Hierarchical structure and the prediction of missing links in networks

LinkPred: a high performance library for link prediction in complex networks

The problem of determining the likelihood of the existence of a link between two nodes in a network is called link prediction. This is made possible thanks to the existence of a topological structure in most real-life networks. In other words, the topologies of networked systems such as the World Wide Web, the Internet, metabolic networks, and human society are far from random, which implies that partial observations of these networks can be used to infer information about undiscovered interactions. Significant research efforts have been invested into the development of link prediction algorithms, and some researchers have made the implementation of their methods available to the research community. These implementations, however, are often written in different languages and use different modalities of interaction with the user, which hinders their effective use. This paper introduces LinkPred, a high-performance parallel and distributed link prediction library that includes the implementation of the major link prediction algorithms available in the literature. The library can handle networks with up to millions of nodes and edges and offers a unified interface that facilitates the use and comparison of link prediction algorithms by researchers as well as practitioners.

SPMLMI: predicting lncRNA-miRNA interactions in humans using a structural perturbation method

Long non-coding RNA (lncRNA)-microRNA (miRNA) interactions are quickly emerging as important mechanisms underlying the functions of non-coding RNAs. Accordingly, predicting lncRNA-miRNA interactions provides an important basis for understanding the mechanisms of action of ncRNAs. However, the accuracy of the established prediction methods is still limited. In this study, we used structural consistency to measure the predictability of interactive links based on a bilayer network by integrating information for known lncRNA-miRNA interactions, an lncRNA similarity network, and an miRNA similarity network. In particular, by using the structural perturbation method, we proposed a framework called SPMLMI to predict potential lncRNA-miRNA interactions based on the bilayer network. We found that the structural consistency of the bilayer network was higher than that of any single network, supporting the utility of bilayer network construction for the prediction of lncRNA-miRNA interactions. Applying SPMLMI to three real datasets, we obtained areas under the curves of 0.9512 ± 0.0034, 0.8767 ± 0.0033, and 0.8653 ± 0.0021 based on 5-fold cross-validation, suggesting good model performance. In addition, the generalizability of SPMLMI was better than that of the previously established methods. Case studies of two lncRNAs (i.e., SNHG14 and MALAT1) further demonstrated the feasibility and effectiveness of the method. Therefore, SPMLMI is a feasible approach to identify novel lncRNA-miRNA interactions underlying complex biological processes.

Relationship Prediction Based on Graph Model for Steam Turbine Control Valve

The control valve is an important piece of equipment in the steam turbine, which frequently suffers from the fault of the dead zone. The graph model is a promising method for dead zone detection, yet establishing an accurate and completed graph topology is not an easy task due to limited mechanism knowledge. Hence, a graph model is proposed to predict the links in the graph and estimate the relationship between variables of related equipment of the control valve. The graph convolution is conducted on the uncompleted graph to learn the low-level representations of the graph nodes, and the score function is used to evaluate the probability of the existence of links between a pair of graph nodes. Results demonstrate a test accuracy of 99.2% for the link prediction, and follow the principles of thermodynamics in the steam turbine. Consequently, the proposed graph model is capable of estimating the relationships for the steam turbine control valve, and other inter-connected industrial systems.

Missing link prediction and spurious link detection based on attractive force and community

With the rapid development of Internet and information technology, networks have become an important media of information diffusion in the global. In view of the increasing scale of network data, how to ensure the completeness and accuracy of the obtainable links from networks has been an urgent problem that needs to be solved. Different from most traditional link prediction methods only focus on the missing links, a novel link prediction approach is proposed in this paper to handle both the missing links and the spurious links in networks. At first, we define the attractive force for any pair of nodes to denote the strength of the relation between them. Then, all the nodes can be divided into some communities according to their degrees and the attractive force on them. Next, we define the connection probability for each pair of unconnected nodes to measure the possibility if they are connected, the missing links can be predicted by calculating and comparing the connection probabilities of all the pairs of unconnected nodes. Moreover, we define the break probability for each pair of connected nodes to measure the possibility if they are broken, the spurious links can also be detected by calculating and comparing the break probabilities of all the pairs of connected nodes. To verify the validity of the proposed approach, we conduct experiments on some real-world networks. The results show the proposed approach can achieve higher prediction accuracy and more stable performance compared with some existing methods.

