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arxiv: 2605.22646 · v1 · pith:K7UHK2Z5new · submitted 2026-05-21 · ❄️ cond-mat.dis-nn · cond-mat.stat-mech· physics.soc-ph

Directed extended-range percolation

Wenbo Liu , Yiwen Zeng , Xueming Liu , Ginestra Bianconi This is my paper

Pith reviewed 2026-05-22 03:55 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn cond-mat.stat-mechphysics.soc-ph
keywords directed percolationextended-range percolationcomplex networksmessage passingcritical phenomenadegree correlationstree-like networksquantum communication networks
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The pith

Directionality in networks with paths of length at least two simplifies percolation, allowing exact thresholds and critical indices on tree-like structures.

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

The paper introduces directed extended-range percolation on networks with non-reciprocal edges. It establishes that when links form only along directed paths of length R at least 2, the absence of backtracking reduces combinatorial complexity compared to undirected percolation. This reduction permits closed message-passing equations that yield the percolation threshold and anomalous exponents exactly when the network is locally tree-like. The results also show that degree correlations alter the critical behavior in a specific, predictable manner. These findings matter for applications such as quantum communication networks where directionality is inherent and long-range trusted connections determine overall connectivity.

Core claim

In directed networks, connectivity ensured by directed paths of length at most R greater than or equal to 2 lets message-passing close exactly on locally tree-like graphs, so the percolation threshold and the anomalous critical indices follow directly from the equations without further approximation, and the location of the transition depends sensitively on the degree correlations present in the network.

What carries the argument

Message-passing equations for Directed Extended-Range Percolation (DERP), which track the probability of directed-path connectivity while forbidding backtracking.

If this is right

  • The percolation threshold becomes an explicit function of the directed degree distribution and the range R.
  • Critical exponents deviate from the standard mean-field values and can be calculated in closed form.
  • Degree correlations shift the threshold and the nature of the transition in a manner captured by the same equations.
  • Monte Carlo simulations on finite random directed graphs match the analytic predictions for both threshold and exponents.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same simplification may apply to other spreading or search processes on directed graphs whenever backtracking is impossible.
  • Real directed networks with moderate clustering could be approximated by adding small corrections to the tree-like formulas.
  • The framework suggests a way to design directed topologies that achieve desired connectivity with lower resource cost.

Load-bearing premise

The networks contain no short loops that would create additional paths outside the tree-like structure.

What would settle it

A numerical measurement of the percolation threshold on a small directed network that contains many triangles or short cycles, compared against the value predicted by the message-passing equations.

Figures

Figures reproduced from arXiv: 2605.22646 by Ginestra Bianconi, Wenbo Liu, Xueming Liu, Yiwen Zeng.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
read the original abstract

While for standard percolation directionality is known to increase the combinatorial complexity of percolation, here we show that when connectivity is ensured by paths of length $R\geq 2$, network directionality, impeding backtracking, can significantly reduce the complexity of percolation. To illustrate this finding, we introduce Directed Extended-Range Percolation (DERP), defined directed networks with non-reciprocal edges, motivated by applications in quantum communication. In this framework, message transmission is enabled between trusted nodes separated by a directed path of length at most $R$. Using a message-passing approach, we show that directionality enables an exact determination of the percolation threshold and the anomalous critical indices on locally tree-like structures. On random directed networks we find that the critical behavior of DERP depends sensitively on degree correlations. These analytical predictions are corroborated by extensive Monte Carlo simulations, highlighting the profound impact of directionality and correlations on long-range connectivity in complex networks.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript introduces Directed Extended-Range Percolation (DERP) on directed networks with non-reciprocal edges. Connectivity is defined by the existence of a directed path of length at most R ≥ 2. On locally tree-like random directed networks, a message-passing formalism is used to obtain exact expressions for the percolation threshold and the anomalous critical exponents; these quantities depend sensitively on degree correlations. The analytical predictions are corroborated by Monte Carlo simulations on finite random directed graphs.

