DyANE: Dynamics-aware node embedding for temporal networks

DyANE: Dynamics-aware node embedding for temporal networks

Koya Sato
University of Tsukuba,
Tsukuba, Japan
koya@osss.cs.tsukuba.ac.jp\AndMizuki Oka
University of Tsukuba,
Tsukuba, Japan
mizuki@cs.tsukuba.ac.jp \AndAlain Barrat
Aix Marseille Univ,
Université de Toulon,
CNRS, CPT, Marseille, France
ISI Foundation, Turin, Italy
Alain.Barrat@cpt.univ-mrs.fr\AndCiro Cattuto
ISI Foundation, Turin, Italy
ciro.cattuto@isi.it
Abstract

Low-dimensional vector representations of network nodes have proven successful to feed graph data to machine learning algorithms and to improve performance across diverse tasks. Most of the embedding techniques, however, have been developed with the goal of achieving dense, low-dimensional encoding of network structure and patterns. Here, we present a node embedding technique aimed at providing low-dimensional feature vectors that are informative of dynamical processes occurring over temporal networks – rather than of the network structure itself –- with the goal of enabling prediction tasks related to the evolution and outcome of these processes. We achieve this by using a modified supra-adjacency representation of temporal networks and building on standard embedding techniques for static graphs based on random-walks. We show that the resulting embedding vectors are useful for prediction tasks related to paradigmatic dynamical processes, namely epidemic spreading over empirical temporal networks. In particular, we illustrate the performance of our approach for the prediction of nodes’ epidemic states in a single instance of the spreading process. We show how framing this task as a supervised multi-label classification task on the embedding vectors allows us to estimate the temporal evolution of the entire system from a partial sampling of nodes at random times, with potential impact for nowcasting infectious disease dynamics.

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Introduction

The ubiquity of network representations of widely different systems has led to a flourishing of methods aimed at the analysis of their structure. Among those, network node embedding methods has recently gained a lot of popularity [Cai, Zheng, and Chang2018, Goyal, Chhetri, and Canedo2019]. Node embedding maps each node of a network into a low-dimensional vector, such that the vectors representing different nodes are close if the network nodes share some similarity or are close in the network. Node embedding thus aims at exposing in the low-dimensional space structural features and relevant patterns of the network that are not necessarily evident in the network representation. Most importantly, the embedding vectors can be used as feature vectors in machine learning applications, and have been shown to yield improved performance for tasks such as node classification, link prediction, clustering, or visualization.

While node embeddings have proven successful in achieving low-dimensional encoding of network structures, networks are also the support of important dynamical processes, such as epidemic or rumor spreading, cascading failures, consensus formation, etc. [Barrat, Barthélemy, and Vespignani2008]. Here we introduce and experiment with node embedding methods tailored to the study of dynamical processes on temporal networks, and in particular to the task of predicting the evolution and outcome of one instance of the dynamics (e.g., an epidemic spread) from partial information and without detailed knowledge of the dynamical process itself. A useful embedding should thus yield low-dimensional vectors that encode information relevant to the dynamics of the process occurring over a temporal network – rather than information about the network structure itself. Since dynamical processes unfold over time-respecting paths determined by the underlying network and by its evolution over time, we argue that the sought embeddings should be informative of these paths – the paths along which information can propagate. Driven by this idea, we propose to map the temporal network to a static graph representation, a so-called supra-adjacency representation, whose nodes are (node, time) pairs of the original temporal network  [Valdano et al.2015]. We modify the original supra-adjacency representation method to only consider nodes at those times when they interact, and we map the original temporal edges to edges between the corresponding (node,time) pairs, so that the static graph representation preserves the temporal paths of the original temporal network (i.e., the paths supporting and constraining the dynamical process at hand). An example of the supra-adjacency representation we use here is shown in Fig. 1. Since the resulting representation is a static graph, we can then apply standard embedding techniques: we focus on embeddings based on random walks as they provide an efficient way to sample the relevant paths.

