Link prediction in dynamic networks using random dot product graphs

Link prediction is defined as the task of predicting the presence of an edge between two nodes in a network, based on latent characteristics of the graph (Liben-Nowell and Kleinberg 2007). The problem has been widely studied in the literature (Lü and Zhou 2011; Menon and Elkan 2011), and has relevant applications in a variety of different fields. In this paper, the discussion about link prediction is motivated by applications in cyber-security and computer network monitoring (Jeske et al. 2018). The ability to correctly predict and associate anomaly scores with the connections in a network is valuable for the cyber-defence of enterprises. In cyber settings, adversaries may introduce changes in the structure of an enterprise network in the course of their attack. Therefore, predicting links in order to identify significant deviations in expected behaviour could lead to the detection of an otherwise extremely damaging network breach. In particular, it is necessary to correctly score new links (Metelli and Heard 2019), representing previously unobserved connections. The task is particularly important since it is common to observe malicious activity associated with new links (Neil et al. 2013), and it is therefore crucial to understand the normal process of link formation in order to detect a cyber-attack.

In this article, it is assumed that snapshots of a dynamic network are observed at discrete time points \(t=1,\dots ,T\), obtaining a sequence of graphs \({\mathbb {G}}_t=(V,E_t)\). The set V represents the set of nodes, which is invariant over time, whereas the set \(E_t\) is a time dependent edge set, where \((i,j)\in E_t\) for \(i,j\in V\), if i connected to j at least once during the time period \((t-1,t]\). Each snapshot of the graph can be characterised by the adjacency matrix \(\mathbf {A}_t\in \{0,1\}^{n\times n}\), where \(n=\left| V \right| \) and for \(1\le i,j\le n\), \(A_{ijt}=\mathbb {1}_{E_t}\{(i,j)\}\), such that \(A_{ijt}=1\) if a link between the nodes i and j exists in \((t-1,t]\), and \(A_{ijt}=0\) otherwise. The graph is said to be undirected if \((i,j)\in E_t \iff (j,i) \in E_t\), implying that \(\mathbf {A}_t\) is symmetric; otherwise, the graph is said to be directed. It will be assumed that the graph has no self-edges, implying \(\mathbf {A}_t\) is a hollow matrix. Similarly, bipartite graphs \({\mathbb {G}}_t=(V_1,V_2,E_t)\) can be represented using two node sets \(V_1\) and \(V_2\), and rectangular adjacency matrices \(\mathbf {A}_t\in \{0,1\}^{n_1\times n_2}\), \(n_1=\left| V_1 \right| , n_2=\left| V_2 \right| \), where \(A_{ijt}=1\) if \(i\in V_1\) connects to \(j\in V_2\) in \((t-1,t]\).

This paper discusses methods for temporal link prediction (Dunlavy et al. 2011): given a sequence of adjacency matrices \(\mathbf {A}_1,\dots ,\mathbf {A}_T\) observed over time, the main objective is to reliably predict \(\mathbf {A}_{T+1}\). In this article, temporal link prediction techniques based on random dot product graphs (RDPG, Young and Scheinerman 2007) are discussed and compared. RDPGs are a class of latent position models (Hoff et al. 2002), and have been extensively studied because of their analytical tractability (Athreya et al. 2018). Each node i is given a latent position \(\mathbf {x}_i\) in a d-dimensional latent space \({\mathbb {X}}\) such that \(\mathbf {x}^\top \mathbf {x}^\prime \in [0,1]\ \forall \ \mathbf {x},\mathbf {x}^\prime \in {\mathbb {X}}\). The edges between pairs of nodes are generated independently, with probability of a link between nodes i and j obtained through the inner product \({\mathbb {P}}(A_{ij}=1)=\mathbf {x}_i^\top \mathbf {x}_j\). In matrix notation, the latent position can be arranged in a \(n\times d\) matrix \(\mathbf {X}=[\mathbf {x}_1,\dots ,\mathbf {x}_n]^\top \in {\mathbb {X}}^n\), and the expected value of a single realised adjacency matrix \(\mathbf {A}\) is expressed as \({\mathbb {E}}(\mathbf {A})=\mathbf {X}\mathbf {X}^\top \).

