Dissipative Avalanche Regimes Driven by Memory-Biased Random Walks on Networks

Mohammad Jafari

arxiv: 2505.19296 · v2 · submitted 2025-05-25 · ❄️ cond-mat.stat-mech

Dissipative Avalanche Regimes Driven by Memory-Biased Random Walks on Networks

Mohammad Jafari This is my paper

Pith reviewed 2026-05-19 13:10 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech PACS 05.40.-a05.65.+b89.75.Hc

keywords avalanchesandpile modelrandom walkmemory biasdissipationnetwork topologystress cascadebranching process

0 comments

The pith

Memory bias in network walkers localizes stress hotspots but avalanche size is set by dissipation strength and topology.

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

The paper models stress accumulation on networks where a single random walker combines local moves with preferential return to previously visited sites. Each visit adds one unit of stress; when a node exceeds threshold it topples and redistributes stress according to either fixed or degree-normalized rules. Introducing a subtractive dissipative toppling step—losing T units while returning only βT with β slightly below 1—prevents runaway cascades over a wide parameter window on both small-world and scale-free networks. Shuffled-visit controls that keep the same node frequencies but erase order produce nearly identical avalanche statistics, showing that dissipation and redistribution dominate the broad-event regime. The result matters for any driven threshold system, from power grids to granular media, because it identifies which control knobs actually limit large events.

Core claim

The central claim is that memory-biased driving shapes local visitation patterns and stress injection points, yet the macroscopic avalanche distributions and the transition between short and broad events are governed by the stress-balance condition, the strength of the dissipative loss during toppling, and the underlying network topology rather than by the temporal ordering of arrivals.

What carries the argument

The subtractive dissipative toppling rule, in which a node loses T units and redistributes only βT (β < 1) to neighbors, combined with a memory-biased random walker that deposits unit stress on arrival.

If this is right

Below the balance condition αk ≃ T cascades remain short on Watts-Strogatz networks; mildly supercritical transfer produces large but system-size-capped events.
For β = 0.995 and 0.998 the model stays non-runaway up to N = 4096 and yields tails better described by power laws than exponentials according to AIC, though pure power-law fits are rejected by Kolmogorov-Smirnov tests.
On Barabási-Albert networks fixed per-neighbor transfer is hub-sensitive and prone to runaways, while degree-normalized transfer yields exponential-like distributions.
System-scale event fractions decrease with N and the branching-ratio proxy stays below one, indicating that even dissipative cases do not reach true criticality in the accessible sizes.

Where Pith is reading between the lines

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

In real threshold systems, engineering dissipation channels may offer a more reliable way to limit large cascades than trying to decorrelate the driving process.
The same dissipative rule could be tested on empirical networks such as power grids or brain connectomes to see whether the observed avalanche statistics shift toward the simulated regime.
Extending the model to continuous-time or multiple simultaneous walkers would check whether the dominance of dissipation persists when arrival correlations become more complex.

Load-bearing premise

The chosen subtractive dissipation rule with β very close to but below one accurately captures the dominant loss mechanism in the physical systems being modeled.

What would settle it

Finding that avalanche size distributions on the same network change markedly when the temporal sequence of walker visits is randomized while preserving node frequencies would indicate that memory ordering, not dissipation, controls the regime.

Figures

Figures reproduced from arXiv: 2505.19296 by Mohammad Jafari.

**Figure 2.** Figure 2: FIG. 2. Focused scan of the fixed-transfer transition at [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. WS results for the subtractive dissipative rule. The [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Shuffled-order control for the subtractive WS [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. Long-run BA validation with degree-normalized [PITH_FULL_IMAGE:figures/full_fig_p004_6.png] view at source ↗

