pith. sign in

arxiv: 2604.20829 · v2 · submitted 2026-04-22 · ⚛️ physics.soc-ph

Network exploration by random walks: A large deviation perspective

Sarvesh K. Upadhyay , Trifce Sandev , Sanjay Kumar , R. K. Singh This is my paper

Pith reviewed 2026-05-09 22:48 UTC · model grok-4.3

classification ⚛️ physics.soc-ph
keywords random walks on networksdistinct nodes visitedcontinuous-time random walkslarge deviationswaiting time distributioncoupon collector problem
0
0 comments X

The pith

At short times the distribution of distinct nodes visited by a random walk depends only on the waiting times between hops, not on network structure.

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

The paper maps the fully connected case to the coupon collector problem and then uses continuous-time random walks with random waiting times at nodes to analyze exploration on general networks. It establishes that the large-deviation limit of the distribution P(S,t) of visited nodes is controlled solely by the analytic properties of the waiting-time density near zero. This independence from topology at early times follows directly from the waiting-time formalism and holds for any network whose links are traversed according to the same waiting-time rule. A reader cares because real networks rarely allow instantaneous hops, so short-time statistics become predictable from local timing alone.

Core claim

Under the sole assumption that the waiting-time distribution ψ(τ) is analytic at small times, the large-deviation function governing P(S,t) at small t becomes independent of network topology and is fixed entirely by the characteristics of ψ(τ); the fully connected coupon-collector limit is recovered as a special case when waiting times are deterministic.

What carries the argument

The large-deviation rate function for P(S,t) obtained from the continuous-time random-walk representation, whose small-time behavior is insensitive to the adjacency matrix once ψ(τ) is analytic near the origin.

Load-bearing premise

The distribution of waiting times at nodes must be analytic at small times.

What would settle it

Compute or measure the small-t form of P(S,t) on two networks with different topologies but identical analytic waiting-time distributions; any statistically significant difference in the large-deviation rate would contradict the claim.

Figures

Figures reproduced from arXiv: 2604.20829 by R. K. Singh, Sanjay Kumar, Sarvesh K. Upadhyay, Trifce Sandev.

Figure 1
Figure 1. Figure 1: Distribution of distinct visited nodes 𝑃 (𝑆, 𝑡) for a discrete time random walk on a complete network (𝑁 = 100). (a) 𝑡 = 30. (b) 𝑡 = 200. This implies that the distribution for the number of distinct nodes visited after 𝑛 jumps is: 𝑃𝑛 (𝑆) = ( 𝑁 − 1 𝑆 − 1)(𝑆 − 1)! { 𝑛 𝑆−1} (𝑁 − 1)𝑛 , 1 ≤ 𝑆 ≤ 𝑁. (4) It is to be noted here that this combinatorial approach works because of the network’s complete symmetry and t… view at source ↗
Figure 2
Figure 2. Figure 2: Distribution of distinct visited nodes 𝑃 (𝑆, 𝑡) for a CTRW on a complete network (𝑁 = 1000, 𝜆 = 2) at small ((a) 𝑡 = 1) and large ((b) 𝑡 = 50) times and following Eq. (8). exhibit a non-degenerate residence time distribution, for example, in particle transport through porous media and other disordered environments [53–55]. Network exploration by a CTRW involves two independent stochastic components: (𝑖) th… view at source ↗
Figure 3
Figure 3. Figure 3: Mean cover time ⟨𝑇cov⟩ for various waiting-time distributions on networks with 𝑁 = 50, 100, 300, 600, 800, 1000 nodes. The distributions are: 𝜓(𝜏) = 𝜆𝑒−𝜆𝜏 with 𝜆 = 1 (exponential); 𝜓(𝜏) = 𝑘 𝜆 ( 𝜏 𝜆 )𝑘−1 𝑒 −(𝜏∕𝜆) 𝑘 with 𝜆 = 1, 𝑘 = 1.5 (Weibull); 𝜓(𝜏) = 1 𝜏𝜎√ 2𝜋 exp[ − (ln 𝜏−𝜇) 2 2𝜎 2 ] with 𝜎 = 0.5, 𝜇 = 0 (log-normal); 𝜓(𝜏) = 𝛼𝜏𝛼 𝑚 𝜏 −1−𝛼 with 𝜏𝑚 = 1 and different 𝛼 values (Pareto). The black dashed line fo… view at source ↗
Figure 4
Figure 4. Figure 4: Small-time behavior of the mean number of distinct nodes visited up to time 𝑡 for a CTRW on an Erdős–Rényi (ER) network with jump rate 𝑟: (left panel) 𝑐0 = 0.6, 𝑟 = 2.5, 𝑁 = 600. Blue circles denote numerical estimates and black dashed lines follow: ⟨𝑆(𝑡)⟩ ∼ 1 + 𝜆𝑡, for 𝑡 small. The right panel corresponds to the Barabási- Albert (BA) network with parameters: 𝑚 = 3, 𝑟 = 2.5, 𝑁 = 600, where 𝑚 is the number … view at source ↗
Figure 5
Figure 5. Figure 5: Probability distribution of the number of distinct nodes visited 𝑃 (𝑆, 𝑡), by a CTRW on (a) an Erdős–Rényi (ER) network with 𝑁 = 1000 and connection probability 𝑐 = 0.01 (sparse and homogeneous), and (b) a Barabási–Albert (BA) network with 𝑁 = 1000 and 𝑚 = 4 ( heterogeneous). Symbols represent CTRW simulations, and dashed line follows Eq. 14. Different waiting time distributions are: exponential (exp): 𝜓(𝜏… view at source ↗
read the original abstract

