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arxiv: 2604.26215 · v1 · submitted 2026-04-29 · ❄️ cond-mat.stat-mech

Complex first-passage transport in ring networks with long-range jumps and stochastic resetting

Oscar Ivan Torres Mena , Francisco J Sevilla This is my paper

Pith reviewed 2026-05-07 12:50 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords ring networksrandom walksmean first-passage timeslong-range jumpsstochastic resettingcommensurabilitynon-monotonic transportspectral methods
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The pith

Ring networks with fixed-length shortcuts show non-monotonic mean first-passage times due to commensurability effects.

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

This paper studies discrete-time random walks on a ring where nodes connect to nearest neighbors and to all nodes a fixed distance r away. Using the eigenvalues and eigenvectors of the transition matrix, the authors obtain exact expressions for the probability of being at any node at any time and for the average time to first reach a target node. They discover that the average travel time between two far-apart nodes does not decrease steadily as r increases. Instead, it rises and falls in a repeating pattern that gets finer as r grows, with the peaks and valleys occurring when r shares common factors with the total number of nodes. Stochastic resetting, which occasionally sends the walker back to the start, makes these oscillations larger and creates uneven long-term occupation probabilities across the ring.

Core claim

The mean first-passage time between distant nodes in the ring develops a highly non-monotonic dependence on the shortcut length r. Beyond a threshold, the MFPT landscape shows a hierarchy of maxima and minima in a self-similar pattern tied to commensurability relations between r and the ring size. This organization produces alternating regimes of enhanced and suppressed transport efficiency.

What carries the argument

Spectral properties of the transition matrix, which permit closed-form expressions for occupation probabilities and mean first-passage times in the presence of deterministic long-range jumps of length r.

If this is right

  • The MFPT between distant nodes varies non-monotonically with shortcut length r.
  • A self-similar hierarchy of MFPT maxima and minima appears due to commensurability between r and ring size.
  • Transport efficiency is strongly enhanced or suppressed at specific scaling regimes of r.
  • Stochastic resetting amplifies the oscillatory MFPT structure and induces nonuniform stationary distributions.
  • Analysis of mean squared displacement shows how the long-range jumps shape diffusive behavior.

Where Pith is reading between the lines

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

  • Network designers could choose shortcut lengths to avoid the MFPT maxima and thereby improve overall transport speed.
  • The self-similar pattern may appear in other periodic lattices or modular networks that include uniform long-range links.
  • The exact positions of the extrema could be predicted from number-theoretic properties of r and N without running full simulations.
  • Analogous non-monotonic search times might arise in biological transport networks or communication graphs with periodic organization.

Load-bearing premise

The long-range connections are all of exactly the same length r and connect every node to its counterparts at distance r, preserving enough symmetry for the transition matrix to be diagonalized explicitly.

What would settle it

Numerical computation of MFPT for increasing values of r on a ring of size N=100 would show monotonic decrease without the predicted oscillations and self-similar maxima if the central claim is false.

Figures

Figures reproduced from arXiv: 2604.26215 by Francisco J Sevilla, Oscar Ivan Torres Mena.

Figure 1
Figure 1. Figure 1: FIG. 1. Three examples of the networks used in this paper, view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Top panel.- Logarithm of MFPT, view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. However, shortcuts stops “accelerating” the first view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Top panel.- The scaling dependence of view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The MSD time dependence view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The stationary probability distribution view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Top panel.- The MFPT view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Top panel.- The self-similarity of the MFPT is shown view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9 view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Monotonic branches of view at source ↗
read the original abstract

The transport properties of discrete-time random walks on ring networks with deterministic shortcuts are investigated through analytical and numerical methods. The network consists of a periodic chain where each node is connected to its nearest neighbors and to nodes located at a fixed distance $r$. Using the spectral properties of the transition matrix, we derive explicit expressions for the occupation probabilities and mean first-passage times (MFPTs). Contrary to the common expectation that shortcuts monotonically enhance transport, we find that the MFPT between distant nodes develops a highly non-monotonic dependence on the shortcut length. Beyond a threshold value, the MFPT landscape exhibits a hierarchy of maxima and minima organized in a self-similar pattern associated with commensurability relations between the shortcut length and the system size. The scaling behavior of these extrema reveals regimes where transport efficiency is either strongly enhanced or suppressed. We further analyze the mean squared displacement and the influence of stochastic resetting, showing that resetting amplifies the oscillatory MFPT structure and induces strongly nonuniform stationary distributions. These results demonstrate that the spatial organization of long-range connections plays a crucial role in determining transport efficiency in 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 investigates discrete-time random walks on a one-dimensional ring with nearest-neighbor connections plus deterministic long-range jumps of fixed length r. Exploiting the circulant symmetry of the transition matrix, the authors obtain closed-form spectral expressions for the occupation probabilities and mean first-passage times (MFPTs). The principal claim is that the MFPT between distant nodes is a highly non-monotonic function of r, displaying a self-similar hierarchy of maxima and minima governed by commensurability conditions between r and the ring size N. The analysis is extended to the mean-squared displacement and to the case with stochastic resetting, where resetting is shown to sharpen the oscillatory MFPT structure and to generate strongly nonuniform stationary distributions.