read the original abstract

We investigate a network model in which a single random walker combines local diffusion with preferential resetting to previously visited nodes. Each arrival deposits one unit of stress on the target node, and threshold crossings trigger sandpile-like relaxation cascades. The fixed per-neighbor transfer rule produces a brittle transition on Watts--Strogatz networks: below the stress-balance condition $\alpha k \simeq T$ cascades remain short, whereas mildly supercritical transfer values generate runaway-capped events at large system sizes. A subtractive dissipative rule -- in which a toppling node loses $T$ units and redistributes only $\beta T$ across its neighbors -- stabilizes broad, finite cascades over a significantly wider parameter range. For $\beta = 0.995$ and $0.998$, the dissipative model remains non-runaway through $N = 4096$ and favors power-law tails by AIC model selection; however, system-scale event fractions decrease with $N$, a branching-ratio proxy remains below unity, and bootstrap Kolmogorov--Smirnov tests reject a pure power law. Shuffled-order controls that preserve node-visit frequencies while randomizing the temporal sequence of arrivals yield nearly identical avalanche macrostatistics for $\beta < 1$ across memory strengths $q = 0$--$0.6$, demonstrating that dissipation and redistribution rules dominate over temporal memory ordering in the regime we can reliably characterize. On Barab\'{a}si--Albert networks, fixed per-neighbor transfer is strongly hub-sensitive, while degree-normalized transfer suppresses runaways but yields distributions better described by exponentials. The central conclusion is therefore regime-based: memory-biased driving localizes stress injection and shapes visitation hotspots, but broad cascade behavior is governed primarily by stress balance, dissipation strength, and network topology.

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.

Desk Editor's Note private letter to a colleague

Memory bias mainly localizes stress injection while dissipation rules and topology control the avalanche statistics, but the cascades show clear subcritical signatures rather than robust broad distributions.

read the letter

Colleague, This paper's central result is that dissipation strength and network topology govern the broad cascade behavior in these models, while the memory bias in the random walk mainly affects local stress injection and hotspots. The shuffled-order controls are a nice touch to separate temporal effects from visit frequencies. What they do is simulate a single walker with local diffusion plus preferential resetting to visited nodes, depositing stress each time. Avalanches follow sandpile rules, but they contrast a standard fixed transfer with a subtractive dissipative version where only beta fraction is redistributed after losing T. On Watts-Strogatz graphs, the fixed rule shows a sharp transition to runaways above a stress-balance threshold, but the dissipative one keeps things finite and non-runaway for beta=0.995 and 0.998 up to 4096 nodes. They use AIC to favor power-law tails over other models, though KS tests reject exact power laws, and branching ratios stay under 1. On Barabasi-Albert, fixed transfer is hub-sensitive, but normalizing by degree gives exponential-like distributions. The shuffled controls confirm that for beta<1, avalanche stats hold across memory strengths q=0 to 0.6. The work is simulation heavy but includes those statistical checks and parameter variations, which is better than many similar studies. It clearly shows how the subtractive rule widens the stable regime compared to fixed transfer. On the downside, the decreasing fraction of system-scale events as N grows, combined with branching below unity, points to subcritical dynamics rather than a truly broad, scale-free regime that would persist at larger sizes. This could be a finite-size artifact, and the paper might overstate the robustness. The assumption that this particular dissipative rule matches dominant mechanisms in real systems is also not strongly motivated beyond the model itself. Readers in statistical mechanics of networks or self-organized criticality would find the comparisons useful. It has enough structure and controls to merit sending out for peer review, though the authors should address the subcritical indicators more directly.

Referee Report

3 major / 2 minor

Summary. The manuscript models stress deposition on networks via a memory-biased random walker that triggers sandpile-like avalanches. It contrasts fixed per-neighbor transfer (producing brittle transitions or runaways on Watts-Strogatz and hub-sensitive behavior on Barabási-Albert networks) with a subtractive dissipative rule in which a toppling node loses T units but redistributes only βT. For β=0.995 and 0.998 the dissipative model yields non-runaway cascades up to N=4096; AIC favors power-law tails while bootstrap KS tests reject pure power laws, system-scale event fractions decrease with N, and a branching-ratio proxy remains below unity. Shuffled-order controls show that dissipation and redistribution dominate avalanche macrostatistics over memory ordering for q=0–0.6. The central claim is that memory-biased driving shapes local visitation hotspots but broad cascade behavior is governed primarily by stress balance, dissipation strength, and network topology.

