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Torrisi, Giovanni Luca; Garetto, Michele; Leonardi, E.

doi:10.1016/j.spa.2018.06.006

Cited by 3 publications

(4 citation statements)

References 35 publications

(100 reference statements)

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“…A different approach is to look at rare events arising from atypical initial conditions such as a low number of "infected" or "damaged" nodes evolving by contact or percolation dynamics to a large connected component [7][8][9][10][11]. Similarly, one can study how an initially large population spread on a network becomes extinct in time [12][13][14][15][16] or how a process transitions, more generally, between macroscopically distinct states [17].…”

Section: Introductionmentioning

confidence: 99%

Large deviations of random walks on random graphs

2019

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We study the rare fluctuations or large deviations of time-integrated functionals or observables of an unbiased random walk evolving on Erdös-Rényi random graphs, and construct a modified, biased random walk that explains how these fluctuations arise in the long-time limit. Two observables are considered: the sum of the degrees visited by the random walk and the sum of their logarithm, related to the trajectory entropy. The modified random walk is used for both quantities to explain how sudden changes in degree fluctuations, similar to dynamical phase transitions, are related to localization transitions. For the second quantity, we also establish links between the large deviations of the trajectory entropy and the maximum entropy random walk.

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Section: Introductionmentioning

confidence: 99%

Large deviations of random walks on random graphs

2019

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“…Torrisi et al [33] established a full large deviations principle in the supercritical case, α > 1, where typically |I * | ∼ n. As discussed in [33], the main step in this regard is establishing sharp tail estimates (as in our Theorem 3 above). The full large deviations principle then follows by "elementary topological considerations."…”

Section: Related Workmentioning

confidence: 79%

Large Deviations for Subcritical Bootstrap Percolation on the Erdős–Rényi Graph

Angel

Kolesnik

2021

J Stat Phys

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We study atypical behavior in bootstrap percolation on the Erdős–Rényi random graph. Initially a set S is infected. Other vertices are infected once at least r of their neighbors become infected. Janson et al. (Ann Appl Probab 22(5):1989–2047, 2012) locates the critical size of S, above which it is likely that the infection will spread almost everywhere. Below this threshold, a central limit theorem is proved for the size of the eventually infected set. In this work, we calculate the rate function for the event that a small set S eventually infects an unexpected number of vertices, and identify the least-cost trajectory realizing such a large deviation.

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“…, in (86) we used (7), and (87) follows by the union bound. Choosing c 1 large enough and arguing as in the proof of relation (59) in [33], for all i, ℓ and n large enough we get…”

Section: Proof Of Theorem 33mentioning

confidence: 95%

“…The following inequalities are proved in Step 4 of the proof of Proposition 4.1 in [33] and hold for any i and all n large enough:…”

Section: Proof Of Theorem 33mentioning

confidence: 99%

Bootstrap percolation on the stochastic block model with k communities

Torrisi,

Garetto,

Leonardi

2018

Preprint

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We analyze the bootstrap percolation process on the stochastic block model (SBM), a natural extension of the Erdös-Rényi random graph that allows representing the "community structure" observed in many real systems. In the SBM, nodes are partitioned into subsets, which represent different communities, and pairs of nodes are independently connected with a probability that depends on the communities they belong to. Under mild assumptions on system parameters, we prove the existence of a sharp phase transition for the final number of active nodes and characterize sub-critical and super-critical regimes in terms of the number of initially active nodes, which are selected uniformly at random in each community.

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A large deviation approach to super-critical bootstrap percolation on the random graph Gn,p

Cited by 3 publications

References 35 publications

Large deviations of random walks on random graphs

Large deviations of random walks on random graphs

Large Deviations for Subcritical Bootstrap Percolation on the Erdős–Rényi Graph

Bootstrap percolation on the stochastic block model with k communities

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