Revealing latent factors of temporal networks for mesoscale intervention in epidemic spread

1.6 MB·English

by  Laetitia Gauvin

Revealing latent factors of temporal networks for mesoscale
intervention in epidemic spread
L. Gauvin1, A. Panisson1, A. Barrat1,2, and C. Cattuto1
1
5 Data Science Lab, ISI Foundation, Torino, Italy
1 2Aix Marseille Universit´e, Universit´e de Toulon, CNRS, CPT, UMR 7332, 13288 Marseille, France
0
2
January 13, 2015
n
a
J

2 Abstract
1

The customary perspective to reason about epidemic mitigation in temporal networks
]
h hinges on the identification of nodes with specific features or network roles. The ensuing
p individual-based control strategies, however, are difficult to carry out in practice and
- ignore important correlations between topological and temporal patterns. Here we
c
adopt a mesoscopic perspective and present a principled framework to identify collective
o
features at multiple scales and rank their importance for epidemic spread. We use
s
. tensor decomposition techniques to build an additive representation of a temporal
s
c network in terms of mesostructures, such as cohesive clusters and temporally-localized
i mixing patterns. This representation allows to determine the impact of individual
s
y mesostructures on epidemic spread and to assess the effect of targeted interventions
h that remove chosen structures. We illustrate this approach using high-resolution social
p network data on face-to-face interactions in a school and show that our method affords
[ the design of effective mesoscale interventions.

1
Keywords: complex networks, temporal networks, tensor decomposition, epidemic spread,
v
8 high-resolution social networks, human contact networks, targeted interventions
5
Many natural and artificial systems are ade- focusing on heterogeneous distributions of inter-
7
2 quately represented in terms of interaction net- event times (“burstiness”), heterogeneous activity
0 works between their components1. The network distributions, causality constraints, etc.5–16. High-
. representations has led to key insights on the re- resolution social network data17–20, in particular,
1
0 lation between the structure of these systems and have opened the way to richer individual-based
5 the dynamics of diverse processes that take place models of epidemic spread18;21–23 and have raised
1 overthenetwork,suchasinformationdiffusion,epi- new questions on designing control strategies for
:
v demic spread, and much more2;3. In recent years, epidemic processes in temporal networks22;24–26.
i the availability of time-resolved network data has However, micro-interventions that operate on indi-
X
pushed network science beyond the static graph vidual nodes, such as targeted vaccination, are dif-
r
a representation and has prompted new research on ficulttoimplement,bothforlackofhigh-resolution
understanding and modeling time-varying graphs, data in general cases and because action at the
commonlyreferredtoastemporalnetworks4. This level of individuals is subjected to many infor-
has spurred an intense activity on studying how mational and operational constraints. Therefore,
dynamical processes are affected by the temporal it is natural to consider interventions that target
evolution of the network on which they unfold, important collective patterns and structural units
1
of temporal networks. Since network communi- able strategy for epidemic mitigation38 but come
ties, i.e., cohesive clusters, have long been recog- with steep socio-economic costs39;40, thus calling
nized as key structural units in the architecture for more targeted approaches. We show that our
of complex networks27, it is natural to think of approach identifies effective strategies for epidemic
interventions at the community level23;28. Time- mitigation that involve the targeted removal of in-
varying networks, however, can exhibit a richer dividual mesostructures. Remarkably, we find that
range of activity/connectivity patterns than static the most important mesostructures for epidemic
networks. Defining and detecting temporal net- spread correspond neither to the strongest struc-
workstructuresattheintermediate(“meso”)scale, tures nor to cohesive structures that can be easily
as well as understanding their relevance for dy- detected by community detection methods.
namical processes occurring on the networks, re-
mains a largely open question with important ap-
Mesoscale structure of temporal networks
plications. Early work on the mesoscale structure
of complex networks has mostly focused on static Latent factor analysis of temporal networks
networks29 and on community detection in tem-
poral networks30. Recently, latent factor analysis Wefirstsummarizeageneralframeworkthatwein-
techniqueshavebeenusedtosimultaneouslydetect troduced in ref. 31 to expose the mesoscale struc-
topological andtemporal activity patterns of time- ture of a temporal network. The approach com-
varying networks31. bines a tensor representation of a temporal net-
Herewetackletheproblemofdesigninginterven- work41 and a dimensionality reduction technique
tions that selectively target the mesoscopic struc- based on tensor decomposition32.
