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Susceptibleinfectedspreadingbased network embedding in static and temporal networks
EPJ Data Science volumeÂ 9, ArticleÂ number:Â 30 (2020)
Abstract
Link prediction can be used to extract missing information, identify spurious interactions as well as forecast network evolution. Network embedding is a methodology to assign coordinates to nodes in a lowdimensional vector space. By embedding nodes into vectors, the link prediction problem can be converted into a similarity comparison task. Nodes with similar embedding vectors are more likely to be connected. Classic network embedding algorithms are randomwalkbased. They sample trajectory paths via random walks and generate node pairs from the trajectory paths. The node pair set is further used as the input for a SkipGram model, a representative language model that embeds nodes (which are regarded as words) into vectors. In the present study, we propose to replace random walk processes by a spreading process, namely the susceptibleinfected (SI) model, to sample paths. Specifically, we propose two susceptibleinfectedspreadingbased algorithms, i.e., SusceptibleInfected Network Embedding (SINE) on static networks and Temporal SusceptibleInfected Network Embedding (TSINE) on temporal networks. The performance of our algorithms is evaluated by the missing link prediction task in comparison with stateoftheart static and temporal network embedding algorithms. Results show that SINE and TSINE outperform the baselines across all six empirical datasets. We further find that the performance of SINE is mostly better than TSINE, suggesting that temporal information does not necessarily improve the embedding for missing link prediction. Moreover, we study the effect of the sampling size, quantified as the total length of the trajectory paths, on the performance of the embedding algorithms. The better performance of SINE and TSINE requires a smaller sampling size in comparison with the baseline algorithms. Hence, SIspreadingbased embedding tends to be more applicable to largescale networks.
1 Introduction
Realworld systems can be represented as networks, with nodes representing the components and links representing the connections between them [1, 2]. The study of complex networks pervades in different fields [3]. For example, with biological or chemical networks, scientists study interactions between proteins or chemicals to discover new drugs [4, 5]. With social networks, researchers tend to classify or cluster users into groups or communities, which is useful for many tasks, such as advertising, search and recommendation [6, 7]. With communication networks, learning the network structure can help understand how information spreads over the networks [2, 8]. These are only a few examples of the important role of analyzing networks. For all these examples, the data may be incomplete. If so, it could be important to be able to predict the links most likely to be missing. If the network is evolving, it could be crucial to forecast the next link to be added. For both of these applications one needs link prediction [9â€“13].
In link prediction, one estimates the likelihood that two nodes are adjacent to each other based on the observed network structure [14]. Methods using similaritybased metrics, maximum likelihood algorithms and probabilistic models are major families of link prediction methods [15]. Recently, network embedding, which embeds nodes into a lowdimensional vector space, has attracted much attention in solving the link prediction problem [15â€“20]. The similarity between the embedding vectors of two nodes is used to evaluate whether they would be connected or not. Different algorithms have been proposed to obtain network embedding vectors. A simplest embedding method is to take the row or column vector in the adjacency matrix, which is called an adjacency vector of the corresponding node, as the embedding vector. Then, the representation space is Ndimensional, where N is the number of nodes. As realworld networks are mostly large and sparse, the adjacency vector of a node is sparse and highdimensional. In addition, the adjacency matrix only contains the firstorder neighborhood information, and therefore the adjacency vector neglects the highorder structure of the network such as paths that are longer than an edge. These factors limit the precision of network embedding based on the adjacency vector in link prediction tasks. Work in the early 2000s attempted to embed nodes into a low dimension space using dimension reduction techniques [21â€“23]. Isomap [21], locally linear embedding (LLE) [22] and Laplacian eigenmap [23] are algorithms based on the knearest graph, where nodes i and j are connected by a link in the knearest graph if the length of the shortest path between i and j is within the kth shortest among the length of all the shortest paths from i to any other nodes. Matrix factorization algorithms decompose the adjacency matrix into the product of two lowdimensional rectangular matrices. The columns of the rectangular matrices are the embedding vectors for nodes. Singular value decomposition (SVD) [24] is one commonly used and simple matrix factorization algorithm. However, the computation complexity of most of the aforementioned algorithms is at least quadratic in terms of N, limiting their applicability to large networks with millions of nodes.
Randomwalkbased network embedding is a promising family of computationally efficient algorithms. These algorithms exploit truncated random walks to capture the proximity between nodes [25â€“27] generally via the following three steps [28â€“30]: (1) Sample the network by running random walks to generate trajectory paths. (2) Generate a node pair set from the trajectory paths: each node on the trajectory path is viewed as a center node, the nearby nodes within a given distance are considered as the neighboring nodes. A node pair in the node pair set is formed by a center node and each of its neighboring nodes. (3) Apply a word embedding model such as SkipGram to learn the embedding vector for each node by using the node pair set as input. SkipGram assumes nodes that are similar in topology or content tend to have similar representations [27]. Algorithms have been designed using different random walks to capture highorder structure on networks. For example, DeepWalk [25] and Node2Vec [28] adopted uniform and biased random walks, respectively, to sample the network structure. In addition, randomwalkbased embedding methods have also been developed for temporal networks, signed networks and multilayer networks [31â€“34].
