Direct Embedding of Temporal Network Edges via Time-Decayed Line Graphs
Authors: Sudhanshu Chanpuriya
Ryan A. Rossi
Sungchul Kim
Tong Yu
Jane Hoffswell
Nedim Lipka
Shunan Guo
Cameron N Musco
Venue: ICLR 2023 poster (2023)
Keywords: temporal
networks
graphs
embedding

Abstract: Temporal networks model a variety of important phenomena involving timed interactions between entities. Existing methods for machine learning on temporal networks generally exhibit at least one of two limitations. First, many methods assume time to be discretized, so if the time data is continuous, the user must determine the discretization and discard precise time information. Second, edge representations can only be calculated indirectly from the nodes, which may be suboptimal for tasks like edge classification. We present a simple method that avoids both shortcomings: construct the line graph of the network, which includes a node for each interaction, and weigh the edges of this graph based on the difference in time between interactions. From this derived graph, edge representations for the original network can be computed with efficient classical methods. The simplicity of this approach facilitates explicit theoretical analysis: we can constructively show the effectiveness of our method's representations for a natural synthetic model of temporal networks. Empirical results on real-world networks demonstrate our method's efficacy and efficiency on both link classification and prediction.

Published as a conference paper at ICLR 2023
DIRECT EMBEDDING OF TEMPORAL NETWORK EDGES VIA TIME-DECAYER LINE GRAPHS
Sudhanshu Champuriya', Ryan A. Rossi', Sungchul Kim', Tong Yu', Jane Hoffswell', Nedim Lipka', Shunan Guo², and Cameron Musco!
Unniversity of Massachusetts Amherst, {schanpuriya, emusco) umass edu 2 Adobe Research, {ryrossi, sukim, tyu, jhoffs, 1ipka, squo com.
ABSTRACT
Temporal networks model variety of important phenomena involving timed interactions between entities. Existing methods for machine learning on temporal networks generally exhibit least one of two limitations. First, many methods assume time to be discretized, so if the time data is continuous, the user must determine the discretization and discard precise time information. Second, edge representations can only be calculated indirectly from the nodes, which may be suboptimal for tasks like edge classification. We present simple method that avoids both shortcomings: construct the line graph of the network, which includes a node for each interaction, and weigh the edges of this graph based on the difference in time between interactions. From this derived graph, edge representations for the original network can be computed with efficient classical methods. The simplicity of this approach facilitates explicit theoretical analysis: we constructively show the effectiveness of our method's representations for a natural synthetic model of temporal networks. Empirical results real-world networks demonstrate our method's efficacy and efficiency on both link classification and prediction.
INTRODUCTION
Temporal networks, which are graphs augmented with a time value for each edge, model variety of important phenomena involving timed interactions between entities, including financial transactions, flights, and web browsing. Common tasks for machine learning on temporal networks include classi- fication of the temporal edges, as well as temporal link prediction, which involves predicting future links given some links in the past. These tasks yield various applications, such recommendation systems (Zhou et al., 2021) and detection of illicit financial transactions (Pareja et al., 2020). As with most machine learning for graphs, the key to learning for temporal networks is creating effective vector representations, also called embeddings, for the network's 's components, namely the nodes and edges. These embeddings can either be made as part of end-to-end framework, or created then passed to off-the-shelf classifiers for downstream tasks; for example, for the edge classification task, logistic regression classifier can be trained using the training edges' embedding vectors and class labels, then applied at inference time on the test edges' vectors.
The node embedding task, in particular, has seen great interest, and many methods for 'static" (i.e., non-temporal) networks have been proposed over the years (Belkin & Niyogi, 2001; Perozzi al., 2014; Grover & Leskovec, 2016). Edge embedding has seen less interest; there are some exceptions (I.i et al., 2017b; Bandyopadhyay et al., 2019), but generally, edge embeddings are created by first making node embeddings, then aggregating them, e.g., by taking the entrywise product of the two endpoint nodes' embeddings. There are wide variety of methods for temporal network embedding, including ones based on matrix/tensor factorization (Dunlavy et al., 201 I; Li et al., 2017a; Zhang et al., 2018), random walks (Yu et al., 2018; Nguyen al., 2018), graph convolutional networks (Pareja et al., 2020), and deep autoencoders (Goyal et al., 2018; Rahman et al., 2018; Goyal et al., 2020), but generally, these methods can be seen variants of those for static networks.
These prior methods overall present some limitations, which we now summarize. With some exceptions, especially in more recent work (Nguyen al., 2018; Trivedi et al., 2019; Rossi et al.,

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NODE RUPRESENTATIO--ASSE
EDGE REPRESENNTATIO--IAAEEI
TIMERS (Zhang et al., 2018)
DISCRETE TIME
EvolveGiNN (Pureja et al., 2020)
DeL (Rahman et al., 2018)
CTDNE (Nguyen et al., 2018)
CONTINUOUS TIME
DyRep (Trivedi et al., 2019)
This work
Table I: Problem studied in this work.
2020; Wang et al., 2021), these methods often do not work directly with the centinuous-valuee times of temporal edges, but rather assume that the times are discretized, yielding a sequence of static graphs. Since most datasets have continuous-timed edges, this assumption requires the user to manually determine the discretization and discard precise time information. Second, prior methods generally return embeddings of nodes rather than edges, so edge embeddings can only be calculated indirectly from the nodes, which may be suboptimal for tasks like edge classification. Given that timestamps are associated with edges rather than nodes, and that there are few public datasets where the nodes rather than edges associated with time-series of attributes or classes, it is intuitively more natural to derive edge embeddings directly.
We present simple but novel framework to address these issues: construct the line graph of the network, which converts each edge (timed interaction) of the network into a node, and connects interactions that share endpoint node. Then, set the edge weights in this line graph based on differences in time between interactions, with interactions that occur closer together in time being connected with higher weights. From this derived graph, which directly represents topological proximity (i.e., adjacency of edges) and remporal proximity, temporal edge representations for the original network can be computed and exploited with efficient classical methods. To our knowledge, ours is the first method that directly forms embeddings for continuous-time temporal edges without supervision and without aggregating node embeddings see Table 1. Our method significantly simpler than recent prior work, particularly compared to deep methods, allowing for more direct theoretical analysis: we propose the union of Gaussian- -aimaadtttmeedd-ee-- stochastic block models (UGT-SBM), which naturally extends the well-known stochastic block model (SBM) for static networks to temporal networks, and we show that our method can exploit time and community information in UGT-SBMs method's simplicity makes it easy to implement,
to form effective representations. Practically, yet it is accurate and efficient: in experiments on five benchmark real-world temporal networks, our approach achieves superior predictive and runtime performance relative prior methods.
METHODOLOGGY
Our approach starts with a sequence of timestamped edges:
Definition 1 (Temporal Graph). A temporal graph = (V,E) is a set of vertices V and set of temporal edges E, where E C V x x IR. t is the time of the temporal edge (u, v,f).