tureofatemporalnetwork,withthegoalofcontrol- We consider an undirected and unweighted tem-
ling a dynamical process such as epidemic spread. poral network, hence the state of the network at
This requires both detecting mesostructures and time t can be represented by a binary-valued ad-
ranking them by their importance for a specific jacency matrix M(t) ∈ RN×N, where the matrix
dynamical process. To detect mesostructures we entry Mij(t) indicates the status of link i-j at time
build on the work of Ref. 31 and use tensor de- t, and N is the number of nodes of the network.
composition techniques32 to represent a temporal Thematricesdescribingthestateofthenetworkat
network as an additive superposition of mesostruc- differenttimescanbecombinedintoa3-modeten-
tures,eachofwhichcanencodecomplexstructural sor, T ∈ RN×N×L, where the first two dimensions
and temporal correlations. To assess the impor- are the customary node indices of the adjacency
tance of individual mesostructures, we take advan- matrix and the third dimension is a temporal in-
tage of the additivity of the tensor decomposition, dex. L is the number of network snapshots, each
which allows us to dissect the original network and for a different point in time. To detect structures,
reassemblealteredtemporalnetworkswherechosen i.e.,correlatedactivity/connectivitypatternsofthe
mesostructures are selectively removed. By com- temporal network, we set up a latent factor anal-
paringthedynamicsofagivenprocessontheorig- ysis based on non-negative tensor factorization31:
inal network and on the altered ones we can then The central idea is to approximate the tensor T
determinethespecificroleofindividualmesostruc- with a sum T˜ of rank-1 non-negative tensors (see
turesandwecanranktheeffectivenessofinterven- Methods):
tions that target those structures for removal. T (cid:39) T˜ =(cid:88)R S . (1)
We illustrate our approach using high-resolution r
socialnetworkdataonface-to-faceinteractionsina r=1
school setting and investigating the importance of with R < min{N,S}. This decomposition al-
individual mesostructures for simple epidemic pro- lows to describe the original temporal network as
cesses. Schools are actually an interesting context a purely additive superposition of a chosen num-
for models of epidemic spread, as they are thought ber of activity/connectivity patterns, encoded by
toplayanimportantroleinthecommunityspread the individual tensor components. Each compo-
of infectious diseases33–37. Macro-scale interven- nent S is regarded as a mesoscale structure of the
r
tions such as school closure are considered a vi- temporal network and can encode cohesive clus-
2
ters (i.e., communities), temporally-localized mix- The school data are naturally represented as a
ing patterns, and more. Such a parts-based repre- temporalnetwork,wherenodesareindividualsand
sentationisageneralandpowerfulfeatureoftensor links are interactions between pairs of nodes. The
decompositions with non-negativity constraints42 networks snapshots of our dataset are recorded ev-
andgreatlyenhancestheinterpretabilityofindivid- ery20seconds,howevertheschoolactivityissched-
ualcomponents. Asimilarapproachbasedonnon- uled on a coarser temporal scale. Hence, we aggre-
negative matrix factorization has been proposed gate the data over consecutive time intervals of 15
to detect overlapping communities in static net- minutes to obtain unweighted network snapshots:
works43. for every 15-minute interval, a link is drawn be-
TheadditivedecompositionofEq.1alsosuggests tween two nodes if the corresponding individuals
a natural way to investigate the role played by in- had at least one contact during that interval. This
dividual mesoscale structures, both on the overall yields a temporal network represented by binary-
network architecture and on the dynamics of pro- valued valued tensor T with N =241 and L=70.