Survey on graph embeddings and their applications to machine learning problems on graphs

Dealing with relational data always required significant computational resources, domain expertise and task-dependent feature engineering to incorporate structural information into a predictive model. Nowadays, a family of automated graph feature engineering techniques has been proposed in different streams of literature. So-called graph embeddings provide a powerful tool to construct vectorized feature spaces for graphs and their components, such as nodes, edges and subgraphs under preserving inner graph properties. Using the constructed feature spaces, many machine learning problems on graphs can be solved via standard frameworks suitable for vectorized feature representation. Our survey aims to describe the core concepts of graph embeddings and provide several taxonomies for their description. First, we start with the methodological approach and extract three types of graph embedding models based on matrix factorization, random-walks and deep learning approaches. Next, we describe how different types of networks impact the ability of models to incorporate structural and attributed data into a unified embedding. Going further, we perform a thorough evaluation of graph embedding applications to machine learning problems on graphs, among which are node classification, link prediction, clustering, visualization, compression, and a family of the whole graph embedding algorithms suitable for graph classification, similarity and alignment problems. Finally, we overview the existing applications of graph embeddings to computer science domains, formulate open problems and provide experiment results, explaining how different networks properties result in graph embeddings quality in the four classic machine learning problems on graphs, such as node classification, link prediction, clustering and graph visualization. As a result, our survey covers a new rapidly growing field of network feature engineering, presents an in-depth analysis of models based on network types, and overviews a wide range of applications to machine learning problems on graphs.

Link predictions for incomplete network data with outcome misclassification

Link prediction is a fundamental problem in network analysis. In a complex network, links can be unreported and/or under detection limits due to heterogeneous sources of noise and technical challenges during data collection. The incomplete network data can lead to an inaccurate inference of network based data analysis. We propose a parametric link prediction model and consider latent links as misclassified binary outcomes. We develop new algorithms to optimize model parameters and yield robust predictions of unobserved links. Theoretical properties of the predictive model are also discussed. We apply the new method to a partially observed social network data and incomplete brain network data. The results demonstrate that our method outperforms the existing latent-link prediction methods.

Inference of edge correlations in multilayer networks

Many recent developments in network analysis have focused on multilayer networks, which one can use to encode time-dependent interactions, multiple types of interactions, and other complications that arise in complex systems. Like their monolayer counterparts, multilayer networks in applications often have mesoscale features, such as community structure. A prominent approach for inferring such structures is the employment of multilayer stochastic block models (SBMs). A common (but potentially inadequate) assumption of these models is the sampling of edges in different layers independently, conditioned on the community labels of the nodes. In this paper, we relax this assumption of independence by incorporating edge correlations into an SBM-like model. We derive maximum-likelihood estimates of the key parameters of our model, and we propose a measure of layer correlation that reflects the similarity between the connectivity patterns in different layers. Finally, we explain how to use correlated models for edge "prediction" (i.e., inference) in multilayer networks. By incorporating edge correlations, we find that prediction accuracy improves both in synthetic networks and in a temporal network of shoppers who are connected to previously purchased grocery products.

Recovery patterns and physics of the network

In a progressively interconnected world, the loss of system resilience has consequences for human health, the economy, and the environment. Research has exploited the science of networks to explain the resilience of complex systems against random attacks, malicious attacks, and the localized attacks induced by natural disasters or mass attacks. Little is known about the elucidation of system recovery by the network topology. This study adds to the knowledge of network resilience by examining the nexus of recoverability and network topology. We establish a new paradigm for identifying the recovery behavior of networks and introduce the recoverability measure. Results indicate that the recovery response behavior and the recoverability measure are the function of both size and topology of networks. In small sized networks, the return to recovery exhibits homogeneous recovery behavior over topology, while the return shape is dispersed with an increase in the size of network. A network becomes more recoverable as connectivity measures of the network increase, and less recoverable as accessibility measures of network increase. Overall, the results not only offer guidance on designing recoverable networks, but also depict the recovery nature of networks deliberately following a disruption. Our recovery behavior and recoverability measure has been tested on 16 distinct network topologies. The relevant recovery behavior can be generalized based on our definition for any network topology recovering deliberately.