In contrast to randomwalkbased embedding, here we propose SIspreadingbased network embedding algorithms for static and temporal networks. We deploy the susceptibleinfected (SI) spreading process on the given network, either static or temporal, and use the corresponding spreading trajectories to generate the node pair set, which is fed to the SkipGram to derive the embedding vectors. The trajectories of an SI spreading process capture the treelike subnetwork centered at the seed node, whereas random walk explores long walks that possibly revisit the same node. We evaluate our static network embedding algorithm, named SINE,^{Footnote 1} and temporal network embedding, TSINE, via a missing link prediction task in six realworld social networks. We compare our algorithms with stateoftheart static and temporal network embedding methods. We show that both SINE and TSINE outperform other static and temporal network embedding algorithms, respectively. In most cases, the static network embedding, SINE, performs better than TSINE, which additionally uses temporal network information. In addition, we evaluate the efficiency of SIspreadingbased network embedding via exploring the sampling size for the SkipGram, quantified as the sum of the length of trajectory paths, in relation to its performance on the link prediction task. We show that high performance of SIspreadingbased network embedding algorithms requires a significantly smaller sampling size compared to randomwalkbased embeddings. We further explore what kind of links can be better predicted to further explain why our proposed algorithms show better performance than the baselines.
The rest of the paper is organized as follows. We propose our method in Sect. 2. In Sect. 2.1, we propose our SIspreadingbased sampling method for static networks and generation of the node pair set from the trajectory paths. SkipGram model is introduced in Sect. 2.2. We introduce an SIspreadingbased sampling method for temporal networks in Sect. 2.3. In Sect. 3, our embedding algorithms are evaluated on a missing link prediction task on realworld static and temporal social networks. The paper is concluded in Sect. 4.
2 SIspreadingbased embedding algorithm
This section introduces SIspreadingbased network embedding methods. Firstly, we illustrate our SIspreadingbased network embedding method for static networks in Sects. 2.1 and 2.2. Section 2.3 generalizes the method to temporal network.
Because we propose the network embedding methods for both static and temporal networks, we start with the notations for temporal networks, of which the static networks are special cases. A temporal network is represented as \(\mathcal{G} = (\mathcal{N}, \mathcal{L})\), where \(\mathcal{N}\) is the node set and \(\mathcal{L}=\{l(i, j, t), t \in [0, T], i, j\in \mathcal{N}\}\) is the set of timestamped contacts. The element \(l(i, j, t)\) in \(\mathcal{L}\) represents a bidirectional contact between nodes i and j at time t. We consider discrete time and assume that all contacts have a duration of one discrete time step.^{Footnote 2} We use \([0, T]\) to represent the observation time window, and \(N = \mathcal{N}\) is the number of nodes. The aggregated static network \(G=(\mathcal{N}, E)\) is derived from a temporal network \(\mathcal{G}\). Two nodes are connected in G if there is at least one contact between them in \(\mathcal{G}\). E is the edge set of G. We formulate the network embedding problem as follows:
Given a network \(G=(\mathcal{N}, E)\), static network embedding aims to learn a lowdimensional representation for each node \(i \in \mathcal{N}\). The node embedding matrix for all the nodes is given by \(\textbf{U}\in R^{d\times N}\), where d is the dimension of the embedding vectors (\(d < N\)). The ith column of U, i.e., \(\overrightarrow{u_{i}}\in R^{d\times 1}\), represents the embedding vector of node i.
2.1 SIspreadingbased static network sampling
The SI spreading process on a static network is defined as follows: each node is in one of the two states at any time step, i.e., susceptible (S) or infected (I); initially, one seed node is infected; an infected node independently infects each of its susceptible neighbors with an infection probability Î² at each time step; the process stops when no node can be infected further. To derive the node pair set as the input for SkipGram, we carry out the following steps as described in Sections 2.1.1 and 2.1.2.
2.1.1 Construction of spreading trajectory paths
In each iteration or realization of the SI spreading process, a node i is selected uniformly at random as the seed. We perform the SI spreading process from seed node i. The spreading trajectory \(\mathcal{T}_{i}(\beta )\) is the union of all the nodes that finally get infected supplied with all the links that have transmitted infection between node pairs. If a susceptible node i becomes infected via more than one infected neighbors at the same time step, we uniformly randomly choose one of these infected neighbors, say j, and assume i gets infected only by neighbor j via the edge in between. Hence, edge \((i, j)\) will be included into the spreading trajectories. The spreading trajectories are exactly trees under this assumption.
From each of the spreading trajectory \(\mathcal{T}_{i}(\beta )\), we construct \(m_{i}\) trajectory paths, each of which is the path between the root node i and a randomly selected leaf node in \(\mathcal{T}_{i}(\beta )\). The number \(m_{i}\) of trajectory paths to be extracted from \(\mathcal{T}_{i}(\beta )\) is assumed to be given by
where \(m_{\max }\) is a control parameter and \(\mathcal{K}(i)\) is the degree of the root node i in the static network (or aggregated network).
The trajectory paths may have different lengths (i.e., number of nodes in the path). For a trajectory path whose length is larger than \(L_{\max }=20\), we only take the first \(L_{\max }\) nodes on the path.
