{"id":"cd2c7924-0a70-412a-8c2d-fd27e8bdb36e","arxiv_id":"1908.07376","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"The distribution of shortest path lengths in the corded directed node-duplication network is a convolution of the seed distribution with Poisson terms, and its mean distance grows logarithmically with network size while the connected fraction vanishes.","lead":"This paper derives closed-form formulas for the distribution of shortest path lengths in a directed network that grows by node duplication, where each new node copies outgoing links of a random mother node. The formulas show that connected pairs of nodes have logarithmically growing distances, but the fraction of connected pairs vanishes, so the network is not small-world.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The closed form (42) is not exact at finite times because it uses the steady-state degeneracy parameter η; near p=1 the degeneracy distribution converges only as t^{-α} with α→0, so Eq. (42) may fail for all practical network sizes.","rationale":"The reader's weakest assumption is exactly the load-bearing point: the master equation solution uses a constant, steady-state, truncated η. Appendix B confirms that this is not merely a technical detail: the convergence exponent α_2 is explicit and vanishes as p→1, so the parameter the whole closed form depends on is nonstationary on the time scales simulated. This does not invalidate the asymptotic logarithmic scaling at infinite time, nor the comparison at p≤0.8 where the simulations are shown, but it does invalidate the 'exact analytical result valid at all times' framing in the abstract and summary. I found no independent fatal flaw in the mean-distance or connected-fraction derivations; Eq. (28) is consistent with a p=1 backbone-ancestor picture in which reachable pairs are direct ancestors, and the algebra of Eqs. (26)–(27) checks out. The paper gives real support via simulation agreement, parameter-free η up to truncation, and a transparent Appendix C with the correction p→η for ℓ=1,2. The concern is therefore best handled as a condition: either prove a uniform bound on the error from replacing η_t by η_∞, or present the time-dependent solution and restrict the exactness claim. Since the authors already have the machinery in Appendix B, this is addressable without changing the main qualitative message.","tokens_in":23999,"tokens_out":21477,"duration_ms":230969,"concrete_test":"Use the time-dependent degeneracy solution in Appendix B (Eq. B.4) to form η_t = 1 - (1-p)P_t(G=1*) - (1-p)^2 P_t(G=2), and solve Eq. (41) numerically with this time-dependent η for p=0.95 and p=0.99, seed s=2, up to N=10^6. Compare the resulting mean distance E_t[L|L<∞] with Eq. (47) evaluated with the steady-state η of Eq. (20). If the two agree within a few percent across the range, the steady-η approximation is safe; if they diverge increasingly as p→1, then Eq. (42) is not the exact finite-time solution and the 'valid at all times' claim must be restricted to p not too close to 1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central formula, Eq. (42), solves the master equation (41) with a constant η taken from the steady-state degeneracy distribution (Eqs. (13) and (20)). But η is dynamical: η_t = 1 - Σ_g (1-p)^g P_t(G=g). Appendix B solves the degeneracy master equation and shows that P_t(G) approaches steady state only as t^{-α_min}, with α_2 = 1+p-2p^2 (Eq. B.5) tending to 0 as p→1. For p=0.99, α_2 ≈ 0.03, so even at N=10^6 the deviation from steady state is only mildly reduced, and replacing η_t by η_∞ from time zero mis-specifies the coefficient (1−η) on P_t(L=ℓ−1) in Eq. (41) throughout the growth. The summary's claim that Eq. (42) is 'valid at all times' is therefore not supported: at best it is an asymptotic statement for p away from 1, since the authors' own Eq. (B.7) gives a convergence time diverging as p→1. The simulations in Figs. 6–7 stop at p=0.8, where α_2=0.52 and the approximation is benign, so the dangerous regime is exactly the one left untested. Separately, Eq. (42) also substitutes η for p in the ℓ=1 and ℓ=2 source terms (Eqs. (31)–(32)), an approximation the authors acknowledge; Appendix C restores p only in those terms but still keeps η time-independent, so it does not repair the finite-time issue.