Review on Learning and Extracting Graph Features for Link Prediction

Link prediction in complex networks has attracted considerable attention from interdisciplinary research communities, due to its ubiquitous applications in biological networks, social networks, transportation networks, telecommunication networks, and, recently, knowledge graphs. Numerous studies utilized link prediction approaches in order sto find missing links or predict the likelihood of future links as well as employed for reconstruction networks, recommender systems, privacy control, etc. This work presents an extensive review of state-of-art methods and algorithms proposed on this subject and categorizes them into four main categories: similarity-based methods, probabilistic methods, relational models, and learning-based methods. Additionally, a collection of network data sets has been presented in this paper, which can be used in order to study link prediction. We conclude this study with a discussion of recent developments and future research directions.

Similarity-based link prediction in social networks using latent relationships between the users

Social network analysis has recently attracted lots of attention among researchers due to its wide applicability in capturing social interactions. Link prediction, related to the likelihood of having a link between two nodes of the network that are not connected, is a key problem in social network analysis. Many methods have been proposed to solve the problem. Among these methods, similarity-based methods exhibit good efficiency by considering the network structure and using as a fundamental criterion the number of common neighbours between two nodes to establish structural similarity. High structural similarity may suggest that a link between two nodes is likely to appear. However, as shown in the paper, the number of common neighbours may not be always sufficient to provide comprehensive information about structural similarity between a pair of nodes. To address this, a neighbourhood vector is first specified for each node. Then, a novel measure is proposed to determine the similarity of each pair of nodes based on the number of common neighbours and correlation between the neighbourhood vectors of the nodes Experimental results, on a range of different real-world networks, suggest that the proposed method results in higher accuracy than other state-of-the-art similarity-based methods for link prediction.

Seven-Layer Model in Complex Networks Link Prediction: A Survey

Link prediction is the most basic and essential problem in complex networks. This study analyzes the observed topological, time, attributive, label, weight, directional, and symbolic features and auxiliary information to find the lack of connection and predict the future possible connection. For discussion and analysis of the evolution of the network, the network model is of great significance. In the past two decades, link prediction has attracted extensive attention from experts in various fields, who have published numerous high-level papers, but few combine interdisciplinary characteristics. This survey analyzes and discusses the existing link prediction methods. The idea of stratification is introduced into the classification system of link prediction for the first time and proposes the design idea of a seven-layer model, namely the network, metadata, feature classification, selection input, processing, selection, and output layers. Among them, the processing layer divides link prediction methods into similarity-based, probabilistic, likelihood, supervised learning, semi-supervised learning, unsupervised learning, and reinforcement learning methods. The input features, evaluation metrics, complex analysis, experimental comparisons, relative merits, common dataset and open-source implementations for each link prediction method are then discussed in detail. Through analysis and comparison, we found that the link prediction method based on graph structure features has better prediction performance. Finally, the future development direction of link prediction in complex networks is discussed.

Multiscale community detection in Cytoscape

Detection of community structure has become a fundamental step in the analysis of biological networks with application to protein function annotation, disease gene prediction, and drug discovery. This recent impact creates a need to make these techniques and their accompanying visualization schemes available to a broad range of biologists. Here we present a service-oriented, end-to-end software framework, CDAPS (Community Detection APplication and Service), that integrates the identification, annotation, visualization, and interrogation of multiscale network communities, accessible within the popular Cytoscape network analysis platform. With novel design principles, CDAPS addresses unmet new challenges, such as identifying hierarchical community structures, comparison of outputs generated from diverse network resources, and easy deployment of new algorithms, to facilitate community-sourced science. We demonstrate that the CDAPS framework can be applied to high-throughput protein-protein interaction networks to gain novel insights, such as the identification of putative new members of known protein complexes.