cesses occurring over the network, e.g., epidemic
spread. In particular, we can assess the impor-
tance of individual mesostructures for dynamical
Following ref. 31, we carry out the non-negative
processes unfolding over the temporal network by
tensor factorization of Eq. 1 and approximate the
selectivelyremovingthosestructures: Wecanstart tensor T with the sum T˜ of R = 14 compo-
from the decomposition of Eq. 1, remove a cho-
nents (see Methods and Supplementary Informa-
sen term (i.e., a chosen structure), and sum all the
tion): theresultsofthedecompositionarereported
other terms, obtaining a modified temporal net-
in Fig. 1. Components S through S and compo-
work where the chosen structure has been selec- 1 9
nent S correspond to the 10 school classes of the
tively erased. We can then simulate a given dy- 11
school. Theycanbevalidatedusingtheknowclass
namicalprocessovertheoriginaltemporalnetwork
attributes for students and teachers, and they can
and over the modified temporal network, and com-
also be detected to some extent using community
pare the dynamics or the outcome of the process,
detection algorithms on the temporally-aggregated
hence learning about the impact of the removed
network31. ThecomponentsS , S , S andS ,
structure. 10 12 13 14
however, mix individuals from different classes and
– contrary to the above class components – they
Case study are only active during lunch breaks. Such compo-
nents cannot be found by community detection of
As a case study we consider an empirical tempo-
the temporally-aggregated networks as they corre-
ral network of human face-to-face interactions in a
spond to temporally-localized mixing patterns of
primaryschool44,measuredbyusingwearablesen-
the temporal network, due to mixing of students
sors19. The dataset describes the interactions (or
atlunchtime. Tensordecomposition,however,can
“contacts”) between 231 children aged 6 to 12 and
naturally detect these patterns as well as the cus-
10 teachers, organized in 10 classes. It comprises
tomary cohesive network communities.
the contacts that occurred at the school premises
during two consecutive days in October 2009, from
8:30am to 5:15pm of each day. We chose this
datasetbecauseitispubliclyavailable23;44andpro- We remark that whereas the original temporal
vides not only a high-resolution temporal network, network T is unweighted, the approximating net-
butalsogroundtruthinformationoncommunities, work T˜ is in general weighted (see Methods). For
namely, class attributes for all nodes. Moreover, anygivennetworksnapshot,correspondingtoa15-
the school schedule is known and can be used to minute time interval, we will interpret the weight
understand the activity patterns that mix multi- of the edge between two nodes as the cumulated
pleclasses,asdoneinRef. 31. Inthefollowingthis time that those nodes spent in contact over that
willallowustorelatetheeffectofremovinganindi- 15-minute interval. In Table 1 we report the frac-
vidual component to specific knowledge about the tionoftotaltensorweightcorrespondingtoeachof
social behavior associated with that component. the R=14 components, (cid:107)S (cid:107)/(cid:107)T˜(cid:107).
r
3
Figure 1: Components obtained by tensor factorization of the school temporal network.
Each row is associated with one component S = a ◦ a ◦ c of the tensor decomposition T˜ in Eq. 5
r r r r
(seeMethods),withR=14. ComponentsareorderedtoptobottomaccordingtotheirEuclideannorm.
Left-hand color-coded matrix: Node weights for each component. Each row corresponds to a vector
a of the decomposition, normalized to unity in Euclidean norm for clarity of visualization. Nodes are
r
orderedtoexposetheblockstructureofthematrixduetoschoolclasses. Right-handplots: Component
activity as a function of time over the 2-day span of the dataset. Each row correspond to a vector c ,
r
normalized to unity in Euclidean norm. Components S to S and component S are school classes,
1 9 11
while components S , S , S and S mix individuals from different classes and are active during
10 12 13 14
lunch breaks.
Mesoscale targeted interventions for epi- Fig. 1) its removal can be regarded as the effect
demic spreading of an intervention strategy that selectively targets
thatbehavior(e.g.,removinglunchbreaksfromthe
Intervention design and evaluation school schedule).
The decomposition of a temporal network into As a first step in exploring mesoscale interven-
a superposition of components, interpreted as tions, here we focus on the case study of the pre-
mesostructures, makes it possible to pinpoint and vious section and investigate the effect of remov-
assess the contribution of individual mesoscopic ing individual mesostructures on the dynamics of
features to the overall structural and functional simpleepidemicprocessesunfoldingovertheschool
propertiesofthetemporalnetwork. Indeed,thead- temporal network. We study how the timing and
ditive decomposition of Eq. 1 allows to selectively size of the epidemic are affected by the removal of
removeachosencomponentsbyexcludingthecor- individual network components, that is, by differ-
responding term, yielding an altered temporal net- enttargetedinterventionsaimedatremovingthem.
