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Distributed stochastic gradient descent for link prediction in signed social networks
EURASIP Journal on Advances in Signal Processing volume 2019, Article number: 3 (2019)
Abstract
This paper considers the link prediction problem defined over a signed social network, where the relationship between any two network users can be either positive (friends) or negative (foes). Given a portion of the relationships, the goal of link prediction is to identify the rest unknown ones. This task resorts to completing the adjacency matrix of the signed social network, which is low rank or approximately low rank. Considering the large scale of the adjacency matrix, in this paper, we adopt lowrank matrix factorization models for the link prediction problem and solve them through asynchronous distributed stochastic gradient descent algorithms. The lowrank matrix factorization models effectively reduce the size of the parameter space, while the asynchronous distributed stochastic gradient descent algorithms enable fast completion of the adjacency matrix. We validate the proposed algorithms using two realworld datasets on a distributed sharedmemory computation platform. Numerical results demonstrate that the asynchronous distributed stochastic gradient descent algorithms achieve nearly linear computional speedups with respect to the number of computational threads, and are able to complete an adjacency matrix of ten billions of entries within 10 s.
Introduction
Social network analysis has received much research interest in the past decade. A social network can be regarded as a directed graph, where nodes represent users and weights of edges represent relationships between users. The weight of an edge (i,j) can be either positive, meaning that node i treats node j as a friend, or negative, meaning that node i treats node j as a foe. A case of particular interest is the signed social network, where the weights are either + 1 or − 1. For example, at Epinions, an online review website, a user can choose “trust” or “distrust” for another user [1]. Since collecting all relationships from the users’ side is often impossible, estimating the unknown ones is critical to applications such as discovering relationships, recommending friends, and identifying malicious users. The target of link prediction over a signed social network is to infer the unknown relationship between users from the known ones.
Related works
Approaches to solving the link prediction problem over a signed social network can be classified into local, global, and the combination of the two. Below, we give a brief review on these approaches.
Local approaches rely on the theory of structure balance of signed (social) networks, which dates back to 1950s [2]. Structure balance means that the triads in a signed network tend to obey the rules such as “a friend of my friend is my friend” and “an enemy of my friend is my enemy.” This observation enables link prediction at a low level. Based on the theory of structure balance, [3] develops a trust (or distrust) propagation framework to predict the trust (or distrust) between pairs of nodes. A logistic regression classifier is proposed in [4], where the features come from only local triad structures of edges. The work of [5] uses additional features that come from longer cycles in a signed network so as to improve the prediction accuracy. Here, a longer cycle means a directed path from a node to itself, whose length is longer than that of a triad. However, the longer cycle method increases the number of features exponentially with respect to the length of cycle and brings difficulties in thecomputation [6]. All these local methods focus on the lowlevel structure of a signed network and ignore the highlevel structure. They are unable to leverage some global information, for example, the cluster information that leads to a lowrank weight matrix [7]. This disadvantage motivates the introduction of global approaches.
Global approaches are built upon the theory of weak balance that characterizes the global structure of a signed (social) network [8]. A complete signed network is weakly balanced, if the nodes can be divided into several clusters such that withincluster edges are positive and betweencluster edges are negative. It is shown in [9] that a weakly balanced signed network has a lowrank adjacency matrix. Thus, the task of link prediction can be formulated as a lowrank matrix completion problem, and many existing algorithms can be applied as solvers. Two popular lowrank matrix completion models are rank minimization and nuclear norm minimization, where the latter is a convex relaxation of the former [10]. They can be solved by singular value hard thresholding [11] and singular value soft thresholding [12] algorithms. However, every iteration of these algorithms requires a costly singular value decomposition step on a square matrix, whose numbers of rows and columns are the same as the number of network users. Therefore, they are not suitable for solving largescale problems. Instead, matrix factorization is a proper model for largescale lowrank matrix completion, which can be solved by alternating least squares [13] and stochastic gradient descent [9] algorithms. At each iteration of stochastic gradient descent, the learner randomly selects a known weight and updates the corresponding row and column of the two factorized matrices, respectively. Hsieh et al. [9] demonstrates that stochastic gradient descent is an efficient algorithm to solve the link prediction problem defined over a largescale signed social network. As for the convergence, [14] declares that for nonconvex problem with strict saddle property, starting from an arbitrary point, stochastic gradient descent converges to a local minimum in a polynomial number of steps.