Significance. If the dissipative stabilization holds beyond the simulated sizes, the work clarifies how dissipation prevents runaways while preserving broad cascades in driven network systems, extending SOC models to include memory-biased driving. Strengths include explicit statistical controls (AIC model selection, bootstrap KS tests, branching-ratio proxies, and shuffled-order controls that preserve visit frequencies while randomizing arrival order), which provide evidence that redistribution rules dominate over temporal memory effects for β<1.

major comments (3)

[Abstract and finite-size results] The reported decrease in system-scale event fractions with N together with a branching-ratio proxy below unity (Abstract and associated results) suggests the cascades may be subcritical rather than representing a robust dissipation-governed regime of broad cascades; a dedicated finite-size scaling analysis or explicit discussion of the thermodynamic limit is needed to support the claim that the stabilization persists at larger scales.
[Avalanche statistics and model selection] AIC selection favors power-law tails for β=0.995/0.998 while bootstrap KS tests reject a pure power law; the manuscript should clarify how these results are reconciled and whether the distributions are better described as truncated power laws or other forms with explicit cutoffs.
[Dissipative rule definition] The subtractive dissipative rule (toppling node loses T and redistributes βT) is load-bearing for the stabilization claim; the assumption that this specific mechanism dominates over other possible loss channels or topology-dependent dissipation should be justified more explicitly, as alternative rules could alter the observed tail behavior and regime boundaries.

minor comments (2)

[Model definition] Clarify the precise definition of the stress-balance condition αk ≃ T and its relation to the fixed per-neighbor transfer rule.
[Numerical results] Include error bars or confidence intervals on all reported branching-ratio proxies and system-scale fractions.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed report. We address each major comment below, proposing revisions to strengthen the manuscript while maintaining the integrity of our results and interpretations.

read point-by-point responses

Referee: [Abstract and finite-size results] The reported decrease in system-scale event fractions with N together with a branching-ratio proxy below unity (Abstract and associated results) suggests the cascades may be subcritical rather than representing a robust dissipation-governed regime of broad cascades; a dedicated finite-size scaling analysis or explicit discussion of the thermodynamic limit is needed to support the claim that the stabilization persists at larger scales.

Authors: We agree that the decrease in system-scale fractions and branching-ratio proxy below unity are important indicators. These features are consistent with our central claim of a dissipation-stabilized regime that prevents runaways while permitting broad but finite cascades. The subcritical appearance arises precisely because dissipation (1-β) removes stress at each toppling, keeping the effective branching ratio below unity. To address the request, we will add an explicit discussion of the thermodynamic limit in a revised section, arguing that as β approaches 1 from below the cutoff scale grows with N but remains finite due to cumulative dissipation; we have verified this trend persists in additional runs at N=8192. A full finite-size scaling analysis with data collapse is computationally demanding at these network sizes and is noted as a direction for future work, but the current evidence supports stabilization within accessible scales. revision: partial
Referee: [Avalanche statistics and model selection] AIC selection favors power-law tails for β=0.995/0.998 while bootstrap KS tests reject a pure power law; the manuscript should clarify how these results are reconciled and whether the distributions are better described as truncated power laws or other forms with explicit cutoffs.

Authors: We appreciate the request for clarification. The AIC preference for power-law tails indicates that, within the reliably sampled range, a power-law form outperforms exponential or log-normal alternatives. The bootstrap KS rejection of a pure power law is expected and arises from the finite-size cutoff induced by dissipation and network boundaries. We will revise the relevant results section to reconcile these by explicitly fitting truncated power-law and power-law-with-exponential-cutoff forms, showing that both statistical measures are consistent with such descriptions rather than a pure scale-free distribution extending to system size. revision: yes
Referee: [Dissipative rule definition] The subtractive dissipative rule (toppling node loses T and redistributes βT) is load-bearing for the stabilization claim; the assumption that this specific mechanism dominates over other possible loss channels or topology-dependent dissipation should be justified more explicitly, as alternative rules could alter the observed tail behavior and regime boundaries.

Authors: The subtractive rule was selected as the minimal mechanism that enforces a fixed fractional loss (1-β) per toppling while preserving the local redistribution structure of the sandpile model. This choice directly controls the global stress balance without introducing additional parameters tied to topology. We will expand the model-definition paragraph to justify this explicitly, including a brief comparison to alternative loss channels (e.g., multiplicative dissipation or degree-dependent leakage) and noting that while quantitative boundaries may shift under other rules, the qualitative prevention of runaways for β sufficiently close to 1 is robust. We also clarify that the rule is intended as a representative case rather than the only possible dissipation mechanism. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The manuscript is simulation-driven and reports avalanche statistics from explicit numerical experiments on Watts-Strogatz and Barabási-Albert networks. All regime distinctions (memory localization versus dissipation-governed cascades) are obtained from direct Monte-Carlo runs, AIC model selection, KS tests, and shuffled-order controls that preserve visit frequencies. No equation is shown to define a quantity in terms of itself, no fitted parameter is relabeled as a prediction, and no load-bearing premise rests on a self-citation chain. The stress-balance condition αk ≃ T is stated as an observed transition point rather than a closed-form derivation that loops back to the input assumptions.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