We study exploration properties of a random walk on a network. For a fully connected network we find that the problem can be mapped to the well known coupon collector problem, thus allowing us to estimate form of $P(S,t)$: the distribution of number of distinct nodes $S$ visited by the random walk upto time $t$. From a practical point of view, however, both the fully connected network and hops taking place after fixed intervals are an idealization. We solve this problem by introducing the formalism of continuous time random walks wherein the random walk spends a random amount of time a node before hopping to its neighboring node. The formalism allows us to study the large deviation limit of $P(S,t)$ under very mild conditions that the distribution of waiting times $\psi(\tau)$ exhibits analyticity at small times. Furthermore, we find that at small times, the properties of $P(S,t)$ are largely independent of the network topology, and are governed solely by the waiting time characteristics.

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 / 2 minor

Summary. The manuscript examines the exploration of networks by random walks, focusing on the distribution P(S, t) of the number of distinct nodes visited by time t. It begins by mapping the problem on fully connected networks with deterministic waiting times to the coupon collector problem. The authors then introduce continuous-time random walks (CTRW) with general waiting time density ψ(τ) to handle more realistic scenarios. Under the assumption that ψ(τ) is analytic at small times, they derive the large-deviation limit for P(S, t) and conclude that for small t, this distribution's properties are independent of the network topology and determined solely by the waiting time characteristics.

Significance. If the large-deviation analysis and the topology-independence result are rigorously established, the work could offer valuable insights into short-time dynamics of random walks on complex networks, with potential applications in modeling diffusion processes where waiting times vary. The mapping to coupon collector and the use of CTRW formalism are standard tools, but their combination for large deviations under analyticity provides a novel perspective. However, the result's impact depends on whether the analyticity condition sufficiently decouples the graph structure from the small-t statistics.

