We consider a discrete-time Markovian random walk with resets on a connected undirected network. The resets, in which the walker is relocated to randomly chosen nodes, are governed by an independent discrete-time renewal process. Some nodes of the network are target nodes, and we focus on the statistics of first hitting of these nodes. In the non-Markov case of the renewal process, we consider both light- and fat-tailed inter-reset distributions. We derive the propagator matrix in terms of discrete backward recurrence time probability density functions, and in the light-tailed case, we show the existence of a non-equilibrium steady state. In order to tackle the non-Markov scenario, we derive a defective propagator matrix, which describes an auxiliary walk characterized by killing the walker as soon as it hits target nodes. This propagator provides the information on the mean first passage statistics to the target nodes. We establish sufficient conditions for ergodicity of the walk under resetting. Furthermore, we discuss a generic resetting mechanism for which the walk is non-ergodic. Finally, we analyze inter-reset time distributions with infinite mean where we focus on the Sibuya case. We apply these results to study the mean first passage times for Markovian and non-Markovian (Sibuya) renewal resetting protocols in realizations of Watts–Strogatz and Barabási–Albert random graphs. We show nontrivial behavior of the dependence of the mean first passage time on the proportions of the relocation nodes, target nodes, and of the resetting rates. It turns out that, in the large-world case of the Watts–Strogatz graph, the efficiency of a random searcher particularly benefits from the presence of resets.
REFERENCES
1.
M. R.
Evans
and S. N.
Majumdar
, “Diffusion with stochastic resetting
,” Phys. Rev. Lett.
106
, 160601
(2011
). 2.
M. R.
Evans
, S. N.
Majumdar
, and G.
Schehr
, “Stochastic resetting and applications
,” J. Phys. A: Math. Theor.
53
, 193001
(2020
). 3.
D.
Das
and L.
Giuggioli
, “Discrete space-time resetting model: Application to first-passage and transmission statistics
,” J. Phys. A: Math. Theor.
55
, 424004
(2022
). 4.
J.
Quetzalcóatl Toledo-Marin
and D.
Boyer
, “First passage time and information of a one-dimensional Brownian particle with stochastic resetting to random positions
,” Physica A
625
, 129027
(2023
).5.
S.
Gupta
and A. M.
Jayannavar
, “Stochastic resetting: A (very) brief review
,” Front. Phys.
10
, 789097
(2022
). 6.
A.
Pal
, A.
Kundu
, and M. R.
Evans
, “Diffusion under time-dependent resetting
,” J. Phys. A: Math. Theor.
49
, 225001
(2016
). 7.
A. S.
Bodrova
, A. V.
Chechkin
, and T. M.
Sokolov
, “Scaled Brownian motion with renewal resetting
,” Phys. Rev. E
100
, 012120
(2019
). 8.
A. S.
Bodrova
, A. V.
Chechkin
, and T. M.
Sokolov
, “Nonrenewal resetting of scaled Brownian motion
,” Phys. Rev. E
100
, 012119
(2019
). 9.
R. K.
Singh
, K.
Górska
, and T.
Sandev
, “General approach to stochastic resetting
,” Phys. Rev. E
105
, 064133
(2022
). 10.
A.
Masó-Puigdellosas
, T.
Sandev
, and V.
Mendez
, “Random walks on comb-like structures under stochastic resetting
,” Entropy
25
(11
), 1529
(2023
).11.
E.
Barkai
, R.
Flaquer-Galmés
, and V.
Méndez
, “Ergodic properties of Brownian motion under stochastic resetting
,” Phys. Rev. E
108
, 064102
(2023
). 12.
P.
Jolakoski
, A.
Pal
, T.
Sandev
, L.
Kocarev
, R.
Metzler
, and V.
Stojkoski
, “A first passage under resetting approach to income dynamics
,” Chaos, Solitons Fractals
175
, 113921
(2023
). 13.
A.
Pal
, V.
Stojkoski
, and T.
Sandev
, “Random resetting in search problems,” arXiv:2310.12057; to be published in Target Search Problems, edited by D. Grebekov, R. Metzler, and G. Oshanin (Springer Nature, 2024), ISBN: 978-3-031-67801-1.14.
A.
Chechkin
and I. M.
Sokolov
, “Random search with resetting: A unified renewal approach
,” Phys. Rev. Lett.
121
, 050601
(2018
). 15.
O.
Bénichou
, C.
Loverdo
, M.
Moreau
, and R.
Voituriez
, “Intermittent search strategies
,” Rev. Mod. Phys.
83
, 81
(2011
).16.
S.
Eule
and J. J.
Metzger
, “Non-equilibrium steady states of stochastic processes with intermittent resetting
,” New J. Phys.
18
, 033006
(2016
). 17.
R.
Flaquer-Galmés
, D.
Campos
, and V.
Méndez
, “Intermittent random walks under stochastic resetting,” arXiv:2401.16849 (2024).18.
G.
Mercado-Vásquez
and D.
Boyer
, “Search of stochastically gated targets with diffusive particles under resetting
,” J. Phys. A: Math. Theor.
