Long paths in first passage percolation on the complete graph II. Global branching dynamics

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

doi:10.48550/arxiv.1512.06145

Cited by 3 publications

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“…In this section we briefly discuss our results and state open problems. For a more detailed discussion of the results in this paper and in our companion paper [11], as well as an extensive discussion of the relations to the literature, we refer to [Part II, Section 1.4]. First passage percolation (FPP) on the complete graph is closely approximated by invasion percolation (IP) on the Poisson-weighted infinite tree (PWIT), studied in [3], whenever s n → ∞.…”

Section: Discussion Of Our Resultsmentioning

confidence: 99%

“…become numerous, the heuristic fails and the connection to IP on the PWIT ceases to hold. Since we are mainly interested in the case that 1/s n ≫ n −1/3 (see in particular [11]), the critical window observed here is wider than the critical window for the Erdős-Rényi random graph; cf. [6].…”

Section: Relation To Invasion Percolation On the Pwitmentioning

confidence: 99%

“…Since s n → ∞, this intuitively explained 'long paths' in our title. This analysis will be carried out in [11] where we are interested in global properties of the FPP process. There also the title will be substantiated, since it will be shown that H n is of order s n log(n/s 3 n ) when s n = o(n 1/3 ).…”

Section: Decomposing Fpp Into Ip and Branching Dynamicsmentioning

confidence: 99%

“…In this paper, we consider FPP on the complete graph and focus on problem (c). In the companion paper [11], we will use these results to investigate problems (a) and (b). We will often refer to results in [11], and write, e.g., [Part II, Section 5.2] to refer to [11,Section 5.2].…”

mentioning

confidence: 99%

“…In the companion paper [11], we will use these results to investigate problems (a) and (b). We will often refer to results in [11], and write, e.g., [Part II, Section 5.2] to refer to [11,Section 5.2]. We also refer to [Part II, Section 1.4] for an extended discussion of the results in these two papers and their relations to the literature.…”

mentioning

confidence: 99%

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

Eckhoff¹,

Goodman²,

Hofstad³

et al. 2015

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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 [9]. 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 investigate the case where s n → ∞, and focus on the local neighborhood of a vertex. We establish that the smallest-weight tree of a vertex locally converges to the invasion percolation cluster on the Poisson weighted infinite tree. In addition, we identify the scaling limit of the weight of the smallest-weight path between two uniform vertices. Model and resultsIn this paper, we continue the program of studying first passage percolation on the complete graph initiated in [9]. We start by introducing first passage percolation (FPP). Given a graph G = (V (G), E(G)), let (Y (G) e ) e∈E(G) denote a collection of positive edge weights. Thinking of Y (G) e as the cost of crossing an edge e, we can define a metric on V (G) by setting d G,Y (G) (i, j) = inf π : i→j e∈π

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Section: Discussion Of Our Resultsmentioning

confidence: 99%

Section: Relation To Invasion Percolation On the Pwitmentioning

confidence: 99%

Section: Decomposing Fpp Into Ip and Branching Dynamicsmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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

Eckhoff¹,

Goodman²,

Hofstad³

et al. 2015

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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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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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Long paths in first passage percolation on the complete graph II. Global branching dynamics

Cited by 3 publications

References 4 publications

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

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

Explosion in weighted hyperbolic random graphs and geometric inhomogeneous random graphs

Takeover times for a simple model of network infection

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