major comments (2)
  1. The central claim that analyticity of ψ(τ) at small τ suffices for the large-deviation limit of P(S,t) to become independent of network topology (as stated in the abstract and likely derived in the main formalism section) requires explicit justification for interchanging the small-t limit with the rate function while restoring the graph-dependent first-return kernel; without this, the mapping from hop count N(t) to distinct sites S remains topology-dependent even for small numbers of steps.
  2. No error bounds, convergence rates, or restrictions on the network (e.g., bounded degree or regularity assumptions) are provided to guarantee that the claimed independence holds for arbitrary graphs once the renewal process defined by ψ is coupled to the adjacency structure.
minor comments (2)
  1. The abstract introduces the analyticity condition on ψ(τ) without specifying the precise class of functions (e.g., whether it includes power-law tails with cutoff or only exponential-like forms) or how it translates to the Laplace-transform representation used in the large-deviation analysis.
  2. Notation for the waiting-time density ψ(τ) and the propagator should be clarified with respect to normalization and moments to aid readability for readers unfamiliar with CTRW literature.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We address each major comment point by point below, providing clarifications on our derivations and indicating revisions to improve rigor and explicitness.

read point-by-point responses

  1. Referee: The central claim that analyticity of ψ(τ) at small τ suffices for the large-deviation limit of P(S,t) to become independent of network topology (as stated in the abstract and likely derived in the main formalism section) requires explicit justification for interchanging the small-t limit with the rate function while restoring the graph-dependent first-return kernel; without this, the mapping from hop count N(t) to distinct sites S remains topology-dependent even for small numbers of steps.

    Authors: We appreciate the referee's call for greater explicitness in the limit interchange. In our approach, N(t) is generated by a renewal process whose small-t statistics are fixed solely by the analytic expansion of ψ(τ) near zero; the large-deviation rate for atypical N(t) follows directly from this expansion. The mapping to S then proceeds via the occupation measure on the graph. While the first-return kernel is indeed graph-dependent, the analyticity condition ensures that the leading exponential cost in the small-t regime is incurred by the waiting-time tail rather than by return events, whose contribution remains sub-exponential for the relevant deviation scale. We will revise the formalism section (and add a short appendix) to spell out this interchange step by step, showing explicitly that the graph-dependent kernel factors out of the leading rate function I(s) for P(S,t) as t→0. revision: yes

  2. Referee: No error bounds, convergence rates, or restrictions on the network (e.g., bounded degree or regularity assumptions) are provided to guarantee that the claimed independence holds for arbitrary graphs once the renewal process defined by ψ is coupled to the adjacency structure.

    Authors: The referee is right that we have not supplied quantitative error bounds or uniform convergence statements. The result is an asymptotic large-deviation statement in the joint limit t→0 under the analyticity hypothesis on ψ(τ); for any fixed finite connected graph the topology independence emerges at leading order. We will add a dedicated paragraph (and a remark in the conclusions) stating the standing assumptions—finite undirected connected graph with bounded maximum degree—and noting that the rate of convergence to the topology-independent limit may depend on graph invariants such as diameter. Precise error bounds would require additional technical estimates on the remainder of the renewal expansion; we therefore mark this as a partial revision and will include a clear statement of the current scope. revision: partial

Circularity Check

0 steps flagged

No circularity; central claim rests on external analyticity assumption.

full rationale

The paper assumes analyticity of the waiting-time density ψ(τ) at small τ as a mild external condition to obtain the large-deviation limit of P(S,t) and the claimed topology independence at small t. This assumption is not derived from or fitted to any quantity inside the paper. The fully-connected case reduces to the standard coupon-collector problem (a known result, not a self-definition), and the CTRW extension does not rename fitted parameters as predictions or invoke self-citations whose content reduces to the target claim. No load-bearing step collapses by construction to the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The only explicit assumption is analyticity of the waiting-time density at the origin. No free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption The waiting-time distribution ψ(τ) is analytic at small times.
    Invoked to obtain the large-deviation limit of P(S,t) for general networks.

pith-pipeline@v0.9.0 · 5478 in / 1253 out tokens · 62053 ms · 2026-05-09T22:48:34.852714+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

61 extracted references · 61 canonical work pages

  1. [1]