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the distribution of shortest directed path lengths (DSPL) in the corded directed node duplication (DND) network model. It derives a master equation for the time evolution of Pt(L=ℓ), solves it in closed form, and obtains Eq. (42), expressing the DSPL as a convolution of the seed DSPL with Poisson terms plus a growth term. The paper further derives the connected fraction Pt(L<∞) ~ ln N_t / N_t and the conditional mean distance Et[L|L<∞] ~ (1−η)/2 ln N_t. The parameter η is computed from a steady-state degeneracy distribution truncated at g=3, and the derivation replaces p by η in the ℓ=1 and ℓ=2 source terms. Simulations for p up to 0.8 support the analytical formulas.","tokens_in":24456,"tokens_out":8937,"duration_ms":85616,"significance":"If the closed-form results are correct, this is a valuable analytical contribution to the DSPL literature for a simple directed growing-network model with applications to gene regulatory and citation networks. The paper's strengths include a transparent master-equation formulation, an explicit closed-form solution, and a parameter η that is computed from the model's own degeneracy statistics rather than fit to the DSPL, so the central result is not circular in the fitting sense. The Poisson convolution structure and the logarithmic scaling of the conditional mean distance are nontrivial and are supported by simulation agreement in the tested regime. The main caveat is that the time-independent steady-state η and the truncation at g=3 mean the word 'exact' in the abstract and summary overstates the status of Eq. (42), particularly for p close to 1.","major_comments":[{"comment":"The Summary states that Eq. (42) is 'valid at all times', but the derivation solves Eq. (41) with a time-independent η taken from the steady-state degeneracy distribution (Eq. (20)). Appendix B shows that the degeneracy distribution converges to steady state only as a power law with exponent α_min = min{1−p+p^2, 1+p−2p^2}, and that α_min → 0 as p → 1 (Eqs. B.5–B.7). Thus for p close to 1, replacing η_t by its steady-state value mis-specifies the coefficient (1−η) in Eq. (41) throughout the growth period, and Eq. (42) is at best an asymptotic result for p bounded away from 1 rather than an exact finite-time solution. The authors should either solve the master equation with a time-dependent η(t), or explicitly restrict the validity claim and quantify the error in the p→1 regime.","section":"§5, Eq. (42) and Appendix B"},{"comment":"The replacement of p by η in the ℓ=1 and ℓ=2 equations is acknowledged in Section 5, but Appendix C, which is presented as the exact form, restores p only in the source terms while still treating η as time-independent. The 'exact' wording in the abstract and summary therefore overstates the status of the closed form: the exactness applies to the master-equation framework and to the solution conditional on a fixed η, not to Eq. (42) for the actual growing network. Please qualify the claims and specify the range of p and t for which the approximation is controlled.","section":"§5, Eqs. (31)–(32) and Appendix C"},{"comment":"The degeneracy distribution is truncated at g=3, and the closeness of the g=2 and g=3 results in Fig. 5 is reassuring but does not provide an error bound for the neglected g≥4 states. In addition, the numerical validation of the DSPL and the logarithmic law is reported only for p ≤ 0.8, whereas the slow convergence identified by Eq. (B.7) becomes severe for p > 0.8 (for example, α_2 = 0.28 at p = 0.9). The finite-time claims near p = 1 are therefore untested. Please add numerical tests in the p > 0.8 regime or explicitly restrict the stated domain of validity.","section":"§3, Eq. (13) and §6, Fig. 7"}],"minor_comments":[{"comment":"The phrases 'exact analytical results' and 'valid at all times' should be harmonized with the acknowledged approximations used in Section 5; 'closed-form analytical results' would be a more accurate description.","section":"Abstract and Summary"},{"comment":"The word 'congergence' in the appendix heading should be corrected to 'convergence'.","section":"Appendix B heading"},{"comment":"These captions do not state the number of simulated network realizations, unlike the caption of Fig. 6; please include this information for reproducibility.","section":"Fig. 