Random dot product graph models and spectral embedding methods are often the first step in the analysis of a graph, because of their simplicity and ease of implementation, since intensive hyperparameter tuning is not required. RDPGs are extensively applied in neuroscience (see, for example, Priebe et al. 2017). Furthermore, they have appealing theoretical statistical properties in terms of consistency of the estimated latent positions. Therefore, it is of interest to understand their performance for link prediction purposes.

RDPGs models for multiple heterogeneous graphs on the same node set have recently been proposed in the literature, but these models have not been formally extended to a dynamic setting for link prediction purposes. Early examples discuss methods for clustering and community detection with multiple graphs (Tang et al. 2009; Shiga and Mamitsuka 2012; Dong et al. 2014). More recently, the focus has been on testing for differences in brain connectivity networks (Arroyo-Relión et al. 2019; Ginestet et al. 2017; Durante and Dunson 2018; Kim and Levina 2019). Levin et al. (2017) propose an omnibus embedding in which the different graphs are jointly embedded into a common latent space, providing distinct representations for each graph and for each node. Wang et al. (2021) propose the multiple random eigen graph (MREG) model, where a common set of d-dimensional latent features \(\mathbf {X}\) is shared between the graphs, and the inner product between the latent positions is weighted differently across the networks, obtaining \({\mathbb {E}}(\mathbf {A}_t)=\mathbf {X}\mathbf {R}_t\mathbf {X}^\top \), where \(\mathbf {R}_t\) is a \(d\times d\) diagonal matrix. Nielsen and Witten (2018) propose the multiple random dot product graph (multi-RDPG), which more naturally extends the RDPG to the multi-graph setting. Their formulation is similar to the MREG of Wang et al. (2021), but \(\mathbf {X}\) is modelled as an orthogonal matrix, and \(\mathbf {R}_t\) is constrained to be positive semi-definite. The model is further extended in common subspace independent edge (COSIE) graphs (Arroyo-Relión et al. 2020), in which \(\mathbf {R}_t\) does not need to be a diagonal matrix. More recently, Jones and Rubin-Delanchy (2021) proposed the Unfolded Adjacency Spectral Embedding (UASE) for the multilayer random dot product graph (MRDPG), which is also applied to a link prediction example within a cyber-security context. In this work, existing methods for RDPG-based inference, for example omnibus embeddings (Levin et al. 2017) and COSIE graphs (Arroyo-Relión et al. 2020), will be analysed for link prediction purposes, and compared to standard spectral embedding techniques.

The main contribution of this work is to adapt the existing methods for multiple RDPG graph inference for temporal link prediction. Furthermore, this article proposes methods to combine the information obtained via spectral methods with time series models, to capture the temporal dynamics of the observed graphs. The proposed methodologies will be extensively compared on real world and simulated networks. It will be shown that this approach significantly improves the predictive performance of multiple RDPG models, especially when the network presents a seasonal or temporal evolution. Overall, this article provides insights into the predictive capability of random dot product graphs, and gives guidelines for practitioners on the optimal choice of the embedding for temporal link prediction.

The article is organised as follows: Sect. 2 discusses related literature around link prediction, and Sect. 3 introduces background on the generalised random dot product graph and adjacency spectral embeddings, the main statistical tools used in this paper. Methods for link prediction based on random dot product graphs are discussed in Sect. 4. Section 5 presents techniques to improve the predictive performance of the RDPG models, based on time series models. Results and applications on simulated and real world networks are finally discussed in Sect. 6.

Many other models other than RDPGs have been proposed in the literature for link prediction. Traditionally, the temporal link prediction task is tackled using tensor decompositions (Dunlavy et al. 2011). Dynamic models have also been proposed in the literature of Poisson matrix factorisation and recommender systems (Charlin et al. 2015; Hosseini et al. 2020), and extended to Bayesian tensor decompositions (Schein et al. 2015). In general, including time has been shown to significantly improve the predictive performance in a variety of model settings, for example stochastic blockmodels (Ishiguro et al. 2010; Xu and Hero III 2014; Xing et al. 2010). More generic latent feature models for dynamic networks have also been extensively discussed in the literature (Sarkar and Moore 2006; Krivitsky and Handcock 2014; Sewell and Chen 2015).

