Link prediction of time-evolving network based on node ranking

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Abstract

Many real-world networks belong to the kind that evolves over time. So it is very meaningful and challenging to predict whether the link will occur in the network of future time. In this paper, both time-evolving scale-free (SF) network and real-world dynamic network are taken into consideration first and then two kinds of methods are respectively proposed for link prediction. Different from many existing similarity-based dynamic network link prediction methods, many of which adopt node-pair similarity such as common neighbors (CN), Adamic–Adar (AA), and so on, we measure the similarity between nodes from a new perspective. With further research into node ranking, some eigenvector-based methods, such as PageRank (PR), Cumulative Nomination (CuN) and so on, can compute the values of node importance which can be regarded as the stationary distribution of Markov chain for all nodes iteratively. Therefore, from a statistical point of view, the importance of a node is like the probability of attracting other nodes to connect with it and the derivative value of a node pair is like the probability of attracting each other. These node-ranking-based approaches are very novel in the field of link prediction in that few researches have paid enough attention to them before. In addition, an adaptively time series forecasting method is proposed in this paper, and it uses the historical similarity series to predict the future similarity between each node pair adaptively. Experimental results show that our proposed algorithms can predict the future links not only for the growing SF network but also for the dynamic networks in the real-world.

Introduction

Link prediction, which is one of the important research areas in link mining, aims at estimating the existence likelihood of a link between two nodes by considering other links or the node attributes. Some applications of link prediction are concerned with discovery of potential interactions in large organizations, friend suggestions in social networks, hidden or sub-graph connections inferring of the real-world networks [1], [2], [3], [4].

A large number of the real-world networks are time-evolving or dynamic. In recent years, more and more literatures are concerned about the link prediction of dynamic networks. They can be classified into similarity-based methods [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], machine/deep learning-based methods [15], [16], [17], [18], [19], [20], [21], [22], [23] , and so on [24].

Many of the similarity-based dynamic link prediction algorithms adopted some existing link prediction methods (such as CN [25], AA [2], Structural Perturbation Method (SPM) [26], Stochastic Block Models (SBM) [27], Similarity-based Future Common Neighbors (SFCN) [28], community similarity feature [29] and so on) which were used for static networks [6], [7]. However, these methods only considered local information or paths of a single snapshot, which is not enough for dynamic network. In order to alleviate this problem, we propose a series of algorithms based on node ranking. It is found that eigenvector based node ranking methods, such as Eigenvector Centrality (EC) [30], CuN [31], PR [32], LeaderRank (LR) [33] and so on, can iteratively compute the importance which can be regarded as the stationary distribution of Markov chain for each node. The main reasons for our adoption of node-ranking-based methods are as follows: Firstly, from a statistical point of view, the importance of a node is like the probability of attracting other nodes to connect with it and the derivative value of a node-pair is like the probability of attracting each other. Secondly, these node ranking methods consider not only the number of neighbors, but also the importance of neighbors when calculating the importance of nodes. Thirdly, the process of solving the importance of each node iteratively in a network is closely related to the process of dynamic network evolution.

Moving Average (MA) [34], Weighted Moving Average (WMA), Random Walk (RW), Autoregression and Moving Average (ARMA) [7] were often used as the forecasting models in the existing literatures. However, the evolution of different dynamic networks are variable and the contributions of known snapshots are different on the future snapshot even for the same network. It is unreasonable to set the weight of each snapshot as a fixed value or based on experience. ARMA can calculate the weight of each snapshot by autoregression, but it takes too much time to optimize. Therefore, we propose an Adaptive Weighted Moving (AWM) model to forecast the node-pair similarities of snapshot to be estimated. For different dynamic networks, the weights of the known snapshots can be learned adaptively through some regression methods, such as least squares (LS), ridge [35], lasso [36] and so on.

