Discrete Hyperbolic Random Graph Model

Abstract

The hyperbolic random graph model (HRG) has proven useful in the analysis of scale-free networks, which are ubiquitous in many fields, from social network analysis to biology. However, working with this model is algorithmically and conceptually challenging because of the nature of the distances in the hyperbolic plane. In this paper, we propose a discrete variant of the HRG model where nodes are mapped to the vertices of a triangulation; our algorithms allow us to work with this model in a simple yet efficient way. We present experimental results conducted on networks, both real-world and simulated, to evaluate the practical benefits of DHRG in comparison to the HRG model.Hyperbolic geometry has been discovered by the 19th century mathematicians wondering about the nature of parallel lines. One of the properties of this geometry is that the amount of an area in the distance d from a given point is exponential in d; intuitively, the metric structure of the hyperbolic plane is similar to that of an infinite binary tree, except that each vertex additionally connects to two adjacent vertices on the same level.Recently, hyperbolic geometry has proven useful in modeling hierarchical data [LRP95,Mun98]. In particular, it has found application in the analysis of scale-free networks, which are ubiquitous in many fields, from network analysis to biology [PKS + 12]. In the hyperbolic random graph model (HRG), we place the nodes randomly in a hyperbolic disk; closer nodes are more likely to be connected. The properties of a HRG, such as its power-law degree distribution or high clustering coefficient, are similar to those of real-world scale-free networks [GPP12]. Due to high clustering coefficients, HRG is more accurate than earlier models such as Preferential Attachment [BA99] in modeling real-world networks.Perhaps the two most important algorithmic problems related to HRGs are sampling (generate a HRG) and MLE embedding: given a real-world network H = (V, E), map the vertices of H to the hyperbolic plane in such a way that the edges are predicted as accurately as possible. The quality of this prediction is measured with log-likelihood, computed with the formula v,w log p(v, w), where p(v, w) is the probability that the model correctly predicts the existence or not of edge (v, w). Those problems are non-trivial: even simply computing the log-likelihood, using a naive algorithm would require time O(|V | 2 ). The 1

Dynamic Distances in Hyperbolic Graphs

“…If G is a regularly generated hyperbolic triangulation, Properties 3 and 6 yield a simple algorithm for computing the distance between two vertices v 1 , v 2 ∈ 0 1 11 21 1 21 21 1 2 1 21 1 21 1 2 1 21 1 2 1 1 2 1 2 1 1 2 1 2 1 1 2 1 1 2 1 2 1 1 2 1 2 11 1 2 11 1 2 1 1 2 1 1 1 21 1 1 21 1 1 2 1 1 21 1 1 2 1 1 1 2 1 1 1 2 1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1 1 2 1 1 1 2 1 1 1 2 1 1 5 4 1 2 3 2 4 6 4 2 3 2 3 2 4 6 4 2 3 2 4 6 1 5 4 1 2 3 2 4 6 1 V (G) [8]. For d = min{δ 0 (v 1 ), δ 0 (v 2 )}, .…”

“…Another benefit is that we get to use the techniques from this paper to easily obtain efficient algorithms for the relevant computations. The experimental results will be discussed in detail in another paper [8] ; here we present it just as an example area of application.…”

“…Our experiments show that, despite using a discrete approximation, we get a better estimate of continuous log-likelihood than the method from [2], and furthermore, we can improve an embedding by locally moving vertices in order to improve the log-likelihood -this is not only more efficient than the O(n 2 ) method given in [2], but also turns out to produce higher quality embeddings when remapped back to the continuous hyperbolic plane. [8] Pseudo-betweenness A major issue in social network analysis is to find the important nodes in the network. This is done using centrality measures -functions f : V → R which say how important v ∈ V is.…”

“…In a tree, the shortest path from a to c is always a subset of U , the union of the shortest path from a to b, and the shortest path from b to c. In a graph of Gromov hyperbolicity δ, (any) shortest path from a to c is always in the δ-neighborhood of U . Our framework generalizes the theoretical ideas underlying our another paper [8], which focuses on the experimental results of applying them to the HRG model.…”