Network Backboning with Noisy Data

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by  Michele Coscia

Network Backboning with Noisy Data
Michele Coscia Frank M. H. Neffke
Center for International Development Center for International Development
Harvard University Harvard University
Cambridge, MA 02138 Cambridge, MA 02138
Email: michele [email protected] Email: frank [email protected]
7
1 Abstract—Networks are powerful instruments to study com-
0 plex phenomena, but they become hard to analyze in data that
2 contain noise. Network backbones provide a tool to extract the
latent structure from noisy networks by pruning non-salient
n
edges. We describe a new approach to extract such backbones.
a
We assume that edge weights are drawn from a binomial
J
distribution,andestimatetheerror-varianceinedgeweightsusing
5 a Bayesian framework. Our approach uses a more realistic null
2 model for the edge weight creation process than prior work. In
particular, it simultaneously considers the propensity of nodes
] to send and receive connections, whereas previous approaches
h
only considered nodes as emitters of edges. We test our model
p
with real world networks of different types (flows, stocks, co-
-
c occurrences, directed, undirected) and show that our Noise-
o Corrected approach returns backbones that outperform other
s approaches on a number of criteria. Our approach is scalable,
s. able to deal with networks with millions of edges.
c
i
s I. INTRODUCTION
y
h In this paper, we present a new algorithm to extract the
p significant edge backbone from a noisy complex network.
Fig.1. Anexampleofnetworkbackbone.
[
Scientists studying complex phenomena have discovered

1 networks as a powerful tool for their analysis [2]. Examples
v are the discovery of functional modules in networks [11], the tools, such as community discovery, link prediction, or other
6 prediction of missing connections [22] and the modeling of network algorithms.
3
information cascades [7]. Such tools have been applied to
3 Consider Figure 1 as an example. This is a network with
a vast and diverse set of applications, ranging from cultural
7 only 151 nodes, where virtually every possible connection is
analysis [30] to viral marketing [9] and the improvement of
0 expressed in the data. However, the vast majority of these
the efficiency of road networks [43].
. connections are weak and do no represent a significant inter-
1
0 However, these applications face a number of challenges. action between the nodes. In other words, these edges reflect
7 One among the most prevalent ones concerns data quality. noise. The noisy connections are represented as thin black
1 In recent years, there has been an exponential increase in lines. A network backboning algorithm aims at evaluating the
: production and recording of human data. This increase has significance of each edge weight for the two nodes sharing
v
enabled the success of large scale network analysis in the the connection. In the figure, the backbone is represented by
i
X first place. However, the combination of data from disparate the wide red edges that were determined to be statistically
r sources with different measurement techniques and biases significant. Once we prune noisy edges from the network,
a means that connections retrieved from these data are noisy, the underlying structure emerges more clearly, and it can be
yielding networks in which many nodes have a large degree used as input of network analysis algorithms. In this case
such that the underlying structure becomes difficult to parse, we used a community discovery algorithm that attempts to
because all entities connect to each other. retrieve the ground truth classes of the nodes, represented
by the node color. Such task cannot be performed in the
This is the scenario that network backboning algorithms
original network, because the density of connections leads the
address. A network backboning algorithm is an algorithm that
community discovery algorithm to classify all nodes into the
takesasinputadenseandnoisynetworkandreturnsareduced
same giant community.
version of it. The reduction is performed with the aim of
highlightingtheunderlyingstructureofthenetwork,removing As an example, we will consider the task of predicting
spuriousconnections.Thisnetworkbackbonecannowbeused inter-occupational flows of jobs-switchers. A way to predict
to provide a clearer image of the phenomenon the network these flows is to build a network that connects occupations
relates to, that can be studied by standard network analysis that require common skills and tasks. However, certain skills
are so generic that they show up in most occupations, leading threeproperties.First,agoodbackbonemusthaveareasonable
to spurious connections. That is, the likelihood of having network topology. What is “reasonable” is hard to define but,
such skills in common is large, even for random pairs of arguably, it is desirable to prevent that nodes get isolated by
occupations. A good backboning algorithm will drop these dropping the entirety of their connections. NC satisfies the
random connections. The resulting backbone will then be a criterion in all networks by retaining, for a given number of
better predictive tool for our task than the unfiltered network. edges, a high number of nodes under most circumstances.
Second,agoodbackbonehastohighlighttheunderlyingprop-
Current approaches to network backboning can be clas-
ertiesoftheoriginaldata,andtherewithmakemodelingefforts
sified into two categories: generalist and specialized. In the
easier. We show that NC backbones improve the predictive
specialized class, the algorithm focuses on solving a specific
powerofsimplemodelsinournetworkdatasets,moresothan
problem and it optimizes the backbone to fit the particular
any alternative. Finally, a good backbone should be stable,
problem definition. This paper provides a generalist back-
as the underlying phenomenon does not change drastically
boning approach, where the algorithm is agnostic about the
between t and t + 1. The NC backbones have a stability
specific application. However, it is particularly suited for ap-
equivalent to the one of DF backbones. These tests have been
plications in which edge strengths have a count-data structure.
run on networks of countries, connected by: business travel,
Generalist network backboning algorithms can use one trade, similarity in export baskets and so on. We chose this
of two approaches. In the first, the algorithm is defined applicationbecausecountrynetworksbelongtodifferenttypes
structurally: the algorithm trusts that there is no noise in the (flows, stocks, co-occurrences) and have different topological
edge weights and it prunes the edges according to a criterion properties (directed, undirected). Moreover, it is easier to
that lets the latent structure of the network emerge. Prominent evaluate the quality of a backbone since we have reasonable
exponents of this class are the High Salience Skeleton (HSS) priors about which countries should be connected with which
[14] and the Doubly-Stochastic Transformation (DST) [37]. others. Finally, country networks are widely used in a variety
This paper belongs to the second category, where we want of applications, e.g., economic development [17] and aid [11].
to assess the statistical significance of an edge weight to We start our analysis by showing on synthetic networks that
decide whether to prune the edge or not. The state of the the NC backbone recovers the original network in presence of
art methodology in this class is the Disparity Filter [34]. In large amounts of noise better than any alternative.
the Disparity Filter (DF), edge weights are tested against a
The NC backbone implementation is able to scale to
null model of the node emitting them: if an edge weight is
millions of edges in less than two minutes. This shows the
significantcomparedtothetotaloutgoingweightsofthenode,
readinessofourmethodtobedeployedtorealworldproblems.
the edge is kept in the backbone.