We study the usefulness of the proposed embeddings in the context of a paradigmatic dynamical process – epidemic spread over temporal networks – in which network nodes exist in few discrete states and the dynamics consists of transitions between such states (e.g., a “susceptible” node becoming “infectious”). We focus on the task of predicting the nodes’ states over time for a single realization of the epidemic process. Specifically, we set up a multi-label supervised classification problem with a training set obtained by sampling the node states at random times, with no information about the mechanics of state transitions nor on the parameters of the epidemic process. Our contributions are as follows:

  • We propose a new method for node embedding tailored to the study of dynamical process on temporal networks, using a modified supra-adjacency representation for temporal networks and building on standard random-walk based embeddings for static graphs.

  • We show that in the important case of epidemic spreading, a satisfactory prediction performance of nodes’ states can be achieved in a supervised multi-label classification setting informed by the proposed embeddings.

  • We show that our method achieves good performance in estimating the temporal evolution of the entire system from sparse observations, consistently across several data sets and across a broad range of parameters of the epidemic model. Our approach requires no fine-tuning of the embedding hyper-parameters and yields consistently superior performance than other embedding methods.

Problem Formulation

Temporal Network

We consider a temporal network in discrete time on the set of timestamps : is defined as the set of the undirected temporal edges , meaning that nodes and are linked at , possibly with a weight (for simplicity we consider only positive weights). At each timestamp , denotes the set of temporal edges at , and is the set of nodes which have at least one temporal edge at : . We define the snapshot network at as the undirected weighted network , and the temporal network can be seen as the succession of snapshot networks . The overall set of nodes is .

For each node , we define its set of active times as the set of timestamps in which it has at least one temporal edge, i.e., such that : . We denote the -th active time of by , with . We then define the set of active copies of each node , that we call ”active nodes”, as . The overall set of active nodes is .

Dynamical process

We consider a dynamical process taking place on the temporal network, such that the nodes can be in one of a finite set of discrete states . Nodes can change state either spontaneously or through interaction along temporal edges. Our definition is thus very general and encompasses in particular models of epidemic propagation, rumor propagation, opinion formation or cascading processes [Barrat, Barthélemy, and Vespignani2008, Castellano, Fortunato, and Loreto2009, Pastor-Satorras et al.2015].

The mapping specifies the state of each node at each of its active times. We assume that a sample of these states is known: we define the set of the corresponding observed active nodes as . Here, for simplicity, we will assume that results from a uniform random sampling of . We also assume that the state of a node can be only be observed when it is active, i.e., in contact with at least another node.

For clarity, here we will focus on a paradigmatic dynamical process taking place on the temporal network, the Susceptible-Infectious-Recovered (SIR) model for epidemic spreading, which has been widely used to model contagious infections such as flu-like diseases [Keeling and Rohani2008]. In this model, each node can be at each time in one of three possible states: susceptible (S), infectious (I), and recovered (R). At the start of the process, all nodes are in state S, except for the epidemic seeds, which are in state I. A contact between an S and an I nodes leads to a contagion event in which the S node becomes infectious with probability per unit time (recall we work in discrete time). Let us denote by the set of infectious nodes at , and consider a susceptible node . We denote its set of neighbours at as , and is the set of its infectious neighbours at . The probability that none of these infectious neighbours transmits the disease to at timestep is , and thus the probability that becomes infectious at time , due to its interactions, is . Recovery from state I to state R occurs also stochastically: each infectious node becomes recovered (R) at each timestamp with probability . Recovered nodes do not change state any more. The parameters of the model are thus the infection and recovery rates and  [Keeling and Rohani2008].

