Analytical results for the distribution of shortest path lengths in directed random networks that grow by node duplication

Chanania Steinbock; Eytan Katzav; Ofer Biham

REVIEW 3 major objections 5 minor 69 references

Analytical results for the distribution of shortest path lengths in directed random networks that grow by node duplication

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Directed duplication networks have an exact shortest-path law: distances among connected pairs grow logarithmically while almost all pairs are disconnected.

desk verdict 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. read the letter →

arxiv 1908.07376 v1 pith:NNB7IFSR submitted 2019-08-16 physics.soc-ph cond-mat.dis-nncond-mat.stat-mech

Chanania Steinbock , Ofer Biham , Eytan Katzav This is my paper · ORCID

classification physics.soc-phcond-mat.dis-nncond-mat.stat-mech

keywords shortestpathlengthsdirectednetworksnodeduplicationmasterequationmeandistancesmall-worldcitationgeneregulatory

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper 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.

What carries the argument

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 $\ln t_s$.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the 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.

The 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.

The 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.

Referee Report

3 major / 5 minor

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.

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 (3)

[§5, Eq. (42) and Appendix B] 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.
[§5, Eqs. (31)–(32) and Appendix C] 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.
[§3, Eq. (13) and §6, Fig. 7] 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.

minor comments (5)

[Abstract and Summary] 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.
[Appendix B heading] The word 'congergence' in the appendix heading should be corrected to 'convergence'.
[Fig. 7 and Fig. 8 captions] These captions do not state the number of simulated network realizations, unlike the caption of Fig. 6; please include this information for reproducibility.
[Eq. (42)] 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.
[Appendix A] The phrase 'betweeness centrality' should be corrected to 'betweenness centrality'.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the closed-form DSPL is a genuine solution of the paper's own master equation, with eta computed from the model's degeneracy statistics rather than fitted to DSPL data.

full rationale

The derivation chain is self-contained. Eq. (41) is a master equation for Pt(L = ell); Eq. (42) is its explicit solution, and Eqs. (47)-(48) follow by summing that solution. The parameter eta is not fitted to shortest-path data: it is defined in Eq. (18) from the degeneracy distribution P(G = g), which is solved separately in Sec. 3 (Eqs. (8)-(13)) and compared with simulations in Fig. 5. The replacements of p by eta in Eqs. (31)-(32) are acknowledged approximations, with the exact version supplied in Appendix C; this affects accuracy at finite times but does not make the result circular. Self-citations to Refs. [56] and [66] provide model context and earlier degree-distribution results, but the DSPL calculation does not depend on any unverified claim imported from those papers. The only real limitation is that eta is the time-independent steady-state value, so near p = 1 the finite-time accuracy is questionable (Appendix B, Eqs. (B.5)-(B.7)); that is a correctness and regime concern, not a circularity.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim depends on a few model inputs: the duplication probability p, the seed network, and the truncation level used to compute the effective parameter η. No new physical entities are introduced.

free parameters (3)

p (duplication probability) = input in (0,1)
Defines the model; all results are functions of p; not fitted to data.
g_max (degeneracy truncation) = 3
The transition matrix for the degeneracy distribution is truncated at g=3 to compute η; convergence between g=2 and g=3 is shown but no bound on higher g.
Seed network size and DSPL = s=2 linear chain used in figures
Initial condition; results depend on P0(L=ℓ) and s; the paper treats general seeds in the formula but uses a specific seed in simulations.

assumptions (4)

domain assumption The degeneracy distribution reaches steady state and can be replaced by its asymptotic value η in the DSPL master equation.
Appendix B shows power-law convergence, but the main solution uses steady-state η from time zero; the transient is neglected.
ad hoc to paper Truncation of the degeneracy transition matrix at g=3 captures the relevant degeneracy statistics for computing η.
Only g=1,2,3 are included; the authors verify closeness of g=2 and g=3 results but provide no formal error estimate.
ad hoc to paper The replacement of p by η in the master equations for ℓ=1 and ℓ=2 is a good approximation.
Sec. 5 justifies it by Pt(L=1) ~ 1/Nt and the small difference p−η; Appendix C gives the exact form but the main text solution uses η.
domain assumption Seed networks are restricted to acyclic oriented graphs with a single sink node.
Sec. 2 imposes this to ensure a single connected component and no bidirectional edges; results for general seeds are not derived.