The implementation of the NC backbone is freely available
In our approach, which we call Noise-Corrected (NC) as a Python module1. To ensure result reproducibility, we
backbone,weassumeedgeweightsaredrawnfromabinomial also release some of the country networks used in this paper.
distribution. Then, we estimate the probability of observing Pursuant to contractual obligations we are unable to provide a
a weight N connecting nodes i and j using a Bayesian release of the full dataset.
ij
framework. This framework enables us to generate posterior
variances for all edges. This posterior variance allows us to II. RELATEDWORK
create a confidence interval for each edge weight. In practice,
wedropanedgeifitsweightislessthanδ standarddeviations Network backboning is a problem related to Principal
stronger than the expectation, where δ is the only parameter Component Analysis (PCA) [18]. PCA is used to reduce the
of the NC algorithm. However, the confidence intervals the dimensionality of a matrix: to decompose it in a smaller
algorithm produces can also be used more generally, for matrixthatstillrepresentstheoriginaldataaswellaspossible.
instance to determine whether two edges differ significantly A network backbone aims at doing the same for network
from one another in strength. structures. The current network backboning approaches were
recently reviewed in [16]. The classic ways to do network
Our approach improves over DF in several respects. First, backboning are: establishing a naive threshold to prune edges
our null model does not only consider the propensity of the with low weights, calculate the maximum spanning tree, or
originnodetosendconnections,butalsothepropensityofthe applyingk-coredecomposition,wherenodeswithdegreelower
destination node to receive connections. In the DF, links con- than k are recursively removed from the network [33].
necting peripheral nodes to hubs are kept, because periphery-
hubconnectionsalwaysseemstrongfromtheperipheralnode’s We can divide network backboning methods into two
perspective, even though the strong attraction of the hub classes: generalist and specialized. In the generalist class we
makes it likely that such edges form randomly. However, have methods that reduce the dimensionality of the edge set
the much less-anticipated peripheral-peripheral connections to let the underlying structure of the network emerge [15].
between nodes are more likely to be dropped in the DF This is usually useful to simplify “hairballs” where the real
backbone. The NC backbone preserves the latter connections structure(beitcommunitystructureratherthancore-periphery)
attheexpenseoftheformer.Second,wewillarguethatweuse is hidden in noise. Under this category we find the Disparity
a more realistic null model for edge weights than DF. Finally, Filter[34],theDoubly-Stochastictransformation[37],andthe
the NC backbone outperforms the DF backbone in real world High Salience Skeleton [14]. This is the literature we are
applications. addressing here, so these methods will be considered more
in depth in Section III-B.
We provide evidence in favor of the last claim in the
experimentalsection.Webelievethatagoodbackbonesatisfies 1http://www.michelecoscia.com/?page id=287
In the second class, we have methods that were developed B. State of the Art
with a very specific problem definition in mind. Here, the
In this section we present the network backboning algo-
analyst is not interested in the overall general structure of the
rithms with which we compare our proposed methodology
network hidden behind the noise, but in a specific application
more in depth. We focus exclusively on methods for generic
ofanalgorithm.Forinstance,onemethodsparsifiesgraphsfor
backbone extractions, which are used more often in network
the task of inferring the influential nodes in a social network
science. The main purpose of our method, as we describe in
[23], sometimes known as “graph simplification” [6]. In other
SectionIV,isanaccurateestimationoftheamountofnoisein
cases, the backbone is discovered from the activities of nodes,
the edge weights. Once noise is estimated, one can filter out
rather than from the edge weights [8]. Related methods focus
the edges whose weight is not sufficiently different from the
on graph compression, which also reduces the set of nodes
random benchmark, thus obtaining a noise-reduced backbone.
for very large graphs [38]. Graph sampling [21] can be used
for local-first algorithms and also carried out on data streams Facing the problem of filtering connections in dense net-
[1]. Finally, there has been some work on defining backbones works, the simplest approach is to remove all edges whose
in the emerging field of multilayer network analysis [12]. weight is lower than an arbitrary threshold δ. We call this the
Multilayer networks are networks in which there are multiple “Naive”approach.TheNaiveapproachhasseveraldownsides.
edgetypes[4],[19],andbackboneshavetoalsotaketheinter- Inmostreal-worldweightednetworks,weightsarebroadlydis-
layer couplings into account. tributed,locallycorrelatedonedgesincidenttothesamenode,
and nontrivially coupled with topology [34], [3]. This is true
To find the backbone of a network is very often the first alsoforthenetworksweuseinSectionV.Broaddistributions
step of the entire network analysis framework. As such, the imply the lack of a characteristic scale, rendering the choice
applicationscenariosofnetworkbackboningaremany,anddi- of δ meaningless; local correlations imply the possibility of
verse.Wecanfindnetworkbackboningmethodsusedinonline discarding valuable information by filtering out entire parts of
systems [42], cultural analysis [30], microbial co-occurrence thenetwork.Forthesereasons,thenetworksciencecommunity
networks[10],query-dependentgraphsummarization[35],and started developing alternative methodologies.
to boost clustering [32].