Problem statement

Given a known temporal network and a partial observation of the dynamical states of the nodes, the problem consists in predicting the dynamical state of all nodes at all their active times. In other words, knowing the state of the subset of observed nodes at some active times, we want to predict the state of all the active nodes at all times. Crucially, we seek to achieve this prediction without any detailed information on the dynamical process at hand, except for the set of possible states of each node. In particular, we do not make assumptions on the allowed state transitions, the parameters governing the dynamical process, nor even the reversibility or irreversibility of the process. We also remark that the above problem statement implies that we will be working on single realizations of the dynamical process, with the goal of predicting the state of a given node at a given time, rather than predicting statistical properties averaged over a sample of simulated or observed dynamics.

Notice that the SIR model – in addition to its relevance to many real-world phenomena – is particularly interesting to study in this context: it features both state transitions occurring upon interaction (hence, along the edges of the temporal network) as well as spontaneous state transitions that can occur at any time, and in particular between successive active times of a node (the infectious-recovered transition).

Our Approach: DyANE

Our approach consists of three steps. First, we map the temporal network to a static network between active nodes through a modified supra-adjacency representation. Second, we apply standard embedding techniques for static graphs to this supra-adjacency network. We will consider embeddings based on random walks as they explore the temporal paths on which transmission between nodes can occur. Finally, we train a classifier to predict the dynamical state of all active nodes based on the vector representation of active nodes and the partially observed states.

Figure 1: Proposed supra-adjacency representation (dyn-supra). The original temporal network (top), whose state is shown at three different times, is mapped to a static representation (bottom) where nodes are (node,time) pairs of the original network.

Supra-adjacency representation

We first map the temporal network to a supra-adjacency representation, i.e., to a new static network whose nodes are the active nodes of the temporal network. We thus define the supra-adjacency network as , where are (weighted, undirected) edges joining active nodes. The mapping from the temporal network to the supra-adjacency network consists of the following two procedures (Fig. 1):

  • For each node , we connect its successive active times: for each active time of , we draw a “self-coupling” edge between and (recall that active times are ordered in increasing temporal order).

  • For each temporal edge , the time corresponds by definition to an active time for both and , that we denote respectively by and . We then map to two undirected edges , namely and . In other words, the active copy of at , , is linked to the next active copy of , and vice-versa.

The first procedure makes each active node adjacent to its adjacent past and future versions (active times), which ensures that a node carrying an information at a certain time can propagate it to its future self along the self-coupling edges, and is useful in an embedding perspective to favor temporal continuity. The second procedure encodes the temporal interactions, and yields the crucial property that any time-respecting path existing on the original temporal network, on which a dynamical process can occur, is also represented in the supra-adjacency representation. Indeed, if an interaction between two nodes and occurs at time and potentially modifies their states, e.g., by contagion or opinion exchange or modification, this can be observed and will have consequences only at their next respective active times: for instance, if transmits a disease to at , can propagate it further to other neighbours only at its next active time, and not immediately at . This is reflected in the supra-adjacency representation we propose.

The edges in are thus of two types, joining two active nodes corresponding either to the same original node, or to distinct ones. For each type, we can consider various ways of assigning weights to the edge. We first consider for simplicity that all self-coupling edges carry the same weight , which becomes thus a hyper-parameter of the procedure. Moreover, we simply report the weight of each original temporal edge on the two supra-adjacency edges and (with ).

In the following, we will refer to the above supra-adjacency representation as dyn-supra. We will moreover consider two variations of this representation. First, we can encode the direction of time of the original temporal network in the supra-adjacency representation by making all links of directed: an edge is then oriented according to the direction of increasing time, i.e., pointing from the active node with the earlier time to the one with the later time . We will refer to this representation as dyn-supra-directed.

Another possible variation consists in encoding the time delay between active nodes into edge weights, with decreasing weights for increasing temporal differences. This decay of edge weights is consistent with the idea that successive active nodes that are temporally far apart are less likely to influence one another (which is the case for many important dynamical processes). In our case, we will consider that the original weight of the edge in the dyn-supra representation (i.e., if , or the original weight of the temporal edge if ) is multiplied by the reciprocal of the time difference between the active nodes if , i.e., . We will refer to this representation as dyn-supra-decay.

