Sato et al. EPJ Data Science(2021) 10:22
https://doi.org/10.1140/epjds/s13688-021-00277-8

REGULAR ARTICLE

Open AccessPredicting partially observed processes on
temporal networks by Dynamics-Aware
Node Embeddings (DyANE)
Koya Sato1 , Mizuki Oka1 , Alain Barrat2,3*

and Ciro Cattuto4,5*

Correspondence:
alain.barrat@cpt.univ-mrs.fr
Turing Center for Living Systems,
Aix Marseille Univ, Université deToulon, CNRS, CPT, Marseille, France
3
Tokyo Institute of Technology,
Tokyo, Japan
Full list of author information isavailable at the end of the article2

Abstract
Low-dimensional vector representations of network nodes have proven successful tofeed graph data to machine learning algorithms and to improve performance acrossdiverse tasks. Most of the embedding techniques, however, have been developedwith 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 – withthe goal of enabling prediction tasks related to the evolution and outcome of theseprocesses. We achieve this by using a lossless 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 single instances of a spreading process. We show how framing thistask 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 ofnodes at random times, with potential impact for nowcasting infectious disease
dynamics.
Keywords: Temporal networks; Node embedding; Dynamical processes

1 Introduction
The ubiquity of network representations of widely diﬀerent systems has led to a ﬂourishingof methods aimed at the analysis of their structure [1, 2] and of processes taking placeon networks, such as information diﬀusion, epidemic spread, synchronization, etc [3, 4].
Recently, these investigations have been extended to the case of temporal networks, inwhich nodes and links can appear and disappear in time [5, 6].
Most works aim in particular at understanding how the network’s features impact theoutcome of processes taking place on top of them, usually considering averages over many
realizations of a stochastic process. A less considered issue concerns the reconstruction© The Author(s) 2021. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use,sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original
author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or otherthird party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line
to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted bystatutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view acopy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Sato et al. EPJ Data Science(2021) 10:22

of partially observed processes taking place on a network. Indeed, given a dynamical process occurring on a network, such that nodes change state over time, in a realistic setting
only partial knowledge of this evolution can in general be envisioned, as for instance indiﬀusion processes such as the spread of contagious diseases or rumors. Recovering thecomplete information on the unfolding of the process from partial observations can thenbe of crucial importance, for instance to estimate the actual impact of a spread whose evolution is only partially known, and whose parameters are a priori unknown. This issue has
been addressed in the speciﬁc case of spreading processes, under various hypothesis. For
instance, some works have put forward methods to recover the state of all nodes and theseeds of a spread from a partial observation of nodes at a given time [7, 8], without attempting to recover the whole temporal evolution of the process. Methods to recover the stateevolution 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 [9–11]. Finally, several
methods using partially observed snapshots have also been proposed [8, 12–14], typically
based on strong assumptions on the nature of the underlying diﬀusion process.
Here, we propose a novel approach to tackle the general issue of recovering all the information about a single instance of a partially observed and unknown process taking place
on a known temporal network, leveraging the recent development of node embedding
methods. Network node embedding methods have indeed recently gained a lot of popularity [15–17] as tools able to explore network structure, and we propose here that embeddings can also be designed in order to recover infomation on dynamical processes on networks. In short, a 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 embeddings thus aim at exposing in the low-dimensional space structural features and relevant patterns of the networkthat are not necessarily evident in the network representation. Most importantly, the embedding vectors can be used as feature vectors in machine learning applications, and havebeen shown to yield improved performance for tasks such as node classiﬁcation, link prediction, clustering, or visualization [16].
Here we show that node embedding methods can also be tailored to the study of dynamical processes on temporal networks, and in particular to the task described above ofpredicting the evolution and outcome of any instance of the dynamics (e.g., an epidemic
outbreak) from partial information and without detailed knowledge of the dynamical process itself. We notice on the other hand that the task at hand concerns a temporal network in discrete time on a ﬁnite interval, assumed to be fully known. 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 thesought embeddings should be informative of these paths – the paths along which information can propagate. Driven by this idea, we propose to ﬁrst map the temporal network
to a static graph representation without losing any temporal information. We note that
this is in contrast with aggregated representations, in which the temporal information islost [18], even if the aggregation procedure can be tailored to speciﬁc contexts such as
epidemic processes on networks [19]. Higher-order aggregate networks, which take into
account that not all the paths present in aggregated representations are actually realizable,

