{"id":"c52c1979-aa1e-447b-87d2-355c95185a0b","arxiv_id":"1908.08163","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Adding any spontaneous activation to a spreading process on networks shifts the critical behavior from directed to undirected percolation, with a derived critical line and a crossover scale of p^{-2/3}.","lead":"Spreading processes that keep starting new spontaneous events while old ones are still running, such as brain activity or zoonotic diseases, are shown to change their critical behavior. The paper derives the new transition line and shows that standard criticality measures fail, which has implications for the critical brain hypothesis.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Universality-class claim rests on the causal-web merging rule; unvalidated labeling convention is the load-bearing assumption.","rationale":"The strongest claim is well-defined and the analytical derivation for k-regular networks is internally consistent; the tree-like PGF calculation, the critical line Eq. (2), and the p^{-2/3} crossover follow from the stated cluster definition. The most load-bearing condition for the claim's physical relevance is that causal-web clusters are the correct observable for spreading without time-scale separation. The paper asserts this, but the Methods description leaves the key case—spontaneous activation with active parents—ambiguous, and the analytical construction resolves that ambiguity by merging. The q=0 endpoint at p=1/(2k-1) is a sharp diagnostic: it exists only because parent links are included symmetrically in the cluster definition; under a strict causal-root definition, activations at q=0 are isolated and no such endpoint would appear. Therefore the universality-class change is not yet separated from the labeling convention. The HMN complications (p-dependent intermediate exponents and a critical-line fit exponent near 2 rather than 3) are real but secondary; they affect the crossover-scaling universality for that topology, not the central analytical result on k-regular networks. Because the reader's weakest_assumption already identifies this same premise, and the appropriate response is to require additional validation rather than to reject the analysis outright, the CONDITIONAL verdict stands unchanged.","tokens_in":34690,"tokens_out":18275,"duration_ms":191311,"concrete_test":"Implement the same ε-SIS dynamics on a 10-regular tree and compare two avalanche definitions at identical parameters: (A) the paper's rule, where spontaneous nodes with active parents merge into parent clusters; (B) a strict causal rule, where every spontaneous activation starts a new root regardless of active parents, and parent-induced nodes inherit exactly one parent label. Measure P(s) and the giant fraction g along the analytically predicted line Eq. (2). If definition B still gives τ_tail ≈ 2.5 and g ∼ (q−q_c)^1 at the same q_c, the universality-class claim is robust to the labeling rule. If B gives τ ≈ 1.5 and no giant below q = 1/k, the undirected class is created by the merging convention. As a second check, implement the original causal-webs algorithm of ref. [30] and verify whether its clusters coincide with the paper's merging rule.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim—that any p>0 moves the transition from directed to undirected percolation—is not a statement about the raw activation dynamics alone; it is a statement about clusters defined by the causal-web merging rule. The analytical PGFs in Appendix B count parent branches for any active node (e.g., Bi in Eq. B18 conditions only on X being active), so a node that activates spontaneously but has active parents is implicitly merged into the parent clusters. The simulation Methods text is exactly ambiguous at this point: spontaneous nodes with no active parents initiate new clusters, and non-spontaneous nodes with active parents inherit labels, but the fate of spontaneous nodes with active parents is never specified. If such nodes instead start new clusters, the q=0 limit has no giant component, the endpoint at p=1/(2k-1) disappears, and the undirected-percolation universality class could be an artifact of the labeling convention. The paper's own Discussion concedes the observable dependence: 'it is essential to know the network structure to resolve the underlying dynamics' and there is 'currently no way to recover the correct exponents without access to the network structure.' Yet no independent validation of the merging rule against a direct measurement, against the original causal-webs algorithm [30], or against a strictly causal definition is provided. Equation (2), the p^{-2/3} crossover, and the undirected exponents are all conditional on this rule, making it the most load-bearing assumption.