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Sign prediction problem aims to predict the signs of links for signed networks. Currently it has been widely used in a variety of applications. Due to the insufficiency of labeled data, transfer learning has been adopted to leverage the auxiliary data to improve the prediction of signs in target domain. Existing works suffer from two limitations. First, they cannot work if there is no target label available. Second, their generalization performance is not guaranteed due to that fact that the solution of their objective functions is not global optimal solution. To solve these problems, we propose a novel sign prediction on unlabeled social networks using branch and bound optimized transfer learning (SP_BBTL) sign prediction model. The main idea of SP_BBTL is to use target feature vectors to reconstruct source domain feature vectors based on relationship projection, which is a complicated optimal problem and is solved by proposed optimization based on branch and bound that can obtain global optimal solution. With this design, the target domain label information is not required for classifier. Finally, the experimental results on the large scale social signed networks validate the superiority of the proposed model.

Sign prediction predicts signs for links of signed networks, in which signed networks are networks whose edges have signs representing the relationship between nodes. The sign of a link is either positive or negative. A link with a positive sign is also called a positive link, which means the two end nodes of this link trust or like each other. A link with a negative sign is also called a negative link, which means the two end nodes of this link distrust or dislike each other. Compared with unsigned networks which only contain values representing the existence of links, signed networks contain more valuable node relationship information. Because of this rich preserved information, signed networks have been widely used in many applications, such as recommender systems [

Most link prediction methods for signed networks are supervised or semisupervised. It is difficult for them to predict unlabeled target networks without any prior target label information. As for those link prediction methods using transfer learning or ensemble learning technologies, there is nonneglectable knowledge loss in knowledge transferring or domain mapping. The main challenge of the link prediction in signed networks is the data insufficiency problem. And this is the motivation to use an auxiliary labeled network to predict signs for unlabeled target network. Signs are manually labeled by experts, which is time consuming and expensive. This leads to the insufficient number of labeled signs in the real applications. Transfer learning, which is able to transfer knowledge from other domains to assist sign prediction, has therefore been used to address this problem [

Though transfer learning based sign prediction methods have good performances on labeled signed networks, they are unable to predict signs on unlabeled signed networks. In [

In addition, sign prediction performances of existing works need further improvement since the optimization of existing objective functions always lead to local optimal solutions or ill-condition solutions. Transferring knowledge between different domains is a complicated process, so the objective functions of sign prediction are usually nonconvex. Most existing works like [

To solve the problems of existing works, we proposed a novel sign prediction model using branch and bound optimized transfer learning (SP_BBTL). SP_BBTL is different from existing works [

There are three main advantages in SP_BBTL. First, it does not require any sign labels in the target domain because of feature vectors mapping. Secondly, the BB based model can be used to compute the global optimal solution of a nonconvex mixed optimization problem with feature vectors in social networks. Third, the proposed method performs well in the imbalance networks compared with existing works because SP_BBTL gets the global optimal solution in the course of source feature vectors reconstruction that has preserved more complete and original transferable knowledge in source domain.

The rest of this paper is organized as follows. Section

In this paper, we propose a novel sign prediction method via transfer learning technology. Thus, the relative works are mainly separated into two parts: sign prediction and transfer learning.

There are mainly three categories for sign prediction approaches. The first type constructs the nonbayesian model based on a set of vertex attributes. The second type derives the joint probability of each sample based on the knowledge of probabilities. The third type leverages linear algebra methods to calculate the similarities between network nodes based on rank-reduced similarity matrices. References [

In addition to model design, another focus of sign prediction is the extraction of useful feature vectors to construct the sign prediction model. There are mainly two types of features: vertex features and edge features. Vertex features consist of neighborhood node based features, path based features, Katz value [

In the real social networks, it is very hard or expensive to obtain the label for our target problem which results in the insufficiency of available data. To solve this problem, transfer learning has been adopted in the sign prediction problem, which tries to utilize the knowledge from source domain to predict the signs in the target domain. Currently transfer learning based sign prediction approaches can be divided into three types: transferring knowledge of instances, transferring knowledge of parameters, and transferring knowledge of feature representations [

To deal with data insufficiency problem, there are some unsupervised transfer learning approaches [

In general, the analysis of related work shows that traditional sign predictions require a sufficient number of sign labels for training. To alleviate this, transfer learning based approaches have been proposed, yet most of these approaches still need some number of sign labels in the target domain. The existing unsupervised sign prediction approaches based on transfer learning do not need any sign labels in the target domain, but they are usually designed to solve a certain sign prediction problem and hard to use as a universal solution. Therefore, a novel transfer learning based approach for sign prediction is required. In this paper, we propose a novel sign prediction model named sign prediction on unlabeled social networks using branch and bound optimized transfer learning (SP_BBTL). The detailed introduction of SP_BBTL is presented in next section.