In addition, few researchers paid attention to link prediction in the light of evolutionary networks with increasing scale. But evolutionary networks can often be used to describe the evolutionary process of the real-world networks. For example, the Internet, WWW, metabolic network and so on conform to the SF evolution mechanism [37]. If link prediction of the evolutionary network can be carried out in advance, artificial intervention can be implemented in its evolutionary process, which is of great significance to preventing the spread of disease, rumors and achieving medical intervention.

In summary, the contributions of this paper are as follows:

(1) For time-evolving network with growing size of nodes and edges, there is every reason to adopt a kind of link prediction algorithms based on node ranking, to which researchers have paid little attention before.

(2) A novel similarity metric based on node ranking is proposed for dynamic networks.

(3) A forecasting model named AWM is proposed to predict the score of each node pair adaptively.

This paper is organized as follows: Section 2 is listed with some related works. Section 3 is concerned with the preliminary. Section 4 deals with the proposed Node-Ranking-based Time-Evolving network Link Prediction (NRTE). Section 5 focuses upon the proposed Node-Ranking-based Dynamic network link prediction (NRDy). Section 6 centers upon the experiments. And Section 7 serves as the conclusion of our algorithms.

Section snippets

Related works

Many researches recently integrated time information into network embedding to make it capable of capturing the dynamic evolution of the network [16], [18], [17], [19]. Ref. [16] imposed triad (i.e., a group of three nodes) to model the dynamic changes of network structures. Ref. [17] employed a deep auto-encoder as its core and leveraged the recent advances in deep learning to generate highly non-linear embeddings. Ref. [18] learned the temporal transitions in the network using a deep

Preliminary

In this section, the notations and methods are given in relation to our proposed works.

Time-evolving SF network link prediction

Many real-world networks are SF networks. The evolution of SF network can be used to simulate the evolution of many real-world networks. Therefore, we propose a series link prediction algorithms for time-evolving SF network in this section. In practice, these algorithms can be extended to real-world networks with SF characteristic, and then used to realize the link prediction and intervention of such real-world networks.

Dynamic network link prediction

Dynamic network typically consists of a sequence of edges with time stamps. Such temporal networks can sometimes be called as graph streams. Each graph stream in the sequence depicts the structure of network at a certain time.

Many existing dynamic link prediction algorithms adopted node similarity-based methods (e.g., CN, AA and so on) which were used for static network link prediction [6], [7]. However, these node similarity-based methods only considered the local information or paths of the

Evaluation criterion

Evaluating link prediction accuracy involves comparing a binary label (whether or not an edge exists) with a real-valued prediction score. We apply two variable-threshold methods such as the area under the Receiver Operating Characteristic curve (AUROC) and the area under the Precision–Recall curve (AUPR) [48], [49]. The AUROC is created by plotting the True Positive Rate (TPR) against False Positive Rate (FPR) and AUPR is created by plotting the precision against recall/TPR at various

Conclusions

In this paper, several link prediction algorithms based on node ranking are proposed for time-evolving networks. Through theoretical and experimental analyses, some conclusions can be drawn as follows. Firstly, some eigenvector-based node ranking methods such as PR and LR which compute the importance of each node iteratively are effective and reasonable for SF evolving network or dynamic network link prediction. Because they consider the structural and temporal information of the time-evolving

CRediT authorship contribution statement

Xiaomin Wu: Conceptualization, Methodology, Software, Data curation, Writing - original draft, Writing - review & editing. Jianshe Wu: Investigation, Supervision, Writing - review & editing. Yafeng Li: Writing - review & editing. Qian Zhang: Writing - review & editing.

Acknowledgments

This work is supported in part by the National Natural Science Foundation of China (No. 61672405, 61572383), the Natural Science Basic Research Program of Shaanxi, China (No. B018360019) and the Funds of the Artificial Intelligence Joint Laboratory of the 20th Research Institute of CETC and the Xidian University, China .

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    No author associated with this paper has disclosed any potential or pertinent conflicts which may be perceived to have impending conflict with this work. For full disclosure statements refer to https://doi.org/10.1016/j.knosys.2020.105740.

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