On the largest component of a hyperbolic model of complex networks

We consider a model for complex networks that was introduced by Krioukov et al. In this model, N points are chosen randomly inside a disk on the hyperbolic plane and any two of them are joined by an edge if they are within a certain hyperbolic distance. The N points are distributed according to a quasi-uniform distribution, which is a distorted version of the uniform distribution. The model turns out to behave similarly to the well-known Chung-Lu model, but without the independence between the edges. Namely, it exhibits a power-law degree sequence and small distances but, unlike the Chung-Lu model and many other well-known models for complex networks, it also exhibits clustering. The model is controlled by two parameters α and ν where, roughly speaking, α controls the exponent of the power-law and ν controls the average degree. The present paper focuses on the evolution of the component structure of the random graph. We show that (a) for α > 1 and ν arbitrary, with high probability, as the number of vertices grows, the largest component of the random graph has sublinear order; (b) for α < 1 and ν arbitrary with high probability there is a "giant" component of linear order, and (c) when α = 1 then there is a non-trivial phase transition for the existence of a linear-sized component in terms of ν

Typical distances in a geometric model for complex networks

We study typical distances in a geometric random graph on the hyperbolic plane. Introduced by Krioukov et al.~\cite{ar:Krioukov} as a model for complex networks, $N$ vertices are drawn randomly within a bounded subset of the hyperbolic plane and any two of them are joined if they are within a threshold hyperbolic distance. With appropriately chosen parameters, the random graph is sparse and exhibits power law degree distribution as well as local clustering. In this paper we show a further property: the distance between two uniformly chosen vertices that belong to the same component is doubly logarithmic in $N$, i.e., the graph is an ~\emph{ultra-small world}. More precisely, we show that the distance rescaled by $\log \log N$ converges in probability to a certain constant that depends on the exponent of the power law. The same constant emerges in an analogous setting with the well-known \emph{Chung-Lu} model for which the degree distribution has a power law tail.Comment: 38 page

Ricci Curvature of the Internet Topology

Analysis of Internet topologies has shown that the Internet topology has negative curvature, measured by Gromov's "thin triangle condition", which is tightly related to core congestion and route reliability. In this work we analyze the discrete Ricci curvature of the Internet, defined by Ollivier, Lin, etc. Ricci curvature measures whether local distances diverge or converge. It is a more local measure which allows us to understand the distribution of curvatures in the network. We show by various Internet data sets that the distribution of Ricci cuvature is spread out, suggesting the network topology to be non-homogenous. We also show that the Ricci curvature has interesting connections to both local measures such as node degree and clustering coefficient, global measures such as betweenness centrality and network connectivity, as well as auxilary attributes such as geographical distances. These observations add to the richness of geometric structures in complex network theory.Comment: 9 pages, 16 figures. To be appear on INFOCOM 201

The diameter of KPKVB random graphs

We consider a model for complex networks that was recently proposed as a model for complex networks by Krioukov et al. In this model, nodes are chosen randomly inside a disk in the hyperbolic plane and two nodes are connected if they are at most 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 and a strictly positive clustering coefficient. The model is specified using three parameters : the number of nodes $N$, which we think of as going to infinity, and $\alpha, \nu > 0$ which we think of as constant. Roughly speaking $\alpha$ controls the power law exponent of the degree sequence and $\nu$ the average degree. Earlier work of Kiwi and Mitsche has shown that when $\alpha < 1$ (which corresponds to the exponent of the power law degree sequence being $< 3$) then the diameter of the largest component is a.a.s.~polylogarithmic in $N$. Friedrich and Krohmer have shown it is a.a.s.~$\Omega(\log N)$ and they improved the exponent of the polynomial in $\log N$ in the upper bound. Here we show the maximum diameter over all components is a.a.s.~$O(\log N)$ thus giving a bound that is tight up to a multiplicative constant.Comment: very minor corrections since the last versio