{"id":"ee30d6f2-9fde-4c5c-a1c3-9039551688dd","arxiv_id":"2508.10140","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Random walks on small-world networks can have two-humped first-passage time distributions, and exact generating-function formulas capture the full distribution without simulation.","lead":"The authors derive exact generating-function formulas for random-walk first-passage statistics on sparse networks built from ring lattices, and use them to show that first-passage time distributions on small-world networks can be bimodal. The result matters because it lets researchers replace simulations with exact calculations and shows that mean first-passage times can hide two distinct search regimes.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Exactness hinges on the quoted defect-determinant formalism and on the completeness of the four-tuple set, but Eq. (5)'s defect count appears too small for a single WS rewiring; the paper supplies no code/list to rule out omitted defects.","rationale":"Read in good faith, the paper does several things well: it recovers k=1 and complete-graph limits, and the analytic curves in Fig. 2/Fig. S1 match simulations for the displayed realizations. Those checks give real support to Eq. (2) and Eq. (4). The load-bearing fragility is not in the ring calculus but in the transfer from the defect-free ring to the SWN: Eqs. (S58)-(S61) are quoted from ref. [10], not re-derived, and the only description of how the ring-to-SWN perturbation is encoded is the four-tuple sentence plus the M=(K+1)pN estimate. That estimate is not obviously consistent with the degree normalization changes a single rewiring produces, so the exactness of the SWN propagator is under-specified. The reader's conditional verdict already captures this; my proposed check would upgrade the conditional to accept if the full four-tuple construction is validated, or force a correction if not. No ad hominem; the issue is testable and independent of the authors' intent.","tokens_in":20991,"tokens_out":17599,"duration_ms":219682,"concrete_test":"Use N=8, K=2, one rewiring generated by the paper's algorithm. Build B (ring) and A (rewired graph), compute X=B-A, and count the minimal four-tuples (unordered pairs with at least one non-zero entry). Then evaluate \\tilde S_{n0}(n,z) from Eq. (S58) using (i) only M=(K+1)pN defects and (ii) the full four-tuple set, invert numerically for t=1..20, and compare both with exact A^t from matrix powers for all n. If (ii) matches and (i) does not, the defect count in Eq. (5) is wrong and the reported exact results must state the full construction; if (ii) also fails, the quoted determinant formula has an additional hidden assumption.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that SWN first-passage statistics are obtained exactly rests on Eq. (6) plus the quoted determinants Eqs. (S58)-(S61): every modification of the ring transition matrix B must be encoded as four-tuples extracted from X=B-A. That representation is exact only if the four-tuple list is complete. The paper states M=(K+1)pN defects, but a single Watts-Strogatz rewiring changes the outgoing transition probabilities of the two nodes whose degree changes (one loses a link, one gains a link): O(K) directed entries, not K+1. For K=2 one rewiring already requires five nonzero pairs in the example above, not three. Furthermore, under the standard WS prescription the expected number of rewired edges is pNK/2, not pN; combined with O(K) entries per rewiring, the claimed sparsity condition K+1<1/p could underestimate the rank of the defect matrices by O(K). No code or explicit four-tuple construction is provided, so the reader cannot check whether Figs. 2-3 used the full X or only the stated M. If the former, the exactness claim survives; if the latter, omitted defects make Eqs. (S58)-(S61) approximate.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops an analytical framework for computing first-passage statistics of a random walk on sparse networks, specifically Watts-Strogatz small-world networks built from a K-neighbor ring lattice. The defect-free ring propagator is derived in closed form (Eq. 2), the ring MFPT is given in Eq. (4), and the inert-defect formalism of Ref. [10] is used to represent the rewired network as a ring plus a set of localized transition-probability modifications. The resulting occupation-probability generating function (Eq. S58) and MFPT formula (Eq. S61) are then used to obtain first-absorption distributions and mean times. The paper reports a bimodal regime in the