Short Paths for First Passage Percolation on the Complete Graph

Eckhoff, Maren; Goodman, Jesse; Hofstad, RW Remco van der; Nardi, FR Francesca

doi:10.1007/s10955-013-0743-7

Cited by 16 publications

(17 citation statements)

References 25 publications

(263 reference statements)

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“…important mathematical model in probability theory that has been studied extensively in grid-like graphs, see e.g. [43,53,73], and on non-spatial (random) graph models as well such as the complete graph [48,37,38,39,56], the Erdős-Rényi graph [14], the configuration model [13,12,50] and the inhomogeneous random graph model [61]. FPP in the 'true' scale-free regime on the configuration model, (i.e.…”

Section: Introductionmentioning

confidence: 99%

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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Section: Introductionmentioning

confidence: 99%

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 first passage percolation on the complete graph equipped with independent and identically distributed positive and continuous edge weights. In contrast to earlier work [11,12,16,20,27], we consider the case where the extreme values of the edge weights are highly separated.…”

Section: Model and Summary Of Resultsmentioning

confidence: 99%

“…Since H n is integer-valued, it follows that (2.10) must fail in this case. First order asymptotics are investigated in [20], and it is shown that H n /(s n log n)…”

Section: The Universality Class In Terms Of S Nmentioning

confidence: 99%

“…In view of ( 2.5 ), for , the first passage percolation problem approximates the graph metric, where the approximation is stronger the faster tends to zero. We distinguish four different scaling regimes according to how fast : Firstly, : the case that for falls into this class with (see [ 20 , Sect. 1.2]) and was investigated in [ 16 ], where it is proved that is concentrated on at most two values.…”

Section: The Universal Picturementioning

confidence: 99%

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Long Paths in First Passage Percolation on the Complete Graph II. Global Branching Dynamics

et al. 2020

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We study the random geometry of first passage percolation on the complete graph equipped with independent and identically distributed positive edge weights. We consider the case where the lower extreme values of the edge weights are highly separated. This model exhibits strong disorder and a crossover between local and global scales. Local neighborhoods are related to invasion percolation that display self-organised criticality. Globally, the edges with relevant edge weights form a barely supercritical Erdős-Rényi random graph that can be described by branching processes. This near-critical behaviour gives rise to optimal paths that are considerably longer than logarithmic in the number of vertices, interpolating between random graph and minimal spanning tree path lengths. Crucial to our approach is the quantification of the extreme-value behavior of small edge weights in terms of a sequence of parameters (s n) n≥1 that characterises the different universality classes for first passage percolation on the complete graph. We investigate the case where s n → ∞ with s n = o(n 1/3), which corresponds to the barely supercritical setting. We identify the scaling limit of the weight of the optimal path between two vertices, and we prove that the number of edges in this path obeys a central limit theorem with mean approximately s n log (n/s 3 n) and variance s 2 n log (n/s 3 n). Remarkably, our proof also applies to n-dependent edge weights of the form E s n , where E is an exponential random variable with mean 1, thus settling a conjecture of Bhamidi et al. (Weak disorder asymptotics in the stochastic meanfield model of distance. Ann Appl Probab 22(1):29-69, 2012). 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 Eckhoff et al. (Long paths in first passage percolation on the complete graph I.

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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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Short Paths for First Passage Percolation on the Complete Graph

Cited by 16 publications

References 25 publications

Explosion in weighted hyperbolic random graphs and geometric inhomogeneous random graphs

Explosion in weighted hyperbolic random graphs and geometric inhomogeneous random graphs

Long Paths in First Passage Percolation on the Complete Graph II. Global Branching Dynamics

Takeover times for a simple model of network infection

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