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Several observations indicate the existence of a latent hyperbolic space behind real networks that makes their structure very intuitive in the sense that the probability for a connection is decreasing with the hyperbolic distance between the nodes. A remarkable network model generating random graphs along this line is the popularity-similarity optimisation (PSO) model, offering a scale-free degree distribution, high clustering and the small-world property at the same time. These results provide a strong motivation for the development of hyperbolic embedding algorithms, that tackle the problem of finding the optimal hyperbolic coordinates of the nodes based on the network structure. A very promising recent approach for hyperbolic embedding is provided by the noncentered minimum curvilinear embedding (ncMCE) method, belonging to the family of coalescent embedding algorithms. This approach offers a high-quality embedding at a low running time. In the present work we propose a further optimisation of the angular coordinates in this framework that seems to reduce the logarithmic loss and increase the greedy routing score of the embedding compared to the original version, thereby adding an extra improvement to the quality of the inferred hyperbolic coordinates.

Network theory has become ubiquitous in the study of complex systems composed of many interacting units1,2,3. Over the last two decades, the overwhelming number of studies using this approach in systems ranging from metabolic interactions to the level of the global economy have shown that the statistical analysis of the underlying graph structure can highlight non-trivial properties and reveal previously unseen relations1,2,3,4,5. Probably the most important universal features of networks representing real systems are the small-world property6,7, the high clustering coefficient8 and the scale-free degree distribution9,10. On the modelling ground, a large number of network models were proposed for capturing one (or several) of these properties in a simple mathematical framework, and a quite notable example among these is provided by the PSO model11, which reproduces all three properties simultaneously in a natural manner. In this approach the nodes are placed one by one on the native disk representation12 of the 2D hyperbolic plane with a logarithmically increasing radial coordinate and a random angular coordinate, and links are drawn with probabilities determined by the hyperbolic distance between the node pairs. In vague terms, the degree of nodes is determined by their radial coordinate (lower distance from the origin corresponds to larger degree), and the angular proximity of the nodes can be interpreted as a sort of similarity, where more similar nodes have a higher probability to be connected.

The idea that hidden metric spaces can play an important role in the structure of complex networks first arose in a study focusing on the self-similarity of scale-free networks13. This was followed by reports showing the signs of hidden geometric spaces behind protein interaction networks14,15, the Internet16,17,18,19,20, brain networks21,22, or the world trade network23, also revealing important connections between the the navigability of networks and hyperbolic spaces16,24,25. In parallel, practical tools for generating hyperbolic networks26 and methods for measuring the hyperbolicity of networks were also proposed27,28. In the recent years the geometric nature of weights29 and clustering30,31 was revealed, and further variants of the original PSO model were proposed for generating random hyperbolic networks with communities32,33.

In the present paper we propose an embedding algorithm combining a coalescent approach with likelihood optimisation based on the E-PSO model. One of the best performing dimension reduction techniques in Ref.36 was corresponding to the non-centered minimum curvilinear embedding (ncMCE)39, which also provides the starting point of our method. However, after obtaining the initial node coordinates based on ncMCE, we also apply an angular optimisation of the coordinates using a logarithmic loss function originating from the E-PSO model. We test the proposed approach on both synthetic and real network data, and compare the results with the outcome of HyperMap, the original ncMCE coalescent embedding and Mercator in terms of the achieved logarithmic loss and the greedy routing score (which is a model-free quality measure of the embeddings).

In the following, we briefly describe the necessary preliminaries together with our angular optimisation algorithm. Since the optimisation uses a logarithmic loss function based on the E-PSO model, we begin with the outline of the PSO and E-PSO models. This is followed by the definition of the loss function and a short description of two state-of-the-art embedding methods, the HyperMap and the Mercator algorithms. Finally, we provide a summary of the coalescent embedding algorithm ncMCE and describe the proposed optimisation of the angular coordinates.

The networks generated according to these rules have the small-world property, are scale-free (with a degree decay exponent equal to \\(1+1/\\beta \\)), and with an appropriate choice of T can be made also highly clustered (lower temperature results in larger average clustering coefficient)11. However, an important criticism raised against the PSO model is that for subgraphs spanning between nodes having a degree \\(k>k_{\\mathrm{min}}\\), we cannot observe the densification law seen in a couple of real networks when \\(k_{\\mathrm{min}}\\) is increased38.

A generalisation of the PSO model circumventing this problem was proposed in Refs.11,34, where the iteration rules listed above are extended by adding extra links also between already existing nodes as follows:

To generalise the concept of internal links further, it is also conceivable that after a while some of the connections are deleted. Along this line, we can extend the generalised PSO model with the deletion of the link between \\(L_-\\) number of connected pairs of old nodes at each time step. But how should the links be selected for deletion If the temperature T is set to 0, when creating new (either external or internal) links, we have to always connect the node pair from the candidates that is characterised by the smallest hyperbolic node to node distance. The opposite of this deterministic connection rule is easy to phrase: for \\(T=0\\) in each deletion step the link connecting the hyperbolically furthermost nodes is split up. Consequently, for \\(T=0\\) the case \\(L_+=L_-\\) gives back exactly the original PSO model.

At \\(T>0\\), a natural extension of the above concept is to assume a link removal process where the probability that a link will not be deleted corresponds to the usual PSO linking probability, and the complementary probability of this is the removal probability, according to which we remove at each time step \\(L_-\\) number of internal links at random. In this way, when \\(L_+=L_-\\) in the generalised PSO model, we add and remove the same number of internal links at each time step, and therefore, the resulting networks become equivalent to the networks generated by the original PSO model.

By taking \\(L=L_+ - L_-\\) as the net number of added and removed internal links per time step, we can also consider the analogous generalised E-PSO model, where all connections are created as external links at the appearance of the new nodes, without any additional link insertion or deletion. In this framework, by adjusting \\({\\bar{m}}_i\\), the expected number of links connected to the new node i at its appearance, the resulting network can be made equivalent to the generalised PSO model with the insertion and the deletion of internal links. The method is straightforward, we can simply use

In order to demonstrate that the introduction of the internal links during the network generation process can solve the problem of the lack of the densification in the subgraphs between nodes having a degree \\(k>k_{\\mathrm{min}}\\) observed in the original PSO model, in Fig. 1 we plot the average internal degree \\(< k_{\\mathrm {internal}}>\\) of the subgraphs spanning between nodes having a degree larger than a certain threshold \\(k_{\\mathrm{min}}\\) as a function of \\(k_{\\mathrm{min}}\\) for both positive and negative L values (indicated by different colours) at different \\(\\beta \\) and T parameters. When L is positive (analogous to generalised PSO networks, where at each time step the number of newly created internal links is larger than the number of deleted internal links), the average internal degree becomes larger as the degree threshold begins to increase. For \\(L=0\\) (corresponding to the case of the original PSO model) the average internal degree remains constant until the degree threshold does not become so large that the subgraphs become extremely small. And finally, for negative L (analogous to generalised PSO networks, where at each time step more internal links are deleted than created) with the increase of the degree threshold the average internal degree decreases even for relatively small values of the threshold. Note that the shape of the \\(< k_{\\mathrm {internal}}>-k_{\\mathrm{min}}\\) curve does not depend on the popularity fading parameter \\(\\beta \\), and thus, neither on the exponent \\(\\gamma \\) of the degree distribution, as opposed to the