Precision as a measure of predictability of missing links in real networks

Random hyperbolic graphs in d+1 dimensions

We consider random hyperbolic graphs in hyperbolic spaces of any dimension d+1≥2. We present a rescaling of model parameters that casts the random hyperbolic graph model of any dimension to a unified mathematical framework, leaving the degree distribution invariant with respect to the dimension. Unlike the degree distribution, clustering does depend on the dimension, decreasing to 0 at d→∞. We analyze all of the other limiting regimes of the model, and we release a software package that generates random hyperbolic graphs and their limits in hyperbolic spaces of any dimension.

The D-Mercator method for the multidimensional hyperbolic embedding of real networks

One of the pillars of the geometric approach to networks has been the development of model-based mapping tools that embed real networks in its latent geometry. In particular, the tool Mercator embeds networks into the hyperbolic plane. However, some real networks are better described by the multidimensional formulation of the underlying geometric model. Here, we introduce D-Mercator, a model-based embedding method that produces multidimensional maps of real networks into the (D + 1)-hyperbolic space, where the similarity subspace is represented as a D-sphere. We used D-Mercator to produce multidimensional hyperbolic maps of real networks and estimated their intrinsic dimensionality in terms of navigability and community structure. Multidimensional representations of real networks are instrumental in the identification of factors that determine connectivity and in elucidating fundamental issues that hinge on dimensionality, such as the presence of universality in critical behavior.

Enhancing Predictability Assessment: An Overview and Analysis of Predictability Measures for Time Series and Network Links

Driven by the variety of available measures intended to estimate predictability of diverse objects such as time series and network links, this paper presents a comprehensive overview of the existing literature in this domain. Our overview delves into predictability from two distinct perspectives: the intrinsic predictability, which represents a data property independent of the chosen forecasting model and serves as the highest achievable forecasting quality level, and the realized predictability, which represents a chosen quality metric for a specific pair of data and model. The reviewed measures are used to assess predictability across different objects, starting from time series (univariate, multivariate, and categorical) to network links. Through experiments, we establish a noticeable relationship between measures of realized and intrinsic predictability in both generated and real-world time series data (with the correlation coefficient being statistically significant at a 5% significance level). The discovered correlation in this research holds significant value for tasks related to evaluating time series complexity and their potential to be accurately predicted.

A Link Prediction Algorithm Based on Weighted Local and Global Closeness

Link prediction aims to identify unknown or missing connections in a network. The methods based on network structure similarity, known for their simplicity and effectiveness, have garnered widespread attention. A core metric in these methods is "proximity", which measures the similarity or linking probability between two nodes. These methods generally operate under the assumption that node pairs with higher proximity are more likely to form new connections. However, the accuracy of existing node proximity-based link prediction algorithms requires improvement. To address this, this paper introduces a Link Prediction Algorithm Based on Weighted Local and Global Closeness (LGC). This algorithm integrates the clustering coefficient to enhance prediction accuracy. A significant advantage of LGC is its dual consideration of a network's local and global features, allowing for a more precise assessment of node similarity. In experiments conducted on ten real-world datasets, the proposed LGC algorithm outperformed eight traditional link prediction methods, showing notable improvements in key evaluation metrics, namely precision and AUC.

Hyperbolic disc embedding of functional human brain connectomes using resting-state fMRI

The brain presents a real complex network of modular, small-world, and hierarchical nature, which are features of non-Euclidean geometry. Using resting-state functional magnetic resonance imaging, we constructed a scale-free binary graph for each subject, using internodal time series correlation of regions of interest as a proximity measure. The resulting network could be embedded onto manifolds of various curvatures and dimensions. While maintaining the fidelity of embedding (low distortion, high mean average precision), functional brain networks were found to be best represented in the hyperbolic disc. Using the 𝕊¹/ℍ² model, we reduced the dimension of the network into two-dimensional hyperbolic space and were able to efficiently visualize the internodal connections of the brain, preserving proximity as distances and angles on the hyperbolic discs. Each individual disc revealed relevance with its anatomic counterpart and absence of center-spaced node. Using the hyperbolic distance on the 𝕊¹/ℍ² model, we could detect the anomaly of network in autism spectrum disorder subjects. This procedure of embedding grants us a reliable new framework for studying functional brain networks and the possibility of detecting anomalies of the network in the hyperbolic disc on an individual scale.

Information evolution in complex networks

Many biological phenomena or social events critically depend on how information evolves in complex networks. However, a general theory to characterize information evolution is yet absent. Consequently, numerous unknowns remain about the mechanisms underlying information evolution. Among these unknowns, a fundamental problem, being a seeming paradox, lies in the coexistence of local randomness, manifested as the stochastic distortion of information content during individual-individual diffusion, and global regularity, illustrated by specific non-random patterns of information content on the network scale. Here, we attempt to formalize information evolution and explain the coexistence of randomness and regularity in complex networks. Applying network dynamics and information theory, we discover that a certain amount of information, determined by the selectivity of networks to the input information, frequently survives from random distortion. Other information will inevitably experience distortion or dissipation, whose speeds are shaped by the diversity of information selectivity in networks. The discovered laws exist irrespective of noise, but noise accounts for disturbing them. We further demonstrate the ubiquity of our discovered laws by analyzing the emergence of neural tuning properties in the primary visual and medial temporal cortices of animal brains and the emergence of extreme opinions in social networks.

Progresses and challenges in link prediction

Link prediction is a paradigmatic problem in network science, which aims at estimating the existence likelihoods of nonobserved links, based on known topology. After a brief introduction of the standard problem and evaluation metrics of link prediction, this review will summarize representative progresses about local similarity indices, link predictability, network embedding, matrix completion, ensemble learning, and some others, mainly extracted from related publications in the last decade. Finally, this review will outline some long-standing challenges for future studies.