Long paths in first passage percolation on the complete graph I. Local PWIT dynamics

Eckhoff, Maren; Goodman, Jesse; Hofstad, Remco van der; Nardi, Francesca R.

doi:10.48550/arxiv.1512.06152

Cited by 4 publications

(6 citation statements)

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“…In this paper, we continue the program of studying first passage percolation on the complete graph initiated in [6]. We often need to refer to results presented in [14], so we choose to cite specific results with the following expression, e.g., [Part I, Lemma 2.14], to say Lemma 2.13 in [14].…”

Section: Model and Resultsmentioning

confidence: 99%

“…Second, Theorem 2.15 allows us to relate FPP on the complete graph (n-independent dynamics run on an n-dependent weighted graph) with an exploration defined in terms of a pair of Poissonweighted infinite trees (n-dependent dynamics run on an n-independent weighted graph). By analyzing the dynamics of B when n and s n are large, we obtain a fruitful dual picture: When the number of explored vertices is large, we find a dynamic rescaled branching process approximation that is essentially independent of n. When the number of explored vertices is small, we make use of a static approximation by invasion percolation found in [14]. In fact, under our scaling assumptions, FPP on the PWIT is closely related to invasion percolation (IP) on the PWIT which is defined as follows.…”

Section: Fpp Exploration Process From Two Sourcesmentioning

confidence: 94%

“…Several of our methods do not extend to the case where s n /n 1/3 → ∞; indeed, we conjecture that the CLT in Theorem 1.7 ceases to hold in this regime. First passage percolation (FPP) on the complete graph is closely approximated by invasion percolation (IP) on the Poisson-weighted infinite tree (PWIT), studied in [1], whenever s n → ∞, see [14].…”

Section: The Universality Class In Terms Of S Nmentioning

confidence: 99%

“…The proof relies on a decomposition of the smallest-weight tree into an initial part following invasion percolation dynamics, and a main part following branching process dynamics. The initial part has been studied in [14]; the current article focuses on the global branching dynamics.…”

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Short Paths for First Passage Percolation on the Complete Graph

et al. 2013

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We study the random geometry of first passage percolation on the complete graph equipped with independent and identically distributed edge weights, continuing the program initiated by Bhamidi and van der Hofstad [6]. We describe our results in terms of a sequence of parameters (s n ) n≥1 that quantifies the extreme-value behavior of small weights, and that describes different universality classes for first passage percolation on the complete graph. We consider both n-independent as well as n-dependent edge weights. The simplest example consists of edge weights of the form E sn , where E is an exponential random variable with mean 1.In this paper, we focus on the case where s n → ∞ with s n

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Section: Model and Resultsmentioning

confidence: 99%

Section: Fpp Exploration Process From Two Sourcesmentioning

confidence: 94%

Section: The Universality Class In Terms Of S Nmentioning

confidence: 99%

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confidence: 99%

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Short Paths for First Passage Percolation on the Complete Graph

et al. 2013

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“…This leads to a number of natural questions. The most extensively studied is the "typical distance," quantified by the total weight and number of edges on the shortest path between a pair of random nodes [49][50][51]. It is also possible to analyze the "flooding time" [37,52], defined as the time to reach the last node from a given source node chosen at random.…”

Section: First-passage Percolationmentioning

confidence: 99%

Takeover times for a simple model of network infection

2017

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We study a stochastic model of infection spreading on a network. At each time step a node is chosen at random, along with one of its neighbors. If the node is infected and the neighbor is susceptible, the neighbor becomes infected. How many time steps T does it take to completely infect a network of N nodes, starting from a single infected node? An analogy to the classic "coupon collector" problem of probability theory reveals that the takeover time T is dominated by extremal behavior, either when there are only a few infected nodes near the start of the process or a few susceptible nodes near the end. We show that for N 1, the takeover time T is distributed as a Gumbel distribution for the star graph, as the convolution of two Gumbel distributions for a complete graph and an Erdős-Rényi random graph, as a normal for a one-dimensional ring and a two-dimensional lattice, and as a family of intermediate skewed distributions for d-dimensional lattices with d 3 (these distributions approach the convolution of two Gumbel distributions as d approaches infinity). Connections to evolutionary dynamics, cancer, incubation periods of infectious diseases, first-passage percolation, and other spreading phenomena in biology and physics are discussed.

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Explosion in weighted hyperbolic random graphs and geometric inhomogeneous random graphs

Komjáthy

Lodewijks

2020

Stochastic Processes and their Applications

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In this paper we study weighted distances in scale-free spatial network models: hyperbolic random graphs, geometric inhomogeneous random graphs and scale-free percolation. In hyperbolic random graphs, n = Θ(e R/2 ) vertices are sampled independently from the hyperbolic disk with radius R and two vertices are connected either when they are within hyperbolic distance R, or independently with a probability depending on the hyperbolic distance. In geometric inhomogeneous random graphs, and in scale-free percolation, each vertex is given an independent weight and location from an underlying measured metric space and Z d , respectively, and two vertices are connected independently with a probability that is a function of their distance and their weights. We assign independent and identically distributed (i.i.d.) weights to the edges of the obtained random graphs, and investigate the weighted distance (the length of the shortest weighted path) between two uniformly chosen vertices, called typical weighted distance. In scale-free percolation, we study the weighted distance from the origin of vertex-sequences with norm tending to infinity.In particular, we study the case when the parameters are so that the degree distribution in the graph follows a power law with exponent τ ∈ (2, 3) (infinite variance), and the edge-weight distribution is such that it produces an explosive age-dependent branching process with power-law offspring distribution, that is, the branching process produces infinitely many individuals in finite time. We show that in all three models, typical distances within the giant/infinite component converge in distribution, that is, no re-scaling is necessary. This solves an open question in [51].The main tools of our proof are to develop elaborate couplings of the models to infinite versions, to follow the shortest paths to infinity and then to connect these paths by using weight-dependent percolation on the graphs, that is, we delete edges attached to vertices with higher weight with higher probability. We realise the percolation using the edge-weights: only very short edges connected to high weight vertices are allowed to stay, hence establishing arbitrarily short upper bounds for connections.(3) High clustering. Local clustering is the formation of locally concentrated clusters in a network and it is measured by the average clustering coefficient: the proportion of triangles present at a vertex versus the total number of possible triangles present at said vertex, averaged over all vertices [76]. Many real-life networks show high clustering, e.g. biological networks such as food webs, technological networks as the World Wide Web, and social networks [30,64,69,70,72].The study of real-world networks could be further improved by the analysis of network models. In the last three decades, network science has become a significant field of study, both theoretically and experimentally. One of the first models that was proposed for studying real-world networks was the Watts-Strogatz model [76], that has the small-...

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Long paths in first passage percolation on the complete graph I. Local PWIT dynamics

Cited by 4 publications

References 15 publications

Short Paths for First Passage Percolation on the Complete Graph

Short Paths for First Passage Percolation on the Complete Graph

Takeover times for a simple model of network infection

Explosion in weighted hyperbolic random graphs and geometric inhomogeneous random graphs

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