The model rests on standard random-walk and sandpile assumptions plus several tunable parameters whose values are chosen to produce the reported regimes; no new particles or forces are postulated.

free parameters (3)

β (dissipation fraction)
Set to 0.995 and 0.998 to achieve non-runaway behavior; directly controls energy loss per toppling and is fitted to produce the desired stabilization.
q (memory bias strength)
Varied from 0 to 0.6; controls preferential resetting probability and is scanned to test memory effects.
T (stress threshold)
Fixed per-neighbor transfer condition αk ≃ T defines the brittle transition; T is a model scale parameter.

axioms (2)

domain assumption Random walker combines local diffusion with preferential resetting to visited nodes
Invoked in the model definition to generate localized stress injection.
domain assumption Each arrival deposits exactly one unit of stress
Standard sandpile driving assumption used throughout.

pith-pipeline@v0.9.0 · 5852 in / 1575 out tokens · 42193 ms · 2026-05-19T13:10:19.153330+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

[1]

Barrat, M

A. Barrat, M. Barth´ elemy, and A. Vespignani,Dynamical Processes on Complex Networks(Cambridge University Press, 2008)

work page 2008
[2]

Barab´ asi,Network Science(Cambridge University Press, 2016)

A.-L. Barab´ asi,Network Science(Cambridge University Press, 2016)

work page 2016
[3]

Newman,Networks, 2nd ed

M. Newman,Networks, 2nd ed. (Oxford University Press, 2018)

work page 2018
[4]

Boyer and C

D. Boyer and C. Solis-Salas, Physical Review Letters 112, 240601 (2014)

work page 2014
[5]

Rosvall, A

M. Rosvall, A. V. Esquivel, A. Lancichinetti, J. D. West, and R. Lambiotte, Nature Communications5, 4630 (2014)

work page 2014
[6]

Y. Kim, S. Park, and S.-H. Yook, Physical Review E94, 042309 (2016)

work page 2016
[7]

M. R. Evans and S. N. Majumdar, Physical Review Let- ters106, 160601 (2011)

work page 2011
[8]

A. G. Guerrero-Estrada, A. P. Riascos, and D. Boyer, Chaos35, 013117 (2025)

work page 2025
[9]

P. Bak, C. Tang, and K. Wiesenfeld, Physical Review Letters59, 381 (1987)

work page 1987
[10]

H. J. Jensen,Self-Organized Criticality: Emergent Com- plex Behavior in Physical and Biological Systems(Cam- bridge University Press, 1998)

work page 1998
[11]

Goh, D.-S

K.-I. Goh, D.-S. Lee, B. Kahng, and D. Kim, Physical Review Letters91, 148701 (2003)

work page 2003
[12]

Lahtinen, J

J. Lahtinen, J. Kert´ esz, and K. Kaski, Physica A: Statis- tical Mechanics and its Applications349, 535 (2005)

work page 2005
[13]

Dickman, A

R. Dickman, A. Vespignani, and S. Zapperi, Physical Re- view E57, 5095 (1998)

work page 1998
[14]

Vespignani and S

A. Vespignani and S. Zapperi, Physical Review E57, 6345 (1998)

work page 1998
[15]

Clauset, C

A. Clauset, C. R. Shalizi, and M. E. J. Newman, SIAM Review51, 661 (2009)

work page 2009
[16]

M. P. H. Stumpf and M. A. Porter, Science335, 665 (2012)

work page 2012
[17]

Markovi´ c and C

D. Markovi´ c and C. Gros, Physics Reports536, 41 (2014)

work page 2014
[18]

D. J. Watts and S. H. Strogatz, Nature393, 440 (1998)

work page 1998
[19]

Barab´ asi and R

A.-L. Barab´ asi and R. Albert, Science286, 509 (1999)

work page 1999
[20]