Combined approaches utilize both local and global information. For example, [15] proposes a probabilistic matrix factorization method, where each user’s feature is determined by its neighbors and social psychology factors. The work of [16] combines features from local triads and global rank minimization to design classification algorithms. The work of [17] builds a probabilistic prediction model, which adaptively adjusts the estimate of adjacency matrix through both local structure balance and global clustering results. However, feature extraction from a largescale signed social network is often computationally expensive.
This paper focuses on the matrix factorization model for link prediction in a largescale signed social network and solves it through the celebrated stochastic gradient descent algorithm. In particular, we take advantages of asynchronous distributed computation to enable fast completion of the largescale adjacency matrix. On a sharedmemory computation platform, Hogwild! distributes stochastic gradient descent steps to multiple computational threads without any locking and achieves a nearly optimal convergence rate when the optimization problem has a sparse structure. On a masterworker computation platform, [18] proposes the distributed stochastic gradient descent algorithm to for largescale matrix factorization, which randomly splits the matrix to blocks and assigns different blocks to different workers. To overcome the data discontinuity problem in Hogwild! and the block imbalance and locking problems in distributed stochastic gradient descent, the fixedpoint stochastic gradient descent algorithm selects samples within a block in order but randomizes the selection of blocks [19]. Also, [20] is proposed to address the data structure issue of Hogwild! by parallelizing the random selections of variable blocks and training samples. Based on Hogwild!, [21] proposes an asynchronous decentralized stochastic gradient descent algorithm, which replaces the global model with a set of local models kept by workers. A decentralized tokenbased protocol is developed to keep the computation synchronized. In this paper, we shall consider sharedmemory computation platform, which fits for distributed implementations of stochastic gradient descent to solve the largescale lowrank matrix completion problem. In addition, linear speedup of asynchronousparallel algorithms has been recently established in both deterministic [22] and stochastic [23] settings. We propose two asynchronous distributed stochastic gradient descent algorithms without locking, one is partially asynchronous and close to Hogwild!, while the other is fully asynchronous which both achieve nearly linear speedup.
Our contributions

(i)
We consider lowrank matrix factorization models for link prediction of a signed social network, and solve them through asynchronous distributed stochastic gradient descent algorithms. The lowrank matrix factorization models effectively reduce the size of the parameter space, while the asynchronous distributed stochastic gradient descent algorithms enable fast completion of the adjacency matrix.

(ii)
We validate the proposed algorithms using two realworld datasets on a distributed sharedmemory computation platform. Numerical results demonstrate that the asynchronous distributed stochastic gradient descent algorithms achieve nearly linear computational speedups with respect to the number of computational threads and are able to complete an adjacency matrix of ten billions of entries within 10 s.
Problem statement
Consider a signed social network with n users, where a user can either like or dislike another user. It can be represented as a directed graph \(\mathcal {G} = (\mathcal {V},\mathcal {E},A)\), where \(\mathcal {V}\) is the node set whose cardinality is \(\mathcal {V}=n\), \(\mathcal {E}\) is the edge set whose cardinality is \(\mathcal {E}=m=n^{2}n\), and \(A \in \mathbb {R}^{n \times n}\) is the signed adjacency matrix of \(\mathcal {G}\). The (i,j)th entry of A, denoted as a_{ij}, is given by
With particular note, the diagonal entries a_{ii} are often set to 1, for all i=1,⋯,n.