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a discrete-time epsilon-SIS spreading model with spontaneous activation probability p and network-borne transmission probability q on directed networks. Because p>0 removes time-scale separation and produces overlapping independent cascades, the authors define avalanches through a causal-web merging rule: nodes without active parents are roots of new avalanches, and nodes with active parents inherit and merge cluster labels. Their central claim is that for any p>0 the transition is no longer in the directed-percolation universality class but instead in the undirected-percolation universality class, with directed-percolation exponents surviving only below a merging crossover scale s_m ~ p^{-2/3}. For k-regular networks they derive, via a pair of coupled probability generating functions, the critical line 0 = k(1-σ)^2 - (k-1)σσ_m (Eq. 2), the exponents β=1, γ=1, ν=1, and the crossover scaling. Numerical simulations on small-world, power-law, and hierarchical modular networks show two power-law regimes in the critical avalanche-size distribution and finite-size scaling of the giant component. The paper concludes that standard criticality markers such as the branching ratio and dynamic susceptibility fail in the absence of time-scale separation, and it draws implications for the critical brain hypothesis and for zoonotic disease spreading.","tokens_in":34892,"tokens_out":5315,"duration_ms":58595,"significance":"If the central claim holds, the paper is significant: it identifies a concrete mechanism—merging of independently initiated avalanches—by which a genuinely non-equilibrium directed-percolation transition is converted, for all p>0, into a transition with undirected-percolation exponents, and it makes a falsifiable prediction about two power-law regimes separated by a p-dependent crossover. The analytical generating-function derivation for k-regular networks is a genuine strength: Eq. (2) is derived from the model rather than fitted, and the resulting exponents are checked against simulations on infinite and finite networks. The paper also gives a clear, parameter-light prediction for the critical line and crossover, and the causal-web susceptibility offers a concrete observable for brain-avalanche analysis. The main limitation is that the universality-class claim is established for the particular avalanche-labeling observable defined by the causal-web merging rule, and the evidence for the generality of the undirected-percolation class across non-regular topologies rests on a small number of exponents, several reported without uncertainties.","major_comments":[{"comment":"The causal-web merging rule is the load-bearing observable of the paper, but its relation to a direct physical separation of avalanches is not validated. The finite-network simulation paragraph specifies that spontaneous nodes with no active parents initiate a new cluster, while the general description in Section II.A states that nodes with active parents inherit labels; however, the label assignment for the case of a spontaneously activated node that also has active parents is not explicitly stated in the Methods paragraph. Since the q=0 endpoint, the critical line Eq. (2), and the p^{-2/3} crossover all depend on this labeling choice, the authors should either state the rule unambiguously and test its sensitivity (for example, by comparing with an alternative rule in which such nodes seed new clusters) or compare the resulting clusters directly with the causal-webs algorithm of reference [30]. The Discussion's admission that 'there is currently no way to recover the correct exponents without access to the network structure' makes the observable-dependence of the claim explicit but does not by itself justify it.","section":"Section II.A and Appendix A (Simulation of model on finite networks)"},{"comment":"The claim that the entire p>0 critical line belongs to the undirected-percolation universality class for each network topology rests on very few fitted exponents. For the small-world and hierarchical modular networks, Table I reports τ≈2.5 and τ≈2.1, β≈1.0 and β≈0.8, with no uncertainties or goodness-of-fit measures; only the k-regular network has a full analytical exponent set. A two-power-law fit of P(s) and a single β estimate are not sufficient to distinguish undirected percolation from an effective exponent, especially on networks where the directed-percolation and undirected-percolation exponents are close (e.g., small-world τ values). The authors should report exponent estimates with uncertainties and provide at least one additional independent exponent (such as γ or the finite-size exponent 1/ν) for at least one non-regular topology before claiming a full universality class.","section":"Table I and Section II.B"},{"comment":"The universal p^{-2/3} merging crossover is derived from the mean-field phase-line exponent a=3, but the numerically fitted phase lines in Table II are not all consistent with a=3. In particular, the hierarchical modular network has a≈1.94, and the same argument that gives s_m ~ p^{-2/3} from δ ~ p^{1/3} would give s_m ~ p^{-2/a} ≈ p^{-1} for this network, while the small-world network with rewire probability 10^{-3} is reported in Appendix F to collapse with s_m ~ p^{-0.75}. The statement in Section II.B that 'all critical-avalanche distributions exhibit a universal curve collapse ... by re-scaling the distribution by p^{-2/3}' is therefore internally inconsistent with the fitted phase lines for these topologies. The authors should either justify the p^{-2/3} collapse independently of Eq. (3), or restrict the p^{-2/3} statement to mean-field networks and state the expected topology-dependent exponents explicitly.","section":"Section II.C, Eq. (3), Fig. 3, and Table II"}],"minor_comments":[{"comment":"The sentence introducing complementary probabilities says 'using the notation p = 1− p', which is a typographical error; it should introduce a distinct symbol such as \\bar p for the complementary probability, since p is already the spontaneous activation rate.","section":"Appendix B, Eq. (B11)"},{"comment":"Reference [64] duplicates reference [30] (Williams-Garcia, Beggs, and Ortiz, 'Unveiling causal activity of complex networks'); one of the two entries should be removed or replaced with the appropriate distinct citation.","section":"References"},{"comment":"Several numerical exponents in Tables I and III are reported without error bars (for example, τ≈1.35 and β≈1.0 for the small-world network, α≈7 for durations), while the paper elsewhere reports uncertainties such as 0.36(2); a consistent policy of reporting uncertainties for all fitted exponents would strengthen the quantitative claims.","section":"Table I and Table III"},{"comment":"The caption of Fig. 4d says 'Solid line is g ∼ (q−qc)β for β=2', but the text in Section II.B reports β=2 for the power-law network while the finite-size scaling in panel d uses N^{0.50} and (q−q_c)N^{0.25}; clarifying whether the solid line is the infinite-size limit or a finite-size collapsed curve would avoid confusion.","section":"Figure 4d and Appendix A"},{"comment":"The phrase 'the σ=1 line scales as p∼(1/k−q)^2' and the later statement that the Widom line scales as p∼(1/k−q) are derived only in the appendices; adding a one-sentence pointer to Appendix E at the first occurrence would help the reader.","section":"Section II.C, paragraph after Eq. (2)"}],"recommendation":"major_revision","confidential_remarks":"This is a thought-provoking paper with a clean analytical core for k-regular networks and a clear, falsifiable claim about universality-class change. My main concern is that the universality claim is tied to the causal-web labeling rule, which is a modeling choice as much as a measurable observable; the authors need to validate or at least bound the sensitivity of the critical exponents to that choice. The internally inconsistent use of p^{-2/3} for non-mean-field topologies also needs to be resolved. Neither issue seems fatal, but both need substantive work before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The thing to know about this paper is that it makes a sharp, testable claim: introduce any spontaneous activation p>0 into a spreading process and the critical transition moves from directed-percolation to undirected-percolation universality, with directed-percolation exponents surviving only below a merging scale sm ~ p^{-2/3}. The second thing to know is that the claim is tied to the causal-web merging rule used to define avalanches, and that observable is load-bearing.\n\nWhat is genuinely new: the analytical critical line for k-regular networks, Eq. (2), and the two-power-law structure with the p^{-2/3} crossover. The generating-function derivation in Appendix B is internally consistent and gives exact exponents (β=1, γ=1, ν=1) that match simulations on infinite and finite graphs. The finite-size scaling using the B-ratio is a solid, reproducible method, and the curve collapses are convincing. The paper also does useful service by showing that the branching ratio and dynamic susceptibility no longer identify the transition once p>0, and it distinguishes the Widom line, σ=1 line, and true critical line with different power laws near the directed-percolation limit. The Discussion is honest about needing network structure to recover the exponents.\n\nThe soft spots are real but not fatal. The main one is the merging rule itself. The Methods text is ambiguous about the fate of a node that activates spontaneously and also has active parents: it says such nodes initiate a new cluster only if they have no active parents, then says nodes with active parents that were not already spontaneously activated are checked for activation. That leaves a gap. The analytical PGFs implicitly merge such nodes into parent clusters, so if the simulations do something different, the theory-simulation agreement would break. Without code or data, this cannot be checked. The stress-test worry that the whole universality-class claim is an artifact of labeling is probably too strong—the q=0 endpoint is unaffected, and the agreement for p>0 is good—but the ambiguity should be fixed. Second, Table I reports several fitted exponents without error bars, and the hierarchical modular network shows a p-dependent Griffiths regime that complicates the universal p^{-2/3} scaling. These are minor-to-moderate issues.\n\nWho gets value: people working on nonequilibrium percolation, epidemic reintroduction, and the critical brain hypothesis. It deserves a serious referee. The analytical core is solid, and the claim is important enough to warrant revision rather than rejection. I would recommend accepting it for review and pushing for a clarified merging rule, error bars on exponents, and code/data release.","headline":"Solid analytical core and a novel universality-class claim, but the result is conditional on the causal-web merging rule, which is not independently validated and is ambiguously described in the Methods.","tokens_in":35495,"tokens_out":4249,"would_cite":true,"duration_ms":42607,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["64.60.ah","05.70.Jk","89.75.Hc"],"model":"deepseek-v4-flash","headline":"Spontaneous