first-absorption distribution, argues that the MFPT can decouple from the statistics of direct trajectories, and claims that these are the first such results for symmetric diffusive systems. The central exactness claim rests on the completeness of the four-tuple defect set used in Eqs. (S58)-(S61).","tokens_in":21300,"tokens_out":16627,"duration_ms":197098,"significance":"If the defect representation is exact and complete, this is a valuable contribution: it provides parameter-free analytical expressions for occupation and first-passage statistics on sparse networks, recovering the k=1 and fully connected limits of the ring, and it makes the bimodality phenomenon accessible without stochastic simulation. The paper explicitly checks several analytic curves against independent stochastic simulations (Figs. 1-2, S1), and the determinant formulas are quoted from a published framework. However, the significance is conditional on the defect-count issue raised below: the stated number of four-tuples in Eq. (5) appears inconsistent with the number of nonzero entries of X=B-A, and because the actual construction is not provided, the reader cannot verify that the figures used the full defect set.","major_comments":[{"comment":"The stated defect count is not consistent with the Watts-Strogatz rewiring described in the Letter. Under the standard prescription, the expected number of rewired edges is pNK/2, not pN. More importantly, a single rewiring (a,b)->(a,c) changes the degree of b from K to K-1 and the degree of c from K to K+1. The nonzero entries of X=B-A are then: two in row a (to b and to c), K in row b (the removed edge plus the K-1 remaining neighbors), and K+1 in row c (the new edge plus the K existing neighbors). Thus X has 2K+3 nonzero directed entries, or 2K+1 pairs (i,j) usable as four-tuples, not K+1. If the actual calculations used the full X, then Eq. (5) is wrong and the sparsity condition K+1<1/p is misleading. If they used M=(K+1)pN, the defect set is incomplete and the 'exact' network results in Figs. 1-3 and S1 are not exact. The manuscript must provide the explicit four-tuple construction","section":"Dynamics on the network, Eq. (5)"},{"comment":"The engineered one-shortcut network is stated to require M=K+1 defects. Adding one shortcut between two vertices raises the degree of both endpoints from K to K+1, so every outgoing probability from each endpoint changes: the K existing edges and the new edge. This requires 2K+1 four-tuples, or 2(K+1) directed entries, not K+1. Unless a special normalization is being used that is not explained, the phase diagrams in Figs. 3(a,b), the distributions in Fig. 3(c), and the mode/MFPT comparison in Fig. 3(d) are computed with an incomplete defect matrix. The bimodality boundaries and the decoupling claim must be re-established with the full defect set.","section":"SM Sec. VI / Fig. 3"}],"minor_comments":[{"comment":"Even if the per-rewiring count were correct, 'each SWN has, on average, pN re-wirings [28]' misrepresents Ref. [28]; the standard Watts-Strogatz construction rewires each edge with probability p, giving pNK/2 expected rewired edges. Clarify which construction is used.","section":"Dynamics on the network, Eq. (5)"},{"comment":"The sentence 'we find the modifications ... as the non-zero elements of X' is ambiguous: a four-tuple (i,j,Xj,i,Xi,j) stores two directed entries, so it should be specified whether M counts nonzero directed entries or nonzero pairs. This ambiguity directly affects the size of the determinant matrices in Eqs. (S58)-(S61).","section":"Defect extraction paragraph after Eq. (6)"},{"comment":"The parametric characterization of the bimodal regime (N/K and rho dependence) is established only for the engineered one-shortcut network; the random-SWN evidence is a single realization (Fig. 2, Graph 4). The abstract and Discussion phrase the result as a property of small-world networks generally. Either provide ensemble-level statistics for random SWNs or explicitly restrict the claim to the engineered defect network.","section":"General discussion / Abstract"},{"comment":"The Letter says predictions are 'exact,' but the time-domain curves in Fig. 3 are obtained by numerical inversion of the generating function. This is a standard and acceptable procedure, but the wording should distinguish exact generating functions from the numerical inversion used for display.","section":"Fig. 3 and time-domain inversion"}],"recommendation":"major_revision","confidential_remarks":"The defect-count issue is potentially fixable: if the authors can show that all figures were produced with the full X and correct the text, the contribution may be strong. However, as written, Eq. (5) and the M=K+1 claim in the engineered network are load-bearing and unverifiable without the explicit four-tuple list or code. I would like to see the corrected manuscript before judging the exactness claims."