work,
We start with the simple case of a susceptible-
T˜s =(cid:88) Sr , (2) infected (SI) process. Each network node can be
r(cid:54)=s in either of two states: susceptible (S) or infected
which can be compared to the original network to (I). A susceptible node in contact with an infected
elucidate the specific role of component s. When onebecomesinfectedwithafixedprobabilityλper
component s can be interpreted in terms of a spe- unit time. Once infected, a node stays indefinitely
cific behavioral pattern (e.g., the lunch breaks of in that state. The system is initialized with all
4
r 1 2 3 4 5 6 7
weight fraction 11.3% 8.6% 8.8% 7.1% 8.3% 6.9% 5.7%
r 8 9 10 11 12 13 14
weight fraction 6.8% 7.5% 7.5% 4.8% 6.8% 5.7% 6%
Table 1: Component weight. Fraction of total tensor weight corresponding to each term S of the
r
decomposition T˜.
nodes in the S state, except for a single infected the epidemics at the end of the process, measured
node(seednode). Thetiminganddurationofcon- as the final number of nodes that are in states I
tacts between nodes is described by the temporal or R. We compute the ratio ρ between the average
network T˜, which is a sequence of weighted net- size Ωr of the epidemic on the altered network T˜r
works: foreachnetworksnapshot, theweightofan and the average size Ω of the epidemic on the full
edge between two nodes gives the total duration of network T˜:
the contacts between those nodes during the cor- (cid:104)Ωr(cid:105)
ρ (r)= . (4)
responding time interval. Although the SI process λ,µ (cid:104)Ω(cid:105)
does not describe any realistic infectious disease, it
The average is computed over all possible seed
represents a paradigmatic dynamical process and
nodes and over different seeding times near the be-
it is frequently used to investigate the structural
ginning of the data set (see SI for details). Note
properties of temporal networks.
that, due to the finite temporal span of our net-
To investigate the effect on epidemic spread of
work data, the epidemic process may not be over
removing a chosen mesostructure S , we simulate
r by the end of the simulation (i.e., some nodes may
the epidemic process on the altered network T˜r,
still be in the I state). In this case Eq. 4 only
and compare its dynamics to that observed on the
yields an estimate of how much the epidemic has
full temporal network T˜. We quantify the effect of
been mitigated during the time interval covered by
removing component r by measuring the epidemic
the data.
delay ratio24
(cid:28)tr−t (cid:29) Targeted interventions: case study
τ = j j , (3)
r t
j Westartfromthedecompositionoftheschooltem-
poral network described in Fig. 1 and study the ef-
where tj is the half-infection time on T˜ when the fect of removing individual mesostructures on the
seed is node j, and trj is the same quantity for the dynamics of an SI process. We consider several
altered temporal network T˜r. The half-infection values of the transmission rate λ: here we present
time is measured from the first time the seed node resultsforλ=0.7andintheSupplementaryInfor-
infects another node, and the average (cid:104)·(cid:105) is com- mation we show that qualitatively similar results
puted over all possible seed nodes j and different hold for other values of the parameter. Figure 2
starting times for the SI process (see Supplemen- shows the delay ratio τ of Eq. 3 obtained by se-
r
tary Information for details). lectivelyremovingeachofthecomponentsS ,with
r
We also study the impact of mesoscale fea- r = 1,... ,14: that is, comparing the dynamics
tures on the dynamics of a more realistic epidemic of the SI process over each of the altered networks
process, the Susceptible-Infected-Recovered (SIR) T˜r with the dynamics observed for the full tem-
model. In this model, S → I transitions occur as poral network T˜. Since the altered networks have
intheSIcasewhenasusceptiblenodeisincontact by definition a smaller total weight (cid:107)T˜r(cid:107) than the
with an infected one, with probability λ per unit full network, we expect epidemic spread to be mit-
time. Infected nodes recover with a constant rate igated for the altered networks. To assess whether
µ, and Recovered (R) nodes no longer take part in agivencomponentr playsastructuralroleforepi-
theepidemicprocess. Toassesstheeffectofremov- demicspreadthatgoesbeyonditsweight,itisthus
ing a given mesostructure r we compare the size of important to compare the delay ratio τ observed
r
5
on removing that component with a suitable null classes and have activity concentrated during the
model. Such a null model is obtained by measur- lunchbreakofthefirstday. Theothercomponents
ingthedelayratioforseveralstochasticrealizations correspondeithertoindividualclasses(components
of altered temporal networks obtained from T˜ by 1 through 9 and 11) or to components, such as 10
removing, at random, a fraction of weight equal to and13,withlittleactivityonthefirstday. Overall,
that of component r (as given by Table 1). The theSIprocessissloweddownthemostbyremoving
distributionsofτ forthenullmodelsareshownby comparativelyweakstructures(intermsofweight)
r
the boxplots of Fig. 2. that mix classes and occur early in the data. We
remarkthatthemosteffectiveinterventionformiti-
gatingepidemicspreadthusinvolvestheremovalof
seeminglyminormesostructuresthatcorrespondto
complexcorrelationsbetweenlinkactivityandnet-
work structure, and cannot be uncovered by stan-
dard community detection approaches.
