Abstract
Hyperbolic geometry appears to be intrinsic in many large real networks. We construct and implement a new maximum likelihood estimation algorithm that embeds scale-free graphs in the hyperbolic space. All previous approaches of similar embedding algorithms require at least a quadratic runtime. Our algorithm achieves quasi-linear runtime, which makes it the first algorithm that can embed networks with hundreds of thousands of nodes in less than one hour. We demonstrate the performance of our algorithm on artificial and real networks. In all typical metrics, such as log-likelihood and greedy routing, our algorithm discovers embeddings that are very close to the ground truth.
Highlights
The study and analysis of complex real-world networks is a rapidly growing field
There are a number of commonly observed properties of complex networks like power-law degree distribution, small clustering coefficient, and small average distances
Most accessible for mathematical analysis is the inhomogeneous random graph model by van der Hofstad [33], which generalizes the models of Chung and Lu [10, 1, 2] and Norros and Reittu [26]
Summary
The study and analysis of complex real-world networks is a rapidly growing field. There are a number of commonly observed properties of complex networks like power-law degree distribution, small clustering coefficient, and small average distances. A general algorithm for embedding a network in a hyperbolic space was later presented by Papadopoulos et al [29] Their HyperMap algorithm is an approximate maximum likelihood estimation (MLE) algorithm. We substantially improve runtime by using the geometric data structure of Bringmann et al [9] that allows traversing nodes of close proximity in expected amortized constant time This enables us to embed significantly larger graphs than before. We investigate the performance of two classical methods of embedding graphs in the Euclidean space, namely spring embedders and maximum variance unfolding, when applied to the hyperbolic space (Sections 3 and 4) We find that both of them can work under some strong assumptions, but generally fail to translate to large real-world graphs
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