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know: this is a mathematically substantive paper with genuinely new exact results for the K-neighbor ring walk, but the advertised exactness on small-world networks is not fully verifiable from the manuscript as written.\n\nWhat's new and solid: Eq. (2) gives the propagator generating function for a K-neighbor walk on a ring; Eq. (4) gives the MFPT. For k=2,3 the SM provides explicit closed forms via Chebyshev polynomials. The k=1 and fully-connected limits are recovered, which is a good internal check. Using the inert defect formalism from their prior work, they write the SWN propagator and MFPT as determinants of M×M matrices. No parameters are fitted; analytic curves line up with stochastic simulations in Figs. 1-2 and S1, at least visually. The bimodal first-absorption distribution is an interesting observation, and the argument that the MFPT loses meaning in strongly heterogeneous environments is a worthwhile challenge to the Godec-Metzler picture.\n\nNow the soft spots, in proportion. The paper says the average number of defects is M=(K+1)pN, based on pN rewires and K+1 defects each. That is wrong on both counts: in a Watts-Strogatz graph the expected number of rewired edges is pNK/2, not pN, and a single rewiring changes outgoing transition probabilities of at least three nodes, yielding at least K+1 nonzero pairs (for K=2, five, not three). The text says the four-tuples are extracted as the non-zero elements of X=B-A, which would make the set complete. If the code did that, the exactness claim holds; if it used the stated M, the determinants are missing defects and the 'exact' curves are approximate. There is no code or explicit list, so a reader cannot tell which.\n\nThe bimodality phase diagram in Fig. 3 is computed on an engineered one-shortcut ring, not on random WS ensembles. The four realizations in Fig. 2 are nice, but they do not establish the phase boundary for the actual model. Time-domain curves come from numerical inversion of generating functions with no error bars or code.\n\nBottom line: this deserves peer review. The ring results alone are worth publishing. The SWN exactness needs the authors to clarify the defect count and ideally release code or a small worked example. I would recommend a serious referee with a request for code/amplification.","headline":"Serious mechanics paper with genuinely new ring-lattice results; the SWN exactness claim needs a sharper accounting of defects before I'd fully trust it.","tokens_in":21778,"tokens_out":7940,"would_cite":true,"duration_ms":79839,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Random walks on sparse small-world networks can be solved exactly by treating each rewired link as a probability-preserving defect, and the solution exposes a bimodal first-absorption regime in which the mean first-passage time loses its us","keywords":["first-passage probability","mean first-passage time","small-world networks","random walks","inert defects","disordered systems","bimodal first-absorption","Watts-Strogatz network"],"falsifier":"Take one fixed Watts-Strogatz realization (for example the $N=100$, $K=6$, $p=0.15$ case of Fig. 2), compute the occupation probability from Eq. (S58) at several times, and compare it with the average of a large number of simulated walks on that same graph; any node-time pair where the difference exceeds Monte Carlo sampling error would show the four-tuple or determinant representation is incomplete. The bi-modality claim can be tested directly by measuring the first-absorption distribution in the predicted $N/K$ window and looking for two resolved peaks.","tokens_in":20907,"feed_emoji":"🎯","tokens_out":12225,"duration_ms":124906,"temperature":0.7,"pith_summary":"This paper claims that the search dynamics of a random walker on a sparse Watts-Strogatz small-world network can be computed exactly, without stochastic simulation. The central move is to view the network as a homogeneous $K$-neighbour ring lattice plus a small number of 'inert' defects: each rewired edge is encoded as a four-tuple of transition-probability changes, and the exact occupation and first-passage probabilities follow from determinants of $M\\times M$ matrices built from