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14
Diameters in supercritical random graphs via first passage percolation
"... We study the diameter of C1, the largest component of the Erdős-Rényi random graph G(n, p) in the emerging supercritical phase, i.e., for p = 1+ε n where ε3n → ∞ and ε = o(1). This parameter was extensively studied for fixed ε> 0, yet results for ε = o(1) outside the critical window were only ob ..."
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Cited by 19 (5 self)
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We study the diameter of C1, the largest component of the Erdős-Rényi random graph G(n, p) in the emerging supercritical phase, i.e., for p = 1+ε n where ε3n → ∞ and ε = o(1). This parameter was extensively studied for fixed ε> 0, yet results for ε = o(1) outside the critical window were only obtained very recently. Prior to this work, Riordan and Wormald gave precise estimates on the diameter, however these did not cover the entire supercritical regime (namely, when ε 3 n → ∞ arbitrarily slowly).Łuczak and Seierstad estimated its order throughout this regime, yet their upper and lower bounds differed by a factor of 1000 7. We show that throughout the emerging supercritical phase, i.e. for any ε = o(1) with ε 3 n → ∞, the diameter of C1 is with high probability asymptotic to D(ε, n) = (3/ε) log(ε 3 n). This constitutes the first proof of the asymptotics of the diameter valid throughout this phase. The proof relies on a recent structure result for the supercritical giant component, which reduces the problem of estimating distances between its vertices to the study of passage times in first-passage percolation. The main advantage of our method is its flexibility. It also implies that in the emerging supercritical phase the diameter of the 2-core of C1 is w.h.p. asymptotic to 2 D(ε, n), and the maximal distance in C1 between
Random graph asymptotics on high-dimensional tori
- Comm. Math. Phys
"... Abstract We investigate the scaling of the largest critical percolation cluster on a large d-dimensional torus, for nearest-neighbor percolation in high dimensions, or when d > 6 for sufficient spread-out percolation. We use a relatively simple coupling argument to show that this largest critica ..."
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Cited by 8 (5 self)
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Abstract We investigate the scaling of the largest critical percolation cluster on a large d-dimensional torus, for nearest-neighbor percolation in high dimensions, or when d > 6 for sufficient spread-out percolation. We use a relatively simple coupling argument to show that this largest critical cluster is, with high probability, bounded above by a large constant times V 2/3 and below by a small constant times V 2/3 (log V ) −4/3 , where V is the volume of the torus. We also give a simple criterion in terms of the subcritical percolation two-point function on Z d under which the lower bound can be improved to small constant times V 2/3 , i.e. we prove random graph asymptotics for the largest critical cluster on the highdimensional torus. This establishes a conjecture by Our method is crucially based on the results in
Flooding in Weighted Random Graphs
"... In this paper, we study the impact of edge weights on distances in diluted random graphs. We interpret these weights as delays, and take them as i.i.d exponential random variables. We analyze the weighted flooding time defined as the minimum time needed to reach all nodes from one uniformly chosen n ..."
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Cited by 7 (1 self)
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In this paper, we study the impact of edge weights on distances in diluted random graphs. We interpret these weights as delays, and take them as i.i.d exponential random variables. We analyze the weighted flooding time defined as the minimum time needed to reach all nodes from one uniformly chosen node, and the weighted diameter corresponding to the largest distance between any pair of vertices. Under some regularity conditions on the degree sequence of the random graph, we show that these quantities grow as the logarithm of n, when the size of the graph n tends to infinity. We also derive the exact value for the prefactors. These allow us to analyze an asynchronous randomized broadcast algorithm for random regular graphs. Our results show that the asynchronous version of the algorithm performs better than its synchronized version: in the large size limit of the graph, it will reach the whole network faster even if the local dynamics are similar on average. 1
The mixing time of the Newman–Watts small world
- in Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM
, 2012
"... Abstract. “Small worlds ” are large systems in which any given node has only a few con-nections to other points, but possessing the property that all pairs of points are connected by a short path, typically logarithmic in the number of nodes. The use of random walks for sampling a uniform element fr ..."