The network operator can collect a portion of the relationships from the users’ side, but is very likely unable to collect all of them. Denote Ω as the set containing the indexes of the known entries of A. That is, for any (i,j)∈Ω, a_{ij} is known to the network operator. It is of practical importance for the network operator to infer the unknown entries of A from known ones for the sake of discovering relationships of users, recommending friends, identifying malicious users, etc. This link prediction task is essentially a special matrix completion problem, where all the matrix entries are either 1 or − 1.
Rank and nuclear norm minimization models
The fact that the adjacency matrix of the signed social network is low rank or approximately low rank motivates the use of lowrank matrix completion techniques in solving the link prediction problem [9]. A commonly used formulation is
where \(Z \in \mathbb {R}^{n \times n}\) is the lowrank matrix to be recovered and rank(Z) denotes the rank of Z. The constraint z_{ij}=a_{ij}, ∀(i,j)∈Ω enforces all known entries in A which exactly appear at the corresponding locations in Z. The constraint z_{ij}∈{1,−1}, ∀i, ∀j guarantees that the recovered Z is indeed an adjacency matrix of a signed network. Since rank(Z) is a highly nonconvex cost function, it is often replaced by the nuclear norm of Z, denoted by ∥Z∥_{∗}. Thus, (2) is relaxed to
Without the discrete constraint z_{ij}∈{1,−1}, ∀i, ∀j, (2) and (3) can be solved by singular value hard thresholding [11] and singular value soft thresholding [12] algorithms, respectively. Every iteration of these algorithms runs a singular value decomposition step on the intermediate estimate of Z, followed by hard or soft thresholding on the singular values. Since the complexity of singular value decomposition is O(n^{3}), these algorithms are not applicable to the link prediction problem as n is often large in a social network.
Matrix factorization models
Besides the rank minimization and nuclear minimization models, matrix factorization models are also popular in lowrank matrix completion, especially when the matrix to be completed is of a large scale. It is known that any matrix \(Z \in \mathbb {R}^{n \times n}\) with rank up to r can be decomposed into a matrix product Z=X^{T}Y, where \(X \in \mathbb {R}^{r \times n}\) and \(Y \in \mathbb {R}^{r \times n}\). Since the upper bound of the rank is often known or can be estimated in advance, it is natural to introduce a matrix factorization model for link prediction, as
Here, x_{i} is the ith column of X, y_{j} is the jth column of Y, and \(x_{i}^{T} y_{j}\) is the (i,j)th entry of the matrix to be completed. For a given user i, the elements of x_{i} measure the extent of interest that user i has in other users on some factors. For a given user j, the elements of y_{j} measure the extent to which user j possesses these factors [13]. The distance of a_{ij} and \(x_{i}^{T} y_{j}\) is measured by the loss function loss(·). We expect \(x_{i}^{T} y_{j}\) to be close to a_{ij} for all (i,j)∈Ω, leading to the minimization of \(\sum _{(i,j)\in \Omega } loss\left (a_{ij}, x_{i}^{T} y_{j}\right)\). On the other hand, the Frobenius regularization term \(\lambda \X\_{F}^{2} + \lambda \Y\_{F}^{2}\), where λ>0 is a regularization parameter, is used to avoid overfitting by penalizing the magnitudes of the matrices X and Y.
The matrix factorization model (4) has 2rn parameters. In comparison, the rank minimization model (2) and the nuclear norm minimization model (3) both have n^{2} parameters. When the matrix to be completed is low rank, we have r≪n. Therefore, the lowrank matrix factorization models effectively reduce the size of the parameter space. This advantage motivates us to apply the matrix factorization model in solving the link prediction problem defined over a largescale signed social network.
In the matrix factorization model (4), the loss function loss(·) has several choices. A commonly used one is the square loss
which enforces \(x_{i}^{T} y_{j}\) to be as close to a_{ij} as possible, for all (i,j)∈Ω. However, in the link prediction problem, we often allow \(x_{i}^{T} y_{j}\) to be biased from a_{ij} in magnitude and care more about the consistency of their signs. This consideration motivates the introduction of sigmoid loss or squared hinge loss, which penalize the inconsistency of signs [9]. The sigmoid loss and the squared hinge loss are defined as
respectively.
