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arxiv: 2604.20770 · v1 · submitted 2026-04-22 · ❄️ cond-mat.stat-mech

Encounter times of random walkers with simultaneous resetting on networks

Cristian D. Suarez-Jimenez , Alejandro P. Riascos , Denis Boyer This is my paper

Pith reviewed 2026-05-09 23:11 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords random walksnetworkssimultaneous resettingfirst-encounter timespectral decompositionMarkov chainscollective search
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The pith

Simultaneous resetting lets multiple random walkers on a network meet faster, with exact formulas from the transition-matrix spectrum.

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

The paper derives closed-form expressions for the mean first-encounter time of several random walkers that all reset together to their starting nodes. These formulas are written directly in terms of the eigenvalues and eigenvectors of the ordinary transition matrix that governs each walker when no reset occurs. A general criterion on finite networks identifies the resetting probabilities that shorten the average meeting time and produce an optimum. The expressions are tested on both regular lattices and heterogeneous graphs for two or more walkers and for local or nonlocal moves.

Core claim

For the collective Markov process of multiple random walkers undergoing simultaneous resetting on a finite network, the mean first-encounter time admits an exact expression built from the eigenvalues and eigenvectors of the transition matrices of the reset-free dynamics. A general criterion on finite networks determines the resetting probabilities that reduce this mean time relative to the no-reset case and yield an optimal value.

What carries the argument

Spectral decomposition of the single-walker transition matrix, which supplies the closed-form mean first-encounter time once simultaneous resetting is imposed on the whole collection.

If this is right

  • The mean first-encounter time can be evaluated from the spectrum without running trajectories.
  • Nonzero resetting shortens encounter times only for targets that are close to the walkers' initial positions.
  • Simultaneous resetting outperforms independent resetting on regular networks.
  • On heterogeneous networks the advantage reverses and independent resetting can be faster.

Where Pith is reading between the lines

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

  • The same spectral criterion could be applied to design collective search protocols on transportation or communication graphs.
  • A hybrid scheme that sometimes resets walkers together and sometimes independently might combine the benefits observed in the two cases.
  • The formulas remain valid for any diagonalizable transition matrix, including those arising from directed or weighted networks.

Load-bearing premise

The underlying process without resetting must be a Markov chain on a finite network whose transition matrix is diagonalizable.

What would settle it

Simulate two walkers on a four-node cycle with resetting probability 0.2, compute the empirical average time until they occupy the same node, and check whether it equals the value obtained from the eigenvalue sum.

Figures

Figures reproduced from arXiv: 2604.20770 by Alejandro P. Riascos, Cristian D. Suarez-Jimenez, Denis Boyer.

Figure 1
Figure 1. Figure 1: FIG. 1. Two walkers move on a street network whose nodes [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Mean first-encounter times of two normal random walke [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Mean first-encounter times [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Mean first-encounter times of three (a,b) and four (c, [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Encounter times for two random walkers with simultan [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Comparison of the MFETs for two normal random walkers [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. ∆ [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
read the original abstract

In this work, we study the dynamics of multiple random walkers on networks subject to a simultaneous resetting protocol, whereby all walkers are synchronously returned to their respective initial nodes. For this collective Markovian process, we derive exact analytical expressions for the mean first-encounter time, defined as the average time required for all walkers to meet for the first time at a given node. These results are formulated in terms of the eigenvalues and eigenvectors of the transition matrices governing the dynamics without resetting, providing a clear spectral interpretation of the impact of resetting on encounter processes. We further establish a general criterion for finite networks that determines when the introduction of a nonzero resetting probability reduces the mean first-encounter time and leads to an optimal resetting strategy. The theoretical predictions are illustrated through numerical results on regular and heterogeneous networks, for encounters involving two or more walkers, and for combinations of local and nonlocal dynamics. Our findings demonstrate that simultaneous resetting can significantly reduce encounter times for specific target nodes and initial conditions, while becoming ineffective for highly exploratory dynamics or distant targets. A comparison with independent resetting shows that simultaneous resetting is more efficient in homogeneous networks, whereas independent resetting can outperform it in heterogeneous structures, thereby revealing a trade-off between synchronization and exploration. The framework provides a unified approach to collective search and encounter problems on networks with resetting.

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

0 major / 3 minor

Summary. The manuscript examines multiple random walkers on finite networks undergoing simultaneous resetting to their initial positions. It derives exact closed-form expressions for the mean first-encounter time (MFET) to a target node in terms of the eigenvalues and eigenvectors of the individual walkers' transition matrices (without resetting), obtained via spectral decomposition of the joint Markov chain. A general criterion is given for when a nonzero resetting probability r reduces the MFET relative to the r=0 case and yields an optimal r. The results are illustrated numerically for two or more walkers on regular and heterogeneous networks, comparing simultaneous versus independent resetting and local versus nonlocal dynamics.