Our approach centers around our proposed notion of 'time-decayed line graphs' (TDLGs) which are derived from temporal graphs. Our notion of TDLGs extends the well-established idea of line graphs, which are derived from static (i.e., non-temporal), undirected graphs; the line graph of an undirected, unweighted graph is another undirected, unweighted graph that represents adjacencies between edges in the original graph. We propose to incorporate time information by using it to set the weights of the resulting line graph. Specifically, given temporal graph G, we construct a TDLG, which a static weighted line graph (G), as follows in Definition 2.
Definition (Time-Decayed Line Graph). Given temporal graph G = (V, E), the associated time- decayed line graph LTD(G) = (VL, EL, w). where VL E, EL { [(u, v, t,t (v. z, t2)) (u, v. t1), (v, 2,t2) E E}, and the edge weight function w EL. + R evaluated on edge ((,, v. L1), (v, z, 2,12)) is given by exp
(-2g,
2 (41 - 12)) for some fixed a, > 0.
Thus, proximity of two temporal edges in the TDLG incorporates both topological proximity in the original graph well proximity in time. The parameter or controls how quickly proximity in the TDLG decays as difference in time grows. For a graph with n nodes and m edges, we can construct the weighted adjacency matrix A Rmxm of the TDL. follows. Given the incidence matrix
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B {0,1]"xm each column vector of which is n-dimensional, corresponds to edge. and is 1 for the edge's 's two endpoint nodes and O elsewhere; and also given the vector of times of the temporal edges t E R":
(tz (4:-tt) t,)² 2
A. = b,) exp
where b, and by denote columns i and j of B. Note that A is symmetric. While self-loops in line graphs are generally removed, we keep them here for simplicity. The choice of Gaussian weight decay is somewhat arbitrary, and, in theory, the TDLG concept would also work with, e.g., Laplacian weight decay. We use Gaussian decay since it is well known and effective in practice.
We now describe our approach for temporal edge embedding, classification, and link prediction.
Temporal edge embedding and classification For downstream tasks, we seek temporal edge vector for each temporal edge that can be provided
embeddings, that is, an informative to classifier. For temporal edge i, we simply return the 7th row of the TDLG adjacency matrix A. This returns an m--dimensional sparse vector as the feature vector. Note that we could reduce the dimensionality of these vectors, e.g., via eigendecomposition, but we find that this is not necessary. We use the standard supervised learning pipeline for edge classification. Given training data comprising set of temporal edges and the class labels of some fraction of these edges, the TDLG matrix A is created using all of the edges, then a logistic regression classifier is trained using the edges for which the classes are known and the corresponding rows of A as feature vectors. After this, the classifier can be used on the remaining rows to make predictions for the remaining edges.
Temporal link prediction We describe the full experimental setup for temporal link prediction in Section 6, but essentially, it amounts to binary temporal edge classification where the classes are +1 for true/positive edges (u,, Us 44) and -1 for false/negative edges (ul, v/z ()). The difference that for temporal link prediction, we must additionally be able to form representations for negative edges, as well as for test edges (which are not given when training the classifier). Let subscripts r and e denote training and test edges, respectively. We deal with negative edges exactly the same positive edges besides the class labels: the positive and negative training edges are concatenated, producing the training incidence matrix B, {0. 1) 1) and times vector f, TR. m. from which construct the training TDLG adjacency matrix Apr E mm,, according to Equation 1. The classifier can then be trained using the rows of A,r feature vectors and the edge labels (i.e., positive/negative edge). Now feature vectors for the test edges must each be -,ddiennsional and consider only proximity to training edges. Letting the test incidence matrix and test times vector be B E {0, 1) x m and te E Iem,,, the test edge feature vectors will be the rows of the matrix Agr E Rm.xm.. the (it entry of which is given by
(lei -,,))
(ba:-ba by 5) exp
20²
which can be passed to the classifier for inference.
DEMONSTRATIVE EXAMPLE
To demonstrate the power of our TDLG method at capturing latent structure in temporal graphs, we introduce simple, natural model for temporal networks based on the stochastic block model (SBM) (Holland et al., 1983). Our model is called the union of Gaussian-time As the name suggests, it comprises the union of several SBMs, where normal distribution is attached to each SBM, and the edges drawn from an SBM are given a time which is sampled from its distribution. The formal defintion follows in Definition 3.
SBMs (UGT-SBM).
Definition 3 (UGT-SBM), Consider set of n. nodes partitioned into two equal-sized communities (U,V), For some integer À and real numbers 0 < 011, Q2 < 1, construct a random temporal edge set over this graph as follows: Let there be As temporal edges in each of two time periods; the 2
times associated with edges in each time period are drawn from the normal distributions N (H1, af) and N Nu2, oï), respectively. Of the edges in time period 1, let (1 - a1) Ar edges be drawn uniformly at random from U x V, and let the remaining o1 -2
An edges be drawn half each from
U x U and V x V. Similarly, of the edges in time period 2, let (1 ag) A edges be drawn from U x V, and the remaining 02 A Ain
edges be drawn half each from U x U and V V.
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Here A is the expected degree of each node in each of the two SBMs (so the total degree of each node is 2 A), and 01 d2 represent the fraction of edges in each SBM which are intra aatra-community as opposed to inter-community. This definition generalize straightforwardly to union of arbitrary number of SBMs, potentially with unequal community sizes. We note some related prior models with more sophisticated temporal dependencies based on point processes: the Hawkes IRM (Blundell et al., 2012), the block point process (BPP) model (Junuthula et al., 2019), and the community Hawkes independent pairs (CHIP) model (Arastuie al., 2020); the latter two in particular are also based on SBMs. Also related is the model from Barbillon et al. (2017), which extends SBMs not to the temporal setting, but rather to the multiplex setting (i.e., multiple graphs over the same set of nodes).
We first construct UGT-SBM where it is impossible to distinguish any community structure if one ignores temporal information. This UGT-SBM visualized in Figure 1. The graph has n 100 nodes, with expected degrees A = 40, and the time distributions of the two SBMs are V (-1, (1/2)) and NV (1/2))) the distributions have the same variance, but occurs earlier on average. We set 01 = 9/10 and 02 3/10, so that the mean fraction of notracommuuntyy edges is 21ta 1/2; this ensures that it is impossible to retrieve community structure without considering time, as is best illustrated Figure in the union of the two SBMs, which simply an Erdós Erdös-Rényi graph.
Early
Late
Union
Distribution
250
200
150
50.
50
100
Earl
Lafe
0
100
Figure The motivating synthetic temporal network. The adjacency matrix the early and late edges, well as their union, then the distribution of times separated by the early and late edges. Note that the early edges are mostly in-group, whereas the late edges are mostly out-group.
We will show that TDLG method can utilize the time information recover the node communi- ties. Since our method produces embeddings for temporal edges, to demonstrate recovery of node communities, must aggregate the edge embeddings into node embeddings. Node embeddings can be straightforwardly constructed from edge embeddings as follows: for each node, return the mean of the embeddings of all edges incident on that node. This is described formally in Definition 4:
Definition 4 (Mean-Edge Node Embeddings). Suppose a network has nodes and m edges, with incidence matrix B E R=""" Let B be the matrix that results from dividing each row of B by its sum. Given a matrix of k-dimensional edge embeddings Y IR. mx produce matrix of node embeddings X E Rank via the matrix product X BY.