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capture more temporal dependencies than usual aggregation, and processes simulated onsuch higher-order representations better reﬂect the true dynamics on the raw data [20–22]. For instance, such higher-order representations yield better estimations of centrality
measures in temporal networks [21, 22]. Recently, two completely lossless static representations of temporal networks have moreover been put forward: the so-called supraadjacency representation, whose nodes are the (node,time) pairs of the original temporal network [23], and the event-graph [24, 25], in which all temporal events are represented as nodes of a static graph. We propose here to modify the original supra-adjacency
representation method: we only consider nodes at those times when they interact, andwe map the original temporal edges to edges between the corresponding (node,time)pairs: all nodes and temporal edges are represented, ensuring that no temporal information is lost. Crucially, this static graph representation preserves all the temporal paths ofthe original temporal network (i.e., the paths supporting and constraining the dynamical
process at hand), so that it does not suﬀer from the limitations of aggregated representations [20]. 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 [26, 27] as they
provide an eﬃcient way to sample the relevant paths.
We show the performance of the proposed embeddings for the prediction of partiallyobserved processes in the context of a paradigmatic dynamical process – epidemic spreadover 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”). As described above, we focus on the task of predicting the nodes’ states overtime for a single realization of the epidemic process. Speciﬁcally, we set up a multi-label

Figure 1 Modiﬁed supra-adjacency representation (dyn-supra). The top panel shows a toy example with atemporal network at four successive times. At each time t we show in bold the nodes of Vt , i.e., the nodes withat least one temporal edge. This network is mapped to a static representation (bottom) where nodes are(node,time) pairs of the original network, keeping for each node of the original network only the times inwhich it is active. In this toy example, node i is active at times t and t + 1 so the corresponding active nodes
are (i, ti,1 = t), (i, ti,2 = t + 1); node j is active at times t, t + 2 and t + 3 so the corresponding active nodes are(j, tj,1 = t), (j, tj,2 = t + 2), (j, tj,3 = t + 3); ﬁnally, node k is active at times t + 1, t + 2 and t + 3 so the corresponding
active nodes are (k, tk,1 = t + 1), (k, tk,2 = t + 2), (k, tk,3 = t + 3)

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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 theparameters of the epidemic process. In summary, 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 forstatic graphs.
• We show that in the important case of epidemic spreading, a good 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 with respect to other embedding methods.
The paper is organized as follows: we ﬁrst formulate in detail the problem at hand in
Sect. 2. We then describe our approach in Sect. 3 and show the results of numerical experiments in Sect. 4. We conclude with some perspectives in Sect. 5.

2 Problem formulation
Let us ﬁrst state in general terms the problem we want to address: given a known temporal
network over a given time interval, an instance of an unknown process unfolding on thisnetwork and a partial observation of the dynamical states of the nodes, we want to recover the dynamical state of all nodes at all times of the temporal network. Crucially, thisprediction must be performed without any information on the details of the dynamical
process taking place on the network, except for the set of possible states of each node. In
particular, we do not make any assumption on the type of transitions, the parameter values, nor even on the reversibility or irreversibility of the process. Moreover, this prediction
does not concern an average over various realizations of the same process, but instead one
single realization, which has been partially observed. Note that this task concerns a fully
known temporal network, and the recovery of the nodes’ states over the time span of thetemporal network: we do not consider a situation in which information on the temporal
network and the state of the nodes would be arriving in real-time in a streaming fashion.
2.1 Temporal network
In more precise terms, we consider a temporal network g in discrete time on a time interval
T = (1, 2, . . . , |T|), i.e., a set V of N = |V | nodes and a set of temporal edges of the form(i, j, t) denoting that nodes i and j are in interaction at time t ∈ T. Note that each temporaledge can also potentially carry a weight wij (t). The set of temporal edges at t is denoted Et ,and Vt is the set of nodes which have at least one temporal edge at t: the snapshot network
at t is the undirected weighted network Gt = (Vt , Et ). We assume in the following that thewhole temporal network is known.
For each node i, we deﬁne its set of active times Ti as the set of timestamps t in which
it is involved in at least one temporal edge (i.e., such that i ∈ Vt ). We denote the a-th
active time of i by ti,a ∈ Ti , with ti,a < ti,a+1 , and we deﬁne the set of active copies of eachnode i, that we call “active nodes”, as Vi = {(i, t)|t ∈ Ti }. An active node is thus of the form

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Sato et al. EPJ Data Science(2021) 10:22(node,time). The overall set of active nodes is the union of all the sets of active nodes,
i.e., V = ∪i∈V Vi .