    Epidemicprocessesincomplexnetworks.Rev

    R.Pastor-Satorras,C.Castellano,P.VanMieghem,andA.Vespignani. Epidemicprocessesincomplexnetworks.Rev. Mod. Phys.,87(3):925– 979, 2015

    work page 2015

  2. [2]

    Bisnik and A

    N. Bisnik and A. A. Abouzeid. Optimizing random walk search algorithms in p2p networks.Computer Networks, 51(6):1499–1514, 2007

    work page 2007

  3. [3]

    N.Perra,A.Baronchelli,D.Mocanu,B.Gonçalves,R.Pastor-Satorras,andA.Vespignani.Randomwalksandsearchintime-varyingnetworks. Phys. Rev. Lett., 109(23):238701, 2012

    work page 2012

  4. [4]

    Sarkar and A

    P. Sarkar and A. W. Moore. Random walks in social networks and their applications: a survey.Social Network Data Analytics, pages 43–77, 2011

    work page 2011

  5. [5]

    V. M. Lopez Millan, V. Cholvi, L. López, and A. Fernandez Anta. A model of self-avoiding random walks for searching complex networks. Networks, 60(2):71–85, 2012

    work page 2012

  6. [6]

    Gkantsidis, M

    C. Gkantsidis, M. Mihail, and A. Saberi. Random walks in peer-to-peer networks. InIEEE INFOCOM 2004, volume 1. IEEE, 2004

    work page 2004

  7. [7]

    L. Lü, D. Chen, X.-L. Ren, Q.-M. Zhang, Y.-C. Zhang, and T. Zhou. Vital nodes identification in complex networks.Phys. Rep., 650:1–63, 2016

    work page 2016

  8. [8]

    Fortunato

    S. Fortunato. Community detection in graphs.Phys. Rep., 486(3-5):75–174, 2010

    work page 2010

  9. [9]

    Ballal, W

    A. Ballal, W. B. Kion-Crosby, and A. V. Morozov. Network community detection and clustering with random walks.Phys. Rev. Research, 4(4):043117, 2022

    work page 2022

  10. [10]

    Carletti, D

    T. Carletti, D. Fanelli, and R. Lambiotte. Random walks and community detection in hypergraphs.J. Phys.: Complexity, 2(1):015011, 2021

    work page 2021

  11. [11]

    de Guzzi Bagnato, J

    G. de Guzzi Bagnato, J. R. F. Ronqui, and G. Travieso. Community detection in networks using self-avoiding random walks.Physica A, 505:1046–1055, 2018

    work page 2018

  12. [12]

    Fortunato and A

    S. Fortunato and A. Flammini. Random walks on directed networks: the case of pagerank.International J. Bifurcation and Chaos, 17(07):2343–2353, 2007

    work page 2007

  13. [13]

    Agarwal and S

    A. Agarwal and S. Chakrabarti. Learning random walks to rank nodes in graphs. InProceedings of the 24th international conference on Machine learning, 2007

    work page 2007

  14. [14]

    Cambridge University Press, Cambridge, 2008

    Alain Barrat, Marc Barthélemy, and Alessandro Vespignani.Dynamical Processes on Complex Networks. Cambridge University Press, Cambridge, 2008

    work page 2008

  15. [15]

    Anultrametricrandomwalkmodelfordiseasespreadtakingintoaccountsocialclusteringofthepopulation

    A.KhrennikovandK.Oleschko. Anultrametricrandomwalkmodelfordiseasespreadtakingintoaccountsocialclusteringofthepopulation. Entropy, 22(9):931, 2020

    work page 2020

  16. [16]

    Draief and A

    M. Draief and A. Ganesh. A random walk model for infection on graphs: spread of epidemics & rumours with mobile agents.Discrete Event Dynamic Systems, 21:41–61, 2011

    work page 2011

  17. [17]

    Iannelli, A

    F. Iannelli, A. Koher, D. Brockmann, P. Hövel, and I. M. Sokolov. Effective distances for epidemics spreading on complex networks.Phys. Rev. E, 95(1):012313, 2017

    work page 2017

  18. [18]