From each iteration of the SI spreading process that starts from a randomly chosen seed node i, we can generate a spreading trajectory \(\mathcal{T}_{i}(\beta )\) and \(m_{i}\) trajectory paths from \(\mathcal{T}_{i}(\beta )\). We stop such iteration to generate new trajectory paths once the total length of the trajectory paths collected reaches the sampling size \(B=NX\), where X is a control parameter. We consider \(X \in \{1,2, 5, 10, 25, 50, 100, 150, 200, 250, 300, 350\}\). We compare different algorithms using the same B for fair comparison [31] to understand the influence of the sampling size. We show how to sample the trajectory paths in Algorithm 1.
2.1.2 Node pair set generation
We illustrate how to generate the node pairs, the input of the SkipGram, from a trajectory path in Fig. 1. Consider a trajectory path, \(1,3,6,8,9,10,7,5\), starting from node 1 and ending at node 5. We set each node, e.g., node 3, as the center node, and the neighboring nodes of the center node are defined as nodes within \(\omega =2\) hops. The neighboring nodes of node 3 are, 1, 6 and 8. We thus obtain ordered node pairs \((3, 1)\), \((3, 6)\), and \((3, 8)\). Thus, we use the union of node pairs centered at each node in each of trajectory path as the input to the SkipGram model.
2.2 SkipGram model
We illustrate how the SkipGram derives the embedding vector for each node based on the input node pair set. We denote by \(N_{SI}(i)\) the neighboring set for a node i derived from the SI spreading process. A neighboring node j of i may appear multiple times in \(N_{SI}(i)\) if \((i, j)\) appears multiple times in the node pair set.
Given node i, let \(p(ji)\) be the probability of observing neighboring node j. We model the conditional probability \(p(ji)\) as the softmax unit parametrized by the product of the embedding vectors, i.e., \(\overrightarrow{u_{i}}\) and \(\overrightarrow{u_{j}}\), as follows:
SkipGram is to derive the set of the N embedding vectors that maximizes the log probability of observing every neighboring node from \(N_{SI}(i)\) for each i. Therefore, one maximizes
Equation (2) can be further simplified to
where
To compute \(Z_{i}\) for a given i, we need to traverse the entire node set \(\mathcal{N}\), which is computationally costly. To solve this problem, we introduce negative sampling [27], which randomly selects a certain number of nodes k from \(\mathcal{N}\) to approximate \(Z_{i}\). To get the embedding vectors for each node, we use the stochastic gradient ascent to optimize Eq. (3).
We name this static SIspreadingbased network embedding algorithm that uses SkipGram model as SINE.
2.3 SIspreadingbased temporal network sampling
We generalize SINE to the SIspreadingbased temporal network embedding by deploying SI spreading processes on the given temporal network, namely, TSINE. For a temporal network \(\mathcal{G} = (\mathcal{N}, \mathcal{L})\), SI spreading follows the time step of the contacts in \(\mathcal{G}\). Initially, node i is chosen as the seed of the spreading process. At every time step \(t\in [0, T]\), an infected node infects each of its susceptible neighbor in the snapshot through the contact between them with probability Î². The process stops at time T. We construct the spreading trajectory starting from node i as \(\mathcal{T}_{i}(\beta )\), which records the union of nodes that get infected together with the contacts through which these nodes get infected. We propose two protocols to select the seed node of the SI spreading. In the first protocol, we start by selecting uniformly at random a node i as the seed. Then, we select uniformly at random a time step from all the times of contacts made by node i as the starting point of the spreading process, i.e., the time when i gets initially infected. We refer to this protocol as TSINE1. In the second protocol, we choose a node i uniformly at random as the seed and start the spreading at the time when node i has the first contact. We refer to this protocol as TSINE2.
Both TSINE1 and TSINE2 generate the node pair set from the spreading trajectory \(\mathcal{T}_{i}(\beta )\) in the same way as described in Sect. 2.1. The node pair set is the input of the SkipGram for calculating the embedding vectors. The SIspreadingbased temporal network embedding uses the information on the time stamps of contacts in addition to the information used by the static network embedding.
3 Results
For the link prediction task in a static network, we remove a certain fraction of links from the given network and predict these missing links based on the remaining links. We apply our static network embedding algorithm to the remaining static network to derive the embedding vectors for the nodes, which are used for link prediction. For a temporal network, we select a fraction of node pairs that have at least one contact. We remove all the contacts between the selected node pairs from the given temporal network. Then, we attempt to predict whether the selected node pairs have at least one contact or not based on the remaining temporal network. We confine ourselves to the prediction of whether the selected node pairs have contact(s) or not, instead of predicting the number of contacts and their associated time stamps between each selected node pair. We use the area under the curve (AUC) score to evaluate the performance of the algorithms on the link prediction task. The AUC quantifies the probability of ranking a random node pair that is connected or has at least a contact higher than a random node pair that is not connected or has no contact.
3.1 Empirical networks
We consider temporal networks, each of which records the contacts and their corresponding time stamps between every node pair. For each temporal network \(\mathcal{G}\), one can obtain the corresponding static network G by aggregating the contacts between each node pair over time. In other words, two nodes are connected in static network G if there is at least one contact between them in \(\mathcal{G}\). The static network G derived from \(\mathcal{G}\) is unweighted by definition. We consider the following temporal social network data sets.