In Figure 2, we show the result of applying our method to this UGT-SBM. In particular, we construct the TDLG with o; = -/2 and show the TDLG adjacency matrix, temporal edge embeddings generated from this matrix, and node embeddings generated by Definition 4. Note that while we generally use the raw rows the TDLG adjacency matrix as edge embeddings (which produces sparse vectors), only in this section, for visualization purposes, we instead use dense eigenvector embeddings. The embedding visualizations show that our method is recovering the community structure: The edge embeddings roughly separate the kinds of edges (i.e., edges in U U, V V, and U x V occurring in each of the two time periods), and further, the node embeddings linearly separate the two node communities U and V.
4 THEORETICAL DISCUSSION
We now discuss our theoretical results, which concern the action of our TDLG method the proposed UGT-SBM family of synthetic graphs. We present these results not only to show the power of the TDLG approach, but also to contrast with deep methods, the complexity of which often precludes explicit analysis. In particular, under certain strong assumptions, our approach's 's simplicity allows us give the exact expectation for the TDLG adjacency matrix A for UGT-SBMs. Further,
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Raw Line Graph
Time Decayed
Edge Embedding
Node Embedding
2000
4000
2000
4000
0.30
2000
2000
0.00
0.15
4000
4000
0.25 0.00
0.25
0.00
0.08
Figure 2: Applying our method to the motivating synthetic network of Figure 1. The line graph is ordered so that the early edges appear first. The embedding plots show the top second and third eigenvectors (by eigenvalue magnitude) of the TDLG adjacency. Note that edge types are roughly separated in the edge embeddings, and node types are linearly separated in the node embeddings.
we can show constructively that the TDLG method is able to preserve and disentangle the node community structure of the UGT-SBM, specifically, taking the temporal edge embeddings returned by our method and aggregating them into node embeddings via averaging over node's 's incident edges (as in Definition 4) yields node embeddings which can distinguish the two node communities. The assumptions for these two results are as follows: we analyze using the expected TDLG adjacency matrix and the expected incidence matrix, and assume zero time variance within time periods of the UGT-SBM; these assumptions all simplify the analysis by reducing variance, though perhaps via concentration bounds could extend such results to the sampled setting (Spielman, 2012). Also, we will consider UGT-SBMs in the limit as n - oo for simplicity of the resulting constant values, but the following analysis would carry through in general with different values.
Proposition 4.1 (Structure of TDLG Adjacency Matrix for UGT-SBMs), Suppose A E R is the expected TDLG adjacency matrix of a UGT-SBM graph with node communities (U,V), time periods {1,2), and zero time variance {o} = o3 = 0). A has a 6 x block structure where each block is constant-valued, that is, multiple of an all-ones matrix J of some dimensionality. Letting subscripts denote time period, the block matrix is indexed by 6 edge sets: (U x U), (V x V)1, (U x V)1, (U x 2)2, (V V)2, and (U 2- In the limit as - oo, the constant values of each block are given by the Kronecker product
UXU
8/1
0
1/2
7
VxV
0
8
UXV
4/n
4/n
4/n
7
1
where 7 = exp (- (12)
20²
Note that writing A as Kronecker product involves some abuse of notation, and the actual sizes of the edge sets (and hence the sizes of the blocks) are as given in Definition 3. In the order of Proposition 4.1, these sizes are a1 4 a1 4 (1 (1-aa a1) A-n A-n or : a2 4
A. An
Am and (1 (1- a2) 2"
Proof. We calculate an entry Aij of this matrix. We first consider the effect of the time decay, then the effect of edge adjacency (i.e., b. by). Given the assumption of zero time variance, there are only two possible values for each temporal edge's time, fn and μ22- This means effect of time decay on each entry of A is multiplication by either 1, if i and j occur at the same time, or otherwise 7.
Second, we calculate the expected value of b. by. that is, the expected number of endpoint nodes shared between edges and j. Suppose both and j are in U x U. Since there are "/2 nodes in U, the probability of any two random nodes in U being the same is 2/n. There are 4 pairs of endpoints in U between and j; in the assumed n + oo limit, the 4 events of these pairs each having the same two nodes tend toward independence, so b b, = 4- 2/n- %/. By similar logic, if i in U x U and j is in U x V, there are 2 pairs of endpoints that could be identical (each of i's nodes in U with the single node of j in U), so b. by = 2/2n- = %/. The same holds if both and j are in U x V. Finally, if i in U x U and j is in V x V. there is zero chance of a shared endpoint. Remaining combinations follow by symmetry. Combining the terms for time decay and edge adjacency, the expected TDLG adjacency matrix has the specified 6 x 6 block structure.
We now state the second result, about the ability to distinguish UGT-SBM node communities using the TDLG method. This result requires that it is not the case that on 02 1/2; in this case of
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UGT-SBM, the SBMs at both time periods are Erdös- Brdōs-Rényi, there is community structure to recover.
Proposition 4.2 (TDLG Embedding Recovers Communities from UGT-SBMs), Suppose A R mm is the expected TDLG adjacency matrix of a UGT-SBM graph with node communities {U, V). time periods {1, 2). and zero time variance {o} 0 = 0). Let X E RRxx be the mean-edge node embedddnggs resulting from treating the rows of A as edge embeddings. Assuming that # and it is not the case that 0 = 02 = 1/2, the node communities are distinguishable in X.
For brevity, the proof is deferred Appendix A.1. It essentially involves the same flavor of counting- based arguments as Proposition 4.1, though it is much more involved. As part of this proof, we provide a description of part of the expected block structure of the resulting node embeddings X, similar to the preceding description of the TDLG adjacency matrix A. When time decay is not applied (i.e., by setting Y = 1), we find that nodes in U and V become indistinguishable in the derived embeddings for UGT-SBMs with + a2 = 1, as opposed to just when a2 1/2. These are UGT-SBMs like the one from Figure I, which do have community structure when considering time, but become Frdös-Rényí Rényi graphs when ignoring time.
5 RELATED WORK
Node embeddings Many methods have been proposed over the years for the node embedding task. The best understood are the classical 'spectral' methods based on eigendccomposiiion of the graph's s adjacency matrix or Laplacian (Roweis & Saul, 2000; Belkin & Niyogi, 2001). Perhaps the most famous non-spectral method is DeepWalk (Perozzi et al., 2014), which takes random walks on the graph, then, using nonlinear, probabilistic objective, increases the similarity of the embeddings of nodes which frequently co-occur in the walks. Similar methods include node2vec (Grover & Leskovec, 2016), which adds bias the random walks, and LINE (Tang al., 2015), which increases scalability. GraRep (Cao et al., 2015) and NetMF (Qiu et al., 2018) incorporate both matrix factorization like the classical methods and nonlinear objectives like the recent ones, yielding good speed and performance. Unsupervised deep learning approaches, like VGAE (Kipf & Welling, 2016) and SDNE (Wang et al., 2016), involve deep autoencoders; besides this, there is a wide array of supervised deep models which implicitly form node representations, including graph convolutional networks (GCNs) (Kipf & Welling, 2017) and graph attention networks (GATs) (Veličković et al., 2018).