Significance. If the central results hold, the work is significant because it demonstrates that directionality can reduce the combinatorial complexity of extended-range percolation by eliminating backtracking, thereby enabling closed-form solutions for thresholds and exponents where standard percolation is typically intractable. The explicit dependence on degree correlations and the direct simulation corroboration strengthen the contribution, with potential relevance to quantum communication networks.

major comments (2)
  1. [§3.1] §3.1, message-passing equations: the claim that the equations close exactly for R ≥ 2 due to the absence of backtracking is central to the exact solvability result; however, the explicit handling of joint in/out-degree correlations in the generating functions is only sketched, and a full derivation showing that no additional cycle corrections arise would strengthen the load-bearing step.
  2. [§4.2] §4.2, critical exponents: the anomalous exponents are obtained by linearizing the message-passing recursion near threshold; the paper should verify that the leading eigenvalue analysis remains valid when the degree distribution has finite variance but non-zero correlations, as this directly supports the reported dependence on correlations.
minor comments (3)
  1. [Simulations section] The abstract states that simulations 'corroborate' the predictions, but the main text does not report the number of network realizations, system sizes, or fitting procedure used to extract exponents; adding these details would improve reproducibility.
  2. [Eq. (7)] Notation for the cavity probabilities in Eq. (7) uses subscripts that are easily confused with degree indices; a short table of symbols would aid clarity.
  3. [Introduction] The motivation linking DERP to quantum communication is mentioned only briefly; expanding this connection by one sentence in the introduction would help contextualize the model for readers outside network science.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful review and recommendation for minor revision. We appreciate the positive assessment of the significance of our work on Directed Extended-Range Percolation. Below, we address each major comment in detail.

read point-by-point responses

  1. Referee: [§3.1] §3.1, message-passing equations: the claim that the equations close exactly for R ≥ 2 due to the absence of backtracking is central to the exact solvability result; however, the explicit handling of joint in/out-degree correlations in the generating functions is only sketched, and a full derivation showing that no additional cycle corrections arise would strengthen the load-bearing step.

    Authors: We agree that providing a more explicit derivation would enhance the clarity of our central result. In the revised manuscript, we will expand Section 3.1 and add a detailed appendix deriving the message-passing equations from first principles. This will explicitly incorporate the joint in/out-degree correlations through the appropriate generating functions and demonstrate that, due to the directed nature and absence of backtracking for R ≥ 2, no additional cycle corrections are needed on locally tree-like networks. revision: yes

  2. Referee: [§4.2] §4.2, critical exponents: the anomalous exponents are obtained by linearizing the message-passing recursion near threshold; the paper should verify that the leading eigenvalue analysis remains valid when the degree distribution has finite variance but non-zero correlations, as this directly supports the reported dependence on correlations.

    Authors: We thank the referee for this suggestion. The linearization procedure and the leading eigenvalue analysis are valid under the stated conditions because the message-passing equations are formulated in terms of the joint degree distributions, which account for correlations explicitly. To address this, we will include in the revised version a brief verification, perhaps through an additional numerical check or analytical argument, confirming that the eigenvalue analysis holds for finite-variance distributions with non-zero correlations, thereby supporting the correlation-dependent critical behavior. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation

full rationale

The central derivation relies on standard message-passing equations applied to locally tree-like directed networks where directionality for paths of length R >= 2 eliminates backtracking and allows exact closure without cycle corrections. Thresholds and anomalous exponents are obtained by solving these equations, with explicit accounting for degree correlations; the results are not defined in terms of themselves nor obtained by fitting then relabeling as prediction. Monte Carlo simulations on finite graphs serve as independent external corroboration rather than circular support. No load-bearing self-citation reduces the argument to an unverified prior claim by the same authors, and the approach remains self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Main structural assumption is local tree-likeness enabling closure of message-passing equations; no free parameters or new entities are named in the abstract.

axioms (1)
  • domain assumption Networks are locally tree-like
    Required for the message-passing approach to deliver exact percolation thresholds and indices without cycle corrections.

pith-pipeline@v0.9.0 · 5695 in / 1134 out tokens · 35476 ms · 2026-05-22T03:55:17.345197+00:00 · methodology

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Reference graph

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