2.2 Dynamical process
We consider a dynamical process taking place on the weighted temporal network, suchthat each node i ∈ V can be at each time in one of a ﬁnite set of discrete states S . Nodes
can change state either spontaneously or through interaction along temporal edges. Ourdeﬁnition is thus very general and encompasses a wide variety of processes on networks,
such as models of epidemic propagation, rumor propagation, opinion formation or cascading processes [3, 4, 28].
While the problem description is very general, we will focus here on a paradigmatic dynamical process of strong relevance, namely the Susceptible-Infectious-Recovered (SIR)
model for epidemic spreading, which is widely used to model contagious infections suchas ﬂu-like diseases [29]. 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, allnodes 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 1 – (1 – β)w at each timestamp, where β is the infection rate and w is theedge weight between the S and I nodes. We assume that such a process takes place on
the considered temporal network between times 1 and |T|. Let us denote by It the set of
infectious nodes at t, and consider a susceptible node i. We denote its set of neighbours
at t as Nt (i) = {j|(i, j, t) ∈ Et }, and Nt (i) ∩ It is the set of its infectious neighbours at t. The
probability that none of these infectious neighbours transmits the disease to i at timestep
t is j∈Nt (i)∩It (1 – β)w(i,j,t) and thus the probability that i becomes infectious at time t, due
to its interactions, is 1 – j∈Nt (i)∩It (1 – β)w(i,j,t) . Recovery from state I to state R occurs also
stochastically: each infectious node becomes recovered at each timestamp with probability μ. Recovered nodes do not change state any more. The parameters of the model arethus the infection and recovery rates β and μ [29].
We note here that the SIR model – in addition to its relevance to many real-world phenomena – is particularly interesting to study in the context of the prediction problem
addressed in this paper: it features indeed not only state transitions occurring upon interaction (hence, along the edges of the temporal network) but also spontaneous statetransitions that can occur at any time, and in particular between successive active timesof a node (the infectious-recovered transition).
2.3 Partial observation of the process and prediction task
We assume that a sample of the dynamical evolution of the process during the time span Tof the temporal network is known. More precisely, we ﬁrst assume that the state of a node
can only be observed when it is active, i.e., in contact with at least another node. Denoting
by f : (i, t) ∈ V → s ∈ S the mapping that speciﬁes the state of each node at each of its active
times, we assume that this mapping is only partially known, through the observation of afraction of the active nodes: we deﬁne the set of the observed active nodes as D ⊂ V . Here,for simplicity, we will assume that D results from a uniform random sampling of V .
The task at hand is then to determine the state of all the unobserved active nodes, i.e.,the state in which each node is at each of its active times. This allows to reconstruct theunfolding of the process both at the local node level and obviously as well at the population

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level. In the example of the SIR process, crucial outcomes of the prediction task are theepidemic curve (i.e., the time series of the fraction of infected nodes over the time span of
the temporal network), including the timing of the epidemic peak, and the ﬁnal epidemic
size, i.e., the actual number of nodes that have been aﬀected by the spreading process.

3 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 that, despite being static, contains the whole information of the temporal network. Second, weapply 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 predictthe dynamical state of all active nodes based on the vector representation of active nodes
and the partially observed states. We now give details on each of these steps.
3.1 Supra-adjacency representation
We ﬁrst map the temporal network to a supra-adjacency representation. The supraadjacency representation has been ﬁrst developed for multilayer networks [30, 31], inwhich nodes interact on diﬀerent layers (for instance diﬀerent communication channels
in a social network). It has been generalized to temporal networks, seen as special multilayer networks in which every timestamp is a layer [23]: each node of the supra-adjacency
representation is identiﬁed by the pair of indices (i, t), corresponding to the node label iand the time frame t of the original temporal network. In this representation, the nodes(i, t) are present for all nodes i and timestamps t, even if i is isolated at t.
We propose here to use a modiﬁed version in which we consider only the active timesof each node. This results in a supra-adjacency representation whose nodes are the active
nodes of the temporal network. More precisely, we deﬁne the supra-adjacency network
as G = (V , E ), where E are (weighted, directed) edges joining active nodes. The mappingfrom the temporal network to the supra-adjacency network consists of the following two
procedures (Fig. 1):• For each node i, we connect its successive active versions: for each active time ti,a of i,we draw a directed “self-coupling” edge from (i, ti,a ) to (i, ti,a+1 ) (recall that active times
are ordered in increasing temporal order).
• For each temporal edge (i, j, t), the time t corresponds by deﬁnition to an active time
for both i and j, that we denote respectively by ti,a and tj,b . We then map (i, j, t) ∈ E totwo directed edges ∈ E , namely ((i, ti,a ), (j, tj,b+1 )) and ((j, tj,b ), (i, ti,a+1 )). In other words,
the active copy of i at t, (i, t), is linked to the next active copy of j, and vice-versa.
The ﬁrst procedure makes each active node adjacent to its nearest past and future versions (i.e., at the previous and next active times). This 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. Crucially, all nodes are represented at all thetimes in which they interact, and all temporal edges are represented: the supra-adjacency
representation does not involve any loss of temporal information, and the initial temporal network can be reconstructed from it. In particular, it yields the crucial property thatany 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,