Edge embeddings The task of creating vector representations for each edge, rather than each node, has seen less exploration. A common strategy is to simply create node embeddings, then process them into edge embeddings by aggregating the embeddings of the two endpoints of edge. In particular, this is done by taking the mean, entrywise product, cross product, or concatenation of the two node embedding vectors. Notably, node2vec uses this strategy for link prediction; Shi et al. (2018) and Verma et al. (2019) are more examples where this is part of a larger framework. More along the lines of our method, though still not involving temporal networks, Bandyopadhyay et al. (2019) produce edge representations in an unsupervised manner by embedding the nodes of the line graph of the original network. The method of Li et al. (2017b) implicitly takes a similar approach by taking a random walk on the edges and iteratively updating edge embeddings, as Deep Walk does for node embeddings. Some graph deep learning approaches (Monti et al., 2018; Gong & Cheng, 2019) include layers which form edge embeddings, but do so as part of a supervised framework; by contrast, Zhou et al. (2018b) propose an unsupervised deep method based on generative adversarial networks, though their approach is fairly complicated. Also of note, though not directly applicable to the standard or temporal edge setting, several works in the area of knowledge graphs form embeddings of relations between entities (Bordes al., 2013; Yang et al., 2014; Chen & Quirk, 2019).
Embeddings for temporal networks Many methods for temporal network embedding can be seen as variants of those for static graphs. Dunlavy et al. (201 1) propose a variant of factorization-bant approaches, intended for dynamic network comprising snapshots of static graphs at consecutive intervals of time. It directly incorporates time information by stacking adjacency matrices different time steps and employing tensor factorization. DANE (Li et al., 2017a) is another factorization approach, though it only factorizes matrices. To boost efficiency, rather than computing node embeddings from scratch for each snapshot, DANE iteratively updates the embeddings from the previous time step according to the change in the network; TIMERS (Zhang et al., 2018) is a similar approach which analyzes the growth of the error over iterative updates to determine when it is

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Table 2: Statistics of datasets used in our experiments.
Dataset
Nodes
# Edges
Time Span (days)
BITCOINAL.PH
3783
24186
1901
BITCOINOTC
5881
35592
1903
ESCORTS
10106
50632
2232
WIKIELECT
7118
107071
1378
EPINIONS
131828
841372
943
necessary to re-embed from scratch. NetWalk (Yu et al., 2018) is variant of random walk-based methods like WeepWalk; along the same lines, it saves time by efficiently updating embeddings at each snapshot using warm starts and reservoir sampling. CTDNE (Nguyen et al., 2018) is another random walk method which incorporates time information using the constraint that the walks must obey time. Notably, like our method, CTDNE is designed for networks with continuous time rather than just snapshots of different time intervals.
Among unsupervised deep approaches, DynGEM (Goyal et al., 2018) is an autoencoder method which again uses warm starts for each snapshot, both for speed and to encourage embeddings to be approximately preserved across time steps; DynGraph2Vec (Goyal et al., 2020) more directly allows information at different snapshots to be integrated by employing an RNN to evolve repre- sentations across time steps. The Aimilarly-named DyLink2 (Rahman et al., 2018) is another deep autoencoder approach, which also integrates information across time steps; notably, like our method, it yields embeddings for links rather than nodes, though it works snapshots rather than continuous time. More recently, EvolveGCN (Pareja et al., 2020) proposes the interesting idea of using an RNN evolve not the node embeddings, but the weights of GCN model which gener- ates embeddings for each snapshot. Some deep methods also handle continuous time: using point processes, Know-Evolve (Trivedi al., 2017), HTNE (Zuo et al., 2018), and DyRep (Trivedi et al., 2019) model the occurrence of temporal edges, while Dynamic-Trian (Zhou et al., 201 8a) aims to capture structural information by modeling the closure of wedges into triangles over time. More recent methods handling continuous-time include TGAT (Xu et al., 2020), TGNs (Rossi et al., 2020), MeTа (Wang et al., 2021), and PINT (Souza et al., 2022); these methods adapt and integrate various recent deep learning modules and concepts into the temporal network setting, such as self-attention, memory, data augmentation, and positional embedding. The PINT paper in particular performs an analysis of the continuous-ting time and discrete-time settings, proving that the latter setting reduces to the former without loss of information, whereas the converse is not always true, and hence the class of methods handling continuous- tittiuuuuu-tmeeeeee directly is theoretically more powerful in this sense.
Of the many approaches for temporal networks that we have discussed, most are principally node embedding methods from which edge embeddings can be generated by aggregation; our knowledge, ours is the first work which directly forms embeddings for continuous cintuouus-immeeee temporal edges without supervision and without aggregating node embeddings.
6 EXPERIMENTS
We evaluate our method on two kinds of tasks, edge classification and temporal link prediction, on collection of five real-world temporal network datasets. The statistics for these networks are provided in Table 2, but we defer discussion of these datasets to Appendix A.2.
6.1 EDGE CLASSIFICATION
Experimental setup We use logistic regression classifier to which the input is a feature vector for each edge of the network and the target is binary edge class. We make random 70%/30% splits of training/test data, and report test AUC of binary classification across 10 trials with 10 random splits.
In addition to results for our TDLG method's s sparse embeddings, we report results for several other methods. We compare to results for some prior methods applied to these datasets: 1) CTDNE (Nguyen et al., 2018): 2) TIMERS (Zhang et al., 2018); 3) EvolveGCN (Pareja et al., 2020); 4) DynGEM (Goyal et al., 2018); 5) GCRN (Seo et al., 2018); and 6) VGRNN (Hajiramezanali et al., 2019). Note that these methods output 128-dimensional dense embeddings, unlike our TDLG method, which outputs m-dimensional sparse embeddings; for additional, direct comparison, also report results
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BitcoinAlpha
BitcoinOTC
Escorts
WikiElect
Epinions
100
90
90
80-
Test AUC
80
70
70
70
60
60
60
50
50
EGC DGM
50
TDL ctD
EGC
DGM
CTD
EGC
40
CTD
Figure 3: Test AUC on the edge classification task for real-world datasets using our TDLG method, CTDNE, TIMERS, EvolveGCN, and DynGEM.