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Pith. "Pith review of Analytical results for the distribution of shortest path lengths in directed random networks that grow by node duplication." pith.science (2026). https://pith.science/paper/NNB7IFSR

@misc{pith2026190807376,
  author       = {Pith},
  title        = {Pith review of: Analytical results for the distribution of shortest path lengths in directed random networks that grow by node duplication},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NNB7IFSR}},
  note         = {Machine review of arXiv:1908.07376}
}

abstract

We present exact analytical results for the distribution of shortest path lengths (DSPL) in a directed network model that grows by node duplication. Such models are useful in the study of the structure and growth dynamics of gene regulatory networks and scientific citation networks. Starting from an initial seed network, at each time step a random node, referred to as a mother node, is selected for duplication. Its daughter node is added to the network and duplicates each outgoing link of the mother node with probability $p$. In addition, the daughter node forms a directed link to the mother node itself. Thus, the model is referred to as the corded directed-node-duplication (DND) model. In this network not all pairs of nodes are connected by directed paths, in spite of the fact that the corresponding undirected network consists of a single connected component. More specifically, in the large network limit only a diminishing fraction of pairs of nodes are connected by directed paths. To calculate the DSPL between those pairs of nodes that are connected by directed paths we derive a master equation for the time evolution of the probability $P_t(L=\ell)$, $\ell=1,2,\dots$, where $\ell$ is the length of the shortest directed path. Solving the master equation, we obtain a closed form expression for $P_t(L=\ell)$. It is found that the DSPL at time $t$ consists of a convolution of the initial DSPL $P_0(L=\ell)$, with a Poisson distribution and a sum of Poisson distributions. The mean distance ${\mathbb E}_t[L|L<\infty]$ between pairs of nodes which are connected by directed paths is found to depend logarithmically on the network size $N_t$. However, since in the large network limit the fraction of pairs of nodes that are connected by directed paths is diminishingly small, the corded DND network is not a small-world network, unlike the corresponding undirected network.

Figures

Figures reproduced from arXiv: 1908.07376 by the authors.

**Figure 1.** Illustration of the corded DND model. A random node, referred to as a mother node, M (gray circle) is selected for duplication. The newly formed daughter node, D (empty circle) deterministically acquires a directed link (solid line) to the mother node. It also acquires, with probability p, a directed link (dashed line) to each one of the outgoing neighbors of M. In this example, D forms a directed link to its grandm… view at source ↗

**Figure 2.** Two instances of corded DND networks of size N = 50, with p = 0.2 (a) and p = 0.5 (b). Both networks were grown from a seed network that consists of two nodes connected by a directed link. For the sake of comparison, both instances are formed around the same backbone tree (solid lines). The sink node, which can be reached from all the other nodes in the network via directed paths is shown by a large circle. The prob… view at source ↗

**Figure 3.** Illustration of possible network structures in the vicinity of a newly formed daughter node D and its mother node M. The grandmother (GM) node, the great-grandmother (GGM) node and two more generations are also marked downstream of M. The deterministic links are represented by straight horizontal arrows while the probabilistic links are represented by arcs. (a) A linear branch of the backbone tree (marked by gM = … view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: The evolution of the local downstream configuration and the degeneracies of shortest paths, from a mother node, M, to its daughter node D upon duplication of M. Top line: a random node, denoted by M, is selected for duplication. The deterministic link from the daughter…

**Figure 5.** Figure 5: The probability η that the shortest path length from the daughter node D to any downstream node T is equal to the shortest path length from its mother node M to T, as a function of p. The analytical results obtained from Eq. (19), which takes into account only single a…

**Figure 7.** Figure 7: Analytical results (solid lines) for the mean shortest path length Et[L|L < ∞], of the corded DND network, as a function of network size Nt, obtained from Eq. (47), for p = 0.2, 0.4, 0.6 and 0.8. The analytical results are in very good agreement with the simulation res…

**Figure 6.** Figure 6: Analytical results (solid lines) for the distribution P C t (L = ℓ) of the corded DND network with (a) p = 0.2; (b) p = 0.4; (c) p = 0.6; and (d) p = 0.8, for network sizes of Nt = 102 , 104 and 106 . The analytical results are found to be in very good agreement with t…

**Figure 8.** Figure 8: Analytical results (solid lines) for the variance Vart(L), of the distribution P C t (L = ℓ) of the corded DND network, as a function of network size Nt, obtained from Eq. (52) for p = 0.2, 0.4, 0.6 and 0.8. The analytical results are in very good agreement with the si…

**Figure 9.** Figure 9: (a) The configuration obtained upon duplication of a node M of configuration gM = 1∗∗, in which neither of the two outgoing links of M was duplicated. The degeneracy of the shortest paths from D to downstream nodes is g = 1. Due to the shortcut from M to GGM, none of t…

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