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if an interaction between two nodes i and j occurs at time t and potentially modiﬁes theirstates, e.g., by contagion or opinion exchange or modiﬁcation, this can be observed andwill have consequences only at their next respective active times: for instance, if i transmits a disease to j at t, j can propagate it further to other neighbours only at its next active
time, and not immediately at t. This is reﬂected in the supra-adjacency representation we
propose.
The edges in E are thus of two types, joining two active nodes corresponding either tothe same original node, or to distinct ones. For each type, we can consider various ways ofassigning weights to the edge. We ﬁrst consider for simplicity that all self-coupling edges
carry the same weight ω, which becomes thus a parameter of the procedure. Moreover,
we simply report the weight wij (t) of each original temporal edge (i, j, t) on the two supraadjacency edges ((i, ti,a ), (j, tj,b+1 )) and ((j, tj,b ), (i, ti,a+1 )) (with t = ti,a = tj,b ).
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 ignore thedirection of time of the original temporal network in the supra-adjacency representation
by making all links of E undirected. We will refer to this representation as dyn-supraundirected. 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, as a simple implementationof this concept, that the original weight of an edge ((i, t), (j, t  )) in the dyn-supra representation is multiplied by the reciprocal of the time diﬀerence between the active nodes,
i.e., |1/(t – t  )|. Each self-coupling edge has thus weight ω/(ti,a+1 – ti,a ), while a temporaledge (i, j, t) with t = ti,a = tj,b yields the edges ((i, ti,a ), (j, tj,b+1 )) with weight wij (t)/(tj,b+1 – ti,a )and ((j, tj,b ), (i, ti,a+1 )) with weight wij (t)/(ti,a+1 – tj,b ). We will refer to this representation as
dyn-supra-decay.

3.2 Node embedding
The central idea of the embedding method we propose for temporal networks, which wecall DyANE (Dynamics-Aware Node Embeddings), is to apply embedding methods developed for static networks to the supra-adjacency representation G of the temporal graph.
Numerous embedding techniques have been proposed for static networks, as surveyed
in recent reviews [15, 16]: Most techniques consider as measure of proximity or similarity between nodes either ﬁrst-order proximity (the similarity of two nodes increases withthe strength of the edge connecting them) or second-order proximity (the similarity of
two nodes increases with the overlap of their network neighborhoods). In particular, a
popular technique to probe the (structural) similarity of nodes relies on random walks
rooted at all nodes. Two of the most well-known embedding techniques, DeepWalk [26]and node2vec [27], are based on such an approach.
Methods based on random walks seem particularly appropriate to our framework as
well: Indeed, in the supra-adjacency representation, random-walks will explore for eachactive node both self-coupling edges, connecting instances of the same node at diﬀerent
times, and edges representing the interactions between nodes. As explained above, theseedges encode the paths along which information can ﬂow over time, meaning that theﬁnal embedding will preserve structural similarities relevant to dynamical processes on

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the original network. Here we will use DeepWalk [26], as it is a simple and paradigmaticalgorithm, and it is known to yield high performance in node classiﬁcation tasks [16]. Notethat DeepWalk does not support weighted edges, but it can easily be generalized so thatthe random walks take into account edge weights [27]. We remark that this choice is driven
by the simplicity and popularity of DeepWalk, but that the embedding methodology we
propose here can readily beneﬁt from any other embedding techniques for static graphs.

3.3 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 ofthe possible states. In our speciﬁc case, the three possible node states are S, I, and R. We
recall 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 active 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 history of the states of all nodes from the observed states of few active
nodes (“sentinel” nodes). The task however does not concern the future evolution of theepidemic after the end of the temporal network data.
3.4 Evaluation
The performance of our method can be evaluated along diﬀerent lines. On the one hand,
we can use standard measures used in prediction tasks, counting for each active node
whether its state has been correctly predicted. We construct then a confusion matrix C,
in which the element Css is given by the number of active nodes that are in state s in thesimulated spread and predicted to be in state s by the classiﬁcation method. The number
TPs of true positives for state s is then the diagonal element Css (and the total numberof true positives is TP = s Css ), while the number of false negatives FNs for state s iss =s Css . Similarly, the number of false positives FPs iss =s Cs s (active nodes predicted
to be in state s while they are in a diﬀerent state in the actual simulation).
The standard performance metrics for each state s, namely precision and recall, are given
respectively by PREs = TPs /(TPs + FPs ) and RECs = TPs /(TPs + FNs ) and the F1-score is
F1s = 2PREs · RECs /(PREs + RECs ). In order to obtain overall performance metrics, it iscustomary to combine the per-class F1-scores into a single number, the classiﬁer’s overall F1-score. There are however several ways to do it and we resort here to the Macro-F1
and Micro-F1 indices, which are widely used for evaluating multi-label node classiﬁcation tasks [16]. Both 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, s∈S F1s /|S |. On theother hand, Micro-F1 is obtained by using the total numbers of true and false positives