Link Prediction through Deep Generative Model

Inferring missing links based on the currently observed network is known as link prediction, which has tremendous real-world applications in biomedicine, e-commerce, social media, and criminal intelligence. Numerous methods have been proposed to solve the link prediction problem. Yet, many of these methods are designed for undirected networks only and based on domain-specific heuristics. Here we developed a new link prediction method based on deep generative models, which does not rely on any domain-specific heuristic and works for general undirected or directed complex networks. Our key idea is to represent the adjacency matrix of a network as an image and then learn hierarchical feature representations of the image by training a deep generative model. Those features correspond to structural patterns in the network at different scales, from small subgraphs to mesoscopic communities. When applied to various real-world networks from different domains, our method shows overall superior performance against existing methods.

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A novel link prediction method based on deep generative models is developed

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This method works for general undirected or directed complex networks

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Leveraging structural patterns at different scales, this method outperforms others

Visualizing the Process of Disaster Mental Health Services in the Joso Flood by Network Analyses of Emails

Joso City, Ibaraki Prefecture, Japan was severely affected by flooding of the River Kinugawa in September 2015. Local psychiatric organizations immediately began providing disaster mental health services (DMHS). In post-disaster settings, DMHS involving organizational interventions by multiple regional institutions are required to support disaster victims. However, little is known about the process of coordinating multiple institutions or determining whether appropriate support has been provided. To elucidate the characteristics of communications that enable effective disaster medical team formation, we conducted network analyses of sender-recipient pairs of emails during the period of DMHS activity. The network analysis is a research method that represents various objects as a network of nodes and edges and explores their structural characteristics. We obtained 2,450 time-series emails from five core members of DMHS, including 32,865 pairs of senders and recipients. The network generated by the emails was scale-free, and its structure changed according to the phases of disaster recovery. In the ultra-acute phase, which lasted about 1 week, spreading information and recruiting people to provide disaster support was given the highest priority. In the acute phase, which lasted about 1 month, support and swift decision-making were essential for directing large numbers of staff. In the mid- to long-term phase, support for staff to share information and experience in small groups was observed. Network analyses have revealed that disaster medical teams must change their communication styles during the mission to adapt to different health needs corresponding to each post-disaster phase.

Stacking models for nearly optimal link prediction in complex networks

Most real-world networks are incompletely observed. Algorithms that can accurately predict which links are missing can dramatically speed up network data collection and improve network model validation. Many algorithms now exist for predicting missing links, given a partially observed network, but it has remained unknown whether a single best predictor exists, how link predictability varies across methods and networks from different domains, and how close to optimality current methods are. We answer these questions by systematically evaluating 203 individual link predictor algorithms, representing three popular families of methods, applied to a large corpus of 550 structurally diverse networks from six scientific domains. We first show that individual algorithms exhibit a broad diversity of prediction errors, such that no one predictor or family is best, or worst, across all realistic inputs. We then exploit this diversity using network-based metalearning to construct a series of "stacked" models that combine predictors into a single algorithm. Applied to a broad range of synthetic networks, for which we may analytically calculate optimal performance, these stacked models achieve optimal or nearly optimal levels of accuracy. Applied to real-world networks, stacked models are superior, but their accuracy varies strongly by domain, suggesting that link prediction may be fundamentally easier in social networks than in biological or technological networks. These results indicate that the state of the art for link prediction comes from combining individual algorithms, which can achieve nearly optimal predictions. We close with a brief discussion of limitations and opportunities for further improvements.