ring-lattice propagators. Applied to small-world networks, the method uncovers a bimodal first-absorption distribution in symmetric diffusive systems. In that regime the mean first-passage time is not controlled by the direct trajectories and does not represent the typical time of the indirect ones. If correct, this closes a gap in target-search theory and changes how mean first-passage-time values on disordered networks should be read.","feed_headline":"Small-world search times now exact, no simulation needed","feed_subtitle":"A defect-by-defect determinant method yields first-passage statistics and uncovers a two-peak regime.","key_machinery":"The inert-defect machinery: any rewiring that preserves total probability is encoded as deviations from the ring-lattice transition matrix, each stored as a four-tuple $(u, v, \\eta_{v,u}, \\eta_{u,v})$, and the exact propagator on the disordered network is the ring-lattice propagator plus the ratio of two $M\\times M$ determinants, Eqs. (S58)--(S60). The companion object is the closed-form $K$-neighbour ring-lattice propagator (Eq. 2) and its mean first-passage time (Eq. 4); every network-level expression is built from these ring quantities and the $\\eta$ parameters. The determinants carry all disorder information, which is why the cost scales with the number of defects $M = (K+1)pN$ rather th","core_discovery":"The central claim is that a random walk on a Watts-Strogatz network is exactly solvable through the inert-defect representation. The difference $X = B - A$ between the ring-lattice transition matrix and the rewired network transition matrix is a sparse correction captured by $M$ four-tuples $(u, v, \\eta_{v,u}, \\eta_{u,v})$; substituting these into the determinant formulas (S58)--(S61) gives the occupation probability, the first-passage probability, and the mean first-passage time for any realization of the disorder. On the defect-free ring the paper derives a new $K$-neighbour propagator generating function (Eq. 2) and mean first-passage time (Eq. 4), recovering known nearest-neighbour and f","pith_inferences":["The same four-tuple determinant construction should carry over to base graphs other than ring lattices, such as regular trees, tori, or Cartesian-product lattices, by replacing Eq. (2) in Eqs. (S58)--(S60); the hard part is only knowing the defect-free propagator.","The bi-modality boundaries in the $(N/K, \\rho)$ plane invite a scaling-collapse test: plotting the two-mode separation against $N/K$ for fixed $\\rho$ should make the coloured regions in Fig. 3(a)--(b) line up onto a single curve.","The paper's own stated boundary is computational rather than conceptual: the determinants are $M\\times M$ with $M = (K+1)pN$, so the method is exact but heavy when rewiring is dense; the practical gain over simulation is in the sparse, high-clustering regime.","The observed decoupling of the mean first-passage time from direct trajectories implies that MFPT-based centrality or target-search rankings on networks with long-range links can be systematically misleading; mode times or the full distribution are safer summary statistics."],"forward_implications":["For a single fixed realization of a small-world network, the full time-dependent occupation probability and first-passage probability are available from closed-form expressions, so Monte Carlo averages are no longer required in the sparse regime.","The exact $K$-neighbour ring-lattice results fill the gap between the known nearest-neighbour chains and the fully connected graph; both limits drop out of the same formulas.","Bi-modality of the first-absorption distribution is a genuine feature of symmetric diffusive systems, and its onset is set by $N/K$ and $\\rho$, so a single mean first-passage time cannot summarize the search process in that regime.","In the bi-modal regime the mean first-passage time is decoupled from direct-trajectory statistics; the early mode reflects local structure around start and target, the late mode reflects global structure.","The method transfers to any graph that is a solvable base structure plus a set of defects; the cost is set by the number of defects, not the network size."],"supporting_citations":[{"why":"Supplies the inert-defect determinant formalism that yields exact occupation and first-passage quantities on the rewired network.","marker":"[10]"},{"why":"Defines the Watts-Strogatz small-world construction whose rewired links are modelled as defects.","marker":"[28]"},{"why":"Provides the method-of-images and Chebyshev ring-lattice propagator basis used for Eq. (2) and the