Latent features are usually obtained via matrix factorisation, considering constant and time-varying components within the decomposition (Deng et al. 2016; Yu et al. 2017a, b). Usually, a Markov assumption is placed on the evolution of the latent positions (Zhu et al. 2016; Chen and Li 2018). Gao et al. (2011) propose to combine node features into a temporal link prediction framework based on matrix factorisation. Nonparametric (Sarkar et al. 2014; Durante and Dunson 2014) and deep learning (Li et al. 2014) approaches have also been considered.

More recently, advancements have been made in the application of deep learning to graph-valued data. In particular, deep learning methods on graphs are classified by Zhang et al. (2020) into five categories: graph recurrent neural networks, graph convolutional networks, graph autoencoders, graph reinforcement learning, graph adversarial methods. A comprehensive survey of existing static network embedding methods, including deep learning techniques, is provided in Cai et al. (2018). Commonly used static embedding methods in machine learning are DeepWalk (Perozzi et al. 2014), SDNE (Wang et al. 2016), node2vec (Grover and Leskovec 2016), GraphSAGE (Hamilton et al. 2017), graph convolutional networks (GCN, Kipf and Welling 2017), and Watch Your Step with Graph Attention (WYS-GA, Abu-El-Haija et al. 2018). Many of such methods could be unified under a matrix factorisation framework (Qiu et al. 2018). A systematic comparison of some of the aforementioned methodologies for unsupervised network embedding is provided in Khosla et al. (2021). Methodologies have also been recently proposed in the dynamic network setting, within the context of representation learning (for example, Nguyen et al. 2018; Kumar et al. 2019; Liu et al. 2019; Qu et al. 2020), and deep generative models (for example, Zhou et al. 2020). The interested reader is referred to the survey of Kazemi et al. (2020) and references therein.

In this section, the generalised random dot product graph (Rubin-Delanchy et al. 2017) and methods for estimation of the latent positions are formally introduced. Suppose \(\mathbf {A}\in \{0,1\}^{n\times n}\) is a symmetric adjacency matrix of an undirected graph with n nodes.

The adjacency spectral embedding (ASE) provides consistent estimates of the latent positions in GRDPGs (Rubin-Delanchy et al. 2017), up to indefinite orthogonal transformations.

If the graph is directed, and the adjacency matrix is not symmetric, it could be implicitly assumed that the generating model is \({\mathbb {P}}(A_{ij}=1)=\mathbf {x}_i^\top \mathbf {y}_j, \mathbf {x}_i,\mathbf {y}_j\in {\mathbb {X}}\), where each node is given two different latent positions, representing the behaviour of the node as source or destination of the link. In this case, the embeddings can be estimated using the singular value decomposition (SVD).

