Machine learning meets network science: dimensionality reduction for fast and efficient embedding of networks in the hyperbolic space (1602.06522v1)

Abstract: Complex network topologies and hyperbolic geometry seem specularly connected, and one of the most fascinating and challenging problems of recent complex network theory is to map a given network to its hyperbolic space. The Popularity Similarity Optimization (PSO) model represents - at the moment - the climax of this theory. It suggests that the trade-off between node popularity and similarity is a mechanism to explain how complex network topologies emerge - as discrete samples - from the continuous world of hyperbolic geometry. The hyperbolic space seems appropriate to represent real complex networks. In fact, it preserves many of their fundamental topological properties, and can be exploited for real applications such as, among others, link prediction and community detection. Here, we observe for the first time that a topological-based machine learning class of algorithms - for nonlinear unsupervised dimensionality reduction - can directly approximate the network's node angular coordinates of the hyperbolic model into a two-dimensional space, according to a similar topological organization that we named angular coalescence. On the basis of this phenomenon, we propose a new class of algorithms that offers fast and accurate coalescent embedding of networks in the hyperbolic space even for graphs with thousands of nodes.

• The paper introduces "coalescent embedding," a novel machine learning approach for fast and efficient embedding of networks in hyperbolic space using topological dimensionality reduction.
• This method employs angular coalescence to approximate node coordinates and integrates Repulsion-Attraction and Equidistant Adjustment strategies to optimize network topology before embedding.
• Empirical results show the proposed techniques significantly outperform state-of-the-art methods like Hypermap in performance and computational efficiency, especially for large networks.

Analyzing Techniques for Hyperbolic Network Embedding through Dimensionality Reduction

The intersection of machine learning and network science, as presented in the paper by Thomas et al., addresses the complex issue of embedding networks into hyperbolic space. The paper introduces an innovative approach to dimensionality reduction that facilitates the efficient embedding of network data through a class of algorithms termed "coalescent embedding."

Hyperbolic spaces are highly effective in representing real-world complex networks due to their ability to preserve fundamental topological features. These spaces are useful in applications such as link prediction and community detection. The Popularity Similarity Optimization (PSO) model forms the core theoretical basis of this paper, as it combines the node popularity and similarity dimensions to map complex networks to a hyperbolic plane. Nodes are represented by polar coordinates, where angular distances relate to similarities and radial distances relate to popularity.

The authors propose a novel machine learning approach involving topological-based algorithms for nonlinear unsupervised dimensionality reduction. This method approximates the network nodes' angular coordinates synonymous with the hyperbolic model, employing a configuration they term "angular coalescence." This phenomena is leveraged in creating efficient embedding algorithms for fast network mappings, even for networks comprising thousands of nodes.

The results of the paper are structured around a comparison of manifold-based (Isomap, Laplacian Eigenmaps) and minimum-curvilinearity-based (MCE, ncMCE) techniques. The noncentered Minimum Curvilinear Embedding (ncMCE) is particularly notable for its ability to linearize nonlinear circular patterns by employing a minimum spanning tree (MST) for hierarchical mapping. This approach is distinct from manifold-based methods, which maintain the circular node similarity pattern and offer better angular coordinate approximations for network embeddings at high temperatures where network organization degenerates.

Critically, the authors introduce two pre-weighting strategies: the Repulsion-Attraction (RA) rule and Edge Betweenness Centrality (EBC). These strategies optimize the perceived connectivity topology preceding the dimensionality reduction process. The integration of equidistant adjustment (EA) for node positioning significantly boosts the precision and performance of the coalescent embedding, as validated by simulations involving varying network conditions.

Empirical validation on synthetic networks generated by the PSO model demonstrates that the proposed RA and EA strategies enhance the performance beyond the existing state-of-the-art, Hypermap. This enhancement is particularly pronounced at lower network temperatures. Furthermore, the paper discerns the computational efficiency of the proposed methods, achieving over a 30% performance increase while requiring only a fraction of Hypermap's computing time.

Real-world applications of this research span across diverse domains such as network medicine and social science. The paper connects manifold learning theory with network geometry, heralding a potential paradigm shift in how hidden network structures are understood and manipulated.

The paper suggests that further exploration of this connection between physics and computational theory could uncover impactful applications and encourage the development of improved network embedding algorithms. Future research should aim to explore and refine these connections to further enrich the representational capacity and applied use of network embeddings within hyperbolic spaces.