for using the top 128 eigenvectors (i.e., those with highest magnitude eigenvalues) of the TDLG adjacency matrix ("TDLG Dense"). With the exception of the methods we introduce and CTDNE, all other methods do not directly handle the continuous time stamps of the datasets; for these methods, we slot the edges into discrete snapshots by evenly dividing the full time span into 20 intervals. All experiments
Xeon Gold 6130 CPU and Tesla v100 16GB GPU; only the latter four methods (i.e., the deep methods) use the GPU, and we are unable to run these four methods on EPINIONS due to GPU memory limitations. We release code in the form of Jupyter notebook (Pérez & Granger, 2007) demo which is available at
Hyperparameter selection For all embedding methods, use the scikit-learn (Pedregosa et al., 2011) 1) implementation of logistic regression; we increase the maximum iterations to 10", and, since the edge classes are generally imbalanced across the datasets, we the "balanced,' which adjusts loss weights inversely in proportion to class frequency. We keep otherwise default options. For our TDLG method, we set the time scale hyperparameter o, as ratio of the standard deviation of the edges' times; calling this standard deviation be σт, we use o, 10-1 ar, which chosen by informal tuning. Appendix A.3, we explore the impact of varying o, We the implementation of Liu et al. (2020) for TIMERS and all deep methods, all with default hyperparameters. For the EPINIONS dataset only. for our TDLG method, we modify the solver for sparse logistic regression to the 'saga' solver, which more scalable than the default 'Ibfgs' solver, and reduce the maximum number of iterations to 100 (the default value).
wew.eiiiht option to
Results We first plot mean AUC and 95% confidence intervals for selected methods TDLG, CTDNE, TIMERS, EvolveGCN, and DynGEM on all datasets in Figure 3. Our TDLG method achieves higher performance than the comparison methods all datasets, often by a large margin.
Temporal Edge Classification
Interpolative Link Prediction
Extrapolative Link Prediction
Koai Tust AUC (36)
40
20
Dyn Figure 4: Test AUC all three tasks, aggregated across datasets: mean performance the evaluated real-world datasets, excluding EPINIONS.
Dym
TOLG
CIONE
Danue
GEN
To compactly compare all of the methods across these datasets, aggregate the performance across the datasets as follows: we plot the mean across all datasets, excluding EPINIONS, of each method's test AUC. We exclude EPINIONS since we are unable run some methods on this dataset. See Figure 4 (left). TDLG achieves the highest performance out of all these methods on all datasets; the dense variant of TDLG, which perhaps provides more direct comparison with other methods, sees reduced performance, but also outperforms all prior methods. Tables of full experimental results are deferred to Appendix A.5.
We evaluate the time efficiency of our approach against the selected baseline methods. For this, we report the mean runtime seconds for one trial of edge classification. Runtime constitutes one trial of learning embeddings, learning logistic regression classifier predicting on test data. See Figure 5. Across all five datasets, our TDLG method the fastest, often by significant margin. We discuss the scalability of method in Appendix A.4.
Runtime
training data, and
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BitcoinAipha
BC
Escorts
WikiElect
Epinions
Runtime (200)
207
101
109
CtD
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TDL ctD
TDL
ciu
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Figure Runtime (seconds) on the edge classification task.
6.2 LINK PREDICTION
Experimental setup Given the dataset of actual, 'positive' edge tuples (u,, U2 t;), we generate an equal-sized set of fake. 'negative" edges by independently shuffling the three columns (i.e., independently shuffling the set of first nodes, the set of second nodes, and the times). Note that this preserves the distribution nodes/times in of each of the columns. The task is to distinguish positive edges from negative ones. We evaluate two different settings for link prediction. In the first, which call temporal link outlier detection, the test edges in the same time interval as the training edges. In the second, which we call temporal link prediction, the test edges are an interval of time following that of the training data. We also call these settings 'interpolative' and 'extrapolative, respectively. Prior work generally focuses on the latter setting, since it is more applicable, e.g., for predicting future events/behavior, but we also evaluate the former mainly out of scientific interest.
Specifically, given that we are splitting the data into 20 intervals, we consider intervals 1-19 to be "early' data and interval 20 to be 'late' data. To share computational effort across the two settings, we train both classifiers on 70% of edges (both positive and negative) from the early intervals. For the interpolative setting, the test data comprises the remaining 30% of edges from the early intervals. For the extrapolative setting, the test data comprises all edges from the late interval. Evaluating the link prediction task has greater computational cost compared to edge classification since the input graph changes across trials rather than just train/test splits of labels, requiring new embeddings to be made for each trial. For this reason, we only test the selected methods from the previous section. We report mean AUC and 95% confidence intervals across 5 trials, each trial having different random shuffles producing different negative edges, as well as different 70%/30% train/test splits.
Hyperparameter selection We generally the same hyperparameters for edge classification, with one exception: since there is an equal number of positive and negative edges, we no longer set the class weight option 'balanced," and all edges are given equal weight.
Results As for edge classification, we plot the aggregated (mean) performance across the datasets in Figure (right), for both the interpolative and extrapolative settings of link prediction. For brevity, we defer non-aggregated and tabular results to Figure in Appendix A.5, In both settings, across all datasets, our TDLG method achieves the highest performance, often by a large margin. We do find that the deep methods EvolveGCN and DynGEM generally perform better than CTDNE and TIMERS, and EvolveGCN in particular is competitive with TDLG on some datasets.
7 CONCLUSION
We present a novel framework for temporal edge embedding called the time-decayed line graph (TDLG). Unlike some prior methods, our method works directly with continuous timestamps rather than requiring discretization of times, and directly generates temporal edge embeddings rather than requiring aggregation of node embeddings. Also contrast many prior methods, ours has a simple concept and high ease of implementation. since it essentially just creates a sparse matrix of proximities between temporal edges. Despite its simplicity, our method achieves superior performance for several downstream tasks on benchmark temporal network datasets, while requiring less runtime. Its simplicity also facilitates theoretical guarantees its effectiveness on our proposed UGT-SBM family of networks. Future directions include extensions of our approach more general scenarios involving, for example, streaming input data or inductive learning; an approach for the latter could be straightforward running an inductive method the TDLG of the input graph opposed the raw graph itself. Broadly, this work highlights the potential of methods involving engineered features, opposed to learned ones, to achieve solid performance by directly invoking simple inductive priors.

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Published as a conference paper at ICLR 2023
ACKNOWLEDGMENTS
Cameron Musco and Sudhanshu Chanpuriya were partially supported by an Adobe Research grant.