HT2009 [35] is a network of facetoface contacts between the attendees of the ACM Hypertext 2009 conference.

Manufacturing Email (ME) [36] is an email contact network between employees in a midsized manufacturing company.

Haggle [37] records the physical contacts between individuals via wireless devices.

Fbforum [38] captures the contacts between students at University of Califonia, Irvine, in a Facebooklike online forum.

DNC [39] is an email contact network in the 2016 Democratic National Committee email leak.

CollegeMsg [40] records messages between the users of an online community of students from the University of California, Irvine.
Table 1 provides some properties of the empirical temporal networks. In the first three columns we show the properties of the temporal networks, i.e., the number of nodes (N), timestamps (T) and contacts (\( \mathcal{L}\)). In the remaining columns, we show the properties of the corresponding aggregate static networks, including the number of links (\(E\)), link density, average degree, and clustering coefficient. The temporal networks are considerably different in size, which ranges from hundreds to thousands of nodes, as well as in the network density and clustering coefficient. Choosing networks with different properties allows us to investigate whether the performance of our algorithms is consistent across networks.
3.2 Baseline algorithms
The performance of network embedding algorithms depends on the sampling process to obtain the (node pair) input data as well as the learning model. In this work, we aim to understand the effect of the sampling process that we proposed on the performance of an embedding algorithm. Hence, we choose as baseline three stateoftheart network embedding algorithms (DeepWalk, Node2Vec and CTDNE) that share the same SkipGram learning model as our algorithms. These baseline algorithms and the algorithms that we proposed differ only in the method to sample trajectory paths and the node pair set, which is used as the input of the SkipGram. DeepWalk [25] and Node2Vec [28] are static network embedding algorithms based on random walks. CTDNE [31] is a temporal network embedding algorithm based on random walks. We also choose tNodeEmbed [41] as a baseline model, which uses a different learning model from the above models. It learns temporal network embedding by combining static node embeddings learned from Node2Vec with Recurrent Neural Networks. Via this baseline, we would like to explore whether the performance improvement via the design of learning model is significantly higher than that via the design of the sampling process.

DeepWalk [25] deploys classic random walks on a given static network.

Node2Vec [28] deploys biased random walks on a given static network. The biased random walk gives a tradeoff between breadthfirstlike sampling and depthfirstlike sampling of the neighborhood, which is controlled via two hyperparameters p and q. We use a grid search over \(p, q \in \{0.01, 0.25, 0.5, 1, 2, 4\}\) to obtain embeddings that achieve the largest AUC value for link prediction.

