Steady state and mean recurrence time for random walks on stochastic temporal networks

Speidel, Leo; Lambiotte, Renaud; Aihara, Kazuyuki; Masuda, Naoki

doi:10.1103/physreve.91.012806

Cited by 21 publications

(31 citation statements)

References 32 publications

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“…That is, in the large time regime, the walker behaves as in a completely homogeneous network, in which jumps were performed independently of the node activity. This result recovers the observation made in [30,45]. In order to check the validity of the time dependence expressed in equation (58), we have performed numerical simulations of the activated random walk on a NoPAD network of size N=10 5 where the activity takes three values, c=0.1,1 or 10, each with probability η(c)=1/3.…”

Section: Non-poissonian Activity-driven Network With Infinite Averagsupporting

confidence: 78%

“…A way to neglect these aging effects is by considering active random walks, in which the inter-event time of a node is reinitialized when a walker lands on it, in such a way that intervent and waiting time distributions coincide. In opposition, in passive random walks the presence of the walker does not reinitialize the inter-event times of nodes or edges, and thus the waiting time depends on the last activation time [45]. The non-Poissoinan scenario has been considered in the general context of a fixed network in which edges are established according to a given inter-event time distribution ψ ij (t) for active walkers [45][46][47] and for passive walkers [45], usually with the assumption of a finite average inter-event time distribution, with the exception of [47].…”

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confidence: 99%

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Random walks in non-Poissoinan activity driven temporal networks

2019

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The interest in non-Markovian dynamics within the complex systems community has recently blossomed, due to a new wealth of time-resolved data pointing out the bursty dynamics of many natural and human interactions, manifested in an inter-event time between consecutive interactions showing a heavy-tailed distribution. In particular, empirical data has shown that the bursty dynamics of temporal networks can have deep consequences on the behavior of the dynamical processes running on top of them. Here, we study the case of random walks, as a paradigm of diffusive processes, unfolding on temporal networks generated by a non-Poissonian activity driven dynamics. We derive analytic expressions for the steady state occupation probability and first passage time distribution in the infinite network size and strong aging limits, showing that the random walk dynamics on non-Markovian networks are fundamentally different from what is observed in Markovian networks. We found a particularly surprising behavior in the limit of diverging average inter-event time, in which the random walker feels the network as homogeneous, even though the activation probability of nodes is heterogeneously distributed. Our results are supported by extensive numerical simulations. We anticipate that our findings may be of interest among the researchers studying non-Markovian dynamics on time-evolving complex topologies. IntroductionTemporal networks [1, 2] constitute a recent new description of complex systems, that, moving apart from the classical static paradigm of network science [3], in which nodes and edges do not change in time, consider dynamic connections that can be created, destroyed or rewired at different time scales. Within this framework, a first round of studies proposed temporal network models ruled by homogeneous Markovian dynamics [4]. A prominent example is represented by the activity-driven model [5] (see also [6]), in which nodes are characterized by a different degree of activity, i.e. the constant rate at which an agent sends links to other peers, following a Poissonian process. The memoryless property implied by the Markovian dynamics greatly simplifies the mathematical treatment of these models, regarding both the topological properties of the time-integrated network representation [7], and the description of the dynamical processes unfolding on activity-driven networks [8][9][10][11][12][13].However, the Markovian assumption in temporal network modeling has been challenged by the increasing availability of time-resolved data on different kinds of interactions, ranging from phone communications [14] and face-to-face interactions [15], to natural phenomena [16,17], biological processes [18] and physiological systems [19][20][21]. These empirical observations have uncovered a rich variety of dynamical properties, in particular that the inter-event times t between two successive interactions (either the creation of the same edge or two consecutive creations of an edge by the same node), ψ(t), follows heavy-tailed distributions...