One such methodology is the calculation of the so-called
“Maximum Spanning Tree” (MST). A MST is the spanning
treeofaconnected,undirectedgraphwhosetotalweightingfor
III. BACKGROUND itsedgesismaximum.Aspanningtreeisatreeconnectingall
verticesofthegraph.IfT istheMSTofG,(cid:64)T(cid:48)s.t.(cid:80)N∗ >
A. Problem Definition (cid:80)N∗. Maximum spanning trees are usually extractedTw(cid:48)ith
T
the Kruskal algorithm [20]. Given its definition, it is easy to
The fundamental data structure in this paper is a weighted
graph G =(V,E,N), where V is the set of vertices; N ⊆R+ see that the MST satisfies the first condition of our backbone
is the set of non-negative real edge weights; and E is a set of extractionproblemdefinition:itpreservesallnodesandthereis
triples(i,j,n)withi,j ∈V andn∈N.Eachtriplerepresents no way to select the same number of edges with a higher sum
an edge. Let N be the weight of the edge connecting nodes of weights. However, maximum spanning trees have several
ij
i and j. Furthermore, let subscript “.” indicate a summation shortcomings. First, if there are repeated edge weights in the
over the omitted dimension, i.e.: N = (cid:80) N is the total graph, there could be multiple MSTs. Second, by definition,
(cid:80) i. j ij a MST is a tree, which might omit fundamental properties of
outgoing weights of i; N = N is the total incoming
.j(cid:80) j ij real world networks such as transitivity and communities [5].
weights of j; and N = N the total weights in the
.. i,j ij
network. Edges can be directed – meaning (i,j,n)(cid:54)=(j,i,n) An alternative to Naive and MST backboning was de-
–, or undirected – meaning (i,j,n)=(j,i,n). veloped with a two-step algorithm [37]. In the first step the
adjacency matrix of the network is transformed into a doubly-
In this paper, we aim at solving the problem of extracting stochasticmatrixbyalternatinglynormalizingtherowandcol-
the backbone from a dense complex network. A backbone umn values with their respective sums. Then edges are sorted
is informally defined as a subset of nodes and edges of the indescendingnormalizedweight.Edgesareaddedtotheback-
original network that contains the largest possible subset of bonefromstrongesttoweakest,untilthebackbonecontainsall
nodes and the smallest possible subset of salient connections. original nodes in V in a single connected component. We call
Formally: this the “Doubly-Stochastic” (DS) method. The DS method
has three downsides. First, it requires the adjacency matrix to
Problem Definition Given a weighted graph G = (V,E,N), be square, thus excluding bipartite networks. Second, it is not
find a graph G∗ =(V∗,E∗,N∗), such that E∗ ⊂E, and: always possible to transform any arbitrary square matrix into
a doubly-stochastic one [36]. Third, it provides no theoretical
• |I |≈|I |, where I is the set of isolates in graph foundation for dealing with noisy weight estimations.
G∗ G G
G (Coverage); Two of the most used algorithms recently developed in
network science are the Disparity Filter [34] and the High
• f(E∗,V∗) > f(E,V), where f(.) is the fit of a
SalienceSkeleton[14](HSS).TheHSSisdefinedstructurally.
prediction task (Quality);
For each node, [14] calculates the shortest path tree (SPT)
• D(E∗,E∗,)<D(E ,E ,), where G and G are two connecting the node to all other nodes in the network. Then,
1 2 1 2 1 2 the HSS is the linear superposition of all SPTs, i.e. HSS =
independent measurements of G and D(.) a distance
(cid:80)
SPT(v). Also the HSS does not explicitly estimate the
metric (Stability).
v∈V
noiseintheedgeweightestimation,andthusnoisyedgescould
degenerate the final HSS. L = Nˆij
ij E[N ]
The Disparity Filter (DF) determines whether an edge ij
carries a significant share of the total edge weights that enter
The lift tells us how unexpectedly high an edge weight is
or depart from a given node. It does so node by node. The
giventheweightsofiandj.Whenliftequalsoneitmeansthat
significance of an edge is hence determined with respect to
the edge has the expected weight. Values from one to infinity
a single node, as opposed to a node-pair. Edges are first
indicateastrongerthanexpectedconnection,fromonetozero
expressed as shares of a node’s total incoming edge weights.
a weaker than expected connection. As one can see, the lift is
Next, these shares are compared to a null-model, yielding p-
anasymmetricskewedmeasure,whereavalueof0.1isequally
values for how far a share exceeds its null-model’s prediction.
far from one on the left as 10 is from the right. We therefore
The null-model assumes that the distribution of node’s k
transform L , using the following monotonic mapping:
edge shares are generated by a random process where k−1 ij
points are randomly drawn from a uniform distribution. These
points divide the unit interval into k pieces. The length of L˜ = Lij −1 = κNˆij −1 (1)
each piece now represents one of the edge weights. Note that ij Lij +1 κNˆij +1
this is equivalent to assuming that edge weights are drawn
from an exponential distribution. For each edge, the filter
where κ = 1 . The transformation in Eq. 1 ensures
determines how likely it is that the k−1 other edges would E[Nij]
that the lift becomes a symmetric measure centered on zero.
leave sufficient edge-weight-share for the edge to assume the
In our example, .1 becomes equal to −0.81 and 10 becomes
weight it has or more. This likelihood acts as a p-value for
equal to 0.81.
howsignificantlyanedgeweightdeviatesfromthenull-model
prediction. These p-values are then used to prune insignificant We now need to compute the variance of L˜ , which is
ij
edges. In practice, an edge is tested twice to verify whether given by:
its weight is significant for either of the connected nodes
as emitters or receivers. However, the null-model ignores the (cid:34) (cid:35)
(cid:104) (cid:105) κNˆ −1
interaction between source and target node. V L˜ =V ij .
ij κNˆ +1
ij
IV. METHODS
Applying the delta method, we get:
WerefertoourmethodasNoise-Corrected(NC)backbone.