Methods
Based on the matrix factorization model (4), in this section, we develop asynchronous distributed stochastic gradient descent algorithms to solve the link prediction problem defined over a largescale signed social network. We first overview the classical stochastic gradient descent algorithm and then discuss its asynchronous distributed implementations on a sharedmemory computation platform.
Stochastic gradient descent
Consider the cost function of (4). Its full (sub)gradients with respect to x_{i} and y_{j} are
respectively. Observe that \(G_{x_{i}}\) (or \(G_{y_{j}}\)) are decoupled across i (or j). Thus, a straightforward approach to solving (4) is running a gradient descent algorithm using the full gradients \(G_{x_{i}}\) and \(G_{y_{j}}\) to update all x_{i} and y_{j}, followed by projecting the products \(x_{i}^{T} y_{j}\) onto the binary set {−1,1}. However, every iteration of this full gradient descent algorithm requires to evaluate multiple partial gradients ∂loss/∂x_{i} and ∂loss/∂y_{j} so as to sum them up. This is timeconsuming when the cardinalities of the sets {j:(i,j)∈Ω} and {i:(i,j)∈Ω} are large.
Stochastic approximation [24] and machine learning theories [25] suggest us to apply the stochastic gradient descent method to handle this issue, through replacing the full gradients by stochastic gradients. To be specific, at every iteration only a random pair of x_{i} and y_{j} are updated using their stochastic gradients
Observe that the stochastic gradients \(G_{x_{i}}\) and \(G_{y_{j}}\) are approximations to the full gradients \(G_{x_{i}}\) and \(G_{y_{j}}\), respectively.
The stochastic gradient descent algorithm has been applied to solve the link prediction problem in [9], as listed in Algorithm 1. The nonnegative step size η is often set to be diminishing; otherwise, the gradient noise, which stands for the gap between the full gradient and the stochastic gradient, could prevent the iterate from being convergent.
Though the stochastic gradient descent algorithm is proven to be a computationally lightweight approach to solving the link prediction problem, completing a largescale adjacency matrix is still timeconsuming. As we shall see in the numerical experiments, for a signed social network of around 100,000 users, completing the adjacency matrix of ten billion entries by stochastic gradient descent takes a couple of minutes. Thus, for timesensitive applications, it is necessary to speed up the optimization process. Below, we shall develop asynchronous distributed implementations of stochastic gradient descent, which take advantages of multithread computing and significantly reduce the running time.
Distributed stochastic gradient descent
We propose two asynchronous distributed stochastic gradient descent algorithms to solve the link prediction problem defined over a largescale signed social network. Both algorithms run on a sharedmemory system, where training data (a_{ij},(i,j)∈Ω) and optimization variables (x_{i} and y_{j}, ∀i, ∀j) are stored in the memory, while multiple computational threads asynchronously pick up the training data and the optimization variables, perform stochastic gradient descent, and save the updated optimization variables in the memory. This sharedmemory structure is particularly fit for the link prediction task due to its high throughput and low latency. In our workstation whose detailed specifications shall be given in Section 4, each computational thread can read and write a contiguous physical memory at a speed of over 4 GB/s with latency in tens of nanoseconds, and ordinal multiple hard disks can load data into the memory at a speed of over 500 MB/s. Since each stochastic gradient descent step for link prediction is also lightweight, the reading, computation, and writing procedures are all efficient. In comparison, if we use the masterworker structure that is also popular in distributed computing, each master shall store a portion of the training data and handle a portion of the optimization variables. According to the stochastic gradient descent algorithm for link prediction, in this case, masters have to frequently read training data from others, which is slow in the masterworker structure.
