Core congestion is inherent in hyperbolic networks

European Journal of Combinatorics, 2002

Given a connected graph G, we take, as usual, the distance xy between any two verticesx , y of G to be the length of some geodesic between x and y. The graph G is said to be δ - hyperbolic, for some δ ≥ 0, if for all vertices x,y , u, v in G the inequality holds, and G isbridged if it contains no finite isometric cycles of length four or more. In this paper, we will show that a finite connected bridged graph is 1-hyperbolic if and only if it does not contain any of a list of six graphs as an isometric subgraph.

arXiv: Combinatorics, 2020

In this paper we present a novel approach in extremal set theory which may be viewed as an asymmetric version of Katona's permutation method. We use it to find more Turan numbers of hypergraphs in the Erdős--Ko--Rado range. An $(a,b)$-path $P$ of length $2k-1$ consists of $2k-1$ sets of size $r=a+b$ as follows. Take $k$ pairwise disjoint $a$-element sets $A_0, A_2, \dots, A_{2k-2}$ and other $k$ pairwise disjoint $b$-element sets $B_1, B_3, \dots, B_{2k-1}$ and order them linearly as $A_0, B_1, A_2, B_3, A_4\dots$. Define the (hyper)edges of $P_{2k-1}(a,b)$ as the sets of the form $A_i\cup B_{i+1}$ and $B_j\cup A_{j+1}$. The members of $P$ can be represented as $r$-element intervals of the $ak+bk$ element underlying set. Our main result is about hypergraphs that are blowups of trees, and implies that for fixed $k,a,b$, as $n\to \infty$ \[ {\rm ex}_r(n,P_{2k-1}(a,b)) = (k - 1){n \choose r - 1} + o(n^{r - 1}).\] This generalizes the Erdős--Gallai theorem for graphs which is the ca...

IEEE Transactions on Information Theory, 2017

Data centers play an important role in today's Internet development. Research to find scalable architecture and efficient routing algorithms for data center networks has gained popularity. The fat-tree architecture, which is essentially a folded version of a Clos network, has proven to be readily implementable and is scalable. In this paper, we investigate routing on a fat-tree network by deriving its global packing number and by presenting explicit algorithms for the construction of optimal, load-balanced routing solutions. Consider an optical network that employs Wavelength Division Multiplexing in which every user node sets up a connection with every other user node. The global packing number is basically the number of wavelengths required by the network to support such a traffic load, under the restriction that each source-to-destination connection is assigned a wavelength that remains constant in the network. In mathematical terms, consider a bidirectional, simple graph, G and let N ⊆ V (G) be a set of nodes. A path system P of G with respect to N consists of |N |(|N |−1) directed paths, one path to connect each of the source-destination node pairs in N. The global packing number of a path system P, denoted by Φ(G, N, P), is the minimum integer k to guarantee the existence of a mapping φ : P → {1, 2,. .. , k}, such that φ(P) = φ(P) if P and P have common arc(s). The global packing number of (G, N), denoted by Φ(G, N), is defined to be the minimum Φ(G, N, P) among all possible path systems P. In additional to Wavelength Division optical networks, this number also carries significance for networks employing Time Division Multiple Access (TDMA). In this paper, we compute by explicit route construction the global packing number of (Tn, N), where Tn denotes the topology of the n-ary fat-tree network, and N is considered to be the set of all edge switches or the set of all supported hosts. We show that the constructed routes are load-balanced and require minimal link capacity at all network links.

