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Systemic Risk Analysis on Reconstructed Economicand Financial NetworksGiulio Cimini1,*, Tiziano Squartini1, Diego Garlaschelli2, and Andrea Gabrielli1,3

1Istituto dei Sistemi Complessi (ISC-CNR) UoS “Sapienza” Universita di Roma, 00185 Rome, Italy2Lorentz Institute for Theoretical Physics, University of Leiden, 9506 Leiden, Netherlands3IMT Institute for Advanced Studies, 55100 Lucca, Italy* giulio.cimini@roma1.infn.it

ABSTRACT

We address a fundamental problem that is systematically encountered when modeling complex systems: the limitedness ofthe information available. In the case of economic and financial networks, privacy issues severely limit the information thatcan be accessed and, as a consequence, the possibility of correctly estimating the resilience of these systems to events suchas financial shocks, crises and cascade failures. Here we present an innovative method to reconstruct the structure of suchpartially-accessible systems, based on the knowledge of intrinsic node-specific properties and of the number of connectionsof only a limited subset of nodes. This information is used to calibrate an inference procedure based on fundamental conceptsderived from statistical physics, which allows to generate ensembles of directed weighted networks intended to represent thereal system—so that the real network properties can be estimated with their average values within the ensemble. Here wetest the method both on synthetic and empirical networks, focusing on the properties that are commonly used to measuresystemic risk. Indeed, the method shows a remarkable robustness with respect to the limitedness of the information available,thus representing a valuable tool for gaining insights on privacy-protected economic and financial systems.

IntroductionThe estimation of the structural properties of a complex network when the available information on the system is incompleterepresents an unsolved challenge,1,2 yet it brings to many important applications. The most typical case is that of financialnetworks, whose nodes represent financial institutions andedges stand for financial ties (e.g., loans or derivative contracts)—the latter indicating dependencies among the institutionsthemselves, allowing for the propagation of financial distress acrossthe network. The resilience of the system to the default or the distress of one or more institutions considerably dependson thetopology of the whole network;3–5 however, because of confidentiality issues, the information on mutual exposures that regula-tors are able to collect is very limited.6 Systemic risk analysis has been typically pursued by reconstructing the unknown linksof the network using maximum entropy approaches.7–9 These methods are also known as “dense reconstruction” techniquesbecause they assume that the network is fully connected—an hypothesis that represents their strongest limitation. In fact, notonly real networks show a largely heterogeneous distribution of the connectivity, but such a dense reconstruction was shownto lead to systemic risk underestimation.2,9 More refined techniques like “sparse reconstruction” algorithms2 allow to obtaina network with arbitrary heterogeneity, however they stillunderestimate systemic risk because of the homogeneity principleused to assign link weights. A more recent approach10,11 instead uses the limited topological information on the networkto be reconstructed in order to generate an ensemble of graphs using theconfiguration model(CM)12—where, however, theLagrange multipliers that define it are replaced byfitnesses, i.e. known intrinsic node-specific features.13 The average valuesof the observables computed on the CM-induced ensemble are then used as estimates for the real network properties. Thelatter approach overcomes the heterogeneity issue described above, yet it only allows to reconstruct systems in which each tieis undirected and unweighted—thus limiting the analysis tounrealistic and oversimplified configurations. Indeed, link direc-tionality has been shown to play an important role in contagion processes and percolation analysis over these systems14,15 by,e.g., speeding up or confining the infection with respect to the undirected case. Since real economic and financial networksare, by their nature, directed, links directionality has tobe taken into account when assessing their robustness to shock andcrashes. Moreover, the connection weights between the entities of these systems often assume heterogeneous values, whichin turn strongly affect the way such entities react to the default or distress of their interacting partners.4

In order to achieve a realistic and faithful reconstructionof economic and financial networks, here we develop an improvedprocedure that allows to reconstruct links directionality, and at the same time we implement a simple yet effective prescriptionto assign link weights. Our method can thus be employed specifically for systemic risk estimation, by assessing those topo-logical properties that have been shown to play a crucial role in contagion processes and in the propagation of distress over a

network: the k-core structure,16 the percolation threshold,17 the mean shortest path length18 and the DebtRank.4 In particular,we perform an extensive analysis in order to quantify the accuracy of our method with respect to the size of the subset of nodesfor which the topological information is available. Validation of the method is carried out on benchmark synthetic networksgenerated through a fitness-induced CM, as well as on two representative empirical systems, namely the International TradeNetwork (WTW)19 and the (E-mid) Electronic Market for Interbank Deposits.20 In both cases, we have full information onthese systems and we can thus unambiguously assess the accuracy of the method in describing them.

MethodBefore explaining our method in detail, let us introduce some notation. We will deal with weighted directed networks,i.e.,graphs composed by a setV of nodes (with|V|= N) and described by a weighted directed adjacency matrix—whose genericelementwi→ j represents the weight of the connection that runs from nodei to nodej. The incoming total weight orin-strengthfor a generic nodei is then defined bysin

i = ∑ j∈V wj→i , whereas, its outgoing total weightout-strengthreadssouti = ∑ j∈V wi→ j .