The method is based on [25], which uses the null-model  (cid:16) (cid:17)2
dmeesacsruibreedmehnetreertroordseianlthweitchonattteexntuoaftiroengrbesiassioens athnaatlyasriiss.eWfrhoemn V (cid:104)L˜ij(cid:105)=V[Nˆij]2 (cid:16)κ+NˆijddN(cid:17)ˆκi2j 
comparing to the state of the art in network backboning, the κNˆij +1
NCbackbonemostcloselyrelatestotheDisparityFilter.Asin
theDF,wecompareedgeweightstoanullmodel,butwefor-
with
mulatethisnullmodelatthelevelofnodepairs,notindividual
nodes.TheNCisconstructedinthreesteps.First,wetransform
dκ 1 Nˆ +Nˆ
edge weights such that they are expressed in deviation from = −Nˆ i. .j .
their null-model prediction. Second, we calculate a standard dNˆij Nˆi.Nˆ.j ..(cid:16)Nˆ Nˆ (cid:17)2
deviation for these transformed edge weights. Third, we use i. .j
these standard deviations to construct p-values that are then
used to prune edges. We now have to estimate V[Nˆ ]. Given that we have
ij
interpreted edge weights as the sum of independent unitary
Letusthinkofedgeweightsasthesumofunitaryinterac-
interactions,N followsaBinomialdistributionwithvariance:
tions that occur with constant, edge-specific, probability P . ij
ij
Furthermore, take the total interactions emitted and received
by a node (i.e., a node’s summed incoming or outgoing edge V [N ]=N P (1−P ) (2)
ij .. ij ij
weights) as given. Using “ˆ” to denote observed quantities,
the expected number of interactions in node pair (i,j) can be P is unknown, but can be estimated as the observed
ij
written as: frequency with which interactions occur:
Nˆ
E[Nij]=Nˆi.NNˆˆ.j Pˆij = Nˆi.j..
..
A problem arises when Nˆ = 0. That is, when edge
ij
That is, we assume that each interaction that starts from weights are zero for certain node pairs. Given that many real-
node i finds destination node j with a probability that equals world networks are sparse, this situation is quite common.
theshareoftotalinteractionsinthenetworkreceivedbyj.We For these node pairs, V [N ] = 0, which would suggest that
ij
compare the observed to expected edge weights to calculate measurementerrorisabsentintheseedges.However,inreality,
an edge weight’s lift [40]: there is simply too little information to estimate Pˆ with
ij
swuefifigchitesntarperezceirsoio,nb.uTthailssoafwfehcetsreniontfoornmlyaticoanseiss wsphaerrsee,eid.ge.e, ges 000...111468 123 ges 000...000567 123
wTTohhaeitmnipfso:rocvuesinognothnist,hweeinetesrtiamctaiotensPˆaijmionnagBnaoydeessiaonflforawmdeewgorerke.. Share of Ed 00000 0.....00001.124682 Share of Ed 0000....00001234
Pr[N =n |N =n ,P =p ]=(cid:18) n.. (cid:19)pnij(1−p )n..−nij 0-2.5 -2 -1.5 -1 -0.5 0 0.5 1 0-4-3.5-3-2.5-2-1.5-1-0.5 0 0.5 1
ij ij .. .. ij ij nij ij ij NC Score NC Score
Using Bayes’ law, we get:
Fig.2. TwoexamplesofthresholdsettingfortheCountrySpace(left)and
Pr[P =p |N =n ,N =n ]= Business(right)networks,whichwillbepresentedinSectionV-B.
ij ij .. .. ij ij
Pr[N =n |N =n ,P =p ]Pr[P =p |N =n ]
ij ij .. .. ij ij ij ij .. ..
(cid:82)1Pr[N =n |N =n ,P =q ]Pr[P =q |N =n ]dq 2 4
0 ij ij .. .. ij ij ij ij .. .. ij
(3)
1 5
Choosing a BETA[α,β] distribution, the conjugate prior
of the Binomial distribution, as a prior for P , the posterior 3 6
ij
distribution is also a BETA distribution. In particular, the
posterior distribution of P becomes:
ij
Fig. 3. A toy example of the difference between the Noise-Corrected and
theDisparityFilterbackbones.
P ∼BETA[n +α,n −n +β]. (4)
ij ij .. ij
of P instead of Pˆ to recalculate variances of edge weights
We still have to choose values for α and β that would ij ij
in Eq. 2. Because the posterior expectation of P is always
give plausible prior expectations for the mean and variance ij
strictly larger than zero, variance estimates do not degenerate.
of P . To do so, assume that the total weight of i and j
ij
is given. In other words, think of edge weights as arising At this point we have an estimation of the expected
from a process in which, each time node i increases its total varianceoftheedgeweightL˜ :V [c ].Anedgeiskeptifand
ij ij
weight by one, it draws a node j at random from the pool (cid:114) (cid:104) (cid:105)
of possible nodes. That is, edge weight generation follows only if its observed weight is higher than δ V L˜ij . i.e. it
a hypergeometric distribution. This gives the following prior surpassestheexpectationbyδstandarddeviations.δistheonly
means and variances for Pij: parameter of the NC backbone. It has to be set accordingly
to the tolerance to noise that the particular application can
have. To visualize the effect of δ, consider Figure 2. Here
(cid:20) (cid:21)
N 1 1 N N (cid:114)
E[Pij]=E Nij = N E[Nij]:= N Ni. .j we show the distribution of L˜ij − δ V (cid:104)L˜ij(cid:105) for different
.. .. .. ..
δ values (one to three). Higher δs shift the distribution to the
left.Theverticalblackbaratzeroshowstheboundarybetween
1 1 N N (N −N )(N −N )
V [P ]= V [N ]:= i. .j .. i. .. .j . accepted and rejected edges. The acceptance area lies to the
ij N.2. ij N.2. N.2.(N..−1) right. Since this is roughly equivalent to a one-tailed test of
statisticalsignificance,commonvaluesofδ are1.28,1.64,and
The:=equalityindicateswherewemakeourassumptions. 2.32, which approximate p-values of 0.1, 0.05, and 0.012.