Given a time series of network adjacency matrices \(\mathbf {A}_1,\mathbf {A}_2,\dots ,\mathbf {A}_T\), the objective is to correctly predict \(\mathbf {A}_{T+1}\). The most common approach in the literature (Sharan and Neville 2008; Scheinerman and Tucker 2010; Dunlavy et al. 2011) is to analyse a collapsed version \(\tilde{\mathbf {A}}\) of the adjacency matrices:where \(\psi _1,\dots ,\psi _T\in {\mathbb {R}}\) is a sequence of weights. Scheinerman and Tucker (2010) propose to consider an average adjacency matrix, setting \(\psi _t=1/T\ \forall \ t=1,\dots ,T\), which corresponds to the maximum likelihood estimate of \({\mathbb {E}}(\mathbf {A}_t)\) if \(\mathbf {A}_1,\dots ,\mathbf {A}_T\) are sampled independently from the same \(\mathrm {Bernoulli}(\mathbf {X}\mathbf {X}^\top )\) distribution. The main limitation of such a model is that it is assumed that the graphs do not display any temporal evolution. Furthermore, if (5) is used, it is assumed that all the possible edges of the adjacency matrix follow the same dynamics. Obtaining the ASE \(\hat{\mathbf {X}}=[\hat{\mathbf {x}}_1,\dots ,\hat{\mathbf {x}}_n]\) of \(\tilde{\mathbf {A}}\) leads to an estimate the scores:For simplicity, the inner product (6) is not weighted by the matrix \(\mathbf {I}(d_+,d_-)\), implicitly assuming \(d_+=d\) and \(d_-=0\). The estimation approaches in (5) and (6) will be used as baselines for comparison with alternative methods for temporal link prediction techniques using RDPGs, which will be proposed and discussed in the remainder of this section. The proposed methods will be classified in the following two categories:First, it is possible to consider the individual ASE for each adjacency matrix \(\mathbf {A}_t\), and calculate an AIP score:A second option is to obtain an averaged embedding \(\bar{\mathbf {X}}\) from \(\hat{\mathbf {X}}_1,\hat{\mathbf {X}}_2,\dots ,\hat{\mathbf {X}}_T\), and use this for predicting the link probabilities. This procedure is slightly more complex than (7), since the embeddings are invariant to orthogonal transformations: given an orthogonal matrix \({\varvec{\Omega }}_t\in {\mathbb {R}}^{d\times d}\), \({\mathbb {E}}(\mathbf {A}_t)=\mathbf {X}_t\mathbf {X}_t^\top =(\mathbf {X}_t{\varvec{\Omega }}_t)(\mathbf {X}_t{\varvec{\Omega }}_t)^\top \). Therefore, the embeddings \(\hat{\mathbf {X}}_1,\dots ,\hat{\mathbf {X}}_T\) only provide estimates of the corresponding latent positions up to an orthogonal transformation, which could vary for different values of t. Consequently, the embeddings must be suitably aligned or registered, before a direct comparison can be carried out. Discussion of a technique to align the embeddings is deferred to Appendix A. Assuming that an averaged embedding \(\bar{\mathbf {X}}\) is obtained, the matrix of IPA scores for prediction of \(\mathbf {A}_{T+1}\) is:Similar scoring mechanisms can be derived for the techniques of multiple graph inference described in Sect. 1, such as the omnibus embedding (Levin et al. 2017), based on the the omnibus matrix:The ASE \(\hat{\mathbf {X}}\) of \(\tilde{\mathbf {A}}\) gives T latent positions for each node. The individual estimates \(\hat{\mathbf {X}}_t=[\hat{\mathbf {x}}_{1t},\dots ,\hat{\mathbf {x}}_{nt}]\) of the latent positions for the t-th adjacency matrix are represented by the submatrix formed by the estimates between the \(((t-1)n+1)\)-th and tn-th row of \(\hat{\mathbf {X}}\). Then, from the time series \(\hat{\mathbf {X}}_1,\hat{\mathbf {X}}_2,\dots ,\hat{\mathbf {X}}_T\) of omnibus embeddings, a matrix of scores can be obtained using either AIP (7) or IPA (8). In this case, the individual embeddings are directly comparable and an alignment step is not required. On the other hand, the omnibus embedding cannot easily be updated when new graphs \(\mathbf {A}_{T+1}, \mathbf {A}_{T+2}, \dots \) become available, since \(\tilde{\mathbf {A}}\) and the embedding must be recomputed for each new snapshot. The idea of an omnibus embedding can be also easily extended to directed and bipartite graphs, constructing the matrix \(\tilde{\mathbf {A}}\) analogously and then calculating the DASE.

Embeddings generated using the more parsimonious COSIE model (Arroyo-Relión et al. 2020) are also considered. In COSIE networks, the latent positions are assumed to be common across the T snapshots of the graph, but the link probabilities are scaled by a time-varying matrix \(\mathbf {R}_t\in {\mathbb {R}}^{d\times d}\): \({\mathbb {E}}(\mathbf {A}_t) = \mathbf {X}\mathbf {R}_t\mathbf {X}^\top . \) The common latent positions \(\mathbf {X}\) and the time series of weighting matrices \(\mathbf {R}_1,\dots ,\mathbf {R}_T\) can be estimated via multiple adjacency spectral embedding (MASE, Arroyo-Relión et al. 2020).

