Metric Embedding, Hyperbolic Space, and Social Networks

On the Hyperbolicity of Large-Scale Networks

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...

Lower Bounds on the Distortion of Embedding Finite Metric Spaces in Graphs

Discrete & Computational Geometry, 1998

The main question discussed in this paper is how well a finite metric space of size n can be embedded into a graph with certain topological restrictions. The existing constructions of graph spanners imply that any n-point metric space can be represented by a (weighted) graph with n vertices and n 1+O(1/r) edges, with distances distorted by at most r. We show that this tradeoff between the number of edges and the distortion cannot be improved, and that it holds in a much more general setting. The main technical lemma claims that the metric space induced by an unweighted graph H of girth g cannot be embedded in a graph G (even if it is weighted) of smaller Euler characteristic, with distortion less than g/4 − 3 2. In the special case when |V (G)| = |V (H)| and G has strictly less edges than H , an improved bound of g/3 − 1 is shown. In addition, we discuss the case χ(G) < χ(H) − 1, as well as some interesting higher-dimensional analogues. The proofs employ basic techniques of algebraic topology.

Clustering in random geometric graphs on the hyperbolic plane

Clustering is a fundamental property of complex networks and it is the mathematical expression of a ubiquitous phenomenon that arises in various types of self-organized networks such as biological networks, computer networks or social networks. In this paper, we consider what is called the global clustering coefficient of random graphs on the hyperbolic plane. This model of random graphs was introduced recently by Krioukov et al. as a mathematical model of complex networks, which naturally embeds the (tree-like) hierarchical structure of a complex network into a hyperbolic space. We give a rigorous analysis of clustering and characterize the global clustering coefficient in terms of the parameters of the model. We show that the global clustering coefficient can be tuned by such parameters and we give an explicit formula for this function.

High Dimensional Hyperbolic Geometry of Complex Networks

Mathematics

High dimensional embeddings of graph data into hyperbolic space have recently been shown to have great value in encoding hierarchical structures, especially in the area of natural language processing, named entity recognition, and machine generation of ontologies. Given the striking success of these approaches, we extend the famous hyperbolic geometric random graph models of Krioukov et al. to arbitrary dimension, providing a detailed analysis of the degree distribution behavior of the model in an expanded portion of the parameter space, considering several regimes which have yet to be considered. Our analysis includes a study of the asymptotic correlations of degree in the network, revealing a non-trivial dependence on the dimension and power law exponent. These results pave the way to using hyperbolic geometric random graph models in high dimensional contexts, which may provide a new window into the internal states of network nodes, manifested only by their external interconnectiv...

A Bound for the Diameter of Random Hyperbolic Graphs

2015 Proceedings of the Twelfth Workshop on Analytic Algorithmics and Combinatorics (ANALCO), 2014

Random hyperbolic graphs were recently introduced by Krioukov et. al. [KPK + 10] as a model for large networks. Gugelmann, Panagiotou, and Peter [GPP12] then initiated the rigorous study of random hyperbolic graphs using the following model: for α > 1/2, C ∈ R, n ∈ N, set R = 2 ln n + C, and define the graph G = (V, E) with |V | = n as follows: For each v ∈ V , i.i.d. polar coordinates (r v , θ v ) are generated using the joint density function f (r, θ), with θ v uniformly chosen from [0, 2π) and r v has density f (r) = α sinh(αr) cosh(αR)−1 for 0 ≤ r < R. Two vertices are then joined by an edge if their hyperbolic distance is at most R. We prove that in the range 1/2 < α < 1 a.a.s. for any two vertices of the same component, their graph distance is O(polylog(n)), thus answering a question raised in [GPP12] concerning the diameter of such random graphs. As a corollary from our proof we obtain that the second largest component has size O(polylog(n)), thus answering a question of Bode, Fountoulakis and Müller . We also show that a.a.s. there exist isolated components forming a path of length Ω(log n), thus yielding a lower bound on the size of the second largest component.

Machine learning meets network science: dimensionality reduction for fast and efficient embedding of networks in the hyperbolic space

arXiv (Cornell University), 2016

Physicists recently observed that realistic complex networks emerge as discrete samples from a continuous hyperbolic geometry enclosed in a circle: the radius represents the node centrality and the angular displacement between two nodes resembles their topological proximity. The hyperbolic circle aims to become a universal space of representation and analysis of many real networks. Yet, inferring the angular coordinates to map a real network back to its latent geometry remains a challenging inverse problem. Here, we show that intelligent machines for unsupervised recognition and visualization of similarities in big data can also infer the network angular coordinates of the hyperbolic model according to a geometrical organization that we term "angular coalescence." Based on this phenomenon, we propose a class of algorithms that offers fast and accurate "coalescent embedding" in the hyperbolic circle even for large networks. This computational solution to an inverse problem in physics of complex systems favors the application of network latent geometry techniques in disciplines dealing with big network data analysis including biology, medicine, and social science.

Embedding Metrics into Ultrametrics and Graphs into Spanning Trees with Constant Average Distortion

SIAM Journal on Computing, 2015

This paper addresses the basic question of how well can a tree approximate distances of a metric space or a graph. Given a graph, the problem of constructing a spanning tree in a graph which strongly preserves distances in the graph is a fundamental problem in network design. We present scaling distortion embeddings where the distortion scales as a function of ǫ, with the guarantee that for each ǫ the distortion of a fraction 1−ǫ of all pairs is bounded accordingly. Such a bound implies, in particular, that the average distortion and ℓq-distortions are small. Specifically, our embeddings have constant average distortion and O( √ log n) ℓ2-distortion. This follows from the following results: we prove that any metric space embeds into an ultrametric with scaling distortion O( 1/ǫ). For the graph setting we prove that any weighted graph contains a spanning tree with scaling distortion O( 1/ǫ). These bounds are tight even for embedding in arbitrary trees. For probabilistic embedding into spanning trees we prove a scaling distortion ofÕ(log 2 (1/ǫ)), which implies constant ℓq-distortion for every fixed q < ∞. *

Results on hyperbolicity in graphs: a survey

2020

Let X be a geodesic metric space, and take x1, x2, x3 ∈ X. A geodesic triangle T = {x1, x2, x3} is the union of the three geodesics [x1x2], [x2x3] and [x3x1] in X. The space X is said to be δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. If X is hyperbolic, we denote by δ(X) the sharp hyperbolicity constant of X, that is, δ(X) := sup{δ(T ) : T is a geodesic triangle in X }. In this paper, we collect some of the main theoretical results on the hyperbolicity constant in graphs.

Machine learning meets complex networks via coalescent embedding in the hyperbolic space

Nature communications, 2017

Hyperbolic Embedding of Internet Graph for Distance Estimation and Overlay Construction

IEEE/ACM Transactions on Networking, 2000

Estimating distances in the Internet has been studied in the recent years due to its ability to improve the performance of many applications, e.g., in the peer-to-peer realm. One scalable approach to estimate distances between nodes is to embed the nodes in some dimensional geometric space and to use the pair distances in this space as the estimate for the real distances. Several algorithms were suggested in the past to do this in low dimensional Euclidean spaces.

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