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and negatives. The total number of true positives is TP = s Css , and, since any classiﬁcation error is both a false positive and a false negative, the total numbers of false posi
tives and of false negatives are both equal to FP = FN = s=s Css . As a result, Micro-F1 iss Css /s,s Css (sum of the diagonal elements divided by sum of all the elements). In thecase of imbalanced classes, Micro-F1 gives thus more importance to the largest classes,
while Macro-F1 gives the same importance to each class, whatever its size. In our speciﬁccase of the SIR model, the three classes S, I, R might indeed be very imbalanced, depending on the model parameters, so that it is important to use both Macro- and Micro-F1 toevaluate the method’s performance in a broad range of conditions.
From an epidemiological point of view, it is also interesting to focus on global measurescorresponding to an evaluation of the correctness of the prediction about the overall impact of the spread, as measured by the epidemic curve and the ﬁnal epidemic size. For
instance, if we denote by Iareal (t) the numbers of infectious active nodes at time t in thepredsimulated spread, and by Ia (t) the number predicted in the classiﬁcation task, we candeﬁne as measure of discrepancy between the real and predicted epidemic curves:

I =

T
1

T t=1

g(t),

g(t) =

⎧
⎨0
⎩

pred
|Ia (t)–Iareal (t)||Vt |if |Vt | = 0,otherwise.

We can also focus on the ﬁnal impact of the spread, as an evaluation of the global impact
on the population, and compute the discrepancy in the ﬁnal epidemic size
size =

I pred (T) + Rpred (T) – I real (T) + Rreal (T) /N .

Note that not all nodes might be active at the last time stamp T, so we can in this caseand for simplicity consider for each node its last active time and assume that it does notchange state until T.

3.5 Comparison with other methods and sensitivity analysis
Our framework entails two choices of procedures: the way in which the temporal network
is represented as a static supra-adjacency object, and the choice of the node embedding
method.
First, we consider a variation of our proposed supra-adjacency representation (dynsupra), using a “baseline” supra-adjacency representation, which we denote by mlayersupra: in this representation, we simply map each temporal edge (i, j, t) to an edge between
active nodes, namely ((i, t), (j, t)), similarly to the original supra-adjacency representation
developed for multilayer networks [31]. Self-coupling edges are drawn as in dyn-supra.
This static representation of the temporal network is also lossless.
Moreover, for both dyn-supra and mlayer-supra, we consider an alternate embedding
method to DeepWalk, namely LINE [32], which embeds nodes in a way to preserve bothﬁrst and second-order proximity.
In addition, we consider four state of the art embedding methods for temporal networks,
which do not use the intermediate step of using a supra-adjacency representation, but directly embed the temporal network, namely: (i) DynamicTriad (DTriad) [33], which embeds the temporal network by modeling triadic closure events; (ii) DynGEM [34], which is

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based on a deep learning model. It outputs an embedding for the network of each timestamp, initializing the model at timestamp t + 1 with the weights found at time t, thustransferring knowledge from t to t + 1 and learning about the changes from Gt to Gt+1 ;(iii) StreamWalk [35], which uses time-respecting walks and online machine learning tocapture temporal changes in the network structure; (iv) Online learning of second order node similarity (Online-neighbor) [35], which optimizes the embedding to match theneighborhood similarity of pairs of nodes, as measured by the Jaccard index of these neighborhoods.
Overall, we obtain eight methods to create an embedding of the temporal network –four variations of DyANE and four methods that directly embed temporal networks –which we denote respectively dyn-supra + DeepWalk, dyn-supra + LINE, mlayer-supra +DeepWalk, mlayer-supra + LINE, DTriad, DynGEM, StreamWalk and Online-neighbor.
Each variation of DyANE has moreover two parameters whose value can be a priori
arbitrarily chosen, namely the weight ω and the embedding dimension d. In each of thesevariations of DyANE, it is also possible as explained above to consider undirected edges
and to take into account the diﬀerence of the times between linked active nodes.
For each obtained embedding, we thus explore the performance of the classiﬁcation task
to explore the robustness of the results and their potential dependency on speciﬁc choicesof the embedding method and of the parameter values.