From the BETA[α,β] distribution, we get:
Now that we have formally defined our method, we can
α directly compare it to the Disparity Filter on a toy example.
E[pij]=µ= α+β (5) Consider Figure 3. Here we have a hub connected to five
nodes. Two of them are also connected to each other. The
edges are undirected and their width is proportional to their
αβ
V [p ]=σ2 = (6) weight. The black edges are selected by both the DF and our
ij (α+β)2(α+β+1) NCbackbones.TheDFbackbonefurthermoreselectstheblue
dashed edges connecting the hub (node 1) to the peripheral
Solving for α and β, we get: nodes. From the perspective of the hub these edges should
be dropped, given that they are expected to arise even under
µ2 random edge formation, but when considering the perspective
α= (1−µ)−µ (7) of the other nodes as emitters, the strength of the connections
σ2
tothehubmakesthemhighlyunanticipated.Ontheotherhand,
(cid:32) (cid:33)
(1−µ)2 2Analternativeapproach,implementedinthepythonpackage,istoskipthe
β =µ +1 −1 (8) transformation step described above. p-values follow directly from the null-
σ2
model’sBinomialdistribution,usingN..asthenumberofdrawsand NˆiNˆ.N.2ˆ..j
as probability of success. However, in this way we are not able to estimate
Eqs.4,5,6,7and8nowdefineaposteriorexpectationfor
the standard deviation of the estimate. This makes impossible, for instance,
Pij for each node pair. We can use this posterior expectation todetermineiftwoedgesaresignificantlydifferentfromeachother.
the NC backbone finds the connection between nodes 2 and 1
3 more important than connections to node 1. That is because
connecting to node 1 is not extraordinary, given its propensity 0.8
toconnecttoeverything,andtodispensehighweightededges. y
er 0.6
However,eventhoughtheconnections1-2and1-3arestronger v
o
than2-3,thelatterismoreunanticipated,becausenodes2and ec 0.4
3 tend to have low edge weights in general. Hence, the fact R
0.2
that these weak nodes connect to each other strongly suggests NT MST DS HSS DF NC
that this edge represents a deviation from randomness.
0
0 0.05 0.1 0.15 0.2 0.25 0.3
We publicly release our implementation as the
backboning Python module. The module contains also the Amount of Noise
implementationsoftheDisparityFilter[34],theHighSalience
Fig.4. RecoveryofthetruebackboneofsyntheticBarabasi-Albertnetworks.
Skeleton [14], the Doubly Stochastic transformation [37], as
well as Maximum Spanning Tree and naive thresholding.
ThesearethemethodstowhichwecomparetheNCbackbone
with U(0,η). In practice, a noisy edge can have at most a
in the following experiment section. The module is available
fraction η of the degrees of i and j.
at the link provided in Section I.
For all methods we set the parameters (if any) so that
the backbone will return the same number of edges as the
V. EXPERIMENTS
underlying actual network. Our quality target is the Jaccard
In this section we perform a series of experiments to show coefficient between the set of edges of the original non-noisy
the effectiveness of the Noise-Corrected backbones. We start network and the backbone. It is equal to one if the two edge
bybuildingsyntheticnetworkswithaddednoise,showinghow sets are identical, and to zero if they do not share a single
the NC backbone is able to recover the original edge set in common edge.
presence of high levels of noise (Section V-A). Then, we
Figure 4 reports the results. For very low amount of noise,
focusonreal-worldnetworks.Thedatacomesfromaseriesof
the two best performing solutions are the Disparity Filter
country-country networks presented in Section V-B. We start
and the naive thresholding. However, our Noise-Corrected
byshowingthatouredgeweightvarianceestimationcorrelates
backbone is more resilient to increasing noise with the best
with the actual variance of the edge weight as observed in
overall performance, while also performing very well in low-
the real world, validating our assumed null model (Section
noise environments. As noise increases, there is no significant
V-C). Then, we compare the NC method with the state of the
difference between DF and naive thresholds.
art over three criteria of success. As stated in the problem
definition, a good backbone is a backbone that: (a) produces
thelargestpossiblenodecoverageofthenetwork(Topology–
B. Data
Section V-D); (b) improves the usefulness of the network data
for prediction tasks (Quality – Section V-E); and (c) reduces Our test set includes six networks. In all networks, nodes
topology fluctuations in time (Stability – Section V-F). We are countries and connections are relationships among them,
validate the scalability of our approach in Section V-G. calculated in six different ways. The six networks, in alpha-
betical and discussion order, are as follows.
A. Synthetic Networks
• Business: two countries are connected through the
In this section we test the performance of each method numberofcorporatecreditcardsissuedinonecountry
in recovering the backbone of synthetic networks. In this (origin) that are utilized for expenditures in another
scenario, we have perfect information about which edge is (destination).Thisisadirectedflownetwork,observed
part of the actual network and which edge reflects noise. in the years 2012, 2013 and 2014. Anonymized and
We generate several Barabasi-Albert random networks with aggregateddatafromMastercardsCenterforInclusive
average degree 3 and 200 nodes. We set η as our noise Growth.3.
parameter. Each actual edge in the Barabasi-Albert network
• Country Space:twocountriesareconnectedwiththe
carries the following weight:
number of products they both export in significant
quantities. This is an undirected co-occurrences net-
N =(k +k )×U(η,1), work, observed in the years 2011, 2012 and 2013.
ij i j
Trade data comes from [13]. To determine whether
exports are significant we use the same criterion of
where k is the degree of node i, and U(η,1) is a number
i [17], based on the concept of Revealed Comparative
extracted from a uniform distribution with minimum η and
Advantage.
maximum 1. In practice, we use a fraction of at least η of the
sum of the degrees of the connected nodes. In this way, we • Flight: two countries are connected through the ex-
ensure broad edge weight distributions locally correlated with isting passenger capacity in flights from airports in
the network topology. Then, the complement of the adjacency one country (origin) to another (destination). This is
matrix is filled with noisy edge weights, which are defined
with the same formula, only changing the uniform element 3http://mastercardcenter.org/
Network NCCorr
0
10 Business .590
ght) 10-1 CFloiguhnttrySpace ..662173
ei Migration .064
W 10-2 Ownership .872
e Trade .162
dg 10-3 TABLEI. THECORRELATIONCOEFFICIENTSBETWEENPREDICTED
E ANDOBSERVEDVARIANCEFORTHENCANDDFBACKBONES.