Node embedding

The central idea of the node embedding method we propose for temporal networks, which we call DyANE (Dynamics-Aware Node Embeddings), is to apply to the supra-adjacency network any of the node embedding methods that have been developed for static networks. In particular, here we will use DeepWalk [Perozzi, Al-Rfou, and Skiena2014], as it is a simple and paradigmatic algorithm, and it is known to yield high performance in node classification tasks [Goyal and Ferrara2018].

DeepWalk is based on an exploration of the neighborhood of a node by truncated random-walks rooted at that node, which makes it particularly appropriate in our framework. Indeed, in the supra-adjacency representation, these random-walks will explore for each active node both the self-coupling edges leading to past and future versions of the same original node, and the edges representing the interactions between nodes. As written above, these edges encode the paths along which dynamical processes occur, meaning that the final embedding will preserve structural similarities relevant to these dynamical processes. Note that DeepWalk does not consider weighted edges, but it can easily be generalized so that the random walks take into account edge weights [Grover and Leskovec2016].

Prediction of dynamical states

Once we have obtained an embedding for the supra-adjacency representation of the temporal network, we can turn to the task of predicting the dynamical states of active nodes. Since we assume that the set of possible states is known, this is naturally cast as a (supervised) classification task, in which each active node should be classified into one of the possible states. In our specific case, the three possible node states are S, I, and R. Note that the classification task is not informed by the actual dynamical process (except knowing the set of possible node states). In particular, no information is available about the possible transitions nor about the parameters of the actual process.

We will use here a one-vs-rest logistic regression classifier, which is customarily used in multi-label node classification tasks based on embedding vectors. Naturally, we could use any other suitable classifier.

We remark that we seek to predict node states for individual realizations of the dynamics. This is relevant to several applications: for example, in the context of epidemic spreading, and given a temporal interaction network, one might use such a predictive capability to infer the state of all nodes from the observed states of few active nodes (“sentinel” nodes).

Experiments

Name
InVS15 22451 217 699 37582 4.164 0.148
LH10 4880 76 342 14870 4.448 0.188
SFHH 10815 403 127 34446 4.079 0.211
Thiers13 32546 327 246 71724 5.256 0.405
LyonSchool 17174 242 104 89640 2.806 0.682
Table 1: Empirical temporal networks. Columns, from leftmost to rightmost: data set name, number of active nodes, number of nodes, number of timestamps, number of temporal edges, average weight of temporal edges, average fraction of timestamps in which a node is active.
, , , , ,
Data set P(S) P(I) P(R) P(S) P(I) P(R) P(S) P(I) P(R) P(S) P(I) P(R) P(S) P(I) P(R)
InVS15 0.171 0.0500 0.778 0.103 0.0534 0.843 0.154 0.0457 0.799 - - - 0.123 0.0769 0.799
LH10 0.0801 0.152 0.767 0.292 0.0975 0.609 0.197 0.120 0.681 - - - 0.0891 0.168 0.742
SFHH 0.0488 0.259 0.692 0.337 0.195 0.467 0.653 0.155 0.191 0.0737 0.151 0.774 0.163 0.333 0.502
Thiers13 0.0340 0.0908 0.875 0.139 0.0959 0.764 0.183 0.0923 0.723 0.240 0.051 0.707 0.111 0.164 0.723
LyonSchool 0.0955 0.155 0.748 0.104 0.191 0.703 0.128 0.156 0.715 0.102 0.0908 0.806 0.107 0.311 0.580
Table 2: Proportion of the three possible states S, I and R among the active nodes, for each data set and each parameter set of the SIR model. “-” indicates the cases for which the epidemic did not spread sufficiently (i.e., ).

We study the effectiveness of the DyANE, in particular with the dyn-supra+DeepWalk combination, in the context of node classification tasks. For our experiments we use temporal networks built from empirical datasets that describe close-range proximity interactions of persons in a variety of real world environments. We simulate the SIR (Susceptible-Infected-Recovered) dynamical process described above over these temporal networks, generating state labels for all active nodes.