4 Numerical experiments and results
In this section, we study the eﬀectiveness of DyANE, in particular with the dyn-supra +DeepWalk combination to predict the nodes’ epidemic states in a single instance of an SIR
spreading process.
To this aim, we use temporal networks built from empirical data sets that describe closerange proximity interactions of persons in a variety of real world environments. We simulate the SIR (Susceptible-Infected-Recovered) dynamical process described above overthese 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 supraadjacency representations and of embedding methods for the static network, and use theresulting embedding vectors as inputs to a supervised multi-label classiﬁcation task. We
compare the results with the ones obtained with the other embedding methods describedin the previous section, and we test the sensitivity of our approach with respect to the
choice of parameters and to the number of sampled active nodes D.
4.1 Data sets and dynamical process
We use publicly available data sets describing the face-to-face proximity of individuals with a temporal resolution of 20 seconds [36]. These data sets were collected by
the SocioPatterns collaboration1 and we speciﬁcally use data sets collected in oﬃces(“InVS15”), a hospital (“LH10”), a highschool (“Thiers13”), a conference (“SFHH”) and aschool (“LyonSchool”) [37]. These data correspond to a broad variety of contexts, with activity timelines, group structures and potential correlations between structure and activityof diﬀerent types. The original data are unweighted temporal networks with a 20 seconds
time resolution. For each data set, in order on the one hand to smoothen the short time1

http://www.sociopatterns.org/

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Table 1 Empirical temporal networks. Columns, from leftmost to rightmost: data set name, numberof 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
Name

|V|

|V|

|T|

|E|

1 
e∈E w(e)|E|

1 v∈V |Vv ||V||T|InVS15
LH10
SFHH
Thiers13
LyonSchool

22,4514880
10,81532,546
17,174

217
76
403
327
242

699
342
127
246
104

37,582
14,87034,446
71,724
89,640

4.164
4.448
4.079
5.256
2.806

0.148
0.188
0.211
0.405
0.682

noisy dynamics, and on the other hand to obtain weighted temporal networks, we aggregate the proximity events on successive time windows of length 600 seconds. Whenevermultiple proximity events were registered between two individuals within a time window,
we used the number of such events as the weight of the corresponding temporal edge. We
thus obtain temporal networks in which each time step represents a 10 minutes time window, and the temporal edges presenting at a given time point encode the proximity events
that occurred during the corresponding time window. Table 1 shows some basic statisticsof the resulting temporal networks for each data set.
We simulated the SIR model on each such weighted temporal network, using the following ﬁve combinations of epidemic parameters: (β, μ) = {(0.25, 0.055), (0.13, 0.1), (0.13,
0.055), (0.13, 0.01), (0.01, 0.055)}. 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 thestochastic nature of the model, in some cases the infectious state barely spreads, with alarge 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., |I|T|/2 | ≥ 1).
We select 50 such simulations for each data set and each parameter set. For each selected
simulation, we assign as ground truth label to each active node (i, ti,a ) the state of node i
at time ti,a .
In each case, we select uniformly at random |D| = ρ|V | active nodes, and build our training set using those active nodes and the corresponding active node states. Unless otherwise noted, ρ = 1 (i.e., each node is observed on average once). 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 the 50 realizations of the SIR model,
over ﬁve realizations of the embeddings and over ﬁve realizations of the random choice of
training data, for each data set and parameter values.

4.2 Implementation of the embedding methods
We used publicly available implementations of all embedding methods, namely the implementation of LINE,2 DynamicTriad,3 DynGEM,4 StreamWalk and Online-neighbor by
the original authors .5 As for DeepWalk, we used an implementation of node2vec6 with
p = q = 1. Unless otherwise noted, we conducted experiments with embedding dimension2

https://github.com/tangjianpku/LINE3

https://github.com/luckiezhou/DynamicTriad4

http://www-scf.usc.edu/~nkamra/

5

https://github.com/ferencberes/online-node2vec

6

https://github.com/aditya-grover/node2vec

Sato et al. EPJ Data Science(2021) 10:22

Figure 2 Prediction performance of each method, for each data set and for various β , at ﬁxed μ = 0.055. The
results at ﬁxed β and for various μ are shown as a Supplementary Figure. The top and bottom row give theobtained values of Macro-F1 and Micro-F1, respectively. Error bars give the standard deviation computed over
the realizations of the SIR process and the embeddings

d = 128 and self-coupling edge weight ω = 1. We used the default values of each implementation of the embedding methods, except for the number of iterations of LINE, which
we took equal to the number of samples of DeepWalk. We used Scikit-Learn [38] to implement one-vs-rest logistic regression.