CTDNE [31]: CTDNE is a temporal network embedding algorithm based on temporal random walks. The main idea is that the timestamp of the next temporal contact on the walk should be larger than the timestamps of previously traversed contacts. Given a temporal network \(\mathcal{G} = (\mathcal{N}, \mathcal{L})\), the starting contact for the temporal random walk is selected uniformly at random. Thus, every contact has probability \(1/\mathcal{L}\) to be selected as the starting contact. Assume that a random walker visits node i at time step t. We define \(\Gamma _{t}(i)\) as the set of nodes that have contacted node i after time t allowing duplicated elements. A node may appear multiple times in \(\Gamma _{t}(i)\) because it may have multiple contacts with node i over the course of time. The next node to walk to is uniformly selected from \(\Gamma _{t}(i)\), i.e., every node in \(\Gamma _{t}(i)\) is chosen with probability \(1/\Gamma _{t}(i)\). Nguyen et al. [31] generalized the starting contact and the successor node of a temporal walk to other distributions beyond the uniform distribution illustrated here. When we compare the performance of the algorithms on link prediction, we explore the embeddings that give the largest AUC value for link prediction of CTDNE by taking into account all possible generalizations proposed by Nguyen et al.

tNodeEmbed [41]: The objective of tNodeEmbed is to preserve the static network neighborhood for each node and the stability of embeddings over time. Given a temporal network observed within \([1,T]\), a set of \(T_{1}\leq T\) coarsegrained network snapshots can be obtained, where each snapshot is the temporal network aggregated over the time window of the snapshot. In each snapshot network, an embedding vector can be obtained for each node using Node2Vec. Recurrent Neural Networks (RNN) is further employed among the embeddings at different time snapshots to obtain a final ddimensional embedding vector for each node. For each dataset, we tune hyperparameters such as the number of snapshots and learning rate to obtain the largest AUC value. The number of snapshots for obtaining optimal AUC for HT2009, ME, Haggle, Fbforum, DNC and CollegeMsg are \(22, 24, 27, 29, 42\) and 34, respectively.
In our SIspreadingbased algorithms for both static and temporal networks, we set \(\beta \in \{0.001, 0.01, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7,0.8, 0.9, 1.0 \}\). We use window size \(\omega =10\) and embedding dimension \(d=128\) for our algorithms and the baseline algorithms.
3.3 Performance evaluation
3.3.1 Training and test sets
In this section, we illustrate how to generate the training and test sets in the link prediction task in temporal and static networks. We run the network embedding algorithms on the corresponding training set and obtain embedding vector for each node, and use AUC to evaluate link prediction performance in the test set.
Given a temporal network \(\mathcal{G}\), we select uniformly at random 75% node pairs among the node pairs that have at least one contact between them in \(\mathcal{G}\) as the training set for temporal embedding algorithms, including all the contacts and their timestamps. The training set for static network embedding algorithms is the aggregation of the training set for temporal embedding algorithms. In other words, for every node pair, there is a link between the two nodes in the training set for static network embedding if and only if they have at least one contact in the training set for temporal embedding algorithms.
We use the remaining 25% node pairs among the node pairs that have at least one contact of \(\mathcal{G}\) as the positive links in the test set. We label these node pairs 1. Then, we uniformly randomly sample an equal number of node pairs in \(\mathcal{G}\) which have no contact between them. These node pairs are used as negative links in the test set, which we label 0. The same test set is used for the link prediction task in both temporal and static networks.
For each temporal network data set, we randomly split the network to obtain the training and test set according to the procedures given above five times. Both random walks and SI spreading processes are stochastic. For each split data, we run each algorithm on the training set and perform the link prediction on the test set for ten realizations. Therefore, we obtain ten AUC scores for each splitting of the data into the training and test sets, evening the randomness stemming from stochasticity of the random walk or SI spreading processes. We obtain the AUC score for each algorithm with a given parameter set as an average over 50 realizations in total.
3.3.2 Evaluation results
We summarize the overall performance of the algorithms on missing link prediction in Table 2. For each algorithm, we tune the parameters and show the optimal average AUC score.
First of all, tNodeEmbed with its advanced design like RNN in the learning model performs worse than the algorithms we proposed in 3 out of the 6 networks. This illustrates the importance of exploring the design of the sampling process in embedding algorithms, not only of the learning model. From now on, we will focus on embedding algorithms based on SkipGram, but not tNodeEmbed any more, to investigate the influence of the sampling process on the performance of the embedding algorithms.
Among the static network embedding algorithms, SINE significantly outperforms DeepWalk and Node2Vec. The improvement in the AUC score is up to 30% on the CollegeMsg dataset. Additionally, we have tested the static embedding algorithms on another two large static networks. The network description and the embedding results are shown in Tables S1 and S2 in the Supporting Information 1. The outperformance of SINE in such large networks is even more evident. Embedding algorithms CTDNE, TSINE1 and TSINE2 are for temporal networks. The SIspreadingbased algorithms (i.e., TSINE1 and TSINE2) also show better performance than randomwalkbased one (CTDNE). Additionally, TSINE2 is slightly better than TSINE1 on all data sets. Therefore, we will focus on TSINE2 in the following analysis. In fact, SINE shows better performance than temporal network embedding methods including TSINE2 on all data sets except for HT2009. It has been shown that temporal information is important for learning embeddings [31, 42, 43]. However, up to our numerical efforts, SINE outperforms the temporal network algorithms although SINE deliberately neglects temporal information.