A key common feature of the two proposed asynchronous distributed stochastic gradient descent algorithms is lockfree updating, which is also adopted in Hogwild! [26]. To be specific, when a computational thread is assigned to update x_{i} and y_{j}, it reads a_{ij} and the current x_{i} and y_{j} from the memory to run a stochastic gradient step, followed by writing the updated x_{i} and y_{j} to the memory, even when another computational thread is using the current x_{i} or y_{i}. This inconsistency in updating the same optimization variable wastes the computational power and may cause errors, especially when the number of computational threads increases. Fortunately, to solve the link prediction problem, each stochastic gradient descent step only modifies a very small part of the optimization variables, such that memory overwrites are rare. Therefore, the lockfree scheme remains to be efficient due to the sparse data access pattern.
The proposed distributed algorithms are partially asynchronous and fully asynchronous, as outlined in Algorithms 2 and 3, and their processes are depicted in Figs. 1 and 2, respectively. Their difference is as follows. In the partially asynchronous algorithm, computations are split into rounds and every round processes all the training data indexed by Ω. Within each round, every computational thread handles one split of the training data in an asynchronous manner. In comparison, the fully asynchronous algorithm does not have the concept of “round.” The computational threads simply sample from Ω and run stochastic gradient descent steps without any coordination.
Suppose that there are M computational threads. In the partially asynchronous distributed stochastic gradient algorithm, the shared memory stores the training set {a_{ij},(i,j)∈Ω}, as well as the initial values of X and Y. Any one of the M computational threads, which we number as 0, knows the set Ω. The parameter λ is known to all the computational threads. In the beginning of every round, computational thread 0 permutes the elements in Ω, equally divides the elements into M subsets Ω_{1},⋯,Ω_{M}, and assigns them to computational threads m=1,⋯,M, respectively. After that, the computational threads handle their training subsets asynchronously. Computational thread m chooses index (i,j)∈Ω_{m} in order, read the training sample a_{ij} and the current values of x_{i} and y_{j} from the shared memory, and then calculates the stochastic gradients \(G_{x_{i}}\) and \(G_{y_{j}}\) as in (8) and (9). With x_{i}, y_{j}, \(G_{x_{i}}\) and \(G_{y_{j}}\) at hand, computational thread m picks a diminishing step size η, updates \(x_{i} \leftarrow x_{i}  \eta g_{x_{i}}\) and \(y_{j} \leftarrow y_{j}  \eta g_{y_{j}}\) through stochastic gradient descent, and then writes the new x_{i} and y_{j} to the shared memory. When a certain stopping criterion is met, computational thread 0 projects \(x_{i}^{T} y_{j}\) onto {−1,1} for all (i,j)∉Ω, which are the estimates to the unknown elements of the adjacency matrix.
In the fully asynchronous distributed stochastic descent algorithm, the shared memory stores the training set {a_{ij},(i,j)∈Ω}, as well as the initial values of X and Y. The index set Ω and the parameter λ are known to all the computational threads, which asynchronously run stochastic gradient descent steps. Every computational thread m uniformly randomly chooses an index from the fullindex set, namely, (i,j)∈Ω. Then, it reads a_{ij}, x_{i} and y_{j}, calculates stochastic gradients \(G_{x_{i}}\) and \(G_{y_{j}}\), updates x_{i} and y_{j}, and then writes x_{i} and y_{j} to the memory. This procedure repeats when a certain stopping criterion is met, without any synchronous coordination. Finally, one of the computational threads, which is numbered as 0, projects \(x_{i}^{T} y_{j}\) onto {−1,1} for all (i,j)∉Ω.
The two proposed asynchronous algorithms differ in the way of using the training data. In every round of the partially asynchronous algorithm, the computational threads use all of the training data. In the fully asynchronous algorithm, it is possible that one training sample is not used while another one has been used for several times. The partially asynchronous algorithm may waste time in the synchronization step, if the computational threads have imbalanced computational powers, or the training set is not evenly partitioned. These issues can be addressed if the sharedmemory system is properly configured. On the other hand, the imbalanced use of training samples may delay the optimization process of the fully asynchronous algorithm. Nevertheless, when the training data are large scale, we observe that the fully asynchronous algorithm performs well. Indeed, both algorithms demonstrate nearly linear computational speedup with respect to the number of computational threads, as we shall show in the numerical experiments.