    Deborah M. Gordon. The development of an ant colony’s foraging range.Animal Behaviour, 49(3):649–659, 1995

    work page 1995

  19. [19]

    Frank den Hollander and George H. Weiss. Aspects of trapping in transport processes. InContemporary Problems in Statistical Physics, pages 147–203. SIAM, 1994

    work page 1994

  20. [20]

    Kiefer, and George H

    Shlomo Havlin, Menachem Dishon, James E. Kiefer, and George H. Weiss. Trapping of random walks in two and three dimensions.Phys. Rev. Lett., 53:407–410, Jul 1984

    work page 1984

  21. [21]

    The range of stable random walks.The Annals of Probability, 19(2):650–705, 1991

    Jean-François Le Gall and Jay Rosen. The range of stable random walks.The Annals of Probability, 19(2):650–705, 1991

    work page 1991

  22. [22]

    Gillis and George H

    Joseph E. Gillis and George H. Weiss. Expected number of distinct sites visited by a random walk with an infinite variance.Journal of Mathematical Physics, 11(4):1307–1312, 04 1970. Upadhyay et al.:Preprint submitted to ElsevierPage 8 of 10 Network exploration by random walks: A large deviation perspective

    work page 1970

  23. [23]

    Number of distinct sites visited by a resetting random walker.Journal of Physics A: Mathematical and Theoretical, 55(24):244001, may 2022

    Marco Biroli, Francesco Mori, and Satya N Majumdar. Number of distinct sites visited by a resetting random walker.Journal of Physics A: Mathematical and Theoretical, 55(24):244001, may 2022

    work page 2022

  24. [24]

    Redner, and Olivier Bénichou

    Léo Régnier, Maxim Dolgushev, S. Redner, and Olivier Bénichou. Complete visitation statistics of one-dimensional random walks.Phys. Rev. E, 105:064104, Jun 2022

    work page 2022

  25. [25]

    Noh and H

    J.D. Noh and H. Rieger. Random walks on complex networks.Phys. Rev. Lett., 92(11):118701, 2004

    work page 2004

  26. [26]

    Masuda, M

    N. Masuda, M. A. Porter, and R. Lambiotte. Random walks and diffusion on networks.Phys. Rep., 716:1–58, 2017

    work page 2017

  27. [27]

    Oxford University Press, 06 1996

    Barry D Hughes.Random Walks and Random Environments. Oxford University Press, 06 1996

    work page 1996

  28. [28]

    Vineyard

    George H. Vineyard. The number of distinct sites visited in a random walk on a lattice.Journal of Mathematical Physics, 4(9):1191–1193, 09 1963

    work page 1963

  29. [29]

    Eugene Stanley, and George H

    Hernan Larralde, Paul Trunfio, Shlomo Havlin, H. Eugene Stanley, and George H. Weiss. Territory covered by n diffusing particles.Nature, 355(6359):423–426, Jan 1992

    work page 1992

  30. [30]

    Numberofdistinctsitesvisitedbynrandomwalkersonaeuclideanlattice.Phys

    S.B.YusteandL.Acedo. Numberofdistinctsitesvisitedbynrandomwalkersonaeuclideanlattice.Phys. Rev. E,61:2340–2347,Mar2000

    work page

  31. [31]

    Records for the number of distinct sites visited by a random walk on the fully connected lattice.Journal of Physics A: Mathematical and Theoretical, 48(44):445001, oct 2015

    Loïc Turban. Records for the number of distinct sites visited by a random walk on the fully connected lattice.Journal of Physics A: Mathematical and Theoretical, 48(44):445001, oct 2015

    work page 2015

  32. [32]

    Santhanam

    Aanjaneya Kumar, Yagyik Goswami, and M.S. Santhanam. Distinct nodes visited by random walkers on scale-free networks.Physica A: Statistical Mechanics and its Applications, 532:121875, 2019

    work page 2019

  33. [33]