Publishing Community-Preserving Attributed Social Graphs with a Differential Privacy Guarantee

We present a novel method for publishing differentially private synthetic attributed graphs. Our method allows, for the first time, to publish synthetic graphs simultaneously preserving structural properties, user attributes and the community structure of the original graph. Our proposal relies on CAGM, a new community-preserving generative model for attributed graphs. We equip CAGM with efficient methods for attributed graph sampling and parameter estimation. For the latter, we introduce differentially private computation methods, which allow us to release communitypreserving synthetic attributed social graphs with a strong formal privacy guarantee. Through comprehensive experiments, we show that our new model outperforms its most relevant counterparts in synthesising differentially private attributed social graphs that preserve the community structure of the original graph, as well as degree sequences and clustering coefficients.

Computational Prediction of Probable Single Nucleotide Polymorphism-Cancer Relationships

Genetic variations such as single nucleotide polymorphisms (SNPs) can cause susceptibility to cancer. Although thousands of genetic variants have been identified to be associated with different cancers, the molecular mechanisms of cancer remain unknown. There is not a particular dataset of relationships between cancer and SNPs, as a bipartite network, for computational analysis and prediction. Link prediction as a computational graph analysis method can help us to gain new insight into the network. In this article, after creating a network between cancer and SNPs using SNPedia and Cancer Research UK databases, we evaluated the computational link prediction methods to foresee new SNP-Cancer relationships. Results show that among the popular scoring methods based on network topology, for relation prediction, the preferential attachment (PA) algorithm is the most robust method according to computational and experimental evidence, and some of its computational predictions are corroborated in recent publications. According to the PA predictions, rs1801394-Non-small cell lung cancer, rs4880-Non-small cell lung cancer, and rs1805794-Colorectal cancer are some of the best probable SNP-Cancer associations that have not yet been mentioned in any published article, and they are the most probable candidates for additional laboratory and validation studies. Also, it is feasible to improve the predicting algorithms to produce new predictions in the future.

Advancing from protein interactomes and gene co-expression networks towards multi-omics-based composite networks: approaches for predicting and extracting biological knowledge

Prediction of biological interaction networks from single-omics data has been extensively implemented to understand various aspects of biological systems. However, more recently, there is a growing interest in integrating multi-omics datasets for the prediction of interactomes that provide a global view of biological systems with higher descriptive capability, as compared to single omics. In this review, we have discussed various computational approaches implemented to infer and analyze two of the most important and well studied interactomes: protein-protein interaction networks and gene co-expression networks. We have explicitly focused on recent methods and pipelines implemented to infer and extract biologically important information from these interactomes, starting from utilizing single-omics data and then progressing towards multi-omics data. Accordingly, recent examples and case studies are also briefly discussed. Overall, this review will provide a proper understanding of the latest developments in protein and gene network modelling and will also help in extracting practical knowledge from them.

Evolution of the Chinese guarantee network under financial crisis and stimulus program

Our knowledge about the evolution of guarantee network in downturn period is limited due to the lack of comprehensive data of the whole credit system. Here we analyze the dynamic Chinese guarantee network constructed from a comprehensive bank loan dataset that accounts for nearly 80% total loans in China, during 01/2007-03/2012. The results show that, first, during the 2007-2008 global financial crisis, the guarantee network became smaller, less connected and more stable because of many bankruptcies; second, the stimulus program encouraged mutual guarantee behaviors, resulting in highly reciprocal and fragile network structure; third, the following monetary policy adjustment enhanced the resilience of the guarantee network by reducing mutual guarantees. Interestingly, our work reveals that the financial crisis made the network more resilient, and conversely, the government bailout degenerated network resilience. These counterintuitive findings can provide new insight into the resilience of real-world credit system under external shocks or rescues.

The systemic risk of real-world financial networks is understudied. Here the authors focused on the guarantee network among Chinese firms and found that the global financial crisis during 2007-2008 and economic policies in the aftermath had significant influence on the evolution of guarantee network structure.