4.3 Results
4.3.1 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, except for
the LH10 data set (the smallest data set, see Table 1). We moreover observe that: (i) for agiven 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.
4.3.2 Epidemic impact and epidemic curves
Figure 3 conﬁrms the results of Fig. 2 from the point of view of the discrepancy measures between predicted and real epidemic curves and from the point of view of the error

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Figure 3 Prediction performance of each method in terms of epidemic impact, for each data set and forvarious β , at ﬁxed μ = 0.055. The results at ﬁxed β and for various μ are shown as a Supplementary Figure.
The top and bottom row show respectively the cumulative discrepancy between the real and predicted
epidemic curves I and the discrepancy between the ﬁnal epidemic sizes size . Error bars give the standard
deviation computed over the realizations of the SIR process and the embeddingsin the predicted ﬁnal epidemic size: I and size are small for most methods, data sets
and parameter values, and in particular the smallest error is most often obtained for the
dyn-supra + DeepWalk method. For this method, Fig. 4 shows moreover scatterplots ofthe predicted vs. real ﬁnal epidemic size, for all data sets: in each plot, epidemic parameters have been varied to yield a large diversity of ﬁnal epidemic sizes, showing that thepredicted and real outcomes are very strongly correlated.
Figures 5 and 6 give a more qualitative illustration of the performance of our method in
the reconstruction of the epidemic curves, highlighting as well the capacity of the method
to recover the timing of epidemic peaks. This is particularly relevant, as heights and timings of peaks in the number of infectious determine the eventual burden on the healthcare
system. Figure 5 ﬁrst shows that the dyn-supra + DeepWalk method recovers well the periods of large and small number of infectious individuals for all data sets and over a wide
range of parameter values. Moreover, Fig. 6 shows that the four methods combining a
supra-adjacency representation with either DeepWalk or LINE yield good results, while
the four other methods strongly underestimate the largest epidemic peak, predicting epidemic curves that spread out the epidemic impact more evenly over the whole timeline,

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Figure 4 Scatterplots of the ﬁnal predicted epidemic size (number of infectious and recovered nodes at theend of the data set), vs. the real one. Each point corresponds to one simulation, and we vary the epidemic
parameters in order to scan a wide range of outcomes. Here we use the dyn-supra + DeepWalk method. The
Pearson correlation coeﬃcients for the predicted vs. real sizes are 0.957 for InVS15, 0.88 for LH10, 0.98 for
SFHH, 0.96 for Thiers13 and 0.95 for LyonSchool

and thus yielding a less accurate information in terms of size and timing of the largest
epidemic burden.
4.3.3 Sensitivity analysis
We now investigate the eﬀect of the hyper-parameters of the supra-adjacency representation (the weight ω of self-coupling edges) and of the embedding (the embedding dimension d). We show in Fig. 7 the results obtained for two performance measures, for the
InVS15 data set and (β, μ) = (0.13, 0.055), but we have conﬁrmed the same tendency forthe other data sets, parameter values and for the Micro-F1 and size measures. The results
show that the performance of dyn-supra + DeepWalk is very stable with respect to changes
in ω. The performance is also stable on a wide range of embedding dimensions, and decreases when it becomes smaller than ≈ 50. Overall, dyn-supra + DeepWalk remains very
eﬀective without the need for ﬁne-tuning ω or d.
Figure 7 also shows the eﬀect of increasing the parameter ρ, i.e., of being able to observe alarger fraction of active nodes. The performance slightly increases with ρ and in particulardyn-supra + DeepWalk consistently yields the best result at all values of ρ.
As mentioned above, we ﬁnally consider two variations if the dyn-supra representation:(i) we regard edges as undirected (dyn-supra-undirected); (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-undirected
and mlayer-supra-decay, respectively. Notice that, in the mlayer-supra method, the supraadjacency edges representing temporal edges are actually not aﬀected by these variations.
We report in Fig. 8 the results for (β, μ) = (0.13, 0.055) and for the DeepWalk embedding,as DeepWalk overall yielded the best results. We checked that the results of Fig. 8 holdsimilarly for the LINE embeddings.

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Figure 5 Example of predicted timelines of the number of active nodes in the infectious state, for several
data sets and (β , μ) parameter values, for the dyn-supra + DeepWalk method. The blue, red and gray lines arerespectively the number of actual active nodes in state I in the test data, the number of predicted active
nodes in state I and the number of active nodes in the test data at time t. Note that the number of active
nodes in the test data is almost the same as the total number of active nodes, as the training data is of small
size (ρ = 1, i.e., |D| = |V|)

Figure 8 indicates that using undirected edges slightly worsens the performance of bothdyn-supra and mlayer-supra methods. Introducing weights that depend 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 directed edges and using
only the weights of the original temporal edges yields the highest prediction performance.