Significance. If the central derivations hold, the work supplies a clean, analytically tractable counter-example to the widespread intuition that long-range shortcuts monotonically improve transport. The explicit dependence on commensurability relations offers a precise mechanism for the observed non-monotonicity and could inform the design of efficient transport or search processes on networks. The treatment of resetting further links the results to the growing literature on resetting random walks. The combination of exact spectral formulas and numerical illustrations constitutes a useful contribution to statistical mechanics of first-passage processes on structured graphs.

major comments (2)
  1. §3, Eq. (7): the MFPT formula is obtained from the standard spectral sum for reversible Markov chains; however, when r is commensurate with N the eigenvalues become degenerate and the paper must explicitly confirm that the multiplicity is correctly accounted for in the sum (otherwise the claimed self-similar extrema could be artifacts of an incomplete mode sum).
  2. §4.2, Fig. 4: the non-monotonic MFPT landscape is demonstrated for a single system size (N = 200); to substantiate the claim of a 'hierarchy of maxima and minima organized in a self-similar pattern' across scales, the authors should provide a scaling collapse or an explicit relation for the locations of the extrema as a function of the reduced variable r/N.
minor comments (3)
  1. The transition probabilities are normalized by degree 4, but the text occasionally refers to 'nearest neighbors and nodes at distance r' without stating whether the long-range links are bidirectional; a single clarifying sentence in §2 would remove ambiguity.
  2. Figure 2 caption omits the precise values of N and the target node pair used for the MFPT surface; adding these parameters would improve reproducibility.
  3. A few typographical inconsistencies appear in the notation for the Fourier modes (m versus k); uniform use of one index throughout the spectral sections would aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive and insightful report. We are pleased that the significance of the work is recognized. We address the two major comments below and will make the corresponding revisions to the manuscript.

read point-by-point responses

  1. Referee: §3, Eq. (7): the MFPT formula is obtained from the standard spectral sum for reversible Markov chains; however, when r is commensurate with N the eigenvalues become degenerate and the paper must explicitly confirm that the multiplicity is correctly accounted for in the sum (otherwise the claimed self-similar extrema could be artifacts of an incomplete mode sum).

    Authors: We thank the referee for highlighting this potential issue with eigenvalue degeneracy. In our analysis the transition matrix is circulant, so its eigenvalues and eigenvectors are known explicitly and the MFPT expression in Eq. (7) is the standard spectral decomposition for reversible Markov chains, summing over all N-1 non-stationary modes. When r is commensurate with N some eigenvalues may coincide, but because the sum enumerates every mode k individually the algebraic multiplicity is automatically included. To address the concern explicitly we will add a clarifying paragraph in Section 3 stating that the formula accounts for multiplicities through the complete enumeration over the Fourier modes, and we will include a numerical check for a representative commensurate case (N=200, r=100) confirming that the spectral result matches the direct solution of the linear system for the mean first-passage times. This verification shows that the reported self-similar structure is genuine. revision: yes

  2. Referee: §4.2, Fig. 4: the non-monotonic MFPT landscape is demonstrated for a single system size (N = 200); to substantiate the claim of a 'hierarchy of maxima and minima organized in a self-similar pattern' across scales, the authors should provide a scaling collapse or an explicit relation for the locations of the extrema as a function of the reduced variable r/N.

    Authors: We agree that an explicit demonstration of the scaling behavior would strengthen the presentation of the self-similar hierarchy. The locations of the extrema are governed by commensurability conditions between r and N, which become apparent when the MFPT is plotted against the reduced variable r/N. To substantiate the claim we will add a new figure (or panel) in Section 4.2 displaying the MFPT versus r/N for several system sizes (N=100, 200, 400). This will exhibit the collapse onto a common pattern with the hierarchy of maxima and minima persisting at rational values of r/N. We will also include a short analytical remark identifying the primary extrema at r/N = 1/2 and the locations of successive sub-extrema at fractions with small denominators. These additions will be incorporated in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The central derivations use the standard spectral decomposition of the circulant transition matrix P for the vertex-transitive ring with fixed-distance shortcuts. Eigenvalues take the explicit form λ_m = (1/2)[cos(2π m/N) + cos(2π m r/N)] (normalized by degree), which is a direct algebraic consequence of the circulant structure for any integer r. The MFPT expressions follow from the well-known spectral formula for hitting times on reversible chains, yielding a closed sum over modes whose r-dependence produces the reported non-monotonicity and commensurability patterns as an exact mathematical outcome. No parameters are fitted to data, no self-citations are invoked as load-bearing uniqueness theorems, and no ansatz or renaming reduces the target results to the inputs by construction. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The analysis rests on standard linear algebra for transition matrices of Markov chains on graphs and the definition of the ring network with fixed r; no new entities are introduced and the only free parameter is the model choice of r itself.

free parameters (1)
  • shortcut distance r
    The fixed jump length r is a model parameter varied to reveal the non-monotonic and self-similar MFPT behavior.
axioms (1)
  • standard math Spectral properties of the transition matrix yield explicit expressions for occupation probabilities and mean first-passage times
    Invoked directly in the analytical derivation described in the abstract for the discrete-time random walk.

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