Insights from the study of complex systems for the ecology and evolution of animal populations

Populations of animals comprise many individuals, interacting in multiple contexts, and displaying heterogeneous behaviors. The interactions among individuals can often create population dynamics that are fundamentally deterministic yet display unpredictable dynamics. Animal populations can, therefore, be thought of as complex systems. Complex systems display properties such as nonlinearity and uncertainty and show emergent properties that cannot be explained by a simple sum of the interacting components. Any system where entities compete, cooperate, or interfere with one another may possess such qualities, making animal populations similar on many levels to complex systems. Some fields are already embracing elements of complexity to help understand the dynamics of animal populations, but a wider application of complexity science in ecology and evolution has not occurred. We review here how approaches from complexity science could be applied to the study of the interactions and behavior of individuals within animal populations and highlight how this way of thinking can enhance our understanding of population dynamics in animals. We focus on 8 key characteristics of complex systems: hierarchy, heterogeneity, self-organization, openness, adaptation, memory, nonlinearity, and uncertainty. For each topic we discuss how concepts from complexity theory are applicable in animal populations and emphasize the unique insights they provide. We finish by outlining outstanding questions or predictions to be evaluated using behavioral and ecological data. Our goal throughout this article is to familiarize animal ecologists with the basics of each of these concepts and highlight the new perspectives that they could bring to variety of subfields.

Immunization strategies in directed networks

Many complex systems can be modeled as directed networks, which can be regarded as a generalization of undirected networks. In this paper, epidemic dynamics and immunization strategies in directed networks are studied. First, a Susceptible-Infected-Susceptible (SIS) model on a directed network is established employing the mean-field method, and its dynamics and epidemic threshold of the network are studied. Then based on the continuous degree technique, namely, considering the degree of a node as a continuous variable, we propose a method to calculate the epidemic threshold of the immunized network. Besides, some immunization strategies, including optimal immunization, random immunization, combined targeted immunization, and combined acquaintance immunization, and three special networks are considered. Finally, through numerical analysis, all immunization strategies are simulated and compared on different types of networks. We find that the nodes with the largest product of in-degree and out-degree are the most worthy of being immunized.

Consistency of community structure in complex networks

The most widely used techniques for community detection in networks, including methods based on modularity, statistical inference, and information theoretic arguments, all work by optimizing objective functions that measure the quality of network partitions. There is a good case to be made, however, that one should not look solely at the single optimal community structure under such an objective function but rather at a selection of high-scoring structures. If one does this, one typically finds that the resulting structures show considerable variation, which could be taken as evidence that these community detection methods are unreliable, since they do not appear to give consistent answers. Here we argue that, upon closer inspection, the structures found are in fact consistent in a certain way. Specifically, we show that they can all be assembled from a set of underlying "building blocks," groups of network nodes that are usually found together in the same community. Different community structures correspond to different arrangements of blocks, but the blocks themselves are largely invariant. We propose an information theoretic method for discovering the building blocks in specific networks and demonstrate it with several example applications. We conclude that traditional community detection does in fact give a significant amount of insight into network structure.

Predictability of real temporal networks

Links in most real networks often change over time. Such temporality of links encodes the ordering and causality of interactions between nodes and has a profound effect on network dynamics and function. Empirical evidence has shown that the temporal nature of links in many real-world networks is not random. Nonetheless, it is challenging to predict temporal link patterns while considering the entanglement between topological and temporal link patterns. Here, we propose an entropy-rate-based framework, based on combined topological–temporal regularities, for quantifying the predictability of any temporal network. We apply our framework on various model networks, demonstrating that it indeed captures the intrinsic topological–temporal regularities whereas previous methods considered only temporal aspects. We also apply our framework on 18 real networks of different types and determine their predictability. Interestingly, we find that, for most real temporal networks, despite the greater complexity of predictability brought by the increase in dimension, the combined topological–temporal predictability is higher than the temporal predictability. Our results demonstrate the necessity for incorporating both temporal and topological aspects of networks in order to improve predictions of dynamical processes.