We now turn to the case of an SIR process and
investigate the effect of mesostructure removal on
the dynamics of an SIR epidemic over the tempo-
ral network. We compute the epidemic size ra-
tio ρ (r) of Eq. 4 for a wide range of values of
λ,µ
the parameters λ and µ, and for all components r.
The results are shown in Fig 3 for selected com-
ponents and in the Supplementary Information for
all the other components. We notice that for large
Figure 2: Epidemic delay ratio as a function λ values and small µ values (bottom right of the
of removed component. The epidemic delay ra- heat maps) the epidemic size ratio ρλ,µ(r) (cid:39) 1:
tioτ ofEq.3isplotted(solidcircles)asafunction in this region the epidemics spreads and finishes
r
of the removed network component and compared fast, hence mesostructure removal cannot mitigate
to the null model described in the main text (box- its size. The parameter region where mesostruc-
plots). In each boxplot, the central band is the ture removal can have a strong effect, therefore,
median, the bottom and top of the box are the is the arc-shaped region visible, e.g., in the heat
first and third quartiles. Whiskers extent to the map for r = 1 (lighter blue/green). The removal
most extreme data point within 1.5 IQR of the in- of class components r = 1 and r = 2 mildly miti-
ner quartiles. Points beyond whiskers are outliers. gates the epidemic for a broad range of parameter
within that region (almost no effect is obtained for
the other class components, as shown in Fig. S4
We observe that the delay ratio obtained by se- of the Supplementary Information). Removing the
lectivelyremovingacomponent(solidcircles)isal- class component 1 has a comparatively higher ef-
most always larger than the one obtained by re- fect because of the higher overall weight of that
moving links at random (boxplots): Such targeted component (Table 1).
interventions have a stronger effect than random On taking a closer look at Fig. 3 we notice that,
removals of links. For the first 9 structures, the depending on the removed component, significant
delay ratio is positively correlated with the com- epidemic mitigation can be achieved in two main
ponent weight of Table 1, as we could naively ex- parameter regions within the arc. The first region
pect. Strikingly, the removal of components 12 fallswithintheblackcontour(highvaluesofλand
or 14 slows down the epidemic spread consider- µ) and corresponds to parameter values yielding
ably. These two mesostructures, despite carrying epidemics that finish by the end of the second day.
asmallertotalweightthanmostothercomponents In this region the epidemic can be strongly mit-
(see Table 1), thus appear to play a more impor- igated (ρ (cid:39) 0.5) by removing mesostructures
λ,µ
tant role for SI spreading. As shown in Fig. 1 both that involve mixing of classes on the first school
components 12 and 14 mix nodes from different day, namely r = 12 and r = 14. This is con-
6
Figure 3: Epidemic size ratio as a function of SIR model parameters. Each heat map corre-
spondstoatargetedinterventionthatselectivelyremovesonecomponent. Foreachremovedcomponent
r the heat map shows the epidemic size ratio ρ (r) as a function of the SIR parameters λ and µ.