DF( 10-4 CountBryu SsFipnlaiegcshest
C OMwnigerarsthioipn
-5 Trade
10
10-4 10-2 100 102 104 106 108 its nodes. Figure 6 plots the average and variance of neighbor
edge weights against an edge’s own weight. The correlation is
Edge Weight
weakestintheFlightnetwork,butatavalueof.42stillhighly
statistically significant (p< 10−15). The strongest correlation
Fig.5. Cumulativeedgeweightdistributionsforournetworks.
– in the Country Space network – equals to .75.
a directed flow network, observed in the years 2010
and 2014. Proprietary data from OAG4. C. Validation
• Migration: two countries are connected through the Before looking at the quality of the actual backbone, we
number of total migrants from one country (origin) validate our methodology against real world data. The Noise-
(cid:104) (cid:105)
currently living in another (destination). This is a Corrected backbone aims at correctly estimating V L˜ , i.e.
ij
directed stock network, observed in the years 1990, thevarianceofthetransformededgeweights.Sinceweobserve
2000, 2010 and 2013. Data from the UN [39]. all the country networks in several points in time, we can
(cid:104) (cid:105)
• Ownership: two countries are connected through the compareourexpectationofV L˜ij withtheactual(observed)
number of total establishments in a country (destina- variance.
tion) reporting to a global headquarter in a different
country (origin). This is a directed stock network, ob- Table I reports the calculated correlation coefficients. All
served in the years 2008, 2011 and 2014. Proprietary correlation coefficients are significant with p<10−9.
data from Dun & Bradstreet5.
• Trade: two countries are connected through the total D. Topology
dollar value of all products exported by one country
As the first quality criterion, we focus on the topology
(origin)andimportedbyanother(destination).Thisis
of the backbone. In general, backbones ideally isolate as few
a directed flow network, observed in the years 2011,
nodes as possible: each node dropped by the backbone is a
2012and2013.Tradedatahavebeencleanedwiththe
node for which we will not have any result from the network
same procedure outlined in [17].
analysis. Thus, we define the Coverage as the ratio between
Additional edge attribute tables used in prediction tasks non-isolatednodesinthebackboneovernon-isolatednodesin
record: the distance between two countries as the weighted the original network, or:
average distance between all pairs of major cities in these
countries[24];populationdataforallcountriesfromtheWorld
|V|−|I∗|
Development Indicators of the World Bank6. Coverage= G .
|V|−|I |
G
One of the main reasons for using a sophisticated back-
boning method instead of a naive threshold is the broad and
Figure 7 reports the number of nodes preserved in each
locally correlated distribution of edge weights. This has been
backbone as a function of the share of edges that was pre-
observed in many weighted networks [3], and it is also the
served. Note that the Doubly Stochastic method is present
case in the country-country networks used here. We report
only for the networks Country Space, Migration and Trade, as
the cumulative edge weight distributions of all six networks
for the other networks the stochastic transformation was not
in Figure 5. Figure 5 shows that all the studied networks
possible.NotethatDSandMSTdonotrequireanyparameter,
have broad degree distributions (although neither of them is a
so they appear in the plot as a point rather than as a line.
power-law), with the possible exception of the Country Space
network. For instance, in the Ownership network the median Many data points overlap because in many instances all
non-zero edge weight is 1.5, while the top 1% of non-zero methods were able to preserve the entirety of the node set.
edges have weights larger than 50k. The Trade network edge However, it is easy to detect the cases in which a particular
weights span ten orders of magnitude. method was not able to achieve perfect coverage. MST, DS
and HSS achieve perfect coverage by definition (the latter
The edge weights are also locally correlated. We calculate
fails only for very strict parameter choices). There is no clear
the log-log Pearson correlation between the weight of an edge
winnerbetweenNCandDF,asinsomenetworksoneachieves
and the average weight of the edges connected to either of
better coverage than the other, while the converse is true for
4http://www.oag.com/ others. However, the DF is the only method underperforming
5http://www.dnb.com/ the naive method in one case (the Ownership network), which
6http://data.worldbank.org/indicator/SP.POP.TOTL is a critical failure.
ght104 ght102 ght107
ei ei ei
W W W
ge ge ge 106
d d d
or E103 or E or E
hb hb hb105
g g g
ei ei ei
N N N
vg 102 vg 101 vg 104
A 100 101 102 103 104 105 106 A 100 101 102 103 A 101 102 103 104 105 106 107 108
Edge Weight Edge Weight Edge Weight
ght105 ght105 ght104
ei ei ei
W W W
ge 104 ge 104 ge 103
d d d
or E or E or E102
eighb103 eighb103 eighb101
N N N
vg 102 vg 102 vg 100
A 100 101 102 103 104 105 106 107 A 100 101 102 103 104 105 106 107 A 100 101 102 103 104 105 106
Edge Weight Edge Weight Edge Weight
Fig.6. Edgeweightvsaverageneighboringedgeweight.Fromlefttoright:(top)Business,CountrySpace,Flight,(bottom)Migration,OwnershipandTrade.
