":"Provides the critique of transfer entropies that motivates the paper's separation of influence from information transfer and its use of partial information decomposition.","marker":"[11]"},{"why":"Introduces the partial information decomposition framework and the I_min method that the paper tests and finds less suitable.","marker":"[12]"},{"why":"Supplies the I_imi method based on secret-key-agreement rate that the paper selects as one of the two most appropriate PID methods.","marker":"[15]"},{"why":"Accounts for the non-reciprocal phase transition framework and the exceptional transition used for the aligned-chiral case.","marker":"[21]"},{"why":"Supplies the I_ccs method based on pointwise common change in surprisal that the paper selects as the other most appropriate PID method.","marker":"[32]"},{"why":"Provides the susceptibility and Binder cumulant techniques used to locate the traditional order-parameter transitions that are compared against the influence and transfer-entropy markers.","marker":"[41]"}],"fun_headline_variants":["Influence and transfer entropy coincide at three transitions","Three phase transitions revealed by influence and transfer entropy","Influence tracks transfer entropy at three key transitions","Same three transitions from influence and transfer entropy","Influence matches transfer entropy at three flocking transitions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Influence and transfer entropy coincide at three transitions","Three phase transitions revealed by influence and transfer entropy","Influence tracks transfer entropy at three key transitions","Same three transitions from influence and transfer entropy","Influence matches transfer entropy at three flocking transitions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00074,"raw_usage":{"total_tokens":3477,"prompt_tokens":1294,"completion_tokens":2183,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":910,"completion_tokens_details":{"reasoning_tokens":2112}},"tokens_in":910,"tokens_out":2183,"duration_ms":16303,"temperature":1.0,"reasoning_tokens":2112,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T22:40:08.027252+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Shankar, A","cited_arxiv_id":null,"evidence_quote":"Supplies the original Vicsek model update rule and the mean-speed order parameter that the modified model is built on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines transfer entropy, the paper's central information-theoretic measure of directed information transfer."},{"cited_title":"Schreiber, Measuring Information Transfer, Phys","cited_arxiv_id":null,"evidence_quote":"Provides the critique of transfer entropies that motivates the paper's separation of influence from information transfer and its use of partial information decomposition."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the partial information decomposition framework and the I_min method that the paper tests and finds less suitable."},{"cited_title":"Salahshour and I","cited_arxiv_id":null,"evidence_quote":"Accounts for the non-reciprocal phase transition framework and the exceptional transition used for the aligned-chiral case."},{"cited_title":"Ginelli, The Physics of the Vicsek model, Eur","cited_arxiv_id":null,"evidence_quote":"Supplies the I_ccs method based on pointwise common change in surprisal that the paper selects as the other most appropriate PID method."},{"cited_title":"Vicsek and A","cited_arxiv_id":null,"evidence_quote":"Provides the susceptibility and Binder cumulant techniques used to locate the traditional order-parameter transitions that are compared against the influence and transfer-entropy markers."}],"review_version":1}