To explore further the difference in performance among the embedding algorithms, we investigate the distribution of the dot product of node embedding vectors. Given a link \((i, j)\) in the test set, we compute the dot product of the two end nodesâ€™ embedding vectors, i.e., \(\overrightarrow{u_{i}}\cdot \overrightarrow{u_{j}}^{T}\). We show the dot product distribution for the positive links and negative links in the test set separately. For each embedding algorithm, we consider only the parameter set that maximizes the AUC, i.e., the parameter values with which the results are shown in Table 2. We show the distribution of the dot product for Haggle in Fig. 2 and for the other data sets in Figure S1â€“S5 in the Supporting Information 1. Compared to the randomwalkbased algorithms, TSINE2 and SINE yield more distinguishable distributions between the positive (grey) and the negative links (pink). This result supports the better performance of SIspreadingbased embeddings than randomwalkbased ones.
The embedding algorithms differ only in the sampling method to generate the node pair set. These algorithms use the same SkipGram architecture, which takes the node pair set as input, to deduce the embedding vector for each node. We explore further how the algorithms differ in the node pair sets that they sampled. The objective is to discover the relation between the properties of the sampled node pairs and the performance of an embedding method. We represent the node pair set generated by an embedding method as a network \(G_{S}=(\mathcal{N}, E_{S})\), so called the sampled network. Two nodes are connected in \(G_{S}\) if they form a node pair in the node pair set. We note that \(G_{S}\) is an unweighted network. For each algorithm, with the parameter set that maximizes the AUC, we show the cumulative degree distribution of its sampled network \(G_{S}\) in Fig. 3. The cumulative degree distribution of the training set for static network is also given. Compared to the cumulative degree distribution of the training set, the sampled networks tend to have a higher node degree. Zhang et al. and Gao et al. [30, 44] have shown that when the degree distribution of \(G_{S}\) is closer to that of the training set, the prediction performance of a randomwalkbased algorithm tends to be better. Even though SIspreading based algorithms perform the best across most of the data sets, we have not found a direct relation between the performance of the embedding algorithm and similarity between the degree distribution of the sampled network and that of the training set.
Similaritybased methods such as the number of \(l=2,3,4\) paths have been used for link prediction problem [10]. An l path between two nodes refers to a path that contains l links. We show examples of \(l=2, 3, 4\) path between a node pair i and j in Fig. 4. KovÃ¡cs et al. [45] have shown that l paths (\(l=3, 4\)) outperform existing link prediction methods in predicting protein interaction. Cao et al. [46] found that network embedding algorithms based on random walks sometimes perform worse in link prediction than the number of \(l=2\) paths or equivalently the number of common neighbors. This result suggests a limit of randomwalkbased embedding in identifying the links between node pairs that have many common neighbors. Therefore, we explore further whether our SIspreadingbased algorithms can overcome this limitation, thus possibly explain their outperformance.
We investigate what kind of network structure surrounding links makes them more easily be predicted. For every positive link in the test set, we study its two end nodesâ€™ topological properties (i.e., the number of \(l=2\), \(l=3\) and \(l=4\) paths) and the dot product of the embedding vectors of its two end nodes. Given a network, the parameters of each embedding algorithm are tuned to maximize the AUC, as given in Table 2. We take data set Haggle as an example. Figure 5 shows the relation between the dot product of the embedding vectors and the number of \(l=2,3,4\) paths of the two end nodes of a positive link in the test set for all the embedding methods. The Pearson correlation coefficient (PCC) between the two variables for all the networks and algorithms is given in Table S3 in the Supporting Information 1. Figure 5 and Table S3 together show that the dot product of the embedding vectors constructed from TSINE2 and SINE is more strongly correlated with the number of l paths, where \(l=2\), 3 or 4, than the randomwalkbased embeddings. This result suggests that SIspreadingbased algorithms may better predict the links whose two end nodes have many lpaths, thus overcoming the limit of randomwalkbased embedding algorithms.
The number of \(l=2,3\) paths has been used to predict links in [10, 45, 46]. The observation and the limit of randomwalkbased embedding algorithms motivate us to use the number of \(l=2,3,4\) paths between a node pair to predict missing links. Take \(l=2\) paths as an example. For every link in the test set, the number of \(l=2\) paths between the two end nodes in the training set is used to estimate the likelihood of connection between them. In the networks we considered, two end nodes of a link tend to be connected by \(l=2\), \(l=3\) and \(l=4\) paths (see Figs. 5). Table 2 (L2, L3, L4 shown in the table correspond to the method of using the number of \(l=2,3,4\) path for link prediction) shows that in such networks, the similaritybased methods do not evidently outperform the SIspreadingbased embedding. Actually, the SIspreadingbased embedding performs better in two out of six networks.
Next, we study the effect of the sampling size, B, on the performance of each algorithm. The sampling size is quantified as the total length of the trajectory paths as defined in Sect. 2.1. Given a network, we set \(B=NX\), where N is the size of the network and \(X \in \{1,2, 5, 10, 25, 50, 100, 150\}\). We evaluate our SIspreadingbased embedding algorithms SINE and TSINE2, and one randomwalkbased embedding algorithm CTDNE, because CTDNE performs mostly the best among all randomwalkbased algorithms. The result is shown in Fig. 6. For each X, we tune the other parameters to show the optimal AUC in the figure. Both SINE and TSINE2 perform better than CTDNE and are relatively insensitive to the sampling size. This means that they achieve a good performance even when the sampling size is small, even with \(X=1\). This is because the node pair set sampled remains relatively the same when X varies. We give the overlap between node pairs set sampled by different X of \(HT2009\) as an explanation in Figures S6â€“S7 in the Supporting Information 1. The randomwalkbased algorithm, CTDNE, however, requires a relatively large sampling size to achieve a comparable performance with SINE and TSINE2.