Electronic Journal of Probability, 2021

In this paper we consider the clustering coefficient, and clustering function in a random graph model proposed by Krioukov et al. in 2010. In this model, nodes are chosen randomly inside a disk in the hyperbolic plane and two nodes are connected if they are at most at a certain hyperbolic distance from each other. It has been previously shown that this model has various properties associated with complex networks, including a power-law degree distribution, "short distances" and a nonvanishing clustering coefficient. The model is specified using three parameters: the number of nodes n, which we think of as going to infinity, and α, ν > 0, which we think of as constant. Roughly speaking, the parameter α controls the power law exponent of the degree sequence and ν the average degree. Here we show that the clustering coefficient tends in probability to a constant γ that we give explicitly as a closed form expression in terms of α, ν and certain special functions. This improves earlier work by Gugelmann et al., who proved that the clustering coefficient remains bounded away from zero with high probability, but left open the issue of convergence to a limiting constant. Similarly, we are able to show that c(k), the average clustering coefficient over all vertices of degree exactly k, tends in probability to a limit γ(k) which we give explicitly as a closed form expression in terms of α, ν and certain special functions. We are able to extend this last result also to sequences (kn)n where kn grows as a function of n. Our results show that γ(k) scales differently, as k grows, for different ranges of α. More precisely, there exists constants cα,ν depending on α and ν, such that as k → ∞, γ(k) ∼ cα,ν • k 2−4α if 1 2 < α < 3 4 , γ(k) ∼ cα,ν • log(k) • k −1 if α = 3 4 and γ(k) ∼ cα,ν • k −1 when α > 3 4. These results contradict a claim of Krioukov et al., which stated that γ(k) should always scale with k −1 as we let k grow.

Through detailed analysis of scores of publicly available data sets corresponding to a wide range of large-scale networks, from communication and road networks to various forms of social networks, we explore a little-studied geometric characteristic of real-life networks, namely their hyperbolicity. In smooth geometry, hyperbolicity captures the notion of negative curvature; within the more abstract context of metric spaces, it can be generalized as d-hyperbolicity. This generalized definition can be applied to graphs, which we explore in this report. We provide strong evidence that communication and social networks exhibit this fundamental property, and through extensive computations we quantify the degree of hyperbolicity of each network in comparison to its diameter. By contrast, and as evidence of the validity of the methodology, applying the same methods to the road networks shows that they are not hyperbolic, which is as expected. Finally, we present practical computational me...

Lecture Notes in Computer Science, 2014

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Applied Mathematics Letters, 2011

If X is a geodesic metric space and x 1 ,

Symmetry

Gromov hyperbolicity is an interesting geometric property, and so it is natural to study it in the context of geometric graphs. In particular, we are interested in interval and indifference graphs, which are important classes of intersection and Euclidean graphs, respectively. Interval graphs (with a very weak hypothesis) and indifference graphs are hyperbolic. In this paper, we give a sharp bound for their hyperbolicity constants. The main result in this paper is the study of the hyperbolicity constant of every interval graph with edges of length 1. Moreover, we obtain sharp estimates for the hyperbolicity constant of the complement of any interval graph with edges of length 1.

Designs, Codes and Cryptography, 2012

Fix integers n ≥ r ≥ 2. A clique partition of [n] r is a collection of proper subsets A 1 , A 2 ,. .. , A t ⊂ [n] such that i Ai r is a partition of [n] r. Let cp(n, r) denote the minimum size of a clique partition of [n] r. A classical theorem of de Bruijn and Erdős states that cp(n, 2) = n. In this paper we study cp(n, r), and show in general that for each fixed r ≥ 3, cp(n, r) ≥ (1 + o(1))n r/2 as n → ∞. We conjecture cp(n, r) = (1 + o(1))n r/2. This conjecture has already been verified (in a very strong sense) for r = 3 by Hartman-Mullin-Stinson. We give further evidence of this conjecture by constructing, for each r ≥ 4, a family of (1 + o(1))n r/2 subsets of [n] with the following property: no two r-sets of [n] are covered more than once and all but o(n r) of the r-sets of [n] are covered. We also give an absolute lower bound cp(n, r) ≥ n r / q+r−1 r when n = q 2 + q + r − 1, and for each r characterize the finitely many configurations achieving equality with the lower bound. Finally we note the connection of cp(n, r) to extremal graph theory, and determine some new asymptotically sharp bounds for the Zarankiewicz problem.

arXiv (Cornell University), 2017

Let G be a graph with the usual shortest-path metric. A graph is δ-hyperbolic if for every geodesic triangle T , any side of T is contained in a δ-neighborhood of the union of the other two sides. A graph is chordal if every induced cycle has at most three edges. A vertex separator set in a graph is a set of vertices that disconnects two vertices. In this paper we study the relation between vertex separator sets, some chordality properties which are natural generalizations of being chordal and the hyperbolicity of the graph. We also give a characterization of being quasi-isometric to a tree in terms of chordality and prove that this condition also characterizes being hyperbolic, when restricted to triangles, and having stable geodesics, when restricted to bigons.