For prediction, the matrix of AIP scores could be obtained from the time series of estimated link probabilities \(\hat{\mathbf {X}}\hat{\mathbf {R}}_1\hat{\mathbf {X}}^\top ,\dots ,\hat{\mathbf {X}}\hat{\mathbf {R}}_T\hat{\mathbf {X}}^\top \):Alternatively, an averaged \(\bar{\mathbf {R}}\) could be equivalently obtained from the time series of estimates \(\hat{\mathbf {R}}_1,\dots ,\hat{\mathbf {R}}_T\). Combining \(\bar{\mathbf {R}}\) with the estimate of the latent positions \(\hat{\mathbf {X}}\) yields the IPA scores:The COSIE model can also be extended to directed and bipartite graphs assuming \({\mathbb {E}}(\mathbf {A}_t)=\mathbf {X}\mathbf {R}_t\mathbf {Y}^\top \). This construction leads to estimates \(\hat{\mathbf {R}}_t=\hat{\mathbf {U}}^\top \mathbf {A}_t\hat{\mathbf {V}}\), where \(\hat{\mathbf {U}}\) and \(\hat{\mathbf {V}}\) are estimates of \(\mathbf {X}\) and \(\mathbf {Y}\) obtained from MASE on the DASE embeddings \(\hat{\mathbf {X}}_1,\dots ,\hat{\mathbf {X}}_T\) and \(\hat{\mathbf {Y}}_1,\dots ,\hat{\mathbf {Y}}_T\), based on two matrices \(\tilde{{\varvec{\Gamma }}}\) constructed from the left and right singular vectors (cf. Definition 4).

In summary, several link prediction schemes based on random dot product graph models have been proposed, corresponding to three different types of spectral embedding:Two scores, denoted AIP and IPA, are calculated for each embedding type. All methods will be compared to the popular collapsed adjacency matrix method in (6).

The methods described in this section are all based on truncated eigendecompositions of some form of the adjacency matrix. The full eigendecomposition of a \(n\times n\) dense matrix requires a cubic computational cost \({\mathcal {O}}(n^3)\), but only d eigenvectors and eigenvalues, where d is in general \({\mathcal {O}}(1)\), are required in the algorithms presented in this section. This reduces the computational effort to \({\mathcal {O}}(n^2)\) (Yang et al. 2021). Also, graph adjacency matrices are binary and normally highly sparse. In this setting, efficient algorithms based on the power method calculate the required decomposition of a matrix \(\mathbf {A}\) in \({\mathcal {O}}\{\mathrm {nnz}(\mathbf {A}){\mathcal {P}}(1/\varepsilon )\}\) (Ghashami et al. 2016), where \(\mathrm {nnz}(\cdot )\) denotes the number of non-zero entries of the matrix, \({\mathcal {P}}(\cdot )\) is a polynomial function and \(\varepsilon \in (0,1)\) is an approximation error parameter. This allows the methodologies in this section to be scalable to large networks with the support of modern computer libraries. For example, calculating the individual embeddings \(\hat{\mathbf {X}}_1,\dots ,\hat{\mathbf {X}}_T\) only has complexity \({\mathcal {O}}\{{\mathcal {P}}(1/\varepsilon )\sum _{t=1}^T\mathrm {nnz}(\mathbf {A}_t)\}\). COSIE adds a further SVD decomposition in the MASE algorithm, which requires further \({\mathcal {O}}(nd^2)\) operations. On the other hand, calculating the omnibus embedding for large graphs might quickly become cumbersome, especially if T is large, since up to \({\mathcal {O}}(n^2T^2)\) operations are required.

The collapsed matrix used in (5) assumes that the underlying dynamics of each link are the same across the entire graph. This assumption is particularly limiting in real world applications, where different behaviours might be associated with different nodes or links. Instead, edge specific matrix parameters \({\varvec{\Psi }}_1,\dots ,{\varvec{\Psi }}_T\in {\mathbb {R}}^{n\times n}\) might be able to more reliably capture the behaviour of each edge. A modification of the collapsed matrix \(\tilde{\mathbf {A}}\) in (5) is therefore proposed:where \({\varvec{\Psi }}_1,\dots ,{\varvec{\Psi }}_T\in {\mathbb {R}}^{n\times n}\) is a sequence of weighting matrices, and \(\odot \) is the Hadamard element-wise product. The matrix in (13) is denoted predicted adjacency matrix. Note that in (13), the weights can only be estimated for those entries such that \(A_{ijt}=1\) for at least one \(t\in \{1,\dots ,T\}\), but the ASE of \(\tilde{\mathbf {A}}\) still allows to estimate non-zero link probabilities even for those edges such that \(A_{ijt}=0\ \forall t\).