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13


Reviews:
[{"review_id": "zjUXC1VdM99", "reviewer": "Reviewer_qEEe", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The authors propose an approach for computing edge embeddings in temporal networks evolving in continuous time. By considering direct embeddings of edges rather than nodes, the authors claim that they can achieve better accuracy on downstream tasks such as edge classification. The proposed edge embedding uses a time-decayed line graph (TDLG) constructed from the original temporal network. The authors demonstrate impressive gains over other temporal network embedding methods for edge classification and link prediction on several real networks. They also present some theoretical analysis on their proposed embedding approach.", "summary_of_the_review": "The authors have proposed a highly novel and simple approach for temporal network embedding. The simplicity enables theoretical analysis of the model. It also has very strong empirical performance. I could see this paper having tremendous impact on the ever-growing community interested in graph embeddings. It could also inspire more useful rigorous analysis of the proposed model. I strongly support this paper for acceptance!\n", "strength_and_weaknesses": "Strengths:\n- Highly creative and simple approach. While line graphs have previously been used in various network science applications, the proposed time-decayed line graph (TDLG) representation appears to be novel. The edge embedding is simply a row of the TDLG adjacency matrix.\n- The authors prove that their proposed embedding can separate nodes in different communities in a simplified temporal stochastic block model (TSBM).\n- Strong performance in experiments compared to recent temporal network embedding methods, with better accuracy while maintaining low runtime due to the simplicity.\n- Top-notch presentation--very well written and a joy to read. Thank you!\n\nWeaknesses:\n- Theoretical analysis applies only to the expected adjacency matrix and also assumes no variance in the edge timestamps. While this is a good start, it is too simplistic to be of practical use. As the authors suggest, the results could likely be extended to the sampled setting using concentration inequalities for matrices. This type of analysis was conducted on the Community Hawkes Independent Pairs (CHIP) model--another type of temporal SBM model involving Hawkes processes. See Arastuie et al. (NeurIPS, 2020) for more details.\n- Naming the demonstrative example in Section 3 a \"temporal SBM\" is somewhat misleading. There are lots of different probabilistic generative models for temporal networks in the literature that can be considered temporal SBMs, including the CHIP model and even older work such as the Hawkes IRM (Blundell et al., NeurIPS 2012). I suggest the authors to pick a more specific name for their proposed simple model and add references to other work on temporal SBMs.\n\nReferences:\n- Arastuie, M., Paul, S., & Xu, K. S. (2020). CHIP: A Hawkes process model for continuous-time networks with scalable and consistent estimation. In Advances in Neural Information Processing Systems 33 (pp. 16983-16996).\n- Blundell, C., Heller, K. A., & Beck, J. (2012). Modelling reciprocating relationships with Hawkes processes. In Advances in Neural Information Processing Systems 25 (pp. 2600-2608).\n\nQuestions:\n1. In the specification of the temporal SBM, it appears that the number of temporal edges is fixed to $\\Delta \\cdot n/2$. This does not match with the usual definition of the SBM, where the number of edges is not fixed but the probability of forming an edge between any two communities are fixed. This is analogous to the difference between a $G(n,m)$ and $G(n,p)$ random graph model. How are you fixing the number of edges while maintaining community structure?\n2. One disadvantage of using line graphs is the large size of the adjacency matrix they generate, which requires lots of memory to store even using sparse matrices. It appears that your TDLG also has this disadvantage. How much main memory is required to conduct your experiments, especially on the large Epinions data? Do you further sparsify your TDLG in any way, e.g., by removing edges with weights below a certain threshold?\n\nTypos and minor issues:\n- A figure showing the time-decayed line graph for a toy example would be helpful to illustrate the construction of a time-decayed line graph. I was already familiar with line graphs before reading this paper, so I had no problem understanding the definition, but I suspect many readers may not have previously seen line graphs. To create some room, I suggest moving the proof of Proposition 4.1 also to the appendix.\n- The use of $\\sigma_1, \\sigma_2$ in the temporal SBM is potentially confusing given the use of $\\sigma_t$ for the Gaussian decay. I would suggest changing $\\sigma_t$ to a different symbol.\n- I think there is some notation inconsistency in Proposition 4.1. $\\gamma$ is defined but never used, and $\\lambda$ is used in the Kronecker product but not defined.\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "pDyPHUmdxD", "reviewer": "Reviewer_HmSu", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper is about edge representation learning in continuous temporal networks. The proposed method (TDLG) starts with the construction of the line graph which converts each edge to a node and connects interactions that share the same endpoint node.Then, the edge weights in the line graph are calculated based on  the time differences between interactions, with interactions that happen closer in time to have higher weights. The line graph then represents both topological proximity and temporal proximity. Efficient classical methods can then be applied to the line graph. The authors of this paper propose the temporal stochastic block model (TSBM), which naturally extends the stochastic block model for static networks to temporal networks. The results contain a comparison of six state of the art models to the proposed TDGL on five real world datasets. The focus is on two downstream tasks: the temporal edge classification task and the {inter/extra}-polative link prediction task. The TDGL in most cases outperforms all other comparison methods. There is also discussion about the runtime improvement, the hyperparameters and in the appendix a full report on the results can be found.", "summary_of_the_review": "Overall, this is a well-written paper, the quality of the work and the presentation is high. The proposed method is simple with incremental novelty, but it outperforms all other comparison state of the art models in five datasets and has a shorter runtime. The authors have done a great job describing the proposed method, and also in the experimental design and implementation.", "strength_and_weaknesses": "Strengths:\n- The paper is well-structured and well-written. The paper is easy to follow and the flow is great.\n- The authors have done a good job with the related works; they have organized most of the state of art methods and they discuss about the limitations and the challenges of each category which also gives motivation for the proposed model and setup.\n- All figures and tables are well-thought and designed. The authors make the best out of the 9 page limitation.\n- The demonstrating example and the theoretical discussion give the reader the extra proofs and background needed for this methodology.\n- The results are very promising and show that the proposed method outperforms the six comparison models in five datasets in both edge classification and edge prediction tasks.\n- The runtime results also show that the proposed method reduces the runtime, while increases the performance, and it also highlights that a simpler method can have comparable or better performance to other more complicated methods.\n\nWeaknesses:\n- The novelty of the work is incremental.\n- The fact that it is proposed to create the line graph using the continuous stream of timestamps information as weights is not fully convincing on how this is continuous and not discrete time related information. Maybe more explanation and justification is needed in that part.\n- It would be interesting to see all the models that appear in Table 1, in the comparison methods in the experimental setup.