    Theaveragenumberofdistinctsitesvisitedbyarandomwalkeronrandomgraphs

    CaterinaDeBacco,SatyaNMajumdar,andPeterSollich. Theaveragenumberofdistinctsitesvisitedbyarandomwalkeronrandomgraphs. Journal of Physics A: Mathematical and Theoretical, 48(20):205004, apr 2015

    work page 2015

  34. [34]

    Majumdar and Mikhail V

    Satya N. Majumdar and Mikhail V. Tamm. Number of common sites visited by𝑛random walkers.Phys. Rev. E, 86:021135, Aug 2012

    work page 2012

  35. [35]

    Universalexplorationdynamicsofrandomwalks.Nature Communications, 14(1):618, Feb 2023

    LéoRégnier,MaximDolgushev,S.Redner,andOlivierBénichou. Universalexplorationdynamicsofrandomwalks.Nature Communications, 14(1):618, Feb 2023

    work page 2023

  36. [36]

    Code-red:acasestudyonthespreadandvictimsofaninternetworm

    DavidMoore,ColleenShannon,andKimberlyClaffy. Code-red:acasestudyonthespreadandvictimsofaninternetworm. pages273–284, 01 2002

    work page 2002

  37. [37]

    Ataxonomyofcomputerworms

    NicholasWeaver,VernPaxson,StuartStaniford,andRobertCunningham. Ataxonomyofcomputerworms. InProceedings of the 2003 ACM Workshop on Rapid Malcode, WORM ’03, page 11–18, New York, NY, USA, 2003. Association for Computing Machinery

    work page 2003

  38. [38]

    Hethcote

    Herbert W. Hethcote. The mathematics of infectious diseases.SIAM Review, 42(4):599–653, 2000

    work page 2000

  39. [39]

    Epidemicspreadinginscale-freenetworks.Phys

    RomualdoPastor-SatorrasandAlessandroVespignani. Epidemicspreadinginscale-freenetworks.Phys. Rev. Lett.,86:3200–3203,Apr2001

    work page

  40. [40]

    Targeting metastasis.Nat Rev Cancer, 16(4):201–218, April 2016

    Patricia S Steeg. Targeting metastasis.Nat Rev Cancer, 16(4):201–218, April 2016

    work page 2016

  41. [41]

    Isaiah J. Fidler. The pathogenesis of cancer metastasis: the ’seed and soil’ hypothesis revisited.Nature Reviews Cancer, 3(6):453–458, Jun 2003

    work page 2003

  42. [42]

    Updateontheenvironmentalandeconomiccostsassociatedwithalien-invasivespecies in the united states.Ecological Economics, 52:273–288, 02 2005

    DavidPimentel,RodolfoZuniga,andDougMorrison. Updateontheenvironmentalandeconomiccostsassociatedwithalien-invasivespecies in the united states.Ecological Economics, 52:273–288, 02 2005

    work page 2005

  43. [43]

    Mack, Daniel Simberloff, W

    Richard N. Mack, Daniel Simberloff, W. Mark Lonsdale, Harry Evans, Michael Clout, and Fakhri A. Bazzaz. Biotic invasions: Causes, epidemiology, global consequences, and control.Ecological Applications, 10(3):689–710, 2000

    work page 2000

  44. [44]

    The spread of true and false news online.Science, 359(6380):1146–1151, 2018

    Soroush Vosoughi, Deb Roy, and Sinan Aral. The spread of true and false news online.Science, 359(6380):1146–1151, 2018

    work page 2018

  45. [45]

    Graham, Donald E

    Ronald L. Graham, Donald E. Knuth, and Oren Patashnik.Concrete Mathematics: A Foundation for Computer Science. Addison-Wesley Longman Publishing Co., Inc., USA, 2nd edition, 1994

    work page 1994

  46. [46]

    Bagui and K

    S. Bagui and K. L. Mehra. The stirling numbers of the second kind and their applications.Alabama Journal of Mathematics, 47(1):1–22,

    work page

  47. [47]