It is also convenient to introduce the binary directed adjacency matrix that describes the binary topology:ai→ j = Θ[wi→ j ] (Θis the Heaviside step function:Θ[a] = 1 for a> 0 andΘ[a] = 0 otherwise). This allows to define nodei’s number of incomingconnections orin-degree kini = ∑ j∈V a j→i and number of outgoing connections orout-degree kout

i = ∑ j∈V ai→ j . Finally, thebinary undirected adjacency matrix—whose elements are obtained asai j ≡ a ji = Θ[wi→ j +wj→i ]—is used to define thenumber of incident connections or undirecteddegreeof nodei: ki = ∑ j∈V ai j ≡ ∑ j∈V a ji .

Given these ingredients, our network reconstruction method works as follows. Let us suppose to have incomplete informa-tion about the topology of a given networkG0. In particular, suppose to know the in-degree and out-degree sequenceskin

i i∈I

andkouti i∈I only for a subsetI ⊂V of all nodes (where|I |= n< N). Moreover, suppose to know a pair of intrinsic properties

χii∈V andψii∈V for all the nodes—that will be ourfitnesses(see below). The method then invokes a statistical procedureto find the most probable estimate for the valueX(G0) of a given propertyX computed on the networkG0, compatible withthe aforementioned constraints. We build on two important hypotheses.

I) The networkG0 is drawn from an ensembleΩ induced by a directed CM21—meaning thatΩ is a set of networks that aremaximally random, except for the ensemble averages of the in/out-degrees〈kin

i 〉Ωi∈V and〈kouti 〉Ωi∈V that are constrained

to the observed valueskini i∈V andkout

i i∈V , respectively.12 The directed CM prescribes that the probability distributionoverΩ is defined via a set of Lagrange multipliersxi ,yii∈V (two for each node), whose values can be adjusted in order tosatisfy the equivalence〈kin

i 〉Ω ≡ kini and〈kout

i 〉Ω ≡ kouti , ∀i ∈V.21 The values ofxi andyi are thus induced by the in- and out-

degree of nodei, respectively. The role ofxi,yii∈V in controlling the topology is better clarified by writing explicitly theensemble probability for a directed connection between anytwo nodesi and j:12

pi→ j ≡ 〈ai→ j〉Ω =x jyi

1+ x jyi, (1)

so thatxi (yi) quantifies the ability of nodei to receive incoming (form outgoing) connections.II) The fitnessesχii∈V andψii∈V are assumed to be linearly correlated, respectively, to thein-degree-induced and

out-degree-induced Lagrange multipliersxii∈V andyii∈V through universal (unknown) parametersα andβ : xi ≡√

αχi

andyi ≡√

βψi , ∀i ∈V. Therefore eq. (1) becomes:

pi→ j =

√αχ j

√

β ψi

1+√

αχ j√

β ψi=

zχ jψi

1+ zχ jψi, (2)

where we have definedz≡√

αβ . Such an hypothesis is inspired by the so-calledfitnessor hidden-variablesmodel,13 whichassumes the network topology to be determined by intrinsic properties associated to each node of the network. Note that thisapproach has been already used in the past to model several empirical economic and financial networks,20,22 possibly withinthe CM framework assuming a connection between fitnesses andLagrange multipliers.23

These two hypotheses allow us to map the problem of evaluatingX(G0) into the one of choosing the optimal CM ensembleΩ induced by the fitnessesχii∈V andψii∈V , that is compatible with the constraints onG0—given by the knowledge ofkin

i i∈I andkouti i∈I . Indeed, because of the limited available information, finding the CM of the real system12 is impossible,

and we thus have to impose it by assigningad hocvalues to the Lagrange multipliers—whence the name of “fitness-induced”CM (FiCM). Once the FiCM ensembleΩ is determined (it is univocally defined by the setxi ,yii∈V , and thus by the setχi ,ψii∈V), statistical mechanics of networks prescribes that the quantity X(G0) typically varies in the range〈X〉Ω ±σΩ

X ,where〈X〉Ω andσΩ

X are respectively average and standard deviation of property X estimated overΩ.12 We can thus use〈X〉Ωas a good estimation forX(G0). In practice, since we know the fitness valuesχi,ψii∈V , in order to determine unambiguouslythe ensembleΩ we need to find the most likely value of the proportionality constantz that definesΩ according to eq. (2). This

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can be done using the partial knowledge of the degree sequences to estimate the appropriate value ofz through a maximum-likelihood argument,22 i.e., by comparing, for the nodes in the setI , the average number of incoming and outgoing connectionsin the ensembleΩ with their in-degrees and out-degrees observed inG0:

∑i∈I

[

〈kini 〉Ω + 〈kout

i 〉Ω]

= ∑i∈I

[

kini + kout

i

]

. (3)

In the above expression,〈kini 〉Ω = ∑ j( 6=i) p j→i and〈kout

i 〉Ω = ∑ j( 6=i) pi→ j contain the unknown parameterz through eq. (2), andsinceχi,ψii∈V andkin

i ,kouti i∈I are known, eq. (3) defines an algebraic equation inz, whose solution allows to build the

FiCM ensemble and, at the end, to obtain an estimation ofX(G0)—even with the knowledge of the in- and out- degree of justa single node.