λ,µ
ρ (r) = 1 indicates that the intervention does not affect epidemic size. The white area is the region
λ,µ
where the epidemic dies out, i.e., it fails to affect more than 1% of the network nodes. The region inside
the black contour line corresponds to parameter values such that the SIR epidemic finishes within the
finite span of the school dataset (2 days), both for the full and the altered temporal networks. That
is, for those parameter values the epidemic size ratio is not affected by the finite temporal span of the
dataset.
sistent with the results of Fig. 2 for the SI case. gionofinterestliesinthelowerpartofthearc(low
Conversely, removing the class-mixing components valuesofλandµ)andisvisibleinFig.3forr =10
r =10 and r =13, which are mostly active on the and r = 13. This region corresponds to parameter
second day (Fig. 1) has a negligible effect in the values for which the SIR process is slower and the
same parameter region. For a sample parameter epidemic does not finish over the 2-day temporal
choicewithinthisregion,inFig.4weillustratethe span of the data. In this case, removing either of
temporal evolution of the epidemics and the effect the class-mixing mesostructures r = 10,12,13,14
of removing individual components: the epidemic as a similar and limited effect on the epidemic size
on the unmodified network peaks on the first day reached at the end of the second day.
and,consistentlywiththeaboveremarks,removing
r =12 or r =14 strongly mitigates the fraction of
Discussion
affected nodes, whereas the removal of r = 10 and
r = 13 has a negligible effect. We remark that re-
We have put forward a methodology to system-
moving the early class-mixing mesostructures also
atically assess the relevance of mesostructures for
mitigates the SIR epidemic for a wide range of pa-
dynamical processes such as epidemic spread in
rametervaluesoutsidetheblackcontour,asvisible
temporal networks. The proposed method only
in Fig. 3, and that removing these mesostructures
uses time-resolved connectivity information with-
is more effective, in general, than removing class
out metadata and allows to uncover complex pat-
components, even though the latter account for a
ternsofcorrelatedlinkactivitywhoseremovalhasa
larger network weight. The second parameter re-
strong impact on simulated epidemic spreading for
7
a broad range of model parameters. The removal
of an individual mesostructure can be regarded as
a targeted intervention at the mesoscale, such as
theremovalofaspecificgroupofnodesorthesup-
pression of a specific activity pattern (e.g., a lunch
break). These targeted interventions have a po-
tentially high relevance, as they do not involve the
system as a whole, nor they need actions at micro-
scopic level of individual nodes. Hence, they might
strikeabalancebetweencostandeffectivenessthat
might make them actionable in a variety of appli-
cations.
Asanillustrationofthemethod,wehaveconsid-
ered a case study where simple but paradigmatic
spreading processes occur on top of an empirical
temporal network describing face-to-face interac-
tions in a school. In this case, the most impor-
tant mesostructures for epidemic spread are not
thoseinvolvingmostofthelinkactivity,butinstead
consist of weaker, temporally-localized mixing pat-
terns corresponding to scheduled social activities.
Itisimportanttoremarkthatthesemesostructures
cannot by found by means of community detection
techniques on static networks, and that our meth-
ods reveals them in an unsupervised fashion. We
alsonoticethattheactionabilityofanintervention
thatsuppressesagivenmesostructurehingesonthe
recognizability of that mesostructure in terms of
relevantcontextualinformation,suchasaknownor
understandable grouping of nodes, temporal activ-
ity localized at specific times of a known schedule,
Figure 4: Fraction of infected nodes as a
oraspecificrelationtoothermetadataforthecon-
function of time for a sample parameter
textathand. Forexample,theselectiveremovalof
choice. Temporal evolution of the average frac-
component S of Fig. 1 only becomes an action-
12
tion of infected nodes (top) and of the cumulative
abletargetedinterventionwhenitstemporalactiv-
fraction of infected nodes (bottom) as a function
ity profile is identified as a mixing pattern associ-
of time for a sample parameter choice of λ = 0.11
ated with the lunch break on the first school day.
and µ = 0.16. The curve marked as none corre-
sponds to the spread on the unmodified temporal
network T˜. The other curves show the effect of se-
lectively removing components r =1,10,11,12,14.
Our work is a first step towards the systematic
For the chosen parameter values, epidemic spread
design of mesoscale interventions in temporal net-
occurs on the first day, hence removing the early
works. Natural future steps include the use of our
class-mixing components r = 12 or r = 14 has a
methodology on different temporal networks, both
strong mitigation effect.
empiricalandsynthetic,theinvestigationoftheim-
pact of incomplete or noisy data, and the study of
other dynamical processes.