7 and Fig. 8 captions"},{"comment":"The notation t_s is defined in Eq. (43) but is used already in Eq. (42); consider defining it immediately before Eq. (42) to avoid confusion.","section":"Eq. (42)"},{"comment":"The phrase 'betweeness centrality' should be corrected to 'betweenness centrality'.","section":"Appendix A"}],"recommendation":"major_revision","confidential_remarks":"The core derivation is coherent and the simulation agreement in the tested regime supports the main formulas, so rejection is not warranted. The load-bearing issue is the finite-time validity of Eq. (42) due to the time-independent steady-state η, combined with the absence of tests near p=1. A major revision that either uses time-dependent η, or carefully qualifies the validity domain and adds simulations in the p→1 regime, would make the paper publishable. The 'exact' language should be toned down in the abstract and summary."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"I think the paper is solid but oversold. The new result is a closed-form DSPL for the corded directed node-duplication model, and the conceptual point, that reachable pairs shrink as ln N/N while the undirected graph stays connected, is a worthwhile clarification of what small-worldness means in directed growing networks. The master-equation derivation is clear, and the simulation agreement up to p=0.8 is convincing.\n\nThe soft spots are two, and they are real. First, the 'exact' in the abstract and summary overstates the case. Equation (42) is built on the approximation p→η in the ℓ=1,2 terms, and the authors acknowledge that. More seriously, η is the steady-state value of the degeneracy distribution, not the time-dependent η_t. The paper's own Appendix B shows that convergence to steady state is power-law, with exponent α_2=1+p−2p^2 that vanishes as p→1. At p=0.99 the convergence is so slow that substituting η_∞ into Eq. (41) is not justified for any practical network size. The simulations in Figs. 6–7 stop at p=0.8, where α_2≈0.52 and the substitution is harmless, so the dangerous regime is exactly the one that is not tested. Appendix C fixes the ℓ=1,2 source terms but keeps η time-independent, so it does not cure this.\n\nThe central claim should therefore be stated as an asymptotic or quasi-static result for p away from 1, not an exact all-time expression. These are fixable issues: temper the language, possibly add a numerical check near p=1, and give a bound on the error introduced by the constant-η approximation. The mathematical core remains a credible contribution to the authors' research program. I would recommend accepting the paper for review with a request for major revision; it deserves to be in the literature after the claims are calibrated.","headline":"A solid continuation of the authors' DSPL program, but the 'exact' claim is overstated and the constant-η approximation needs a caveat near p→1.","tokens_in":24902,"tokens_out":3528,"would_cite":true,"duration_ms":34161,"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":"Directed duplication networks have an exact shortest-path law: distances among connected pairs grow logarithmically while almost all pairs are disconnected.","keywords":["shortest path lengths","directed networks","node duplication","master equation","mean distance","small-world networks","citation networks","gene regulatory networks"],"falsifier":"Simulate the corded DND model with $p=0.9$ from a two-node seed and measure the full distribution $P_t(L=\\ell)$ at network sizes $10^2$, $10^3$, and $10^4$; if the closed form with steady-state $\\eta$ visibly misses the early-time distribution while the exact Appendix C equations with a time-dependent effective probability match it, the constant-$\\eta$ solution is only asymptotic rather than the finite-time law claimed.","tokens_in":23832,"feed_emoji":"🕸️","tokens_out":11663,"duration_ms":103801,"temperature":0.7,"pith_summary":"This paper establishes the exact distribution of shortest directed path lengths in a network that grows by node duplication: each new daughter node points to its mother and, with probability $p$, to each of the mother's outgoing neighbors. The main result is a closed-form solution of the master equation for the probability $P_t(L=\\ell)$ that a shortest directed path has length $\\ell$, valid