In the last decades, the study of models for large real-world networks has been a very popular and active area of research. A reasonable model should not only replicate all the structural properties that are observed in real world networks (for example, heavy tailed degree distributions, high clustering and small diameter), but it should also be amenable to mathematical analysis. There are plenty of models that succeed in the first task but are hard to analyze rigorously. On the other hand, a multitude of proposed models, like classical random graphs, can be studied mathematically, but fail in creating certain aspects that are observed in real-world networks.Recently, Papadopoulos, Krioukov, Boguñá and Vahdat [INFOCOM'10] introduced a random geometric graph model that is based on hyperbolic geometry. The authors argued empirically and by some preliminary mathematical analysis that the resulting graphs have many of the desired properties. Moreover, by computing explicitly a maximum likelihood fit of the Internet graph, they demonstrated impressively that this model is adequate for reproducing the structure of real graphs with high accuracy.In this work we initiate the rigorous study of random hyperbolic graphs. We compute exact asymptotic expressions for the expected number of vertices of degree k for all k up to the maximum degree and provide small probabilities for large deviations. We also prove a constant lower bound for the clustering coefficient. In particular, our findings confirm rigorously that the degree sequence follows a power-law distribution with controllable exponent and that the clustering is nonvanishing.
In this paper we show how to use simple partitioning lemmas in order to embed spanning
In his seminal paper from 1952 Dirac showed that the complete graph on n ≥ 3 vertices remains Hamiltonian even if we allow an adversary to remove ⌊n/2⌋ edges touching each vertex. In 1960 Ghouila-Houri obtained an analogue statement for digraphs by showing that every directed graph on n ≥ 3 vertices with minimum in-and out-degree at least n/2 contains a directed Hamilton cycle. Both statements quantify the robustness of complete graphs (digraphs) with respect to the property of containing a Hamilton cycle.A natural way to generalize such results to arbitrary graphs (digraphs) is using the notion of local resilience. The local resilience of a graph (digraph) G with respect to a property P is the maximum number r such that G has the property P even if we allow an adversary to remove an r-fraction of (in-and out-going) edges touching each vertex. The theorems of Dirac and Ghouila-Houri state that the local resilience of the complete graph and digraph with respect to Hamiltonicity is 1/2. Recently, this statements have been generalized to random settings. Lee and Sudakov (2012) proved that the local resilience of a random graph with edge probability p = ω (log n/n) with respect to Hamiltonicity is 1/2 ± o(1). For random directed graphs, Hefetz, Steger and Sudakov (2014+) proved an analogue statement, but only for edge probability p = ω (log n/ √ n). In this paper we significantly improve their result to p = ω log 8 n/n , which is optimal up to the polylogarithmic factor.A Hamilton cycle in a graph or a directed graph is a cycle that passes through all the vertices of the graph exactly once, and a graph is Hamiltonian if it contains a Hamilton cycle. Hamiltonicity is one of the central notions in graph theory, and has been intensively studied by numerous researchers. It is well known that the problem of whether a given graph contains a Hamilton cycle is N P-complete. In fact, Hamiltonicity was one of Karp's 21 N P-complete problems [12].Since one can not hope for a general classification of Hamiltonian graphs, as a consequence of Karp's result, there is a large interest in deriving properties that are sufficient for Hamiltonicity. A classic result by Dirac from 1952 [7] states that every graph on n ≥ 3 vertices with minimum degree at least n/2 is Hamiltonian. This result is tight as the complete bipartite graph with parts of sizes that differ by one, K m,m+1 , is not Hamiltonian. Note that this theorem answers the following question: Starting with the complete graph on n vertices K n , what is the maximal integer ∆ such that for any subgraph H of K n with maximum degree ∆, the graph K n − H obtained by deleting the edges of H from K n is Hamiltonian? This question not only asks for a sufficient condition for a graph to be Hamiltonian, it also asks for a quantification for the "local robustness" of the complete graph with respect to Hamiltonicity.A natural generalization of this question is to replace the complete graph with some arbitrary base graph. Recently, questions of this type have drawn a lot of attention under...
Abstract. The internal diffusion limited aggregation (IDLA) process places n particles on the two dimensional integer grid. The first particle is placed on the origin; every subsequent particle starts at the origin and performs an unbiased random walk until it reaches an unoccupied position. In this work we study the computational complexity of determining the subset that is generated after n particles have been placed. We develop the first algorithm that provably outperforms the naive step-by-step simulation of all particles. Particularly, our algorithm has a running time of O(n log 2 n) and a sublinear space requirement of O(n 1/2 log n), both in expectation and with high probability. In contrast to some speedups proposed for similar models in the physics community, our algorithm samples from the exact distribution. To simulate a single particle fast we have to develop techniques for combining multiple steps of a random walk to large jumps without hitting a forbidden set of grid points. These techniques might be of independent interest for speeding up other problems based on random walks.
We say that a graph $G$ has the Ramsey property w.r.t.\ some graph $F$ and some integer $r\geq 2$, or $G$ is $(F,r)$-Ramsey for short, if any $r$-coloring of the edges of $G$ contains a monochromatic copy of $F$. R{\"o}dl and Ruci{\'n}ski asked how globally sparse $(F,r)$-Ramsey graphs $G$ can possibly be, where the density of $G$ is measured by the subgraph $H\subseteq G$ with the highest average degree. So far, this so-called Ramsey density is known only for cliques and some trivial graphs $F$. In this work we determine the Ramsey density up to some small error terms for several cases when $F$ is a complete bipartite graph, a cycle or a path, and $r\geq 2$ colors are available
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.