The idea could be easily extended to all the other prediction settings proposed in Sect. 4, replacing the average link probability or average embedding with an autoregressive combination. For example, from the sequence of standard embeddings \(\hat{\mathbf {X}}_1,\hat{\mathbf {X}}_2,\dots ,\hat{\mathbf {X}}_T\), it could be possible to obtain the scores as:Alternatively, it could be possible to use a similar technique to obtain an estimate \(\tilde{\mathbf {X}}_{T+1}=\sum _{t=1}^T ({\varvec{\Psi }}_{T-t+1}\odot \hat{\mathbf {X}}_t)\) of the embedding \(\mathbf {X}_{T+1}\), where in this case \({\varvec{\Psi }}_t\in {\mathbb {R}}^{n\times d}\). The scores are then obtained as:Similarly, for COSIE, the scores could be based on a linear combination of \(\hat{\mathbf {R}}_1,\dots ,\hat{\mathbf {R}}_T\) to estimate \(\mathbf {R}_{T+1}\).

The equations (14) and (15) define two different methodologies to extend multiple RDPG inference models to a dynamic setting. Following the same nomenclature used in Sect. 4, the scores based on time series models have been respectively denoted as PIP for the predicted inner product (14), and IPP for the inner product of the prediction (15). The PIP scores have a particularly interesting property: the node-specific embeddings are combined with time series models at the edge levels, fusing information at two different network resolutions.

Note that, for COSIE scores, \(\mathbf {S}_\text {AIP}=\mathbf {S}_\text {IPA}\) from (11) and (12), but \(\mathbf {S}_\text {PIP}\ne \mathbf {S}_\text {IPP}\). This is because for \(\mathbf {S}_\text {PIP}\), the scores are obtained directly from a prediction based on the estimated link probabilities, whereas for \(\mathbf {S}_\text {IPA}\), the prediction is based on an estimate of \(\mathbf {R}_{T+1}\) from \(\hat{\mathbf {R}}_1,\dots ,\hat{\mathbf {R}}_T\).

For estimation of the weighting matrices \({\varvec{\Psi }}_1,\dots ,{\varvec{\Psi }}_T\), the time series of link probabilities or node embeddings will be modelled independently. Seasonal autoregressive integrated (SARI) processes represent a flexible modelling assumption. A univariate time series \(Z_1,\dots ,Z_T\in {\mathbb {R}}\), is a \(\text {SARI}(p,b)(P,B)_s\) with period s if the series is a causal autoregressive process defined by the equationwhere \(\phi (v)=1-\phi _1v-\ldots -\phi _pv^p\), \(\Phi (v)=1-\Phi _1v-\ldots -\Phi _Pv^P\), and L is the lag operator \(L^kZ_t = Z_{t-k}\) (Brockwell and Davis 1987). For example, consider a process \(\text {SARI}(1,0)(0,1)_s\). The model equation (16) becomes \({{\tilde{Z}}}_t=\phi _1{{\tilde{Z}}}_{t-1}+\varepsilon _t\) with \({{\tilde{Z}}}_t=Z_t-Z_{t-s}\). The process is causal if and only if \(\phi (v)\ne 0\) and \(\Phi (v)\ne 0\) for \(\left| v \right| \le 1\). The parameters of the process p, b, P, B and s are all integer-valued, and can be interpreted as follows: s is the seasonal periodicity; b corresponds to the differencing required to make the process stationary in mean, variance, and autocorrelations, and B refers to the seasonal differencing; p is to the number of autoregressive terms appearing in the equation, and similarly P refers to the number of seasonal autoregressive terms. More details about SARI models and their interpretation are given in Brockwell and Davis (1987).

The value of s usually depends on the application domain. For example, in computer networks with daily network snapshots, it is reasonable to assume \(s=7\), which represents a periodicity of one week. The remaining parameters, p, b, P and B, could be estimated using information criteria. For small values of T, the corrected Akaike information criterion (AICc) is preferred. The corresponding coefficients of the polynomials \(\phi (v)\) and \(\Phi (v)\), and the variance \(\sigma ^2\), can be estimated via maximum likelihood (Brockwell and Davis 1987). For a discussion on automatic selection of the parameters in SARI models, see Hyndman and Khandakar (2008).