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Figure 6 Example of predicted timelines of the number of active nodes in the infectious state, for the SFHH
(conference) data set for the various methods. Here (β , μ) = (0.13, 0.055) and ρ = 1. Line colors have the same
meaning as in Fig. 5

Figure 7 Results of the hyper-parameter sensitivity on the Macro-F1 and I performance measures for the
InVS15 data set. Bars indicate the standard deviation. Here (β , μ) = (0.13, 0.055)

5 Conclusion
We have introduced a new method to recover the dynamical evolution of a single instanceof a process that has taken place on a known temporal network, from partial observations

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Figure 8 Eﬀect of the variations in the supra-adjacency representation, for the DeepWalk embedding. Here(β , μ) = (0.13, 0.055). Top: eﬀect of using undirected edges. Bottom: eﬀect of introducing weights dependingon time diﬀerence. Bars indicate the standard deviation

and without information on the nature of the process itself except from the set of possible states of the nodes. Our strategy is based on leveraging the ﬁeld of node embedding
techniques and on the introduction of a new method for embedding nodes of temporal
networks aimed at providing low-dimensional feature vectors that are informative of dynamical processes occurring over temporal networks.
Our method ﬁrst maps the temporal network to a modiﬁed supra-adjacency representation, which, despite being a static network, contains all the temporal information of thetemporal network, and in particular 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 andparadigmatic algorithm based on random walks and thus particularly suited to explore theneighborhood 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 stateof all active nodes from a set of observations as a supervised classiﬁcation task.
We have shown the performance of our method on the 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 acrossdata sets and process parameters, without ﬁne-tuning hyper-parameters.
Our results show that it is possible, without any knowledge of the precise nature of theprocess nor of its parameters, to recover crucial information on its outcome, even witha very limited number of observations (for most of our results, each node is observed on
average once). Note in particular that our method assumes no knowledge of which transitions between states actually occur in the real dynamics: this means that the predictedsequence of states of each individual node might yield “forbidden” transitions (e.g., in the

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Sato et al. EPJ Data Science(2021) 10:22

SIR example, transitions from I to S or from R to I). Nevertheless, we have shown thatthe outcome of the classiﬁcation task gives a good estimation of the actual dynamics, as
quantiﬁed both by usual measures of prediction task performance and measures focusing
more on the epidemic burden, such as the cumulative discrepancy between predicted andreal epidemic curves and the diﬀerence between predicted and real ﬁnal epidemic sizes.
We have also shown that the height and timing of the epidemic curve, which in ﬁne determines the period of worst expected burden on the healthcare system during an epidemic,
are also well reproduced in our framework, while the other embedding methods for temporal networks predict a more spread out epidemic over the whole temporal window, withan underestimation of the epidemic peak height.
Our method has 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 thecase. 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 orless detailed) set of statistics of the temporal network is known. Moreover, a related limitation of our investigations concerns our preprocessing of the original temporal network:
indeed, we have chosen to aggregate the network data on consecutive 600 seconds timeintervals, in order to smoothen out short term noisy dynamics. As far as SIR processes areconcerned, this aggregation can arguably have an eﬀect on the epidemic dynamics and on
prediction performance, as the aggregated data has less noise than the raw data, and loses
some information on which paths are time-respecting. It would thus be interesting for
instance to systematically monitor the performance of the procedure as a function of thetemporal aggregation window length. 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, orof a whole snapshot of the system but only at a speciﬁc time. This could yield interesting
insights on how to optimize surveillance strategies in more realistic settings. Another interesting direction for further work would be to consider a dynamical process unfolding
over a streamed temporal network, where information on both the temporal edges and on
the partially observed nodes’ states are available in a streaming fashion. This would yield
more complex prediction tasks based on past information, only.
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.

Funding
This study was partially supported by the Lagrange Project of the ISI Foundation funded by CRT Foundation to CC. It waspartially supported by the ANR project DATAREDUX (ANR-19-CE46-0008-01) to AB.
Availability of data and materials
We have used publicly available data sets.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
AB, CC, MO designed the study. KS implemented the embedding methods, the numerical simulations and analyzed theresults. All authors wrote and approved the ﬁnal version of the manuscript.

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Author details
1
University of Tsukuba, Tsukuba, Japan. 2 Turing Center for Living Systems, Aix Marseille Univ, Université de Toulon, CNRS,
CPT, Marseille, France. 3 Tokyo Institute of Technology, Tokyo, Japan. 4 University of Turin, Turin, Italy. 5 ISI Foundation,
Turin, Italy.

Publisher’s NoteSpringer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations.
Received: 25 August 2020 Accepted: 21 April 2021
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