8
Methods To compute the decomposition of Eq. 5 we need
to solve an optimization problem that minimizes
A 3-mode tensor T, with entries Tijk, can be ap- the residual (cid:15). A standard way to do so is to con-
proximatedbyasumT˜ ofRrank-1tensors32;45;46, vertthe3-modeproblemyieldedbyEq.5intothree
coupled 2-mode sub-problems: This is done by un-
R R
T˜ =(cid:88) S =(cid:88) a ◦b ◦c , (5) foldingtheoriginaltensorT alongitsthreemodes,
r r r r
a technique also known as matricization45. Using
r=1 r=1
this representation, the original tensor approxima-
where each component tensor S is the outer prod-
r tion problem is cast into 3 coupled matrix approx-
ucts of three vectors, S = a ◦ b ◦ c . The
r r r r imation problems that involve the factor matrices
Frobenius norm of the difference between T and
T˜, (cid:15) = (cid:107)T − T˜(cid:107), quantifies the error in approx- A, B, and C. Since the corresponding optimiza-
imating the original tensor with the sum T˜. The tion problems are convex with respect to either of
the matrices – but not with respect to them all –
numberofcomponentsRisinterpretedasthenum-
it is possible to solve the optimization through a
ber of sought mesostructures in the decomposition
method known as Alternative Non-negative Least
of the temporal network. Its value needs to be set
Squares48. The minimization of (cid:15) is usually car-
by balancing the conflicting goals of minimizing (cid:15),
ried out with non-negativity and/or sparsity con-
i.e., recovering as much as possible of the original
straintsonthefactormatrices32;45,asthisisknown
tensor, and avoiding overfitting31;32;47. The sets
to yield decompositions that can be interpreted as
of vectors a , b , c can be
{1,2,...,R} {1,2,...,R} {1,2,...,R} parts-based representations of the original data42.
combined into matrices A ∈ RN×R, B ∈ RN×R
andC∈RS×R. ThesematricesallhaveRcolumns, Here we carry out the decomposition with
one for each component of the decomposition, i.e., non-negativity constraints, using Alternating Non-
one for each sought mesoscale structure. negative Least Squares and a block-coordinate de-
Foranundirectedtemporalnetwork(suchasthe scent method to improve convergence speed49;50.
one of our case study) the adjacency matrix repre- Our implementation builds on the MATLAB code
sentedoneachtensorsliceissymmetricandthede- of ref. 51. The number of components R is set
compositionyieldsA(cid:39)B. Inpracticeitispossible by using the core consistency diagnostic31;47 opti-
to directly seek a decomposition with A ≡ B, i.e., mized over 5 stochastic realizations of the decom-
withcomponenttensorsoftheformS =a ◦a ◦c . position that differ for the random initial condi-
r r r r
Hence in the main text we only refer to the matri- tions of the factor matrices. The components S
r
ces A and C: the matrix elements a of A asso- are ordered so that the Euclidean norm of S de-
ir r
ciate each component r =1,...,R with individual creases for increasing r. That is, the first column
nodesofthenetwork,whilethematrixelementsc of matrices A, B and C correspond to the compo-
kr
of C encode the temporal activity pattern of each nent with highest Euclidean norm, and successive
component, with time indexed by k. The decom- columns encode weaker and weaker components.
position of Eq. 5 can be written as
R
(cid:88)
T = a a c . (6)
ijk ir jr kr
r=1
Acknowledgements
Thematrixelementsa indicatenotionofmember-
ir
ship of node i to component r, so this representa-
tion is suitable to describe overlapping mesostruc- The authors acknowledge support from the La-
tures and complex correlations between connectiv- grange Project of the ISI Foundation funded by
itypatternsandactivitypatternsovertime. Notice theCRTFoundationfromtheQ-ARACNEproject
that even when the original temporal network is fundedbytheFondazioneCompagniadiSanPaolo,
unweighted, that is, the tensor T is binary-valued, and from the FET Multiplex Project (EU-FET-
the component tensors S correspond in general to 317532)fundedbytheEuropeanCommission. The
r
weighted networks, and the approximating tensor authors acknowledge help from Marco Quaggiotto
T˜ is in general real-valued. for the design of figure panels.