for all times and for any acyclic seed network. The solution is a convolution of the seed's path-length distribution with Poisson factors, plus a seed-independent sum of Poisson terms. It implies that among the node pairs connected by directed paths, the mean distance grows logarithmically with network size, while the fraction of connected pairs shrinks as $(\\ln N_t)/N_t$. The upshot is that such networks are not small-world as directed networks, a fact with direct bearing on gene regulatory and citation networks, where most pairs are out of directed reach.","feed_headline":"Duplication-grown directed networks obey one exact distance formula","feed_subtitle":"Connected pairs stay logarithmically close, but the fraction with any directed path shrinks to zero.","key_machinery":"The engine of the calculation is a linear master equation for the probability masses $P_t(L=\\ell)$, obtained by counting what happens to every ordered pair when a random mother node $M$ is duplicated into a daughter $D$. A target at distance $\\ell$ from $M$ sits at distance $\\ell$ from $D$ if at least one edge on a shortest path from $M$ is copied, otherwise at distance $\\ell+1$. The paper encodes this by a single effective probability $\\eta$, defined by the degeneracy sum in Eq. (18), that a shortest-path length is preserved; $\\eta$ is computed from the steady-state distribution of the degeneracy $g$ of first steps on shortest paths, with the hierarchy truncated at $g=3$. Replacing $p$ by $\\eta$ in the $\\ell=1$ and $\\ell=2$ equations turns the hierarchy into a solvable system whose solution is the Poisson-convolution formula. The logarithmic terms enter because each generation of the deterministic backbone tree advances the relevant time scale by a factor captured by $\\ln t_s$.","core_discovery":"The paper's central claim is that the master equation for $P_t(L=\\ell)$ admits a closed-form solution. For $\\ell\\ge 2$, with $t_s=(t+s+1)/(s+1)$ and $\\Delta_0$ the seed diameter, the solution is $$P_t(L=\\ell) = \\frac{1}{$t_s^{{2-\\eta}}$} \\sum_{\\ell'=1}^{\\min\\{\\ell,\\Delta_0\\}} \\frac{(1-\\eta)^{\\ell-\\ell'}}{(\\ell-\\ell')!} (\\ln t_s)^{\\ell-\\ell'} P_0(L=\\ell') + \\frac{1}{(1-\\eta)(s+1)$t_s^{{2-\\eta}}$} \\sum_{\\ell'=\\ell}^{\\infty} \\frac{(1-\\eta)^{\\ell'}}{\\ell'!} (\\ln t_s)^{\\ell'}.$$ The first term propagates the seed network's own shortest-path distribution into the growing network as a Poisson convolution in $\\ln t_s$; the second term is a seed-independent Poisson sum describing paths formed entirely during growth. Companion closed forms for $P_t(L=1)$ and $P_t(L=\\infty)$ complete the distribution. From these the paper obtains $P_t(L<\\infty)\\sim(\\ln N_t)/N_t$ and $\\mathbb{E}_t[L|L<\\infty]\\sim((1-\\eta)/2)\\ln N_t$, so among the vanishing fraction of connected ordered pairs distances are logarithmically short, yet the directed network as a whole is not small-world.","pith_inferences":["The formula's stated validity for any acyclic seed means the same Poisson-convolution form should hold for richer seeds than the two-node chain used in the figures; this is an extrapolation the paper does not test numerically.","The slow power-law relaxation of $\\eta(t)$ for $p$ close to 1 suggests that real growing systems with strong duplication would spend a long time in a transient regime where the exact Appendix C master equation, rather than the steady-state closed form, is the better description.","For citation data, the model predicts a measurable signature: the fraction of ordered paper pairs connected by citation chains should decline roughly as the corpus grows, while the mean chain length among connected pairs climbs only logarithmically; fitting both curves would estimate $\\eta$ from data."],"forward_implications":["At any finite time, the full shortest-path-length distribution is fixed by the seed DSPL, the duplication probability $p$, and the degeneracy parameter $\\eta$; no other microstructural detail of the growth history enters.","Among connected ordered pairs, the mean distance tends to $\\frac{1-\\eta}{2}\\ln N_t$, and the variance tends to $\\frac{(1-\\eta)^2}{12}(\\ln N_t)^2$, so the connected subpopulation has a widening logarithmic distance profile.","The connected fraction tends to $(\\ln N_t)/N_t$, so almost all ordered pairs are disconnected