\n- Do the train/test split percentages refer to the number of days? Can later days appear in the train set and earlier days in the test set? How do the authors do 10 trials with 10 random splits given that the time has the concept of order in it?", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "nU8-OavLDZa", "reviewer": "Reviewer_CTmU", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "Edge embeddings and continuous time have not received as much attention as the alternatives between these two categories, and this paper takes an edge-first approach to temporal graph edge embeddings. The paper introduces a simple strategy to obtain these, justifying several useful aspects of their strategy. A demonstrative example is given to share intuition with the readers, which is followed by theoretical insights on the methodolgy. Following the related work section, empirical analysis is performend, where the proposed method beats several baseline methods. Given that this paper is focusing on a niche, I'm not sure if any method can be considered SoTA. Empirically, the proposed method ran to conlusion over all datasets, whereas some baseline methods did not. ", "summary_of_the_review": "This paper introduces a straightforward approach to edge embeddings of temporal graphs. The proposed approach works very well according to the empirical results. I also find the similicity of the approach to be a strength rather than a weakness of the paper, though, not being an expert in this area, I would be surprised if this didn't exist elsewhere. I appreciated the theoretical analysis, but I am wary of some aspects, particularly that the assumptions (esp. the zero time variance) means the analysis is more suitable discrete rather than continuous contexts, which is not the focus of this paper. If other reviews, or the authors, can ratify the theory and novelty of the contribution I would be willing to increase my recommendation. ", "strength_and_weaknesses": "Pros: \n+ Simple methodology that seems to make a lot of intuitive sense. I found the demonstrative example rather compelling given its simplicity and clear outcomes. I think I would have valued (for comparative purposes) examples of baselines\n+ Experimental results look good, both in terms of accuracy and runtime. They seem to dominate the baselines considered. \n\nCons: \n- The edge weight function, being an RBF kernel, is 'opinionated' as to the nature of the weighting function. I would imagine that a richer function class (even a NN) could allow much richer relationships to be learnt in a data driven manner. Is this a weakness? I would have liked abalation studies with other weight functions, such as those mentioned previously. \n- I didn't see a strong connection between the theoretical analysis and evaluation. In particular, the reduction of the time variance to 0 in the analysis seems to me to implicitly discretise the context of the analysis, and its conclusions have (I believe) very weak links to the continuous time problem. \n- I am curious about spectral interpretations of this construction, particularly regarding the specturm of the edge adjacency matrix. \n- In hyperparam selection section, I would value clarification on what fraction of training data was used for valudation. The authors explicitly train and test sets, but not validation. \n\nAdditional questions: \n- Why is escorts sensitive to the time scale hyperparameter and the others aren't? \n- Why is it that normalisation hinders the classification performance? \n- The method seems so simple that I'm surprised it hasn't been elucidated before. \n- The structure of your adjacency matrix (Eq (1)) looks similar to the product of two kernels (linear kernel and an squared exponential kernel). I wonder if theory could be borrowed from the kernel and / or Gaussian processes literature to help with the understanding of the domain here? ", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "_T6SursJu2b", "reviewer": "Reviewer_wNXK", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper provides an interesting approach for generating link embeddings in temporal graphs, which is converting the graph into a line graph and connecting edges (now nodes), based on the temporal difference they have in the initial graph. They show that with this representation, the adjacency matrix representation is enough to achieve very strong performance in edge prediction and classification. The method is simple and seems to outperform the state of the art representation learning models by a significant margin.  ", "summary_of_the_review": "Simple way for edge representation in temporal graph, with no learning, that performs significantly better than the baselines. It seems not practical given the size of representation is number of edges in the graph.", "strength_and_weaknesses": "The performance gain is impressive and the proposed model is simple and intuitive.\n\nHowever, key related works are missing, e.g. TGN: TEMPORAL GRAPH NETWORKS FOR DEEP LEARNING ON DYNAMIC GRAPHS\n\nThere are also not much discussion on DTDGs and how this method performs on them.  \nSome of the claims might also not be correct [although said differently in different parts of the paper]. For example, there is a statement in the abstract that \"First, time is assumed to be discretized, so if the time data is continuous, the user must determine the discretization and discard precise time information\", while there are many methods for continues times graphs and some recent work explicitly discuss the relation CTDG and DTDG models (this is recent so of course not listing it as a missing reference but more as a reference to see https://arxiv.org/pdf/2209.15059.pdf). This is later expanded and corrected in text, but the sentence in abstract can be rephrased. \n\nThe argument that node to edge embedding is suboptimal seems weak given the proposed method needs a vector of length m, m being the number of edges. There is an attempt to make it smaller with SVD but performance drops. \n\nThe only parameter the model has is a hyper parameter tuned informally, so the although performance is good, this is not much a representation learning. ", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "X8qOOXybPk", "reviewer": "Reviewer_Vkhz", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper proposes to directly model feature embeddings from a line graph. The line graph is comprised of nodes as the original temporal edges, and new set of edges between the line graph nodes are captured by interactions between the original temporal edges.\n\nThe paper addresses critical ML application in the context of dynamic graph structures that is changing as time evolves, and how to conduct representation learning for such dynamic structures. The limitation of discretization of the time dimension is addressed and a great reduction in runtime compared to other state of the art methods are also demonstrated. The resultant method is simple yet effective, and resembles classical methods in recommendation systems which also adopts the weight adjacency matrix to derive embeddings.", "summary_of_the_review": "At the present form of the paper, I am discouraged to recommended it with high ratings due to the limitation of its reliance on the full graph structure being consistent for training and inference time. Granted, there are applications scenarios that favors this setup, but it is indeed a limitation for generalizable representation learning.\n\nThe experiments and evaluation process are also compared against more generalizable methods which could operate on unseen graphs during test time, thus creating an unfair comparison. I would be more convinced if the compared methods are also operating on the same graph throughout and leveraging the global graph structure during its learning process.", "strength_and_weaknesses": "Strength:\n- Well written paper. The paper states its core contributions clearly and directly aims to address related works limitations. As a result, the readers would have a relatively easy time to understand the concepts and proposals written in the article.\n- Extensive experimental studies. The author supported their work with a large set of experiments centering around node classification on a variety of datasets. I appreciate the effort.\n- Ablation study. I appreciate the model study conducted on synthetic dataset, and showcasing TDLG's ability to learn the global structure of the graph.\n- Very fast runtime of this method compared to the related works.\n\nWeakness:\n\n- Graph structure's enormous scale. The original graph with O(NT) nodes, where N is number of nodes and T is number of time steps, would be constructed as O(N^2 T) nodes, which is a quadratic scale. As is shown in table 2, the #edges are typically much larger than the #nodes, thus rendering heavy memory pressure of the model runtime. However, this weakness is partially offset by its fast runtime. But this limitation makes the model hard to deploy on edge devices.\n\n- Reliance of global graph structure. Based on section 2's experimental setup explanation (Temporal edge embedding and classification, Temporal link prediction), the model proposed in this paper relies on adjacency matrix A's row as its input feature to predict the edge class. This creates a limitation that the training and inference has to be operated on the same graph. Whereas, the test data coming from a different graph structure will make this method infeasible. My main concern is that the neural net training/testing on the same graph is not actually learning signatures of edges, and that how those edges are interacting with its important neighbors, but rather implicitly learning the training graph’s full topological information, thus making the inference on test set much easier, at the cost of poor generalizability on unseen graphs. Note, the models ability on learning the global graph structure is supported by the author's model study experiment in section 3,4, see figure 2.\n\n- Reliance of global graph structure cond. However, I understand certain applications may actually favors this setup, i.e. recommender systems. As such, let us relax my argument by a little bit and assume we always operate on the same global graph. Then, if we have an unseen edge during test time, and we have no such row in the matrix A. In order to compute its feature, we have to rely on the set of training edges to acquire a relationship row vector between the test edge to all train edges. Our feature set is then limited/bounded by those training edges. However, the test edges can formulate interactions with edges that are not found in the training set, and we will not be able to acquire those features. This is a critical limitation, and this is also the core reason why cold start is always a tricky and heated topic in recommender systems.