Mobile online social network (mOSN) is a burgeoning research area. However, most existing works referring to mOSNs deal with static network structures and simply encode whether relationships among entities exist or not. In contrast, relationships in signed mOSNs can be positive or negative and may be changed with time and locations. Applying certain global characteristics of social balance, in this paper, we aim to infer the unknown relationships in dynamic signed mOSNs and formulate this sign inference problem as a low-rank matrix estimation problem. Specifically, motivated by the Singular Value Thresholding (SVT) algorithm, a compact dictionary is selected from the observed dataset. Based on this compact dictionary, the relationships in the dynamic signed mOSNs are estimated via solving the formulated problem. Furthermore, the estimation accuracy is improved by employing a dictionary self-updating mechanism.

Over the past few years, a number of mobile applications that allow users to enjoy networking have emerged. Correspondingly, there has been a proliferation in mobile online social networks (mOSNs). With the ubiquitous use of mobile devices and a rapid shift of technology, it is worthy to investigate the mOSNs from a privacy or security standpoint [

One of the most fundamental theories that are applicable to signed social networks is social structural balance [

Structural balance and weak structural balance have been shown to be valid to analyze signed networks. For instance, the sign inference problem, which aims to infer the unknown relationship between two objects, can be solved by mining balance information of signed networks from local and global perspectives [

Actually, there exist several inherent qualities of mOSNs that are challenging to reliably sense the global states of relationships for the large networks in practice [

The aim of this paper is to develop algorithms for the sign inference in signed mOSNs in global and spatiotemporal evolvement perspectives. In particular, we assume that the signed mOSN possesses an underlying dynamic weakly balanced complete network structure. Suppose that we are given an incomplete networking observation tensor (or 3-dimensional array), which consists of the adjacency matrices corresponding to the snapshots of the underlying dynamic weakly balanced complete network at times

Here are the key contributions we make in this paper.

A dictionary selection approach based on group sparsity has been designed to generate a set containing minimal sizes of features to increase computational efficiency. Specifically, the observation tensor is considered to be the raw materials for feature extraction.

The sign inference problem referring to the weakly balanced mOSNs is formulated as a low-rank matrix reconstruction from the selected dictionary. Under certain mild conditions, a low-rank matrix reconstruction algorithm is applied to solve the sign inference problem, and it turns out to be much more accurate and efficient than other inference methods in the literature. A dictionary self-updating mechanism is also introduced to adjust the dynamic characteristics of the network and improve the sensing accuracy.

The rest of this paper is organized as follows. In Section

Formally, a dynamic undirected signed network is represented as a dynamic graph

Particularly, for each time

Illustration of the adjacency tensor, the cube units symbolize the data of relationships: (a) the adjacency tensor of the observed network and (b) the adjacency tensor of the underlying complete network.

While the above kind of signed networks is called homogeneous, that is, relationships of the networks are between the same kinds of entities, a signed network can also be heterogeneous. In a heterogeneous signed network, there can be more than one kind of entities, and relationships between same or different entities can be positive and negative, such as YouTube with two kinds of entities—users and videos. Moreover, this three-dimensional network adjacency tensor can increase dimensions (e.g., spatial dimension, etc.) to adapt to a wider range of scenarios. In this paper, we mainly focus our attention on three-dimensional homogeneous signed networks.

Structural balance theory was first formulated by Heider [

Signed undirected connectivity configurations mentioned in Section

The formal definition of weakly balanced networks is as follows.

A (possibly incomplete) network is weakly balanced if and only if it is possible to obtain a weakly balanced complete network by filling the missing edges in its adjacency matrix. Furthermore, in terms of patterns of global structure, a complete network is weakly balanced if and only if the vertex set can be divided into

There exists the literature discussing the approaches of clustering and sign prediction with respect to signed networks. Ideas derived from local balance of signed networks can be successfully used to yield algorithms for sign inference [

The adjacency matrix

Actually, since the global viewpoint of weak balance stated in Definition

Let

In this section, we focus on a solution of the sign inference to estimate connection statuses via dictionary learning. As the preparation, we propose a large-scale dictionary selection method to generate the dictionary for inferring. Assume that we are given a (usually incomplete) network observation tensor

The method we propose to handle the dictionary selection is motivated by the Singular Value Thresholding (SVT) algorithm, which is a simple and efficient algorithm for nuclear norm minimization problems proposed by Cai et al. [

Considering the problem

We address how to select the dictionary given an initial candidate feature pool in this subsection. To this end, we first extract an initial candidate feature pool from

Due to massive data of the initial feature pool

Motivated by SVT, we have the equivalent problem of (

For all

The proof of Theorem

For each

As a matter of fact, considering the following optimization problem:

Also, the following result can be deduced based on the properties of Moreau-Yosida regularization [

For any

Since

Since

In this subsection, we develop an efficient optimization algorithm to solve the dual problem (

We recall Nesterov’s method with an adaptive line search scheme as follows. Take the unconstrained smooth convex minimization problem

To overcome this drawback, an adaptive line search scheme for Nesterov’s method is proposed in [

A pair of sequences

The estimate sequence defined in Definition

Let

We further specify the estimation sequence in [

Then Algorithm 2 in [

Note that Theorem

(1)

(2)

(3)

(4)

(5) compute

(6) compute

(7) compute

(8)

(9) goto Step

(10)

(11)

(12)

(13)

(14) set

(15) set

(16)

In Algorithm

This subsection details how to use the dictionary to solve the sign inference problem. Actually, this problem bears similarity to the sign prediction problem in the static signed networks or the unsigned networks varying periodically [

Furthermore, assume that we are given a sequence input samples

In this section, we perform experiments on synthetic networks and show that our low-rank model and dictionary learning method outperform other methods on the task of the sign inference for dynamic signed networks. To ensure that our results are reliable, we conduct all experiments 20 times and average out the results from all of the trials.

To construct synthetic networks, we first consider a weakly balanced complete network

With the dictionary

Accuracy of sign inference algorithms on synthetic datasets. In general, we can see that dictionary learning outperforms ALS and SVP.

An example of the sign inference. (a) illustrates the original matrix. Given the matrix with 98% lost-rate, (b) is the result inferred by dictionary learning method. The similarity of inferred matrix is 0.9347.

In this paper, we establish a low-rank tensor model for the dynamic weakly balanced signed networks. With this model, we first extract the feature pool and propose an approach to extract the compact dictionary from pool. To improve the performance of the selection approach, we derive the corresponding dual problem and introduce an accelerated thresholding algorithm to solve the dual problem. Consequently, the optimal solution of the primary problem can be readily obtained from optimizing the dual problem. In addition, combined with the compact dictionary generation method, the sign inference approach is provided for estimating missing relationships of the dynamic weakly balanced signed networks at a certain time slice. Also, the approach is endowed with the function of the dictionary updating if relationship statuses change.

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

This research was jointly supported by National 973 Program of China (no. 2013CB329204), National 863 Program of China (no. 2011AA01A104), National Natural Science Foundation of China (no. 61100206), Research Fund for Doctoral Program of Higher Education of China (no. 20120005130001), China, and Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications), China.

^{2})