For prediction of future values \(Z_{t+1}\) conditional on \(Z_1,\dots ,Z_t\), the general forecasting equation is obtained from (16), setting \(\varepsilon _{t+1}\) to its expected value \({\mathbb {E}}(\varepsilon _{t+1})=0\), and obtaining an estimate \({{\hat{Z}}}_{t+1}\) solving from the known terms of the equation. Analogously, k-steps ahead forecasts for \(Z_{t+k}\) can also be obtained.

In this article, the univariate time series \(Z_1,\dots ,Z_T\) modelled using SARI are of four different types:The coefficients for each entry of the weighting matrices \({\varvec{\Psi }}_1,\dots ,{\varvec{\Psi }}_T\) are obtained by matching (13), (14) and (15) with the model equation (16).

In this work, the binary time series \(\mathbf {A}_1,\mathbf {A}_2,\dots ,\mathbf {A}_T\) is also modelled using indipendent SARI models for estimation of (13). Such a modelling approach might not be entirely technically suited for binary-valued time series, but this choice has relevant practical advantages: most programming languages have packages for automatic estimation of the parameters in SARI models, whereas the choice of initial values and estimation of the parameters in most generalised process for binary time series (MacDonald and Zucchini 1997; Kauppi and Saikkonen 2008; Benjamin et al. 2003) is notoriously difficult, which is not desirable when the estimation task should be performed automatically and in parallel over a large set of time series.

The time series modelling extensions presented in this section are computationally expensive, since up to \(n^2\) or nd time series model are fitted, each with complexity \({\mathcal {O}}(Tk)\), where k is the number of models compared for the purpose of model selection using AICc. This results in a computational cost of \({\mathcal {O}}(n^2Tk)\) for PIP scores or \({\mathcal {O}}(ndTk)\) for IPP scores, difficult to manage for large networks. Therefore, with the exception of the IPP COSIE scores, the methodologies proposed in this section do not scale well to large networks, unlike the techniques proposed in Sect. 4. In the case of the IPP COSIE score (12), the cost for prediction of future values of \(\mathbf {R}_t\) is only \({\mathcal {O}}(d^2Tk)\), independent of n.

The methodologies for constructing link prediction scores discussed in this work are summarised in Fig. 1 in a flowchart. In summary, three choices must be made:Clearly, the same methodology could be applied to any embedding method, not necessarily based on the RDPG. Some examples will be given in Sect. 6.2.2.

The proposed methods were tested on synthetic data and on real world dynamic networks from different application domains: transportation systems, cyber-security, and co-authorship networks. For all examples, the parameter d was selected using the profile likelihood elbow criterion of Zhu and Ghodsi (2006), unless otherwise specified.

In this paper, link prediction techniques based on random dot product graphs have been presented, discussed and compared. In particular, link prediction methods based on sequences of embeddings, COSIE models, and omnibus embeddings have been considered. Applications on simulated and real world data have shown that one of the most common approaches used in the literature, the decomposition of a collapsed adjacency matrix \(\tilde{\mathbf {A}}\), is usually outperformed by other methods for multiple graph inference in random dot product graphs.

Estimating link probabilities with the average of the inner products from sequences of individual embeddings of different snapshots of the graph has given the best performance in terms of AUC scores across multiple datasets. This result is particularly appealing for practical applications: calculating the individual ASEs is computationally inexpensive using algorithms for large sparse matrices, and the method seems particularly suitable for implementation in a streaming fashion, since the average inner product could be easily updated on the run when new snapshots of the graph are observed, as demonstrated in Sect. 6.3.

The methods discussed in the article have then been further extended to include temporal dynamics, using time series models. The extensions have shown improvements over standard random dot product graph based link prediction techniques, especially when the graph exhibits a strong dynamic component. The techniques presented in this article could also be readily extended to any embedding method for static graphs, following the framework for calculating AIP and IPA scores presented in Section 4, and the PIP and IPP extensions in Section 5. Therefore, our methodology is not only applicable to RDPG embeddings, particularly attractive for their theoretical properties and ease of implementation, but also to modern static embedding methods for graph adjacency matrices, for example graph neural network techniques like GCN or GraphSAGE. Overall, this article provides valuable guidelines for practitioners for using random dot product graphs as tools for link prediction in networks, providing insights into the predictive capability of such statistical network models.