    Analytical results for the distribution of cover times of random walks on random regular graphs

    Ido Tishby, Ofer Biham, and Eytan Katzav. Analytical results for the distribution of cover times of random walks on random regular graphs. Journal of Physics A: Mathematical and Theoretical, 55(1):015003, dec 2021

    work page 2021

  48. [48]

    David J. Aldous. On the time taken by random walks on finite groups to visit every state.Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 62(3):361–374, Sep 1983

    work page 1983

  49. [49]

    Herbert S. Wilf. The editor’s corner: The white screen problem.The American Mathematical Monthly, 96(8):704–707, 1989

    work page 1989

  50. [50]

    William Feller.An introduction to probability theory and its applications. Vol. I.Wiley, Oxford, England, 1950

    work page 1950

  51. [51]

    The coupon-collector’s problem revisited.Journal of Applied Probability, 40, 06 2003

    Ilan Adler, Shmuel Oren, and Sheldon Ross. The coupon-collector’s problem revisited.Journal of Applied Probability, 40, 06 2003

    work page 2003

  52. [52]

    Marián Boguñá, Dmitri Krioukov, and K. C. Claffy. Navigability of complex networks.Nature Physics, 5(1):74–80, Jan 2009

    work page 2009

  53. [53]

    De Arcangelis, J

    L. De Arcangelis, J. Koplik, S. Redner, and D. Wilkinson. Hydrodynamic dispersion in network models of porous media.Phys. Rev. Lett., 57(8):996, 1986

    work page 1986

  54. [54]

    M. Sahimi. Dispersion in porous media, continuous-time random walks, and percolation.Phys. Rev. E, 85(1):016316, 2012

    work page 2012

  55. [55]

    Varloteaux, S

    C. Varloteaux, S. Békri, and P.M. Adler. Pore network modelling to determine the transport properties in presence of a reactive fluid: From pore to reservoir scale.Adv. Water Resources, 53:87–100, 2013

    work page 2013

  56. [56]

    Montroll and G.H

    E.W. Montroll and G.H. Weiss. Random walks on lattices. II.J. Math. Phys., 6(2):167–181, 1965

    work page 1965

  57. [57]

    Therandomwalk’sguidetoanomalousdiffusion:afractionaldynamicsapproach.Physics Reports,339(1):1– 77, 2000

    RalfMetzlerandJosephKlafter. Therandomwalk’sguidetoanomalousdiffusion:afractionaldynamicsapproach.Physics Reports,339(1):1– 77, 2000

    work page 2000

  58. [58]

    Packets of diffusing particles exhibit universal exponential tails.Phys

    Eli Barkai and Stanislav Burov. Packets of diffusing particles exhibit universal exponential tails.Phys. Rev. Lett., 124:060603, Feb 2020

    work page 2020

  59. [59]

    Upadhyay and R

    Sarvesh K. Upadhyay and R. K. Singh. Continuous time random walks on networks.Phys. Rev. E, 112:014313, Jul 2025

    work page 2025

  60. [60]

    Naftali R. Smith. Full distribution of the number of distinct sites visited by a random walker in dimension𝑑≥2, 2025

    work page 2025

  61. [61]

    Temporal networks.Physics Reports, 519(3):97–125, 2012

    Petter Holme and Jari Saramäki. Temporal networks.Physics Reports, 519(3):97–125, 2012. Upadhyay et al.:Preprint submitted to ElsevierPage 9 of 10 Network exploration by random walks: A large deviation perspective Appendix Form of the rate function𝐼(𝑡∕𝑛) Starting from𝑄𝑡(𝑛)given in Ref. [58]: 𝑄𝑡(𝑛) large𝑛∕𝑡 ∼ [(𝐶𝐴Γ(𝐴+ 1)) 1∕(𝐴+1) 𝑡]𝑛(𝐴+1) Γ(𝑛(𝐴+ 1) + 1 ) e...

    work page 2012