The AUC as a function of the infection probability Î² is shown in Fig. 7. For each Î², we tune the other parameters to show the optimal AUC. The SIspreadingbased algorithms achieve high performance with a small infection probability (\(0.001 \leq \beta \leq 0.1\)) for all the data sets. The high performance of SIspreadingbased embedding algorithms with the small value of X and Î² across different networks motivates the further study whether one can optimize the performance by searching a smaller range of the parameter values. Finally, we test how the AUC values change with the embedding dimension d for embedding algorithm SINE [47]. The results are shown in Fig. 8. For each of the datasets, we only change the value of d, the other parameters are set the same as these of Table 2. We choose \(d \in \{2,4,8,16,32,64,128,256\}\). With the increase of d, the AUC value firstly slightly increases and then stays stable for all the network datasets. Figure 8 shows that \(d=128\) as the default value is sufficient to produce high AUC values for missing link prediction. Figure 8 also shows the possibility that a similar performance can be achieved by a smaller dimension for the networks we have considered.
The outperformance and efficiency of the algorithms we proposed lie in SI spreading process which samples the information that can not be captured by randomwalkbased sampling process. We take SINE as an example. Each trajectory path generated by the SIspreadingbased sampling method doesnâ€™t have any repeat node on the path. All the nodes on the path are thus unique. The biased randomwalk sampling method used in Node2Vec can practically interpolate between depthfirstsampling and breadthfirstsampling via tuning the parameters. Still, the walks generated, even tuned towards the DFS may contain repeat nodes. Consider the SI spreading trajectory obtained from a random seed node i on a given static network with a given infection probability Î². Links in the static network close to the seed node (root) are more likely to be included in the spreading trajectory. When \(\beta =1\), all links incident to the seed will be included in the trajectory. Links incident to the neighbors of the seed will be included in the trajectory if they link to new nodes but not the seed nor the neighbors of the seed. In this case, the trajectory path between the seed node and a uniformly randomly selected node is the shortest path in the static network. When Î² is small, the trajectory tends to be large in depth (the largest hopcount between the seed and any other node) and the trajectory path tends to be longer, as shown in Figure S6 in the Supporting Information 1. The links around high degree nodes tend to appear in a trajectory tree. The infection probability balances the preference of links around high degree nodes and links close to the seed to appear in the trajectories thus trajectory paths. Such kind of sampling over the neighborhood of the seed and the links of high degree nodes without repeat node in trajectory paths cannot be captured by random walk type sampling.
The optimal performance is obtained when Î² is small. As shown in Figure S6 in the Supporting Information 1, the average length of a trajectory path is around 3.5 for HT2009 and ME, which is far smaller than the length of walks derived from the randomwalksampling strategy. Such a small length of trajectory paths without redundant nodes in each path, and the tendency of links around high degree nodes to appear in a trajectory paths may explain why a small sampling size (the total number of nodes in the trajectory paths) is sufficient to capture the essential information thus obtain high performance.
4 Conclusions
In this paper, we proposed network embedding algorithms based on SI spreading processes in contrast to the previously proposed embedding algorithms based on random walks [48, 49]. We further evaluated the embedding algorithms on the missing link prediction task. The key point of an embedding algorithm is how to design a strategy to sample trajectories to obtain embedding vectors for nodes. We used the SI model to this end. The algorithms that we proposed are SINE and TSINE, which use static and temporal networks, respectively.
On six empirical data sets, the SIspreadingbased network embedding algorithm on the static network, i.e., SINE, gains much more improvement than stateoftheart randomwalkbased network embedding algorithms across all the data sets. The SIspreadingbased network embedding algorithms on the temporal network, TSINE1 and TSINE2, also show better performance than the temporal randomwalkbased algorithm. Temporal information provides additional information that may be useful for constructing embedding vectors [31, 42, 43]. However, we find that SINE outperforms TSINE, which uses timestamps of the contacts. This result suggests that temporal information does not necessarily improve the embedding for missing link prediction. Moreover, when the sampling size of the SkipGram is small, the performance of the SIspreadingbased embedding algorithms is still high. Sampling trajectory paths takes time especially for largescale networks. Therefore, our observation that the SIspreadingbased algorithms require less samples than other algorithms promises the applicability of the SIspreadingbased algorithms to larger networks than the randomwalkbased algorithms. Finally, we show insights of why SIspreadingbased embedding algorithms performs the best by investigating what kind of links are likely to be predicted.
We deem the following future work as important. We have already applied susceptibleinfectedsusceptible (SIS) model and evaluated the SISspreadingbased embedding. However, this generalization has not improved the performance in the link prediction task. Therefore, one may explore whether or not sampling the network information via the other spreading processes, such as susceptibleinfectedrecovered (SIR) model, further improves the embedding. It is also interesting to explore further the performance of the SIspreadingbased algorithms in other tasks such as classification and visualization. Moreover, the SIspreadingbased sampling strategies can also be generalized to other types of networks, e.g., directed networks, signed networks, and multilayer networks.
Notes
The abbreviation SINE of our SusceptibleInfected Network Embedding on static networks should be distinguished from SiNE, a signed network embedding model proposed in [50].
Realworld temporal networks are measured or sampled at discrete time steps with possibly different sampling frequencies thus different durations of a time step, limited by the measurement technique as well as the targeting research question. Moreover, a contact may actually last less than a unit time step, which is not observable due to the limited sampling frequency. These two factors motivate us to consider a wide range of the infection probability per unit time step \(\beta \in \{0.001, 0.01, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7,0.8, 0.9, 1.0 \}\) in the SIspreadingbased embedding algorithms to get the optimal embedding vector for every node.
Abbreviations
 SI:

susceptibleInfected
 SINE:

SusceptibleInfected network embedding on static networks
 TSINE:

SusceptibleInfected network embedding on temporal networks
 LLE:

locally linear embedding
 SVD:

Singular value decomposition
 AUC:

area under the curve
 CTDNE:

ContinuousTime Dynamic Network Embeddings
 tNodeEmbed:

Node Embedding over Temporal Graphs
 BFS:

Breadthfirst Sampling
 DFS:

Depthfirst Sampling
 PCC:

Pearson correlation coefficient
 SIS:

susceptibleinfectedsusceptible
 SIR:

susceptibleinfectedrecovered
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Acknowledgements
We thank the SocioPatterns collaboration (http://www.sociopatterns.org) for providing the data sets.
Availability of data and materials
We use open data which can be downloaded on http://www.sociopatterns.org and https://snap.stanford.edu/data/. The source code will be available from the first author based on reasonable request.
Funding
XZ is supported by the China Scholarship Council (CSC). NM acknowledges support from AFOSR European Office (under Grant No. FA95501917024). PH was supported by JSPS KAKENHI Grant Number JP 18H01655 and by the Grant for Basic Science Research Projects by the Sumitomo Foundation. HW would like to thank Netherlands Organisation for Scientific Research NWO (TOP Grant no. 612.001.802).
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All authors planed the study; XZ and ZL performed the experiments and prepared the figures. All authors analyzed the results and wrote the manuscript. All authors read and approved the final manuscript.
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Zhan, XX., Li, Z., Masuda, N. et al. Susceptibleinfectedspreadingbased network embedding in static and temporal networks. EPJ Data Sci. 9, 30 (2020). https://doi.org/10.1140/epjds/s13688020002485
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DOI: https://doi.org/10.1140/epjds/s13688020002485