Noise-Corrected Disparity Filter High-Salience Skeleton Doubly-Stochastic Maximum Spanning Tree Naive Threshold
1 1 1
0.8 0.8 0.8
e e e
ag 0.6 ag 0.6 ag 0.6
er er er
v 0.4 v 0.4 v 0.4
o o o
C C C
0.2 0.2 0.2
0 0 0
1 0.1 0.01 1 0.1 0.01 1 0.1 0.01
Share of Edges in Backbone Share of Edges in Backbone Share of Edges in Backbone
1 1 1
0.8 0.8 0.8
e e e
ag 0.6 ag 0.6 ag 0.6
er er er
v 0.4 v 0.4 v 0.4
o o o
C C C
0.2 0.2 0.2
0 0 0
1 0.1 0.01 1 0.1 0.01 1 0.1 0.01 0.001
Share of Edges in Backbone Share of Edges in Backbone Share of Edges in Backbone
Fig.7. Coverageperbackboneforvaryingthresholdvalues.Fromlefttoright:(top)Business,CountrySpace,Flight;(bottom)Migration,Ownership,Trade.
E. Quality In the second model M , we restrict the observations edges
bb
that are contained in the network backbone. Quality is then
In this section we argue that a good backbone lets the
defined as:
underlying properties of the data emerge from the noisy data.
To prove this point we create a series of models for OLS
R2
regressions. These models all have the same structure: Quality = Mbb ,
R2
Mfull
log(N +1)=βX +(cid:15) .
ij ij ij which is the ratio of the quality prediction (the R2 of the
OLS model) obtained using the backbone to restrict observa-
Here, N is the edge weights of the network, X is a
ij ij tions over the baseline quality that uses all edges. A value of
collection of variables that are supposedly good predictors of
1 means that the two regressions have equivalent predictive
N ,and(cid:15)istheerrorterm.Inpractice,weproposeamodelto
ij power, while a value higher than 1 means we improve over
explain the connection strength between countries. Then, for
the full network.
each backbone methodology we run two regressions. In the
firstregressionM ,weusethecompletesetofobservations. To allow for a fair comparison of different backbone
full
Method Business CountrySpace Flight Migration Ownership Trade
DoublyStochastic n/a 2.0975 n/a 1.5153 n/a 0.9287
NaiveThreshold 0.7766 0.6834 0.5196 1.1616 1.2384 0.3935
DisparityFilter 0.9315 1.4082 0.8569 2.0715 0.5374 0.9024
HighSalienceSkeleton 1.1341 1.6549 0.9447 1.2597 0.9744 0.8662
MaximumSpanningTree 1.1183 1.9180 0.7981 1.0036 0.9288 0.9532
Noise-Corrected 1.1767 2.2437 1.4676 2.1493 1.4165 1.1037
TABLEII. THEIMPROVEMENTINPREDICTIVEPOWERWHENUSINGBACKBONESINOURSIXNETWORKS.
methodologies, we fix the number of edges we include in the 5
10
backbone. We usually choose the number of edges obtained 4 NC
10 DF
with low threshold values for the High Salience Skeleton, 3 DS
10 NT
because it is the most strict backbone methodology in our s 2 MST
d 10
collection, always returning the fewest number of edges. Note n 1
o 10
that this does not apply to the MST and DS backbones, since ec 100
S
they do not have parameters and thus the number of edges -1
10
cannot be tuned. -2
10
-3
One predictor used in each network is geographical dis- 10
4 5 6 7 8
10 10 10 10 10
tance:nearbycountriesareassumedtobemoreconnected.For
almostallnetworks(exceptCountrySpaceandOwnership)we # Edges
also use population-size as a control: since we are recording
flows and stocks, large countries are expected to have larger Fig.9. Runningtimescalability.
connections. Finally, we add the following network-specific
predictors: Business: trade between the countries. Trading
wherecorristheSpearmancorrelationcoefficientbetween
partners are expected to maintain a high intensity of business
thetwovectors.Inprinciple,anydistancemetricisappropriate
travels. Country Space: economic complexity (ECI [17]) of
for this task, but we prefer the nonparamatric nature of the
thetwocountries.Twocountrieswithsimilartechnologylevels
Spearmancoorelation,whichmimicsourtaskofrankingedges
are expected to export similarly advanced products. Flight:
according to their significance. We calculate the correlation
no additional variable. Country size and distance suffice for
using only the edges present in the backbones. This means
a typical gravity model: we expect many travelers between
that a perfectly stable backbone will have a stability of 1,
nearbylargecountries.Migration:culturalvariableslikecom-
while a value of 0 implies that there is no relation between
mon language and common history [24]. More migrants are
thebackbonesoftimetandt+1.Wecalculatestabilityacross
expected if the destination country has the same language
different thresholds.
and customs as the origin country. Ownership: foreign direct
greenfield investments7 (FDIs). Each greenfield establishment Figure8depictstheresults.Thereisnoclearwinnerinthis
inaforeigncountryhasbeencreatedwithFDI,thustotaldollar quality criterion. All backbones are very stable, with stability
investment between the countries should predict the number always exceding .84.
of establishments. Trade: business travels between countries.
Two countries that visit each other for business are expected G. Scalability
to trade frequently with each other.
We implemented the NC backbone using the pandas8
Table II reports the results. We highlight the best perform- Python library. In this section we provide a quantification of
ing methodology in boldface. Note that in some cases the the running time of this naive implementation, showing that it
doubly stochastic transformation was not possible, thus we isreadyforrealworldtasks.WegenerateasetofErdo¨s-Re´nyi
label these instances as “n/a”. In all cases, the NC backbone graphs, with uniform random weights. We set the average
performsbetterthananyotherbackbone.Whatismore,theNC degree of a node to three and we generate networks from 25
backboneisalsotheonlymethodthatalwaysreturnsaquality thousandto6.5millionnodes.Foreachnetworksize,werepeat
valuehigherthanone,meaningthatthebackboneoutperforms the backbone computation ten times and report the average.
the original, unfiltered, network in all cases.