MFPT limits.","marker":"[42]"},{"why":"Contains the derivations of the ring-lattice propagator, the MFPT, and the determinant formulas used in the main text.","marker":"[43]"},{"why":"Gives the reactive-defect relation (Eq. 3) connecting the occupation propagator to the first-absorption probability.","marker":"[24]"},{"why":"Provides the renewal relations used to extract the mean first-passage time from the first-passage generating function.","marker":"[45]"},{"why":"Gives the network mean-return-time formula $R_n = E/\\chi_n$ and steady state used in the MFPT and absorption expressions.","marker":"[22]"},{"why":"Supplies the prior conclusion that the MFPT is completely specified by direct-trajectory statistics, which the paper's bi-modal regime is designed to overturn.","marker":"[50]"},{"why":"Provides the earlier approximate scaling law for the SWN MFPT that the exact formulas improve on for small networks.","marker":"[38]"}],"fun_headline_variants":["Exact first-passage stats for small-world networks at last","Random walks on small-world graphs solved exactly","Bimodal search times revealed in exact network walk theory","New method gives exact search dynamics on sparse networks","Small-world network search: exact solution exposes two peaks"],"cache_read_input_tokens":2816,"weakest_assumption_plain":"Everything rests on the claim that the rewired network's transition matrix differs from the ring lattice's only through the listed four-tuples, and that the determinant formula of Eq. (S58) inherited from the defect formalism remains exact for that rewired graph; if a rewiring introduces a transition change the four-tuple list misses, the exact small-world results fail.","fun_headline_variants_meta":{"raw":{"variants":["Exact first-passage stats for small-world networks at last","Random walks on small-world graphs solved exactly","Bimodal search times revealed in exact network walk theory","New method gives exact search dynamics on sparse networks","Small-world network search: exact solution exposes two peaks"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000236,"raw_usage":{"total_tokens":1332,"prompt_tokens":730,"completion_tokens":602,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":474,"completion_tokens_details":{"reasoning_tokens":526}},"tokens_in":474,"tokens_out":602,"duration_ms":7138,"temperature":1.0,"reasoning_tokens":526,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T20:38:24.576685+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take one fixed Watts-Strogatz realization (for example the $N=100$, $K=6$, $p=0.15$ case of Fig. 2), compute the occupation probability from Eq. (S58) at several times, and compare it with the average of a large number of simulated walks on that same graph; any node-time pair where the difference exceeds Monte Carlo sampling error would show the four-tuple or determinant representation is incomplete. The bi-modality claim can be tested directly by measuring the first-absorption distribution in the predicted $N/K$ window and looking for two resolved peaks.","supporting_citations":[{"cited_title":"Sarvaharman and L","cited_arxiv_id":null,"evidence_quote":"Supplies the inert-defect determinant formalism that yields exact occupation and first-passage quantities on the rewired network."},{"cited_title":"Giuggioli","cited_arxiv_id":null,"evidence_quote":"Provides the method-of-images and Chebyshev ring-lattice propagator basis used for Eq. (2) and the MFPT limits."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Contains the derivations of the ring-lattice propagator, the MFPT, and the determinant formulas used in the main text."},{"cited_title":"Giuggioli and S","cited_arxiv_id":null,"evidence_quote":"Gives the reactive-defect relation (Eq. 3) connecting the occupation propagator to the first-absorption probability."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the renewal relations used to extract the mean first-passage time from the first-passage generating function."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the network mean-return-time formula $R_n = E/\\chi_n$ and steady state used in the MFPT and absorption expressions."},{"cited_title":"Godec and R","cited_arxiv_id":null,"evidence_quote":"Supplies the prior conclusion that the MFPT is completely specified by direct-trajectory statistics, which the paper's bi-modal regime is designed to overturn."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the earlier approximate scaling law for the SWN MFPT that the exact formulas improve on for small networks."}],"review_version":1}