Based on the above temporal networks and node labels, we run DyANE with different combinations of supra-adjacency representations and of embedding methods for the static network, and use the resulting embedding vectors as inputs to a supervised multi-label classifications tasks. We test the sensitivity of our approach with respect to the choice of parameters and to the number of sampled active nodes . Finally, we also compare classifiers based on DyANE embeddings with methods that directly embed the nodes of a temporal network without relying on a supra-adjacency representation.

Data sets and dynamical process

We used publicly available data sets describing the face-to-face proximity of individuals with a temporal resolution of 20 seconds  [Cattuto et al.2010]. These datasets were collected by the SocioPatterns collaboration111http://www.sociopatterns.org/ and we specifically use data sets collected in a variety of contexts, namely in offices (”InVS15”), a hospital (”LH10”), a highschool (”Thiers13”), a conference (”SFHH”) and a school (”LyonSchool”) [Génois and Barrat2018]. We built a temporal network from each data set by aggregating the data on 600 seconds time windows. Whenever multiple proximity events were registered between two individuals within a time window, we used the number of such events as the edge weight. Table 1 shows some basic statistics for each data set.

We simulated the SIR model on each data set (with the original temporal resolution of 20 seconds), using the following five combinations of epidemic parameters: , , , , . In each case, we consider as initial state a single randomly selected node as seed, setting its state as infectious, with all others susceptible. Given the stochastic nature of the model, in some cases the infectious state barely spreads, with a large majority of the nodes remaining susceptible. The prediction task would then be trivial, and we restrict our study to non-trivial cases in which there is still at least one infectious node when more than half of the total data set time span has elapsed (i.e., ). We thus run the SIR model up to times for each data set until we obtain a simulation in which this condition is met. If the condition is not met in any of the simulations, we discard the corresponding case (see Table 2). For each selected simulation, we assign as ground truth label to each active node the state of node at time . Table 2 shows the proportion of each label among active nodes for each case.

We select uniformly at random active nodes, and build our training data using those nodes and the corresponding node states. Unless otherwise noted, . We evaluate the prediction performance on a test data consisting of the remaining active nodes and their states. We report the prediction performance averaged over five realizations of the random choice of training data, for each data set and parameter values.

Evaluation

We quantified the prediction performance of each method by the Macro-F1 and Micro-F1 indices, which are widely used for evaluating multi-label node classification tasks [Goyal and Ferrara2018]. These indices range between 0 and 1, with higher values indicating better performance. Macro-F1 is an unweighted average of the F1 scores of each label, while Micro-F1 gives more importance to the labels which are more represented. When classes are imbalanced, Macro-F1 and Micro-F1 give a measure of the effectiveness of the method on respectively the smaller and the larger classes. As the number of active nodes in each state might indeed be imbalanced here, we measure both Macro- and Micro-F1 to evaluate models by the prediction performance against both the smaller and the larger classes.

Baseline methods

We considered variations of our method both at the level of the supra-adjacency representation and of the static embedding method. First, as a variation of our proposed supra-adjacency representation (dyn-supra), we consider a “baseline” supra-adjacency representation, which we denote by mlayer-supra, in which we simply map each temporal edge to a single edge between active nodes, namely , similarly to the original supra-adjacency representation developed for multilayer networks [Kivelä et al.2014]. Self-coupling edges are drawn as in dyn-supra. We also considered two alternative embedding methods for the static networks obtained by each supra-adjacency representation: (i) LINE [Tang et al.2015], which embeds nodes in a way to preserve both first and second-order proximity; (ii) Graph Factorization (GF) [Ahmed et al.2013], which considers first-order proximity.