Figure 9 depicts the time complexity scaling in terms of
F. Stability number of edges. Running time increases only slightly super-
linearlywiththenumberofedges.Empirically,weestimatethe
Finally, we are interested in the stability of a backbone. time complexity of our implementation to be ∼ O(|E|1.14).
Because most of our networks should be relatively stable, we The average running time of the algorithm for a network
consider wild year-on-year fluctuations in an edge’s weight containing 20 million edges was 82 seconds on a Xeon(R)
as a sign that the edge weight is imprecisely measured. Our CPU E5-2630 at 2.30GHz.
backbone should contain fewer noisy edges and therefore
be more stable than the orginal network. The stability of a For completeness, we compare with equivalent implemen-
backbone method can be calculated as: tations of the other methods. The NC backbone scales with a
rate indistinguishable from the linear naive thresholding and
DF,whichdifferonlybyaconstantmultiplicativefactor.Other
Stability =corr(Nt,Nt+1),
ij ij methodssuchasHSSandDSaremuchlesstimeefficientand
7https://www.fdimarkets.com/ 8http://pandas.pydata.org/
Noise-Corrected Disparity Filter High-Salience Skeleton Doubly-Stochastic Maximum Spanning Tree Naive Threshold
1 1 1
0.98 0.98 0.98
0.96 0.96 0.96
0.94 0.94 0.94
bility 0 0.9.92 bility 0 0.9.92 bility 0 0.9.92
Sta 0.88 Sta 0.88 Sta 0.88
0.86 0.86 0.86
0.84 0.84 0.84
0.82 0.82 0.82
0.8 0.8 0.8
1 0.1 0.01 1 0.1 0.01 1 0.1 0.01
Share of Edges in Backbone Share of Edges in Backbone Share of Edges in Backbone
1 1 1
0.98 0.98 0.98
0.96 0.96 0.96
0.94 0.94 0.94
bility 0 0.9.92 bility 0 0.9.92 bility 0 0.9.92
Sta 0.88 Sta 0.88 Sta 0.88
0.86 0.86 0.86
0.84 0.84 0.84
0.82 0.82 0.82
0.8 0.8 0.8
1 0.1 0.01 1 0.1 0.01 1 0.1 0.01 0.001
Share of Edges in Backbone Share of Edges in Backbone Share of Edges in Backbone
Fig.8. Stabilityperbackboneforvaryingthresholdvalues.Fromlefttoright:(top)Business,CountrySpace,Flight;(bottom)Migration,Ownership,Trade.
wecouldnotrunthemonnetworkslargerthanafewthousand databaselinksanoccupationtoaskillortaskwithtwoscores:
edges. how important the skill or task is to perform the occupation
andatwhichleveltheskillmustbemastered(expert,medium,
beginner). We keep the occupation-skill association if both
VI. CASESTUDY
scores are higher than the average importance and level of
In this section we provide a more in-depth discussion of the skill or task across all occupations. Next, the similarity
anapplicationofthenetworkbackboningtechniques.Theaim of two occupations is derived from the number of skills
is to show how an analysis task can be improved by filtering the occupations have in common. This yields a weighted
out noisy connections from a network. The topic we study is undirected co-occurrence network.
the estimation of skill relatedness [26]. In [26], the authors
Wecalculatetwobackbonesofthisco-occurrencenetwork.
connected cells in an industry space with the aim of capturing
Figure10depictsthebackboneextractedwithourNCmethod,
similarities in industry skill requirements, assuming that such
while Figure 11 shows the one generated with the Disparity
similarities would be reflected in inter-industry labor flows.
Filter. High Salience Skeleton and Doubly Stochastic are not
Here we explore the validity of this assumption. We reported,becausetheDStransformationwasnotpossible,and
investigate the relationship between a skill-co-occurrence net- theHSScouldnotgeneratethebackboneinareasonabletime.
work and inter-occupational labor flows. Next, we restrict the Each node is an occupation. Nodes are connected if they have
connections, using a backboning technique and show how this a significant number of skills and tasks in common, using
improvesthequalityofpredictions.Thereareafewdifferences the different backboning criteria to filter connections. The two
between this case study and [26] that should be noted. First, networks have roughly the same number of connections. The
we use occupations instead of industries as nodes. Second, in nodes are colored using the first digit of the occupation clas-
[26], the authors use a confidential dataset from Sweden that sification. Node size is proportional to the amount of people
cannotbeshared.Forthisreason,wefollow[41]inusingdata working in an occupation. Edge colors are proportional to the
for the US. The measurement of skill relatedness is based on significance of the link according to the backbone technique,
O*Net data [28]. O*Net records the use of hundreds of skills from black (very significant) to gray (least significant).
and tasks in hundreds of occupations [29]. The labor flows
between occupations in the US are derived from the Census NotehowtheDFbackboneinFigure11looks,andindeed
Bureau’sCurrentPopulationSurvey9.Bothdatasetsarepublic is,moredense.ThereasonisthattheDFdroppedmorenodes
and shared at the URL provided in Section I. than NC, around fifty – a significant part of the network. This
is further proof that NC methodology extracts backbones of a
For each pair of occupations we have the number of higher quality topology.
workers who changed jobs from one occupation in 2009 to
anotheroccupationin2010.Jobswitcherswhodidnotchange From the figures we can see that the NC backbone is
occupationarecountedinthediagonalofthematrix.Theskill able to capture an underlying structure in the data. The DF
relatednessbetweenoccupationsisassessedasfollows.O*Net backbone has two dishomogeneous clusters, where seemingly
providesestimatesoftherelevanceofeachskillandtasktoan connectionsareestablishedbetweenalmostanypairofoccupa-
occupationusingbothdomainexpertsandworkersurveys.The tions. In the NC backbone, we can clearly identify a number
of potential modules, which seem to be correlated with the
9http://www.census.gov/programs-surveys/cps.html occupation classification.