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Cited by 3 (0 self)
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Abstract. “Small worlds ” are large systems in which any given node has only a few con-nections to other points, but possessing the property that all pairs of points are connected by a short path, typically logarithmic in the number of nodes. The use of random walks for sampling a uniform element from a large state space is by now a classical technique; to prove that such a technique works for a given network, a bound on the mixing time is re-quired. However, little detailed information is known about the behaviour of random walks on small-world networks, though many predictions can be found in the physics literature. The principal contribution of this paper is to show that for a famous small-world random graph model known as the Newman–Watts small world, the mixing time is of order log2 n. This confirms a prediction of Richard Durrett, who proved a lower bound of order log2 n and an upper bound of order log3 n. 1.
ANATOMY OF THE GIANT COMPONENT: THE STRICTLY SUPERCRITICAL REGIME
"... Abstract. In a recent work of the authors and Kim, we derived a complete description of the largest component of the Erdős-Rényi random graph G(n, p) as it emerges from the critical window, i.e. for p = (1+ε)/n where ε 3 n → ∞ and ε = o(1), in terms of a tractable contiguous model. Here we provide ..."
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Cited by 3 (2 self)
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Abstract. In a recent work of the authors and Kim, we derived a complete description of the largest component of the Erdős-Rényi random graph G(n, p) as it emerges from the critical window, i.e. for p = (1+ε)/n where ε 3 n → ∞ and ε = o(1), in terms of a tractable contiguous model. Here we provide the analogous description for the supercritical giant component, i.e. the largest component of G(n, p) for p = λ/n where λ> 1 is fixed. The contiguous model is roughly as follows: Take a random degree sequence and sample a random multigraph with these degrees to arrive at the kernel; Replace the edges by paths whose lengths are i.i.d. geometric variables to arrive at the 2-core; Attach i.i.d. Poisson Galton-Watson trees to the vertices for the final giant component. As in the case of the emerging giant, we obtain this result via a sequence of contiguity arguments at the heart of which are Kim’s Poisson-cloning method and the Pittel-Wormald local limit theorems. 1.
Diameters of random circulant graphs
, 2013
"... The diameter of a graph measures the maximal distance between any pair of vertices. The diameters of many small-world networks, as well as a variety of other random graph models, grow logarithmically in the number of nodes. In contrast, the worst connected networks are cycles whose diameters incre ..."
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Cited by 3 (1 self)
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The diameter of a graph measures the maximal distance between any pair of vertices. The diameters of many small-world networks, as well as a variety of other random graph models, grow logarithmically in the number of nodes. In contrast, the worst connected networks are cycles whose diameters increase linearly in the number of nodes. In the present study we consider an intermediate class of examples: Cayley graphs of cyclic groups, also known as circulant graphs or multi-loop networks. We show that the diameter of a random circulant 2k-regular graph with n vertices scales as n1/k, and establish a limit theorem for the distribution of their diameters. We obtain analogous results for the distribution of the average distance and higher moments.
On the Spread of Random Graphs
, 2012
"... The spread of a connected graph G was introduced by Alon, Boppana and Spencer [1] and measures how tightly connected the graph is. It is defined as the maximum over all Lipschitz functions f on V (G) of the variance of f(X) when X is uniformly distributed on V (G). We investigate the spread for cer ..."
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The spread of a connected graph G was introduced by Alon, Boppana and Spencer [1] and measures how tightly connected the graph is. It is defined as the maximum over all Lipschitz functions f on V (G) of the variance of f(X) when X is uniformly distributed on V (G). We investigate the spread for certain models of sparse random graph; in particular for random regular graphs G(n, d), for Erdős-Rényi random graphs Gn,p in the supercritical range p> 1/n, and for a ‘small world’ model. For supercritical Gn,p, we show that if p = c/n with c> 1 fixed then with high probability the spread of the giant component is bounded, and we prove corresponding statements for other models of random graphs, including a model with random edge-lengths. We also give lower bounds on the spread for the barely supercritical case when p = (1+o(1))/n. Further, we show that for d large, with high probability the spread of G(n, d) becomes arbitrarily close to that of the complete graph Kn.
Traveling in randomly embedded random graphs
, 2015
"... We consider the problem of traveling among random points in Euclidean space, when only a random fraction of the pairs are joined by traversable connections. In particular, we show a threshold for a pair of points to be connected by a geodesic of length arbitrarily close to their Euclidean distance, ..."
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We consider the problem of traveling among random points in Euclidean space, when only a random fraction of the pairs are joined by traversable connections. In particular, we show a threshold for a pair of points to be connected by a geodesic of length arbitrarily close to their Euclidean distance, and analyze the minimum length Traveling Salesperson Tour, extending the Beardwood-Halton-Hammersley theorem to this setting.