in the large-network limit even though the underlying undirected network is a single component.","As a minimal citation-network model, the result predicts that citation chains connect only a shrinking fraction of paper pairs, and that the chains among connected pairs are logarithmically long.","The exact formula applies from the seed onward, not merely asymptotically, so it can be compared with finite-time simulation or empirical data at any network size."],"supporting_citations":[{"why":"It supplies the undirected corded node-duplication shortest-path master equation that the paper adapts to directed links.","marker":"[56]"},{"why":"It establishes the in-degree and out-degree distributions and the logarithmic upstream and downstream counts of the corded DND model, which yield the vanishing connected fraction.","marker":"[66]"},{"why":"It introduces the undirected corded node duplication model and its sparse and dense regimes, serving as the baseline against which the directed version is compared.","marker":"[50, 51]"},{"why":"It develops the analytical shortest-path-length method on classical random graphs that motivates the master-equation approach.","marker":"[34]"},{"why":"It fixes the power-law degree exponent and the p-regimes of the uncorded node duplication model, which the directed model inherits as context.","marker":"[46]"},{"why":"It characterizes the random recursive tree formed by the deterministic mother-daughter links, whose logarithmic depth is the source of the Poisson terms.","marker":"[67, 68, 69]"}],"fun_headline_variants":["Exact shortest-path distribution for directed node-duplication networks","Duplication-grown directed networks: exact distances, vanishing connectivity","Corded DND model yields closed-form path lengths, but not small-world","One formula for shortest paths in duplication networks, with a catch"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption is that the effective probability $\\eta$ that a duplicated link preserves a shortest-path length is constant and equal to its steady-state value throughout the growth, even though the true $\\eta(t)$ converges to that value only as a power law, and very slowly when $p$ is close to 1.","fun_headline_variants_meta":{"raw":{"variants":["Exact shortest-path distribution for directed node-duplication networks","Duplication-grown directed networks: exact distances, vanishing connectivity","Corded DND model yields closed-form path lengths, but not small-world","One formula for shortest paths in duplication networks, with a catch"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000248,"raw_usage":{"total_tokens":1688,"prompt_tokens":1227,"completion_tokens":461,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":843,"completion_tokens_details":{"reasoning_tokens":388}},"tokens_in":843,"tokens_out":461,"duration_ms":5243,"temperature":1.0,"reasoning_tokens":388,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:52:56.814033+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate the corded DND model with $p=0.9$ from a two-node seed and measure the full distribution $P_t(L=\\ell)$ at network sizes $10^2$, $10^3$, and $10^4$; if the closed form with steady-state $\\eta$ visibly misses the early-time distribution while the exact Appendix C equations with a time-dependent effective probability match it, the constant-$\\eta$ solution is only asymptotic rather than the finite-time law claimed.","supporting_citations":[{"cited_title":"Steinbock, O","cited_arxiv_id":null,"evidence_quote":"It supplies the undirected corded node-duplication shortest-path master equation that the paper adapts to directed links."},{"cited_title":"Steinbock, O","cited_arxiv_id":null,"evidence_quote":"It establishes the in-degree and out-degree distributions and the logarithmic upstream and downstream counts of the corded DND model, which yield the vanishing connected fraction."},{"cited_title":"Katzav, M","cited_arxiv_id":null,"evidence_quote":"It develops the analytical shortest-path-length method on classical random graphs that motivates the master-equation approach."},{"cited_title":"Ispolatov, P.L","cited_arxiv_id":null,"evidence_quote":"It fixes the power-law degree exponent and the p-regimes of the uncorded node duplication model, which the directed model inherits as context."}],"review_version":1}