Significance. The spectral approach yields parameter-free analytical predictions for MFET under simultaneous resetting and identifies network-dependent trade-offs with independent resetting. The exact expressions and the optimality criterion constitute a unified framework for collective search problems; the numerical verification on both regular and heterogeneous graphs strengthens the claims. These tools are likely to be useful for analyzing resetting-enhanced encounter processes on networks.

minor comments (3)
  1. [§2] §2 (or wherever the joint transition matrix is introduced): the tensor-product construction of the unperturbed joint chain and the rank-one update for simultaneous resetting should be written explicitly, including the precise form of the resetting operator, to make the subsequent resolvent derivation fully self-contained.
  2. [§4] The optimality criterion (presumably in §4) is stated for finite networks; a short remark on whether the same spectral argument extends immediately to infinite or continuous-state limits would clarify the scope.
  3. [Numerical results section] Figure captions for the numerical panels should state the number of independent realizations used to estimate the MFET and whether the plotted points are averages or single-run values.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and accurate summary of our manuscript, as well as for recognizing the significance of the spectral approach and the optimality criterion for resetting. The recommendation for minor revision is noted. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation relies on standard spectral decomposition of the joint Markov chain (tensor product of individual transition matrices) for the unperturbed dynamics, followed by a rank-one update for simultaneous resetting to obtain the mean first-encounter time via the fundamental matrix or resolvent in the same eigenbasis. These steps are self-contained applications of linear algebra to finite-state Markov chains with the stated diagonalizability assumption; no parameter is fitted to data and then renamed as a prediction, no self-citation supplies a load-bearing uniqueness theorem or ansatz, and the optimality criterion follows directly from comparing the resulting closed-form expressions. The numerical illustrations on example networks serve only as verification, not as inputs to the analytic results.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the Markov property of the reset process and the spectral properties of the no-reset transition matrix on finite networks; no new entities are introduced.

free parameters (1)
  • resetting probability r
    Tunable parameter whose optimal value is identified by the derived criterion; not fitted post hoc but varied to find minima.
axioms (2)
  • domain assumption Dynamics without resetting obey a time-homogeneous Markov chain on a finite network
    Invoked to justify the use of eigenvalues and eigenvectors for the encounter-time expressions.
  • domain assumption The transition matrix is diagonalizable
    Required for the closed-form spectral representation stated in the abstract.

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Reference graph

Works this paper leans on

85 extracted references · 85 canonical work pages

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    Alternatively, the dy- namics under resetting can be described by the transition probability matrix ˆΠ (⃗ r;γ), defined as ˆΠ (⃗ r;γ) ≡ (1 − γ) ˆW +γ ˆΘ (⃗ r)

    accounts for the collective hops governed by the transition probabilities ˆW, while the second term rep- resents the resetting process to ⃗ r. Alternatively, the dy- namics under resetting can be described by the transition probability matrix ˆΠ (⃗ r;γ), defined as ˆΠ (⃗ r;γ) ≡ (1 − γ) ˆW +γ ˆΘ (⃗ r). (4) Here, ˆΘ (⃗ r) describes resetting to the nodes ⃗ r...

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    can be written in the more compact form P(⃗i, ⃗j,γ ;t + 1) = ∑ ⃗l∈V S P(⃗i, ⃗l,γ ;t) ˆΠ (⃗ r;γ)⃗l→ ⃗j . (6) B. Mean first encounter times The matrix ˆΠ (⃗ r;γ) fully characterizes the collective dynamics with resetting, ensuring that all walkers can visit every node of the network as long as the resetting probability satisfies 0 ≤ γ < 1. The matrix Π (⃗ r;γ...

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    has a direct analogy with the dynamics of a single walker [ 62]. This similarity al- lows for an analytical treatment of the problem of S non- interacting random walkers, particularly for calculating the mean number of steps required to first reach a given configuration. To this end, we begin by introducing a compact notation for the eigenvalues and eigenve...

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    we have, P(⃗i, ⃗j,γ ;t) = ⟨⃗i|ˆΠ (⃗ r;γ)t|⃗j⟩ = ∑ ⃗l∈V S ⟨⃗i|ˆΠ (⃗ r;γ)t|ψ ⃗l⟩⟨ ¯ψ ⃗l|⃗j⟩ = ∑ ⃗l∈V S ζt ⃗l ⟨⃗i ⏐ ⏐ψ ⃗l ⟩ ⟨ ¯ψ ⃗l ⏐ ⏐⃗j⟩. (13) 4 The stationary distribution is then given by P ∞ ⃗j (⃗i,γ ) ≡ lim T →∞ 1 T T∑ t=0 P(⃗i, ⃗j,γ ;t) = ∑ ⃗l∈V S δζ⃗l, 1 ⟨⃗i ⏐ ⏐ψ ⃗l ⟩ ⟨ ¯ψ ⃗l ⏐ ⏐⃗j⟩, (14) where only the largest eigenvalue ζ = 1 contributes in the lon...

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    We assume that each individual transition matrix W(s) is diagonalizable

    in a more explicit way, involving the resetting probabilityγ. We assume that each individual transition matrix W(s) is diagonalizable. Under this assumption, we introduce the lth eigenvector of the walker s W(s)|φ (s) l ⟩ =λ (s) l |φ (s) l ⟩, s = 1, 2,...,S, (19) where {λ (s) l }N l=1 denote the eigenvalues of the transition matrix W(s), with the correspo...

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    ( 1) for synchronous random walkers, the eigenvalues of ˆW are obtained as λ⃗l ≡ S∏ s=1 λ (s) ls

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    (36) Using the definition of z(⃗i, ⃗j,γ ) introduced in Eq

    becomes z2(⃗i, ⃗j, 0)> 1 + 1 ⟨T (⃗i, ⃗j, 0)⟩ , ⃗i ̸=⃗j. (36) Using the definition of z(⃗i, ⃗j,γ ) introduced in Eq. ( 33), let us define Λ ≡ ⟨T 2(⃗i, ⃗j, 0)⟩ − ⟨T (⃗i, ⃗j, 0)⟩ 2 ⟨T (⃗i, ⃗j, 0)⟩ 2 − 1 ⟨T (⃗i, ⃗j, 0)⟩ − 1. (37) As a consequence of Eq. ( 36), the value taken by Λ in Eq. ( 37) for a given system determines whether the in- troduction of resettin...

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