Results and discussion
In the numerical experiments, we validate the effectiveness of the proposed asynchronous distributed stochastic gradient descent algorithms in link prediction on two realworld datasets, Slashdot, and Epinions [1]. The algorithms are coded in C++ and run in a sharedmemory system on a workstation using pthread. The workstation has a 128 GB RAM, a 3.2TB hard disk and a dual Xeon E52630 CPUs. Each CPU has 10 cores and each core has maximally 2 computational threads. Thus, the maximum number of computational threads is 40.
Datasets
Slashdot is a technologyrelated news platform. Users in Slashdot can tag each other as friend (positive) or foe (negative). They can also write news articles and other users make comments on these articles. Epinions is a whotrustwhom consumer review site, in which users can decide to trust (positive) or distrust (negative) other users. They can also post reviews for products and other users express opinions on these reviews with their ratings.
Table 1 shows the statistics of these two datasets, from which we observe that (i) the matrices to be completed are of a large scale, with billions of entries; (ii) the known entries in the adjacency matrices are very sparse and most of the known edges are positive.
Experimental settings
In the numerical experiments, we compare the following three algorithms: stochastic gradient descent in [9], partially asynchronous distributed stochastic gradient descent, and fully asynchronous distributed stochastic gradient descent. For the two asynchronous distributed algorithms proposed in this paper, we vary the number of computational threads to quantify their computational speedups. Particularly, when the number of computational threads is 1, the fully asynchronous distributed stochastic gradient descent algorithm reduces to the classic stochastic gradient descent algorithm. In the matrix factorization model (4), we test the square loss in (5), the sigmoid loss in (6), and the squared hinge loss in (7).
The performance of the three algorithms is evaluated by the prediction accuracy. We randomly sample 90% of the data as the training set and the remaining 10% as the test set. The estimates are compared with the values in the test set, and the accuracy is defined as the percentage of the correct estimates. We run all the experiments for 10 times and average the results to ensure reliability.
In the numerical experiments, the initial values of X and Y are randomly generated and every entry follows a standard Gaussian distribution. The step size η is set to be η_{0}/(1+k)^{0.1}, where η_{0} is the initial step size and hand tuned to be the best, and k is the number of epochs. By one epoch, we mean that Ω samples have been used during that period. Note that in the partially asynchronous distributed algorithm, one epoch corresponds to one round where every sample is used for one time. However, in one epoch of the classic stochastic gradient descent algorithm and the fully asynchronous distributed algorithm, some samples may be used for more than one time, while some other samples may be never used. We vary the estimated rank r of the adjacency matrix and the regularization parameter λ in the numerical experiments.
Comparisons on accuracies
Figures 3, 4, and 5 depict the prediction accuracies of the algorithms on the Slashdot dataset using sigmoid (λ=0.01), squared hinge (λ=0.4), and square (λ=0.4) losses, respectively. Observe that as long as the number of epochs is fixed, increasing the number of computational threads does not change the accuracy too much. The partially asynchronous and fully asynchronous algorithms also perform similarly. After 100 epochs, the three losses yield similar accuracies, though the curves of the sigmoid loss are not as steady as those of the square loss and the squared hinge loss.
On the Epinions dataset, we also show the prediction accuracies using sigmoid (λ=0.01), squared hinge (λ=0.4), and square (λ=0.4) losses in Figs. 6, 7, and 8, respectively. Similar to the results of the Slashdot dataset, the prediction accuracies with respect to the number of epochs are not sensitive in the partially or fully asynchronous algorithms, the number of computational threads, as well as the choice of loss function. Different to the results of the Epinions dataset, the curves of the squared hinge loss and the squared loss are steady, while those of the sigmoid loss are slightly fluctuant. The steadystate prediction accuracy of the Epinions dataset is around 0.945, higher than that of the Slashdot dataset, which is around 0.875. Our conjecture is that, the percentage of positive known edges in the Slashdot dataset is 76.7% and that in the Epinions dataset is 85.3%, such that estimating an unknown edge to be positive is a quite safe choice in the Epinions dataset. Therefore, it is relatively easier to achieve a higher prediction accuracy in the Epinions dataset.