\n\n- Reliance of global graph structure cond. That being said, I would be interested to see if the authors could compute localized embedding computation based on neural functions between nodes, and use them for edge classification tasks. For example, one could use graph attention networks [1], or spectral graph convolution kernels [2][3] to compute node embeddings on the line graph, thus constituting a valid representation of the temporal edge in the original graph. When this method is equipped with localized embeddings, we can train on a set of graphs and eval on unseen graph structures without issues. \n\n- Experimental results are not completely fair. Most of the related work, for example evolve GCN is not depending on the global graph structure, thus being able to generalize to unseen graphs. When comparing to those kind of more generalizable work, this paper intrinsically has an advantage: it leverages the global graph structure and will not work if it is tested on an unseen graph. This is probably a good reason why this method is performing on 90% level across the board while the related works only performs at 70% level. In addition, figure 4 hints that when the access to global graph structure is impaired (when we only use 128 eigenvector's projections), the model's performance is severely hindered.\n\n\n[1] Graph attention networks. https://arxiv.org/abs/1710.10903\n[2] Spectral graph convnets. https://arxiv.org/abs/1606.09375\n[3] Spectral localized kernels on point clouds. https://arxiv.org/abs/1803.05827\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "Not applicable"}, "extra_text": {}}, {"review_id": "JKH1UCCTnT", "reviewer": "Reviewer_C63A", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper aims to directly model the edges in temporal networks instead of indirectly inferring edge embeddings by computations from nodes. To achieve so, the paper constructs Time-Decayed Line Graphs (TDLGs) to use each node to represent the edges, and weigh the edges between resultant nodes based on differences in time. Such modeling avoids the assumption of discretized time info. Theoretical analysis is provided to ensure the proposed edge modeling can recover original nodes based on the stochastic block model (SBM). \n", "summary_of_the_review": "Overall, the paper is well written and the contents are self-contained. The motivation and the lack of more recent related work are the two main issues. \n", "strength_and_weaknesses": "Strength:\n\nOverall the paper has good clarity in the problem statement. The novelty is stated clearly.  \n\nWeakness:\n\nW1: The motivation of why edge embedding needs to be improved is unclear to me. It is stated that edge embedding has seen less interest, while it does not make a strong and effective motivation. \n\nW2: The related work and baseline approaches are papers proposed in year 2019 or before. It would be better to include more recent papers to have a better reflection on the novelty of this paper.  \n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "64qVTqe1hQ", "reviewer": "Reviewer_Pn6V", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper discusses how to embed the edges of temporal networks. It proposes to create line graphs weighed by Gaussian weight decay and to use the row vectors of the adjacency matrix of the line graph as the embeddings of the edges. The experimental results show that the proposed method performs better in edge classification tasks and link prediction tasks than other methods that discretize the time stamps of the datasets.", "summary_of_the_review": "This paper is well-organized and clearly written. Despite the idea to utilize the row vectors of the line graphs with time-decayed weight being straightforward, there has been no work that shows such a simple method surprisingly works well. A downside is that this paper does not provide enough discussion and evaluation of time/space complexity.", "strength_and_weaknesses": "Strengths:\n\n- The idea to utilize line graphs in data mining tasks has been discussed in a lot of papers such as [1], and it seems straightforward to weigh the line graph according to time decay between edges; however, to my knowledge, there has been no work that shows such simple method can surprisingly outperform other methods that discretize the time stamps.\n- This paper is well-organized and clearly written. It is easy for readers to follow.\n\nWeaknesses:\n\n- The discussion and evaluation related to the time/space complexity are not provided enough. Because the embedding vectors have fundamentally extremely high dimensionality, the computation of the downstream tasks will be affected by the sparsity of the vectors, which depends on the size and fraction of hub nodes in the original graph. Moreover, the computation cost of eigendecomposition should be discussed more including how to update the eigendecomposition in the time-evolving graphs.\n\n[1] Ahn, YY., Bagrow, J. & Lehmann, S. Link communities reveal multiscale complexity in networks. Nature 466, 761-764 (2010).\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "5ZIr7YLf1Yq", "reviewer": "Reviewer_GdB7", "summary": "", "questions": "", "limitations": "", "rating": 8, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The authors present a simple framework to address two issues with learning on temporal networks: (1) manual discretization of time disregards precise time information, and (2) in general, representations of nodes are returned instead of edges, making it suboptimal for edge-classification tasks. They propose a method that constructs the line graph of the network and set the edge weights based on the differences in time between the edge-interactions. This new graph represents topological proximity (existence of the edges) and temporal proximity (weights of the edges). This appears to be the first method that directly constructs embeddings for continuous-time temporal edges without aggregating node embeddings. The method is simple and intuitive. For theoretical analysis, the authors introduce a data model, called the temporal stochastic block model (TSBM), a temporal variant of the popularly studied SBM. The method is also easy to implement and is shown to perform better than prior methods in experiments on five benchmark real-world temporal networks.", "summary_of_the_review": "Solid empirical results. Slightly lacking theoretical arguments with very strong assumptions (zero time-variance). I believe this could be a good contribution.", "strength_and_weaknesses": "### Strengths\n1. The paper is easy to follow and well-written.\n2. The proposed method is easy to implement and simple to understand, yet, is shown to perform better than prior methods via an extensive set of experiments.\n3. Theoretical discussion is apt and provides the relevant intuition behind the method very cleanly.\n\n### Weaknesses\n1. The theoretical discussion is limited to a very bare-bones TSBM. The requirement of zero time variance in the results is not well-justified.\n2. I believe there is a missing discussion on spectral methods for multi-graph community detection where multiple graphs are given, and one is supposed to figure out the community structure. From a theoretical standpoint, I don't see a difference between the temporal problem and the multi-graph problem. A brief discussion with references would be nice. In particular, the survey \"Abbe, E., 2017. Community detection and stochastic block models: recent developments. JMLR, 18(1), pp.6446-6531\" is a good reference.\n\n### Questions and Feedback\n1. [Def 3] Could you justify taking the time values in the two periods to be Gaussian? It seems that a primary motivation was to not have to discretize time. However, saying \"two\" periods but taking the values to be Gaussian still in some sense makes me think that the time values are discretized. I'm thinking that: Gaussians have nice tails, so they are essentially their mean (concentration), then in this regime how is the model different from one where time is discretized?\n2. [Fig 1] The toy example of TSBM is very intuitive. However, it might be good to add why spectral methods are not applicable to this problem. In the setup of the figure, looking at, for example, the second eigenvector of the adjacency matrix of the graphs in both time periods separately might also work (will give community labels, where one can combine the two sets of labels in a clever way to obtain more confidence). Does the sparsity of the graph play a role here? I'm thinking that if the graph is very sparse, then spectral methods won't work.\n3. [Proposition 4.1] Do $\\lambda$ and $\\gamma$ denote the same thing?\n4. [**Important**] Why do we need time variance $0$ for the theoretical results, i.e., both Prop 4.1 and 4.2? Can you provide a brief sketch for how the proof will go for non-zero variance? I believe something can be done since it's Gaussian, otherwise, why take Gaussian to begin with? Shouldn't the definition simply say that you are taking two discrete times $\\mu_1,\\mu_2$ for the two periods instead of Gaussian?", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}]

Author rebuttal:

Decision: Accept: poster