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Section: Non-poissonian Activity-driven Network With Infinite Averagsupporting

confidence: 78%

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confidence: 99%

Random walks in non-Poissoinan activity driven temporal networks

2019

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“…In the taxonomy of RWs on networks, this corresponds to the popular edge-centric passive RW [1]. In particular, we explore in detail the implications of an apparently paradoxical situation [17,18]: despite the fact that edges are independent processes, they cease to be independent along the path of a walker when the inter-event time distribution is non-exponential, which may lead to biases in its dynamics and non-Markovian trajectories.…”

Section: Introductionmentioning

confidence: 99%

“…This question has been considered by means of numerical simulations, by simulating a diffusive process on empirical temporal network data [14], and comparing its speed with the same process run on randomized null models [15]. A theoretical approach, which we also adopt here, consists in neglecting correlations between activations of different edges, and modelling their dynamics as independent renewal processes [16,17]. In the taxonomy of RWs on networks, this corresponds to the popular edge-centric passive RW [1].…”

Section: Introductionmentioning

confidence: 99%

Backtracking and Mixing Rate of Diffusion on Uncorrelated Temporal Networks

Gueuning

Lambiotte

Delvenne

2017

Entropy

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Abstract:We consider the problem of diffusion on temporal networks, where the dynamics of each edge is modelled by an independent renewal process. Despite the apparent simplicity of the model, the trajectories of a random walker exhibit non-trivial properties. Here, we quantify the walker's tendency to backtrack at each step (return where he/she comes from), as well as the resulting effect on the mixing rate of the process. As we show through empirical data, non-Poisson dynamics may significantly slow down diffusion due to backtracking, by a mechanism intrinsically different from the standard bus paradox and related temporal mechanisms. We conclude by discussing the implications of our work for the interpretation of results generated by null models of temporal networks.

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“…For pre-determined temporal network structure, such as empirical contact lists with time stamps, temporal greedy walks are entirely deterministic once the initial conditions (first node, time) have been set, as long as nodes only participate in a single event at a time, which is mainly the case with our empirical data. Note that in some temporal-network models of random walks [18][19][20], the walks themselves are in fact temporally greedy, and randomness only comes from a stochastic model of the underlying temporal network. Further, in studies of random walks on temporal networks the focus has mainly been on issues such as effects of burstiness on mean first passage and relaxation times [15,17], models that generate temporal networks [21], and identification of timescales [22,23].…”

Section: Introductionmentioning

confidence: 99%

Exploring temporal networks with greedy walks

Saramäki

Holme

2015

Eur. Phys. J. B

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Temporal networks come with a wide variety of heterogeneities, from burstiness of event sequences to correlations between timings of node and link activations. In this paper, we set to explore the latter by using greedy walks as probes of temporal network structure. Given a temporal network (a sequence of contacts), greedy walks proceed from node to node by always following the first available contact. Because of this, their structure is particularly sensitive to temporal-topological patterns involving repeated contacts between sets of nodes. This becomes evident in their small coverage per step as compared to a temporal reference model -in empirical temporal networks, greedy walks often get stuck within small sets of nodes because of correlated contact patterns. While this may also happen in static networks that have pronounced community structure, the use of the temporal reference model takes the underlying static network structure out of the equation and indicates that there is a purely temporal reason for the observations. Further analysis of the structure of greedy walks indicates that burst trains, sequences of repeated contacts between node pairs, are the dominant factor. However, there are larger patterns too, as shown with non-backtracking greedy walks. We proceed further to study the entropy rates of greedy walks, and show that the sequences of visited nodes are more structured and predictable in original data as compared to temporally uncorrelated references. Taken together, these results indicate a richness of correlated temporal-topological patterns in temporal networks.

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Steady state and mean recurrence time for random walks on stochastic temporal networks

Cited by 21 publications

References 32 publications

Random walks in non-Poissoinan activity driven temporal networks

Random walks in non-Poissoinan activity driven temporal networks

Backtracking and Mixing Rate of Diffusion on Uncorrelated Temporal Networks

Exploring temporal networks with greedy walks

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