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On the Hyperbolicity of SmallWorld and TreeLike Random Graphs
"... Abstract. Hyperbolicity is a property of a graph that may be viewed as being a “soft ” version of a tree, and recent empirical and theoretical work has suggested that many graphs arising in Internet and related data applications have hyperbolic properties. Here, we consider Gromov’s notion of δhype ..."
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Abstract. Hyperbolicity is a property of a graph that may be viewed as being a “soft ” version of a tree, and recent empirical and theoretical work has suggested that many graphs arising in Internet and related data applications have hyperbolic properties. Here, we consider Gromov’s notion of δhyperbolicity, and we establish several positive and negative results for smallworld and treelike random graph models. In particular, we show that smallworld random graphs built from underlying grid structures do not have strong improvement in hyperbolicity, even when the rewiring greatly improves decentralized navigation. On the other hand, for a class of treelike graphs called ringed trees that have constant hyperbolicity, adding random links among the leaves in a manner similar to the smallworld graph constructions may easily destroy the hyperbolicity of the graphs, except for a class of random edges added using an exponentially decaying probability function based on the ring distance among the leaves. Our study provides the first significant analytical results on the hyperbolicity of a rich class of random graphs, which shed light on the relationship between hyperbolicity and navigability of random graphs, as well as on the sensitivity of hyperbolic δ to noises in random graphs.
F.: Quantum networks: the anticore of spin chains
 Quantum Inf. Process
, 2014
"... The purpose of this paper is to exhibit a quantum network phenomenon—the anticore—that goes against the classical network concept of congestion core. Classical networks idealized as infinite, Gromov hyperbolic spaces with leastcost path routing (and subject to a technical condition on the Gromov ..."
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The purpose of this paper is to exhibit a quantum network phenomenon—the anticore—that goes against the classical network concept of congestion core. Classical networks idealized as infinite, Gromov hyperbolic spaces with leastcost path routing (and subject to a technical condition on the Gromov boundary) have a congestion core, defined as a subnetwork that routing paths have a high probability of visiting. Here, we consider quantum networks, more specifically spin chains, define the socalled maximum excitation transfer probability pmax(i, j) between spin i and spin j, and show that the central spin has among all other spins the lowest probability of being excited or transmitting its excitation. The anticore is singled out by analytical formulas for pmax(i, j), revealing the number theoretic properties of quantum chains. By engineering the chain, we further show that this probability can be made vanishingly small. 1
Random regular graphs are not asymptotically Gromov hyperbolic
, 2012
"... Abstract. In this paper we prove that random d–regular graphs with d ≥ 3 have traffic congestion of the order O(n log3d−1(n)) where n is the number of nodes and geodesic routing is used. We also show that these graphs are not asymptotically δ–hyperbolic for any non–negative δ almost surely as n→∞. 1 ..."
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Abstract. In this paper we prove that random d–regular graphs with d ≥ 3 have traffic congestion of the order O(n log3d−1(n)) where n is the number of nodes and geodesic routing is used. We also show that these graphs are not asymptotically δ–hyperbolic for any non–negative δ almost surely as n→∞. 1.
Central European Journal of Mathematics Non Hyperbolicity in Random Regular Graphs and their Traffic Characteristics Article category
"... USA. Abstract: In this paper we prove that random d–regular graphs with d ≥ 3 have traffic congestion of the order O(n log3d−1(n)) where n is the number of nodes and geodesic routing is used. We also show that these graphs are not asymptotically δ–hyperbolic for any non–negative δ almost surely as n ..."
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USA. Abstract: In this paper we prove that random d–regular graphs with d ≥ 3 have traffic congestion of the order O(n log3d−1(n)) where n is the number of nodes and geodesic routing is used. We also show that these graphs are not asymptotically δ–hyperbolic for any non–negative δ almost surely as n→∞.
SCALING OF CONGESTION IN SMALL WORLD NETWORKS
"... Abstract. In this report we show that in a planar exponentially growing network consisting of N nodes, congestion scales as O(N2 / log(N)) independently of how flows may be routed. This is in contrast to the O(N3/2) scaling of congestion in a flat polynomially growing network. We also show that with ..."