In addition, we considered two methods for temporal network embedding, which thus do not use the intermediate supra-adjacency representation, namely: (i) DynamicTriad (DTriad) [Zhou et al.2018], which embeds the temporal network by modeling triadic closure events; (ii) DynGEM [Goyal et al.2018], which is based on a deep learning model. It outputs an embedding for the network of each timestamp, initializing the model at timestamp with the parameters found at time , thus transferring knowledge from to and learning about the changes from to . Overall, we obtain eight methods – six variations of DyANE and two methods that directly embed temporal networks – which we denote respectively dyn-supra+DeepWalk, dyn-supra+LINE, dyn-supra+GF, mlayer-supra+DeepWalk, mlayer-supra+LINE, mlayer-supra+GF, DTriad, and DynGEM.

We used publicly available implementations of all embedding methods, namely the implementation of LINE222https://github.com/tangjianpku/LINE, DynamicTriad333https://github.com/luckiezhou/DynamicTriad, and DynGEM444http://www-scf.usc.edu/~nkamra/ by the original authors, and the implementation of GF by GEM555https://github.com/palash1992/GEM. As for DeepWalk, we used an implementation of node2vec666https://snap.stanford.edu/node2vec/ with . Unless otherwise noted, we conducted experiments with embedding dimension and self-coupling edge weight . We used the default values of each implementation of the embedding methods, except for the number of iterations of LINE and GF, which we took equal to the number of samples of DeepWalk. We used Scikit-Learn [Pedregosa et al.2011] to implement one-vs-rest logistic regression, with the default hyper-parameter setting.

Figure 2: Prediction performance of each method, for each data set and spreading parameter set. The top and bottom row indicate results of Macro-F1 and Micro-F1, respectively.

Prediction performance

Figure 2 shows the prediction performance of the eight methods considered, for all data sets and SIR parameters considered. The dyn-supra representation combined with DeepWalk yields almost always the highest value both for Macro-F1 and Micro-F1. While dyn-supra+DeepWalk is consistently the best method for both Macro-F1 and Micro-F1 across all considered data sets and SIR parameters, DynGEM yields a higher Micro-F1 value in a few cases. However, DynGEM’s Macro-F1 score is rather low in such cases.

We moreover observe that: (i) for a given static embedding method, the dyn-supra supra-adjacency representation gives better results than the baseline (mlayer-supra) one; and (ii) for a given supra-adjacency representation, DeepWalk performs better than LINE, which in turn outperforms GF.

Finally, we show in the supplementary material that the dyn-supra+DeepWalk prediction recovers the overall temporal evolution of the spreading process.

Sensitivity analysis

We first investigate here the effect of the hyper-parameters, of the supra-adjacency representation (the weight of self-coupling edges) and of the embedding (the embedding dimension ). We show in Fig. 3 the results obtained for the Macro-F1, for the InVS15 data set and , but we have confirmed the same tendency for the other data sets, parameters and also for Micro-F1 values (the omitted results are included in the Supplementary Materials). The results show that the performance of dyn-supra+DeepWalk is very stable with respect to changes in , while other methods depend weakly on . The performance of all methods is stable on a wide range of embedding dimensions, and decrease when it becomes smaller than . Overall, dyn-supra+DeepWalk remains the most effective method without the need for fine-tuning or .

Figure 3 also shows the effect of increasing the parameter , i.e., of being able to observe a larger number of active nodes. The performance increases with for most methods, and in particular for dyn-supra+DeepWalk, which consistently yields the best result at all values of .

Figure 3: Results of the hyper-parameter sensitivity on the InVS15 data set. Bars indicate the standard deviation.

As mentioned above, we moreover considered two variations if the dyn-supra representation: (i) we regard edges as directed towards increasing timestamps (dyn-supra-directed); (ii) we let the weight of an edge decay with increasing temporal lag between the active nodes it links, e.g., we modulate the edge weight according to the reciprocal of the lag (dyn-supra-decay). We also consider these variations for mlayer-supra representation, yielding mlayer-supra-directed and mlayer-supra-decay, respectively. Notice that, in the mlayer-supra, the supra-adjacency edges representing temporal edges are not affected by both variations. We report in Fig. 4 the results for and for the DeepWalk embedding, as DeepWalk overall yielded the best results. We checked that the results of Fig. 4 hold similarly for the LINE and GF embeddings.