9
Author Contributions [9] Rocha,L.E.C.&Blondel,V.D.Burstsofver-
tex activation and epidemics in evolving net-
All authors contributed to the manuscript. LG, works. PLoSComputBiol 9,e1002974(2013).
AB and CC designed the study. AB and CC su-
[10] Masuda, N. & Holme, P. Predicting and con-
pervisedthestudy. ABandCCcollectedandpost-
trollinginfectiousdiseaseepidemicsusingtem-
processedthedata. LGandAPanalyzeddataand
poralnetworks.F1000PrimeReports 5(2013).
prepared figures. LG carried out computer simu-
lations. LG, AB, CC wrote the manuscript. All [11] Takaguchi, T., Masuda, N. & Holme,
authors reviewed the manuscript. P. Bursty communication patterns facilitate
spreading in a threshold-based epidemic dy-
Additional Information namics. PLoS ONE 8, e68629 (2013).
[12] Horv´ath, D. X. & Kert´esz, J. Spreading dy-
The authors declare no competing financial inter-
namics on networks: the role of burstiness,
ests.
topology and non-stationarity. New Journal
of Physics 16, 073037 (2014).
References
[13] Holme, P. & Liljeros, F. Birth and death
of links control disease spreading in empirical
[1] Vespignani,A. Modellingdynamicalprocesses
contact networks. Sci. Rep. 4 (2014).
in complex socio-technical systems. Nature
Physics 8, 32–39 (2012). [14] Scholtes, I., Wider, R., N andPfitzner, Garas,
A., Tessone, C. & Schweitzer, F. Causality-
[2] Barrat, A., Barthelemy, M. & Vespignani, driven slow-down and speed-up of diffusion
A. Dynamical processes on complex net- in non-markovian temporal networks. Nat.
works, vol. 1 (Cambridge University Press Comm 5, 5024 (2014).
Cambridge, 2008).
[15] Backlund, V.-P., Saram¨aki, J. & Pan, R. K.
[3] Onnela,J.-P.etal. Structureandtiestrengths Effects of temporal correlations on cascades:
in mobile communication networks. Proceed- Threshold models on temporal networks.
ingsoftheNationalAcademyofSciences 104, Phys. Rev. E 89, 062815 (2014).
7332–7336 (2007).
[16] Perotti,J.I.,Jo,H.-H.,Holme,P.&Saram¨aki,
[4] Holme,P.&Saram¨aki,J. Temporalnetworks. J. Temporal network sparsity and the slowing
Physics Reports 519, 97125 (2012). down of spreading. preprint arXiv:1411.5553
(2014).
[5] Vazquez,A.,R´acz,B.,Luk´acs,A.&Barab´asi,
[17] Eagle,N.&Pentland,A.Realitymining: sens-
A.-L. Impact of non-poissonian activity pat-
ingcomplexsocialsystems.Personalandubiq-
terns on spreading processes. Phys. Rev. Lett.
uitous computing 10, 255–268 (2006).
98, 158702 (2007).
[18] Salath´e, M. et al. A high-resolution human
[6] Karsai, M. et al. Small but slow world: How
contact network for infectious disease trans-
network topology and burstiness slow down
mission. Proceedings of the National Academy
spreading. Phys. Rev. E 83, 025102 (2011).
of Sciences 107, 22020–22025 (2010).
[7] Perra, N., Gon¸calves, B., Pastor-Satorras, R. [19] Cattuto, C. et al. Dynamics of person-to-
& Vespignani, A. Activity driven modeling of person interactions from distributed rfid sen-
time varying networks. Sci. Rep. 2 (2012). sor networks. PLoS ONE 5, e11596 (2010).
[8] Gauvin, L., Panisson, A., Cattuto, C. & Bar- [20] Stopczynski, A. et al. Measuring large-scale
rat, A. Activity clocks: spreading dynamics social networks with high resolution. PLoS
on temporal networks of human contact. Sci- ONE 9,e95978(2014). URLhttp://dx.doi.
entific reports 3 (2013). org/10.1371%2Fjournal.pone.0095978.
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