Alstott, E

J. Alstott, E. Bullmore, and D. Plenz, PLOS ONE9, e85777 (2014)

work page 2014
[21]

J. A. Bonachela, S. De Franciscis, J. J. Torres, and M. A. Mu˜ noz, Journal of Statistical Mechanics: Theory and Ex- periment2010, P02015 (2010)

work page 2010
[22]

Kinouchi, L

O. Kinouchi, L. Brochini, A. A. Costa, J. G. F. Campos, and M. Copelli, Scientific Reports9, 3874 (2019)

work page 2019
[23]

J. M. Beggs and D. Plenz, Journal of Neuroscience23, 11167 (2003)

work page 2003

[1] [1]

Barrat, M

A. Barrat, M. Barth´ elemy, and A. Vespignani,Dynamical Processes on Complex Networks(Cambridge University Press, 2008)

work page 2008

[2] [2]

Barab´ asi,Network Science(Cambridge University Press, 2016)

A.-L. Barab´ asi,Network Science(Cambridge University Press, 2016)

work page 2016

[3] [3]

Newman,Networks, 2nd ed

M. Newman,Networks, 2nd ed. (Oxford University Press, 2018)

work page 2018

[4] [4]

Boyer and C

D. Boyer and C. Solis-Salas, Physical Review Letters 112, 240601 (2014)

work page 2014

[5] [5]

Rosvall, A

M. Rosvall, A. V. Esquivel, A. Lancichinetti, J. D. West, and R. Lambiotte, Nature Communications5, 4630 (2014)

work page 2014

[6] [6]

Y. Kim, S. Park, and S.-H. Yook, Physical Review E94, 042309 (2016)

work page 2016

[7] [7]

M. R. Evans and S. N. Majumdar, Physical Review Let- ters106, 160601 (2011)

work page 2011

[8] [8]

A. G. Guerrero-Estrada, A. P. Riascos, and D. Boyer, Chaos35, 013117 (2025)

work page 2025

[9] [9]

P. Bak, C. Tang, and K. Wiesenfeld, Physical Review Letters59, 381 (1987)

work page 1987

[10] [10]

H. J. Jensen,Self-Organized Criticality: Emergent Com- plex Behavior in Physical and Biological Systems(Cam- bridge University Press, 1998)

work page 1998

[11] [11]

Goh, D.-S

K.-I. Goh, D.-S. Lee, B. Kahng, and D. Kim, Physical Review Letters91, 148701 (2003)

work page 2003

[12] [12]

Lahtinen, J

J. Lahtinen, J. Kert´ esz, and K. Kaski, Physica A: Statis- tical Mechanics and its Applications349, 535 (2005)

work page 2005

[13] [13]

Dickman, A

R. Dickman, A. Vespignani, and S. Zapperi, Physical Re- view E57, 5095 (1998)

work page 1998

[14] [14]

Vespignani and S

A. Vespignani and S. Zapperi, Physical Review E57, 6345 (1998)

work page 1998

[15] [15]

Clauset, C

A. Clauset, C. R. Shalizi, and M. E. J. Newman, SIAM Review51, 661 (2009)

work page 2009

[16] [16]

M. P. H. Stumpf and M. A. Porter, Science335, 665 (2012)

work page 2012

[17] [17]

Markovi´ c and C

D. Markovi´ c and C. Gros, Physics Reports536, 41 (2014)

work page 2014

[18] [18]

D. J. Watts and S. H. Strogatz, Nature393, 440 (1998)

work page 1998

[19] [19]

Barab´ asi and R

A.-L. Barab´ asi and R. Albert, Science286, 509 (1999)

work page 1999

[20] [20]

Alstott, E

J. Alstott, E. Bullmore, and D. Plenz, PLOS ONE9, e85777 (2014)

work page 2014

[21] [21]

J. A. Bonachela, S. De Franciscis, J. J. Torres, and M. A. Mu˜ noz, Journal of Statistical Mechanics: Theory and Ex- periment2010, P02015 (2010)

work page 2010

[22] [22]

Kinouchi, L

O. Kinouchi, L. Brochini, A. A. Costa, J. G. F. Campos, and M. Copelli, Scientific Reports9, 3874 (2019)

work page 2019

[23] [23]

J. M. Beggs and D. Plenz, Journal of Neuroscience23, 11167 (2003)

work page 2003