A

The main idea of the proposed SP_BBTL model is presented in Figure

Collaborative representation of the source domain knowledge and the target domain knowledge for sign prediction.

The key step of SP_BBTL is to achieve domain adaption from

The detailed architecture of the proposed SP_BBTL model is shown in Figure

The architecture of the proposed SP_BBTL model.

As shown in Figure

Link positive outdegree

Link positive indegree

Link embeddedness

Domain reconstruction is the key part of SP_BBTL model. It will build up the latent relationship between source feature vectors and target feature vectors collaboratively. Reconstructing domain from

To minimize the error of (

Calculating (

Input:

Output: Global optimal value of

Parameters: Constraint parameter

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With the extracted global optimal solution

Four datasets extracted from the real-world applications are used in the experiments to verify the performances of the proposed method. These datasets (

Detail information of the experimental datasets.

OTC | ALP | EPI | SLA | |
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Number of nodes | 5881 | 3783 | 131828 | 82140 |

Number of links | 35592 | 24186 | 841372 | 549202 |

Average degree | 12.1041 | 12.7867 | 13.3723 | 12.7647 |

Number of negative links | 11981 | 8890 | 123705 | 124130 |

Negative link ratio | 33.66% | 36.76% | 14.7% | 22.6% |

Two baseline methods are used in this paper to compare with SP_BBTL. The first method is Source-Only (SO) model. It predicts signs of links with data merely from source domain [

Sign prediction performances of SP_BBTL are firstly measured with various network sizes. In the experiments, the number of samples in the target domain is fixed to 3000, while the number of source domain samples varies from 3500 to 9500. The proportion of positive link to negative link is set to be 7:3 (

Sign prediction performances with the various network sizes.

As shown in Figure

Sign prediction performances of SP_BBTL are then measured with the various negative link ratios. In the experiments, the number of samples in the target domain is set to be 3000, while the number of samples in the source domain is set to be 4500 (OTC-ALP and ALP-OTC) and 6500 (EPI-SLA and SLA-EPI) respectively. The ratio of negative links varies from 10% to 90%. The experimental results are given in Figure

Sign prediction performances with the various negative link ratio.

The influence of the constraint parameter

The influence of the constraint parameter on sign prediction performances.

In this paper, a novel method named sign prediction on unlabeled social networks using branch and bound optimized transfer learning (SP_BBTL) is proposed to solve a sign prediction problem via feature vectors projections. In SP_BBTL, labeled source feature vectors are mapped into unlabeled target feature vectors and then the relationship between two domains can be established so that the classifier can be trained without any target label. In addition, the proposed optimization based on branch and bound (BB) performs efficient on social networks because the branch and bound optimization method adapted in the proposed model can ensure the global optimal solution of the objective function. Branch and bound can get global optimal solution by highly efficient searching and iteration. It can maximize the transferable knowledge of the source domain, while minimize the transfer loss. Experimental evaluation validates the superior effectiveness and stability of SP_BBTL in real social networks. At last we give the suggested value for parameter

In the future, we will try to improve the proposed method from several aspects. Firstly, we will try to develop a generalized algorithm, which could not only minimize the influence of negative transfer, but also discover transferable knowledge with different categories of source domains, such as the text data and the image data. Secondly, we will improve the model to minimize the number of the source domain instances used for knowledge transfer, only with little cost in link prediction performances. Lastly, we will extend our model from solving binary sign prediction problem to multilabel sign prediction problem.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there is no conflict of interest regarding the publication of this paper.

This research was supported by Nature Science Foundation of China (Grant No. 61672284), Natural Science Foundation of Jiangsu Province (Grant No. BK20171418), China Postdoctoral Science Foundation (Grant No. 2016M591841), Jiangsu Planned Projects for Postdoctoral Research Funds (No. 1601225C). The authors extend their appreciation to the Deanship of Scientific Research at King Saud University for funding this work through research group no. RGP-VPP-264.