A Scalable Similarity-Popularity Link Prediction Method

Link prediction is the task of computing the likelihood that a link exists between two given nodes in a network. With countless applications in different areas of science and engineering, link prediction has received the attention of many researchers working in various disciplines. Considerable research efforts have been invested into the development of increasingly accurate prediction methods. Most of the proposed algorithms, however, have limited use in practice because of their high computational requirements. The aim of this work is to develop a scalable link prediction algorithm that offers a higher overall predictive power than existing methods. The proposed solution falls into the class of global, parameter-free similarity-popularity-based methods, and in it, we assume that network topology is governed by three factors: popularity of the nodes, their similarity and the attraction induced by local neighbourhood. In our approach, popularity and neighbourhood-caused attraction are computed directly from the network topology and factored out by introducing a specific weight map, which is then used to estimate the dissimilarity between non-adjacent nodes through shortest path distances. We show through extensive experimental testing that the proposed method produces highly accurate predictions at a fraction of the computational cost required by existing global methods and at a low additional cost compared to local methods. The scalability of the proposed algorithm is demonstrated on several large networks having hundreds of thousands of nodes.

Protein Interface Complementarity and Gene Duplication Improve Link Prediction of Protein-Protein Interaction Network

Protein-protein interactions are the foundations of cellular life activities. At present, the already known protein-protein interactions only account for a small part of the total. With the development of experimental and computing technology, more and more PPI data are mined, PPI networks are more and more dense. It is possible to predict protein-protein interaction from the perspective of network structure. Although there are many high-throughput experimental methods to detect protein-protein interactions, the cost of experiments is high, time-consuming, and there is a certain error rate meanwhile. Network-based approaches can provide candidates of protein pairs for high-throughput experiments and improve the accuracy rate. This paper presents a new link prediction approach "Sim" for PPI networks from the perspectives of proteins' complementary interfaces and gene duplication. By integrating our approach "Sim" with the state-of-art network-based approach "L3," the prediction accuracy and robustness are improved.

Hyperlink regression via Bregman divergence

A collection of U(∈N) data vectors is called a U-tuple, and the association strength among the vectors of a tuple is termed as the hyperlink weight, that is assumed to be symmetric with respect to permutation of the entries in the index. We herein propose Bregman hyperlink regression (BHLR), which learns a user-specified symmetric similarity function such that it predicts the tuple's hyperlink weight from data vectors stored in the U-tuple. BHLR is a simple and general framework for hyper-relational learning, that minimizes Bregman-divergence (BD) between the hyperlink weights and estimated similarities defined for the corresponding tuples; BHLR encompasses various existing methods, such as logistic regression (U=1), Poisson regression (U=1), link prediction (U=2), and those for representation learning, such as graph embedding (U=2), matrix factorization (U=2), tensor factorization (U≥2), and their variants equipped with arbitrary BD. Nonlinear functions (e.g., neural networks), can be employed for the similarity functions. However, there are theoretical challenges such that some of different tuples of BHLR may share data vectors therein, unlike the i.i.d. setting of classical regression. We address these theoretical issues, and proved that BHLR equipped with arbitrary BD and U∈N is (P-1) statistically consistent, that is, it asymptotically recovers the underlying true conditional expectation of hyperlink weights given data vectors, and (P-2) computationally tractable, that is, it is efficiently computed by stochastic optimization algorithms using a novel generalized minibatch sampling procedure for hyper-relational data. Consequently, theoretical guarantees for BHLR including several existing methods, that have been examined experimentally, are provided in a unified manner.

A framework for second-order eigenvector centralities and clustering coefficients

We propose and analyse a general tensor-based framework for incorporating second-order features into network measures. This approach allows us to combine traditional pairwise links with information that records whether triples of nodes are involved in wedges or triangles. Our treatment covers classical spectral methods and recently proposed cases from the literature, but we also identify many interesting extensions. In particular, we define a mutually reinforcing (spectral) version of the classical clustering coefficient. The underlying object of study is a constrained nonlinear eigenvalue problem associated with a cubic tensor. Using recent results from nonlinear Perron-Frobenius theory, we establish existence and uniqueness under appropriate conditions, and show that the new spectral measures can be computed efficiently with a nonlinear power method. To illustrate the added value of the new formulation, we analyse the measures on a class of synthetic networks. We also give computational results on centrality and link prediction for real-world networks.