Figure 4: Effect of the variations in the supra-adjacency representation. Top: effect of using directed edges. Bottom: effect of introducing weights depending on time difference.

Figure 4 indicates that using directed edges worsens the performance of both dyn-supra and mlayer-supra methods. Introducing weights depending on the time difference between active edges also worsens the performance for mlayer-supra, with little effect on dyn-supra. Overall, the original dyn-supra method with undirected edges and using only the weights of the original temporal edges yields the highest prediction performance.

Related Work

Network node embedding

Numerous embedding techniques have been proposed for static networks, and we refer to the recent reviews [Cai, Zheng, and Chang2018, Goyal and Ferrara2018] for detailed descriptions. Most techniques encode as proximity or similarity between nodes either a first-order proximity (two nodes are more similar if they are connected by an edge with larger weight) or a second-order proximity (nodes are more similar if their neighborhoods are similar). A popular way of exploring the (structural) similarity of nodes consists in using random walks rooted at the nodes, which thus explore their neighborhoods. Two of the most well-known embedding techniques, DeepWalk and node2vec [Grover and Leskovec2016] are indeed based on such random walks.

As temporal network data has become more widely accessible in a range of contexts, the issue of embedding dynamically evolving networks has emerged and some methods have been put forward. As nodes’ relationships evolve with time, the structure of their neighborhoods and the similarity or proximity between nodes indeed evolve as well. The typical approach consists then (in discrete time, although see [Nguyen et al.2018] for a case of continuous time embedding) in defining a distinct embedding for each temporal snapshot and to ensure some continuity between the embeddings of successive snapshots [Zhou et al.2018, Goyal et al.2018, Goyal, Chhetri, and Canedo2019]. For a comprehensive review of recent embedding methods for dynamically evolving networks see  [Kazemi et al.2019].

None of these methods however consider, as we do here, supra-adjacency representations of the temporal network as a whole in order to then take advantage of static network techniques for the embedding of temporal networks.

Supra-adjacency representation

The supra-adjacency matrix representation has been developed for multilayer networks [Gómez et al.2013, Kivelä et al.2014], in which nodes interact on different layers (for instance different communication channels in a social network). Using the pairs (node, layer) as elementary entities, this representation builds a graph between these pairs, consisting in (i) the links within each layer, such as if nodes and are linked in layer , and (ii) the links between different copies of the same node in different layers, such as between layers and . The adjacency matrix of this new graph is the supra-adjacency matrix. This representation has proven very convenient in the study of processes on multilayer networks [Kivelä et al.2014] as it makes it possible to use the methods and theoretical results developed using adjacency matrix of simple graphs.

It has been generalized to temporal networks, seen as special multilayer networks in which every timestamp is a layer: in the supra-adjacency representation, each node is identified by the pair of indices , corresponding to the node label and the time frame , respectively [Valdano et al.2015]. Most importantly, (i) each node is linked by a directed edge to its successive copy , but not to other future timestamps and (ii) intra-layer edges are absent: instead, if and are connected at time , in the supra-adjacency representation this is replaced by two directed edges respectively from to and from to . This representation was put forward in the context of spreading processes on temporal networks as it indeed preserves the information relevant for the spreading process [Valdano et al.2015]: if a contagion event occurs from to at time , will be contaminated (and could propagate the disease further) at , but not yet at .

Note that the nodes are in this representation present for all nodes and timestamps , even if is isolated at , and that the links and also exist even if or have no links at or . The modified supra-adjacency representation we propose considers instead only the pairs such that is not isolated at , in order to avoid potentially long chains if remains isolated for some time, and draws links only between and such that (resp. ) is not isolated at (resp. ).