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Abstract. In this report we show that in a planar exponentially growing network consisting of N nodes, congestion scales as O(N2 / log(N)) independently of how flows may be routed. This is in contrast to the O(N3/2) scaling of congestion in a flat polynomially growing network. We also show that without the planarity condition, congestion in a small world network could scale as low as O(N1+), for arbitrarily small . These extreme results demonstrate that the small world property by itself cannot provide guidance on the level of congestion in a network and other characteristics are needed for better resolution. Finally, we investigate scaling of congestion under the geodesic flow, that is, when flows are routed on shortest paths based on a link metric. Here we prove that if the link weights are scaled by arbitrarily small or large multipliers then considerable changes in congestion may occur. However, if we constrain the linkweight multipliers to be bounded away from both zero and infinity, then variations in congestion due to such remetrization are negligible. 1.
TRAFFIC ANALYSIS IN RANDOM DELAUNAY TESSELLATIONS AND OTHER GRAPHS
"... Abstract. In this work we study the degree distribution, the maximum vertex and edge flow in nonuniform random Delaunay triangulations when geodesic routing is used. We also investigate the vertex and edge flow in ErdösRenyi random graphs, geometric random graphs, expanders and random k–regular g ..."
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Abstract. In this work we study the degree distribution, the maximum vertex and edge flow in nonuniform random Delaunay triangulations when geodesic routing is used. We also investigate the vertex and edge flow in ErdösRenyi random graphs, geometric random graphs, expanders and random k–regular graphs. Moreover we show that adding a random matching to the original graph can considerably reduced the maximum vertex flow. 1.
TRAFFIC CONGESTION IN EXPANDERS, (p, δ)–HYPERBOLIC SPACES AND PRODUCT OF TREES
"... Abstract. In this paper we define the notion of (p, δ)–Gromov hyperbolic space where we relax Gromov’s slimness condition to allow that not all but a positive fraction of all triangles are δ–slim. Furthermore, we study maximum vertex congestion under geodesic routing and show that it scales as Ω(p2n ..."
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Abstract. In this paper we define the notion of (p, δ)–Gromov hyperbolic space where we relax Gromov’s slimness condition to allow that not all but a positive fraction of all triangles are δ–slim. Furthermore, we study maximum vertex congestion under geodesic routing and show that it scales as Ω(p2n2/D2n) where Dn is the diameter of the graph. We also construct a constant degree family of expanders with congestion Θ(n2) in contrast with random regular graphs that have congestion O(n log3(n)). Finally, we study traffic congestion on graphs defined as product of trees. 1.
Wenjie Fang
"... Hyperbolicity is a property of a graph that may be viewed as being a “soft ” version of a tree, and recent empirical and theoretical work has suggested that many graphs arising in Internet and related data applications have hyperbolic properties. Here, we consider Gromov’s notion of δhyperbolicity, ..."
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Hyperbolicity is a property of a graph that may be viewed as being a “soft ” version of a tree, and recent empirical and theoretical work has suggested that many graphs arising in Internet and related data applications have hyperbolic properties. Here, we consider Gromov’s notion of δhyperbolicity, and we establish several positive and negative results for smallworld and treelike random graph models. First, we study the hyperbolicity of the class of Kleinberg smallworld random graphs KSW (n, d, γ), where n is the number of vertices in the graph, d is the dimension of the underlying base grid B, and γ is the smallworld parameter such that each node u in the graph connects to another node v in the graph with probability proportional to 1/dB(u, v)γ with dB(u, v) being the grid distance from u to v in the base grid B. We show that when γ = d, the parameter value allowing efficient decentralized routing in Kleinberg’s smallworld network, with probability 1 − o(1) the hyperbolic δ is Ω((log n) 11.5(d+1)+ε) for any ε> 0 independent of n. Comparing to the diameter of Θ(log n) in this case, it indicates that hyperbolicity is not significantly improved comparing to graph diameter even when the longrange connections greatly improves decentralized navigation. We also show that for other values of γ the hyperbolic δ is either at the same level or very close to the graph diameter, indicating poor hyperbolicity in these graphs as well.