Local-ring network automata and the impact of hyperbolic geometry in complex network link-prediction

Local-ring network automata and the impact of hyperbolic geometry in complex network link-prediction Alessandro Muscoloni1, Umberto Michieli1,2 and Carlo Vittorio Cannistraci1,3,* 1Biomedical Cybernetics Group, Biotechnology Center (BIOTEC), Center for Molecular and Cellular Bioengineering (CMCB), Center for Systems Biology Dresden (CSBD), Department of Physics, Technische Universität Dresden, Tatzberg 47/49, 01307 Dresden, Germany 2Department of Information Engineering, University of Padova – Via Gradenigo, 6/b, 35131 Padova, Italy 3Brain bio-inspired computation (BBC) lab, IRCCS Centro Neurolesi “Bonino Pulejo”, Messina, Italy *Corresponding author: kalokagathos.agon@gmail.com

Abstract Methods for topological link-prediction are generally referred as global or local. The former exploits the entire network topology, the latter adopts only the immediate neighbourhood of the link to predict. Global methods are ‘believed’ to be the best performing. Is this common belief well-founded? Stochastic-Block-Model (SBM) is a global method believed as one of the best link-predictors and widely accepted as reference when new methods are proposed. But, our results suggest that SBM, whose computational time is high, cannot in general overcome the Cannistraci-Hebb (CH) network automaton model that is a simple local-learning-rule of topological self- organization proved by multiple sources as the current best local-based and parameter-free deterministic rule for link-prediction. In order to elucidate the reasons of this unexpected result, we formally introduce the notion of local-ring network automata models and their tight relation with the nature of common-neighbours’ definition in complex network theory. In addition, after extensive tests, we recommend Structural-Perturbation-Method (SPM) as the new best global method baseline. However, even SPM overall does not outperform CH and in several evaluation frameworks we astonishingly found the opposite. In particular, CH was the best predictor for synthetic networks generated by the Popularity-Similarity-Optimization (PSO) model, and its performance in PSO networks with community structure was even better than using the original internode-hyperbolic-distance as link-predictor. Interestingly, when tested on non-hyperbolic synthetic networks the performance of CH significantly dropped down indicating that this rule of network self-organization could be strongly associated to the rise of hyperbolic geometry in complex networks. In conclusion, we warn the scientific community: the superiority of global methods in link- prediction seems a ‘misleading belief’ caused by a latent geometry bias of the few small networks used as benchmark in previous studies. Therefore, we urge the need to found a latent geometry theory of link-prediction in complex networks.

Keywords: topological link-prediction, stochastic block model, Cannistraci-Hebb model and Cannistraci-Resource-Allocation (CRA) rule, local-ring network automata, local-community-paradigm and epitopological learning, network models and latent geometry. 2

1. Introduction The aim of topological link-prediction is to detect, in a given network, the non-observed links that could represent missing information or that may appear in the future, only exploiting features intrinsic to the network topology. It has a wide range of real applications, like suggesting friendships in social networks or predicting interactions in biological networks [1]– [3]. Although this study is focused on monopartite networks, link-prediction has recently been successfully implemented also in different types of network topologies such as bipartite [4], [5] and multilayer networks [6]. The link-prediction methods, according to the type of topological information exploited, can be broadly classified in two main categories: global and local. Global methods take advantage of the entire network topology in order to assign a likelihood score to a certain non-observed link. On the contrary, local approaches take into consideration only information about the neighbourhood of the link under analysis [1], [3]. In 2009, Guimerà et al. proposed a new global inference framework based on stochastic block model (SBM) in order to identify both missing and spurious interactions in noisy network observations [7]. The general idea of a block model is that the nodes are partitioned into groups and the probability that two nodes are connected depends only on the groups to which they belong. The framework introduced is a global approach where, assuming that there is no prior knowledge about which partition is more suitable for the observed network, the likelihood of a link can be computed theoretically considering all the possible partitions of the network into groups. Since this is not possible in practice, the Metropolis algorithm, which is based on a stochastic procedure, is exploited in order to sample only a

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