54
, 444002
(2021
).19.
G.
Mercado-Vásquez
et al., “Reducing mean first passage times with intermittent confining potentials: a realization of resetting processes
,” J. Stat. Mech. Theor. Exp.
2022
, 093202
. 20.
A.
Pal
and S.
Reuveni
, “First passage under restart
,” Phys. Rev. Lett.
118
, 030603
(2017
). 21.
R.
Metzler
and J.
Klafter
, “The random walk’s guide to anomalous diffusion : A fractional dynamics approach
,” Phys. Rep.
339
, 1
–77
(2000
). 22.
V. V.
Palyulin
, G.
Blackburn
, M. A.
Lomholt
, N. W.
Watkin
, R.
Metzler
, R.
Klages
, and A. V.
Chechkin
, “First passage and first hitting times of Lévy flights and Lévy walks
,” New J. Phys.
21
, 103028
(2019
). 23.
A. P.
Riascos
and J. L.
Mateos
, “Long-range navigation on complex networks using Lévy random walks
,” Phys. Rev. E
86
, 056110
(2012
). 24.
T. M.
Michelitsch
, B. A.
Collet
, A. P.
Riascos
, A. F.
Nowakowski
, and F. C. G. A.
Nicolleau
, “Recurrence of random walks with long-range steps generated by fractional Laplacian matrices on regular networks and simple cubic lattices
,” J. Phys. A: Math. Theor.
50
, 505004
(2017
). 25.
L.
Kusmierz
, S. N.
Majumdar
, S.
Sabhapandit
, and G.
Schehr
, “First order transition for the optimal search time of Lévy flights with resetting
,” Phys. Rev. Lett.
113
, 220602
(2014
). 26.
B.
Żbik
and B.
Dybiec
, “Lévy flights and Lévy walks under stochastic resetting
,” Phys. Rev. E
109
, 044147
(2024
).27.
A. P.
Riascos
, D.
Boyer
, P.
Herringer
, and J.-L.
Mateos
, “Random walks on networks with stochastic resetting
,” Phys. Rev. E
101
, 062147
(2020
). 28.
F. H.
González
, A. P.
Riascos
, and D.
Boyer
, “Diffusive transport on networks with stochastic resetting to multiple nodes
,” Phys. Rev. E
103
, 062126
(2021
).29.
O.
Tal-Friedman
, Y.
Roichman
, and S.
Reuveni
, “Diffusion with partial resetting
,” Phys. Rev. E
106
, 054116
(2022
). 30.
M.
Dahlenburg
, A. V.
Chechkin
, R.
Schumer
, and R.
Metzler
, “Stochastic resetting by a random amplitude
,” Phys. Rev. E
103
, 052123
(2021
). 31.
R.
Van Der Hofstad
, S.
Kapodistria
, Z.
Palmowski
, and S.
Shneer
, “Unified approach for solving exit problems for additive-increase and multiplicative-decrease processes
,” J. Appl. Probab.
60
(1
), 85
–105
(2023
). 32.
G.
D’Onofrio
, P.
Patie
, and L.
Sacerdote
, “Jacobi processes with jumps as neuronal models: A first passage time analysis
,” SIAM J. Appl. Math.
84
(1
), 189
–214
(2024
).33.
J. K.
Pierce
, “An advection-diffusion process with proportional resetting,” arXiv:2204.07215 (2022).34.
M.
Biroli
, Y.
Feld
, A. K.
Hartmann
, S. N.
Majumdar
, and G.
Schehr
, “Resetting by rescaling: Exact results for a diffusing particle in one dimension,” arXiv:2406.08387 (2024).35.
D.
Boyer
and C.
Solis-Salas
, “Random walks with preferential relocations to places visited in the past and their application to biology
,” Phys. Rev. Lett.
112
, 240601
(2014
). 36.
H.
Meyer
and H.
Rieger
, “Optimal non-Markovian search strategies with n-step memory
,” Phys. Rev. Lett.
127
, 070601
(2021
). 37.
H.
Chen
and Y.
Ye
, “Random walks on complex networks under time-dependent stochastic resetting
,” Phys. Rev. E
106
, 044139
(2022
). 38.
A.
Pachon
, F.
Polito
, and C.
Ricciuti
, “On discrete-time semi-Markov processes
,” Discrete Contin. Dyn. Syst. Ser. B
26
(3
), 1499
–1529
(2021
). 39.
V. S.
Barbu
and N.
Limnios
, Semi-Markov Chains and Hidden Semi-Markov Models Toward Applications, Lecture Notes in Statistics Vol. 191 (Springer, New York, 2008).40.
T. M.
Michelitsch
, F.
Polito
, and A. P.
Riascos
, “On discrete time Prabhakar-generalized fractional Poisson processes and related stochastic dynamics
,” Physica A
565
, 125541
(2021
). 41.
T. M.
Michelitsch
, F.
Polito
, and A. P.
Riascos
, “Asymmetric random walks with bias generated by discrete-time counting processes
,” Commun. Nonlinear Sci. Numer. Simul.