Rank estimate and regularization parameter
Now we analyze the sensitivity of the proposed asynchronous distributed algorithms to the rank estimate and the regularization parameter. We only consider the squared hinge loss in this set of experiments.
In Fig. 9, we show the impact of rank estimate r on the prediction accuracy. The number of computational threads is M=10, and the regularization parameter is λ=0.2. For both partially and fully asynchronous algorithms and both Slashdot and Epinion datasets, the prediction accuracies are not sensitive to the choice of r, and r≃30 yields the best results. This makes sense because a small r is insufficient to recover the highdimensional adjacency matrix, while a large r lacks the generalization ability.
As depicted in Fig. 10, the prediction accuracy is also insensitive to the regularization parameter λ. A moderate value of λ≃0.2 helps both algorithms reach the best accuracies in both datasets.
Nearly linear computational speedups
In the last set of the numerical experiments, we shall show that the proposed asynchronous distributed algorithms are computationally efficient, in the sense that they achieve nearly linear computational speedups with respect to the number of computational threads.
Figures 11 and 12 demonstrate the computational speedups of the partially and fully asynchronous distributed stochastic gradient descent algorithms, respectively. In both datasets, both algorithms are able to achieve nearly linear computational speedups. When the number of computational threads is 10, the speedup is around 5 for the partially asynchronous algorithm and around 6 for the fully asynchronous algorithm. When the number of computational threads is 40, the speedup becomes around 9 and around 10 for the partially and fully asynchronous algorithms, respectively. The Hogwild! style partially asynchronous distributed stochastic gradient descent algorithm requires coordination before every round, which results in the additional cost in terms of computational time. In comparison, the fully asynchronous distributed stochastic gradient descent algorithm, though may suffer from the imbalanced use of training samples, still performs well for over the largescale datasets.
Table 2 gives the running times of the asynchronous distributed algorithms for 100 epochs on the two datasets. When the number of computational threads is M=1, the fully asynchronous distributed stochastic gradient descent algorithm degenerates to the classic stochastic gradient descent algorithm, which needs around 1 min on the Slashdot dataset and around 2 min on the Epinions dataset. In comparison, when the number of computational threads increases to M=40, the running times are around 7 s and 10 s, respectively. These significant savings show that the proposed algorithms are particularly fit for timesensitive applications.
Conclusions
This paper proposes two asynchronous distributed stochastic gradient descent algorithms to solve the link prediction problem defined over a largescale signed network. The link prediction problem is formulated a matrix factorization model, which aims to complete the lowrank adjacency matrix. The two proposed distributed stochastic gradient descent algorithms, one is fully asynchronous and the other is partially asynchronous, are shown to be powerful tools to solve this problem on a sharedmemory computation platform. Numerical experiments on two realworld largescale datasets demonstrate that the two proposed asynchronous distributed algorithms have nearly linear computational speedups, and are able to complete an adjacency matrix of ten billions of entries within 10 s.
Abbreviations
 CPU:

Central processing unit
 RAM:

Random access memory
 SGD:

Stochastic gradient descent
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Acknowledgements
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Funding
Qing Ling is supported in part by the NSF China under Grant 61573331 and the NSF Anhui under Grant 1608085QF130.
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Source codes are available upon request. Datasets are available on http://snap.stanford.edu.
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HZ developed the algorithm, implemented the algorithm, and drafted the paper. GW revised the paper. QL developed the algorithm and drafted the paper. All authors read and approved the final manuscript.
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Correspondence to Qing Ling.
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Zhang, H., Wu, G. & Ling, Q. Distributed stochastic gradient descent for link prediction in signed social networks. EURASIP J. Adv. Signal Process. 2019, 3 (2019). https://doi.org/10.1186/s1363401906010
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Keywords
 Signed social network
 Link prediction
 Lowrank matrix completion
 Asynchronous distributed optimization
 Stochastic gradient descent