Recovering dynamics on a network

Given a dynamical process occurring on a network, such that nodes change state over time, only partial knowledge of this evolution can in general be realistically envisioned, as for instance in diffusion processes such as the spread of contagious diseases or rumors. The task of recovering the complete knowledge of a diffusion process from partial observations has thus been addressed in a variety of settings.

Some works have put forward methods to recover the state of all nodes and the seeds of a spread from a partial observation of nodes at a given time [Sundareisan, Vreeken, and Prakash2015, Xiao, Aslay, and Gionis2018], without attempting to recover the process whole temporal evolution. Methods to recover the state evolution of all nodes have also been proposed, using as input snapshots of the whole system, i.e., the knowledge of the state of all the nodes at a certain time [Sefer and Kingsford2016, Feizi et al.2018, Chen, Tong, and Ying2019]. Bayesian inference methods from partially observed snapshots have also been proposed [Altarelli et al.2014]. These works make use of strong assumptions on the nature of the underlying diffusion model.

Some works have also proposed to recover the time evolution of node states without detailed knowledge of the diffusion model [Rozenshtein et al.2016, Xiao et al.2018, Xiao, Aslay, and Gionis2018]. In particular, Rozenshtein et al. map the temporal network to a supra-adjacency representation almost identical to the mlayer-supra representation described above, in order to preserve the temporal paths on the network, and solve the recovery of the node states as a Steiner tree problem. This relies on the fact that each node changes state only once, hence on the irreversibility of the process. Finally, we mention a method based on tensor decomposition to recover coexisting information diffusion processes from partial knowledge of the node states, without detailed knowledge of the processes, by exploiting the fact that these processes occur in synergy [Sun et al.2017]. The fraction of unknown states is however limited, while we consider a very small fraction of active node states as known.

Conclusion

Here we introduced a method for embedding nodes of temporal networks suited for the task of recovering the dynamical evolution of a single instance of a process occurring on the network, from partial observations and without information on the nature of the process itself except from the set of possible states of the nodes. Our method first maps the temporal network to a modified supra-adjacency representation, which preserves the paths on which the process unfolds. As this representation yields a static graph among the active nodes, which are pairs of the form (node of the temporal network, time of interaction), it enables the use of embedding techniques for static networks. We choose to use DeepWalk, as it is a simple and paradigmatic algorithm based on random walks and thus particularly suited to explore the neighborhood of the nodes of the supra-adjacency representation in a way relevant to the dynamical process on the network. We finally frame the inference of the dynamical state of all active nodes from a set of observations as a classification task.

We have shown the performance of our method on the concrete case of an epidemic-like model on empirical temporal networks and compared it with other state of the art methods. Our method consistently yields very good classification performance in a robust way across data sets and process parameters, without fine-tuning hyper-parameters.

Our results show that it is possible, without any knowledge of the precise nature of the process nor of its parameters, to recover crucial information on its outcome. Note in particular that our method assumes no knowledge of which transitions between states actually occur in the real dynamics: this means that the predicted sequence of states of each individual node might yield ”forbidden” transitions (e.g., in the SIR example, transitions from I to S or from R to I). Nevertheless, we have shown that the outcome of the classification task gives a good estimation of the actual dynamics.

Our method has however the clear limitation that we assume the whole temporal network to be known. Although a full observation of the contact patterns of individuals could be envisioned in some specific controlled settings such as hospitals, this is not generally the case. Further work will address this limitation by considering the effect of noise and errors in the temporal network data, and by considering the case in which only a (more or less detailed) set of statistics of the temporal network is known. Noise could also impact the quality of the sampling (e.g., observational errors), and we will check its impact on our method’s performance. Further work will also address different sampling strategies such as a sampling concentrated at early times, or focused on few specific “sentinel” nodes followed at all times, or of a whole snapshot of the system but only at a specific time. This could yield crucial insights on how to optimize surveillance strategies in concrete settings.

Finally, since our method is largely agnostic with respect to the specific dynamical process, we will consider other processes such as other models of disease propagation, complex contagion phenomena or opinion formation.

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