The different factors involved in the growth process of complex networks imprint valuable information in their observable topologies. How to exploit this information to accurately predict structural network changes is the subject of active research. A recent model of network growth sustains that the emergence of properties common to most complex systems is the result of certain trade-offs between node birth-time and similarity. This model has a geometric interpretation in hyperbolic space, where distances between nodes abstract this optimisation process. Current methods for network hyperbolic embedding search for node coordinates that maximise the likelihood that the network was produced by the afore-mentioned model. Here, a different strategy is followed in the form of the Laplacian-based Network Embedding, a simple yet accurate, efficient and data driven manifold learning approach, which allows for the quick geometric analysis of big networks. Comparisons against existing embedding and prediction techniques highlight its applicability to network evolution and link prediction. The gradual addition of nodes and edges to a network, a common representation of the relationships between complex system components, imprints valuable information in its topology. Consequently, development of techniques to mine the structure of networks is crucial to understand the factors that play a role in the formation of their observable architecture. Strong clustering and scale-free node degree distribution, properties common to most complex networks, have served as the basis for the establishment of tools for link prediction 1 , community detection 2 , identification of salient nodes 3 , and so on. In addition, several models that aim to mimic the evolution and formation of networks with the above-mentioned characteristics have been introduced 4 . Of special interest are a series of models that assume the existence of a hidden geometry underlying the structure of a network, shaping its topology 5-12 (we refer the reader to 13 for an extensive review on the subject). This is justified by the fact that complex networks possess characteristics commonly present in geometric objects, like scale invariance and self-similarity 7, [14] [15] [16] . One of such models is the so-called Popularity-Similarity (PS) model, which sustains that clustering and hierarchy are the result of an optimisation process involving two measures of attractiveness: node popularity and similarity between nodes 12 . Popularity reflects the property of a node to attract connections from other nodes over time, and it is thus associated with a node's seniority status in the system. On the other hand, nodes that are similar have a high likelihood of getting connected, regardless of their rank. The PS model has a geometric interpretation in hyperbolic space, where the trade-off between popularity and similarity is abstracted by the hyperbolic distance between nodes 9,12 . Short hyperbolic distances between them correlate strikingly well with high probabilities of link formation 12 . This means that mapping a network to hyperbolic space unveils the value of the variables in charge of shaping its topology (popularity and similarity in this case), allowing for a better understanding of the dynamics accountable for the system's growth process. Current efforts to infer the hyperbolic geometry of complex networks bet for a Maximum Likelihood Estimation (MLE) approach, in which the space of PS models with the same structural properties as the network of interest is explored, in search for the one that better fits the network topology 12, 17, 18 . This search is computationally demanding, which means that these methods require of correction steps or heuristics in order to make them suitable for big networks 18 . In this paper, a different strategy is followed. Inspired by the well-established field of non-linear dimensionality reduction in Machine Learning
For example, REF used Laplacian Eigenmaps to estimate the angular coordinates of nodes in the PS model.
Efficient embedding of complex networks to hyperbolic space via their Laplacian
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Computer Science Medicine