activation moves spreading transitions from directed to undirected percolation.","keywords":["spreading processes","directed percolation","undirected percolation","universality class","spontaneous activation","critical brain hypothesis","causal webs","avalanche merging"],"falsifier":"On a $k$-regular network with $p=10^{-4}$, measure the critical avalanche size distribution at $q=q_c(p)$ out to sizes well above $p^{-2/3}$: if the central claim is right, the tail must follow $s^{-5/2}$ after an $s^{-3/2}$ regime with a sharp crossover near $s_m \\sim p^{-2/3}$; observing a single pure directed-percolation power law at all scales would falsify the universality shift.","tokens_in":34442,"feed_emoji":"🧠","tokens_out":5237,"duration_ms":53803,"temperature":0.7,"pith_summary":"This paper argues that any nonzero rate of spontaneous activation changes the universality class of a network spreading transition: instead of directed percolation, the transition belongs to undirected percolation, while directed-percolation exponents survive only below a crossover scale set by $p^{-2/3}$. The authors prove this for the mean-field $k$-regular network with a pair of coupled generating functions, give the critical line $0 = k(1-\\sigma)^2 - (k-1)\\sigma\\sigma_m$, and support it numerically on small-world, power-law, and hierarchical modular networks. If right, common markers of criticality, such as a branching ratio of one and a diverging dynamic susceptibility, misidentify the transition whenever spontaneous events occur, which matters for epidemic thresholds and for the critical brain hypothesis.","feed_headline":"Spontaneous activity flips spreading to undirected percolation","feed_subtitle":"With any self-activation, cascades merge and the epidemic threshold shifts to the ordinary-percolation universality class.","key_machinery":"The central object is the causal-web avalanche definition: nodes with no active parents start new clusters, and overlapping streams of activity merge their cluster labels, mapping the dynamics onto percolation clusters. The analytic argument uses two coupled probability generating functions, $H_p$ and $H_d$, for cluster sizes reached along daughter and parent branches; their divergence condition yields the critical line $0 = k(1-\\sigma)^2 - (k-1)\\sigma\\sigma_m$, where $\\sigma$ is the branching ratio and $\\sigma_m$ the merging number. Singly rooted, mergeless avalanches are counted by Fuss-Catalan trees, giving an exponential cutoff $s_m$; combining $s_m \\sim (1/k - q)^{-2}$ with the critical-line scaling $(1/k - q)^3 \\sim p$ produces the $s_m \\sim p^{-2/3}$ crossover that organizes the universal curve collapses.","core_discovery":"In the $\\varepsilon$-SIS model with spontaneous activation probability $p$ and spreading probability $q$, adding any $p>0$ destroys the pure directed-percolation transition; the genuine critical line, traced by the divergence of the causal-web susceptibility, belongs to the undirected percolation universality class, with giant-component exponent $\\beta=1$, susceptibility exponent $\\gamma=1$, and correlation-length exponent $\\nu=1$ on $k$-regular networks. Along this line the avalanche size distribution exhibits two power laws: the directed-percolation exponent $\\tau=3/2$ for small clusters and the undirected value $\\tau=5/2$ above a merging scale $s_m \\sim p^{-2/3}$. Only the singular point $p=0$, $q=1/k$ remains in the directed-percolation universality class. The crossover between the two regimes is governed by merging of initially independent cascades, and standard criticality measures such as the branching ratio and the dynamic susceptibility no longer mark the true transition once $p>0$.","pith_inferences":["If the causal-web observable is the right one, whole-brain recordings paired with tractography should show the $p^{-2/3}$ crossover directly: as the spontaneous rate rises, the tail exponent of the avalanche distribution should move from approximately $3/2$ toward $5/2$ at a size controlled by $p^{-2/3}$.","The merging mechanism is general: any driven branching process with a background initiation rate should exhibit the same shift, so the result likely extends beyond this model to rumor spreading, malware propagation, and neural cultures with external input.","A direct way to test the claim against alternative observables is to reanalyze the same simulated activity using avalanche definitions based on temporal bins or global quiet periods; if those definitions recover pure directed-percolation exponents at all scales for $p>0$, the universality shift is a property of the causal-web clustering rather than of the underlying dynamics."],"forward_implications":["For any nonzero spontaneous activation rate, the epidemic or activity threshold is lowered, and a local reproduction number below one can still coexist with a giant component.","Directed-percolation exponents appear only in avalanches smaller than $s_m \\sim p^{-2/3}$; larger avalanches and the giant component follow undirected percolation exponents.","The branching-ratio line $\\sigma=1$ and the dynamic-susceptibility Widom line scale differently near the directed-percolation point, so they cannot locate the true critical point once $p>0$.","Neuronal avalanche exponents in the range 1.2 to 2.5 can arise from the two-regime distribution near criticality, so