109
, 106121
(2022
). 42.
T. M.
Michelitsch
, F.
Polito
, and A. P.
Riascos
, “Squirrels can remember little: A random walk with jump reversals induced by a discrete-time renewal process
,” Commun. Nonlinear Sci. Numer. Simul.
118
, 107031
(2023
). 43.
T. M.
Michelitsch
, F.
Polito
, and A. P.
Riascos
, “Semi-Markovian discrete-time telegraph process with generalized Sibuya waiting times
,” Mathematics
11
, 471
(2023
). 44.
C.
Godrèche
and J. M.
Luck
, “Statistics of the occupation time of renewal processes
,” J. Stat. Phys.
104
, 489–524
(2001
).45.
46.
M. E. J.
Newman
, Networks: An Introduction
(Oxford University Press
, Oxford
, 2010
).47.
T.
Michelitsch
, A. P.
Riascos
, B. A.
Collet
, A.
Nowakowski
, and F.
Nicolleau
, Fractional Dynamics on Networks and Lattices
(ISTE-Wiley
, London
, 2019
).48.
J. D.
Noh
and H.
Rieger
, “Random walks on complex networks
,” Phys. Rev. Lett.
92
, 118701
(2004
). 49.
P.
Van Mieghem
, Graph Spectra for Complex Networks
(Cambridge University Press
, New York
, 2011
).50.
51.
L.
Lovász
, “Random walks on graphs,” in Combinatorics, Paul Erdös Is Eighty (János Bolyai Mathematical Society, 1993), Vol. 2, pp. 1–46, see https://www.bibsonomy.org/bibtex/2f4e1da99c7143a3e6b56ab7ef264766b/folke52.
E. B.
Dynkin
, “Some limit theorems for sums of independent random quantities with infinite mathematical expectations
,” Izv. Akad. Nauk SSSR Ser. Mat.
19
(4
), 247
–266
(1955
).53.
G.
D’Onofrio
, T. M.
Michelitsch
, F.
Polito
, and A. P.
Riascos
, “On discrete-time arrival processes and related random motions,” arXiv:2403.06821 (2024).54.
I. M.
Gel’fand
and G. E.
Shilov
, Generalized Functions
(Academic Press
, New York
, 1968
), Vol. 1.55.
W.
Wang
, J. H. P.
Schulz
, W.
Deng
, and E.
Barkai
, “Renewal theory with fat-tailed distributed sojourn times: Typical versus rare
,” Phys. Rev. E
98
, 042139
(2018
). 56.
P.
Julián-Salgado
, L.
Dagdug
, and D.
Boyer
, “Diffusion with two resetting points
,” Phys. Rev. E
109
, 024134
(2024
).57.
F.
Huang
and H.
Chen
, “Random walks on complex networks with first-passage resetting
,” Phys. Rev. E
103
, 062132
(2021
). 58.
M.
Kac
, “On the notion of recurrence in discrete stochastic processes
,” Bull. Amer. Math. Soc.
53
, 1002
(1947
). 59.
N.
Masuda
, M. A.
Porter
, and R.
Lambiotte
, “Random walks and diffusion on networks
,” Phys. Rep.
716–717
, 1
–58
(2017
). 60.
J. G.
Kemeny
and J. L.
Snell
, Finite Markov Chains
(D. Van Nostrand
, Princeton, NJ
, 1960
).61.
A.-L.
Barabási
and R.
Albert
, “Emergence of scaling in random networks
,” Science
286
, 509
(1999
).62.
D. J.
Watts
and S. H.
Strogatz
, “Collective dynamics of ‘small-world’ networks
,” Nature
393
, 440
(1998
). 63.
S.
Redner
, A Guide to First-Passage Processes
(Cambridge University Press
, 2001
).64.
A.
Bassola
and V.
Nicosia
, “First-passage times to quantify and compare structural correlations and heterogeneity in complex systems
,” Commun. Phys.
4
, 76
(2021
).65.
G. D.
Birkhoff
, “Proof of the ergodic theorem
,” Proc. Natl. Acad. Sci. U.S.A.
17
(12
), 656
–660
(1931
). 66.
67.
C. M.
Grinstead
and J. L.
Snell
, Introduction to Probability
, 2nd ed. (American Mathematical Society
, 2006
).68.
M.
Sibuya
, “Generalized hypergeometric, digamma, and trigamma distributions
,” Ann. Inst. Stat. Math.
31
, 373
–390
(1979
). 69.
A.
Fronczak
and P.
Fronczak
, “Biased random walks in complex networks: The role of local navigation rules
,” Phys. Rev. E
80
, 016107
(2009
). 70.
T.
Sandev
and A.
Iomin
, “Fractional heterogeneous telegraph processes: Interplay between heterogeneity, memory, and stochastic resetting
,” Phys. Rev. E
110
, 024101
(2024
). © 2025 Author(s). Published under an exclusive license by AIP Publishing.
2025
Author(s)
You do not currently have access to this content.