fitting a single power-law exponent to data is an unreliable test of sub- or super-critical state.","A variant with permanent immunity (SIR-like) also shows the two-power-law structure, extending the merging mechanism to diseases with no reinfection."],"supporting_citations":[{"why":"Supplies the causal-web cluster definition used to separate independent avalanches in the presence of spontaneous activity.","marker":"[30]"},{"why":"Provides the $\\varepsilon$-SIS model with nodal self-infection that is the starting point of the paper's spreading process.","marker":"[31]"},{"why":"Shows that aggregating all activity into one avalanche shifts the size-distribution exponent, motivating the network-based avalanche construction.","marker":"[28]"},{"why":"Gives the directed-percolation avalanche and spreading exponents used for the small-size regime.","marker":"[39]"},{"why":"Gives undirected percolation exponents on Bethe-like networks used for the large-size and giant-component regime.","marker":"[40]"},{"why":"Provides directed-percolation exponents on directed scale-free networks compared with the numerical results.","marker":"[36]"},{"why":"Provides undirected percolation exponents on scale-free networks used to identify the tail universality class.","marker":"[37]"},{"why":"Defines the quasi-critical Widom line in brain dynamics that the paper shows fails to identify the true critical line.","marker":"[44]"},{"why":"Provides the hierarchical modular network used as a brain-connectome analogue and the Griffiths-phase observations the paper contrasts with its critical line.","marker":"[22]"},{"why":"Supplies the finite-size scale-invariant ratio $B$ used to locate the critical point numerically for each network topology.","marker":"[92]"}],"fun_headline_variants":["Self-activation breaks directed-percolation universality","No time-scale separation flips spreading to undirected percolation","Merging cascades shift percolation threshold and exponents","Branching ratio fails to detect true critical onset","Spontaneous activity moves epidemics to ordinary percolation"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that avalanches defined by the causal-web merging rule, where nodes with no active parents start new clusters and overlapping clusters merge, are the physically correct way to separate independent cascades; if that construction mislabels causal structure, the critical line and universality class describe the clustering algorithm rather than the spreading dynamics.","fun_headline_variants_meta":{"raw":{"variants":["Self-activation breaks directed-percolation universality","No time-scale separation flips spreading to undirected percolation","Merging cascades shift percolation threshold and exponents","Branching ratio fails to detect true critical onset","Spontaneous activity moves epidemics to ordinary percolation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000411,"raw_usage":{"total_tokens":2180,"prompt_tokens":1050,"completion_tokens":1130,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":666,"completion_tokens_details":{"reasoning_tokens":1050}},"tokens_in":666,"tokens_out":1130,"duration_ms":10283,"temperature":1.0,"reasoning_tokens":1050,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:48:28.431585+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"On a $k$-regular network with $p=10^{-4}$, measure the critical avalanche size distribution at $q=q_c(p)$ out to sizes well above $p^{-2/3}$: if the central claim is right, the tail must follow $s^{-5/2}$ after an $s^{-3/2}$ regime with a sharp crossover near $s_m \\sim p^{-2/3}$; observing a single pure directed-percolation power law at all scales would falsify the universality shift.","supporting_citations":[{"cited_title":"Ponce-Alvarez, A","cited_arxiv_id":null,"evidence_quote":"Supplies the causal-web cluster definition used to separate independent avalanches in the presence of spontaneous activity."},{"cited_title":"Del Papa, V","cited_arxiv_id":null,"evidence_quote":"Provides the $\\varepsilon$-SIS model with nodal self-infection that is the starting point of the paper's spreading process."},{"cited_title":"Di Santo, P","cited_arxiv_id":null,"evidence_quote":"Gives the directed-percolation avalanche and spreading exponents used for the small-size regime."},{"cited_title":"Pinheiro Neto, F","cited_arxiv_id":null,"evidence_quote":"Gives undirected percolation exponents on Bethe-like networks used for the large-size and giant-component regime."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides directed-percolation exponents on directed scale-free networks compared with the numerical results."},{"cited_title":"Dalla Porta and M","cited_arxiv_id":null,"evidence_quote":"Provides undirected percolation exponents on scale-free networks used to identify the tail universality class."},{"cited_title":"Van Mieghem and E","cited_arxiv_id":null,"evidence_quote":"Defines the quasi-critical Widom line in brain dynamics that the paper shows fails to identify the true critical line."},{"cited_title":"Pruessner, Self-organised criticality: theory, mod- els and characterisation (Cambridge University Press, 2012)","cited_arxiv_id":null,"evidence_quote":"Supplies the finite-size scale-invariant ratio $B$ used to locate the critical point numerically for each network topology."}],"review_version":1}