Summing up, the algorithm works as follows. Given a networkG0, two fitness valuesχ andψ for each of theN nodes,and the in-degrees and out-degrees only for a subsetI of |I |= n< N nodes:

• we compute the sum of the in-degrees and out-degrees of the nodes inI and use it together with the fitnessesχi,ψii∈V

to obtain the value ofzby solving eq. (3);

• using such estimatedzand the fitnessesχi ,ψii∈V , we generate the ensembleΩ by placing a directed link from a givennodei to a given nodej with probabilitypi→ j of eq. (2);

• we compute the estimate ofX(G0) as〈X〉Ω ±σΩX in the FiCM ensemble, either analytically or numerically (i.e., by

measuring it on networks drawn fromΩ).

Note that the numerical generation of a sample network fromΩ consists in building a binary directed adjacency matrix, sothat its generic elementai→ j = 1 with probabilitypi→ j (i.e., the existence probability for the link fromi to j given by eq. (2))andai→ j = 0 otherwise. Finally, once the network is generated, in order to obtain a weighted topology we place∀i, j a weightwi→ j on the directed link fromi to j (provided its existence,i.e., ai→ j = 1) according to the following prescription:

wi→ j =χ j ψi

W pi→ jai→ j ≡

1W

(z−1+ χ j ψi)ai→ j , (4)

where the last equality comes from eq. (2). In this expression, the normalizationW represents theinduced total weight ofthe network, defined as the geometric mean of the sum of the fitnesses:W =

√

(∑i χi)(∑i ψi). Indeed,W corresponds to theensemble average of the total network weight:W ≡ ∑i j 〈wi→ j 〉Ω, where〈wi→ j 〉Ω = (χ j ψi)〈ai→ j 〉Ω/(W pi→ j ) = (χ j ψi)/W isthe ensemble average for the weight of the link fromi to j.

Remarkably, thanks to this procedure to assign link weights, the ensemble averages of a nodei’s total in-strength〈sini 〉Ω =

∑ j∈V〈wj→i〉Ω and out-strength〈souti 〉Ω =∑ j∈V〈wi→ j 〉Ω turn out to be directly proportional toχi andψi , respectively and∀i ∈V.

This suggests a natural way for selecting appropriate quantities to play the role of fitnesses in our method: the straightforwardinterpretation forχ and ψ is that of nodes in-strengths and out-strengths observed inthe real networkG0. Indeed, theassumptionχi = sin

i andψi = souti , ∀i ∈V, brings toW = ∑i∈V χi ≡ ∑i∈V ψi , and finally to the important equivalences〈sin

i 〉Ω ≡sini and〈sout

i 〉Ω ≡ souti . This means that we successfully preserve, on average, the strength sequences of the real networkG0

(and thus its total weight). In other words, our network reconstruction method calibrated on in/out strengths is based on anull model constraining the in-degree and out-degree sequence of a subset of nodes, together with the relative in-strength andout-strength sequence (see Figure1).

Empirical DatasetIn order to test our network reconstruction method, we use two representative empirical systems of economic and financialnature. The first one is the international trade network of the World Trade Web (WTW),19 i.e., the network whose nodes arethe countries and links represent trade volumes between them: thus,wi→ j is the monetary flux from countryi to country j (the“amount” of the export fromj to i). The second one is the (E-mid) Electronic Market for Interbank Deposits:20 in this case,the nodes are banks and a linkwi→ j from banki to bank j represents the amount of the loan thati granted toj.

In the following analysis we will use and show results for WTWtrade volume data of year 2000, and E-mid aggregatedtransaction data of year 1999 (both temporal snapshots correspond to the largest size of the network). Analyses for other annualsnapshots are reported in the Supplementary Information, and bring to comparable results. In the light of the discussionat the end of the previous section, we will use as fitnessesχi (ψi) the real node in-strengthsin

i = ∑ j∈V wj→i (out-strengthsouti = ∑ j∈V wi→ j ), i.e., the total import (export) volumes of countries for WTW, andwith the total liquidity borrowed (lent)

by banks for E-mid. Note that the goodness of any choice for the fitness values must be first validated according to hypothesisII of our method (see the first part of section Results).

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(a) (b)

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Figure 1. Conservation of the strength sequences. Scatter plots of node in-strengthssin and out-strengthssout observed forthe real networkG0 and their ensemble averages obtained from eq. (4). Upper panels (a,b) refer to WTW, lower panels (c,d)to E-mid.

Topological PropertiesAs stated in the introduction, we will test our network reconstruction method focusing on the network properties (each playingthe role ofX in the discussion of section Methods) which are commonly regarded as the most significant for describing thenetwork resilience to systemic shocks and crashes. We first consider two properties defined for undirected networks (in orderto reconstruct these properties, we use the undirected version of the method10):

• Degree of the main corekmain and size of the main coreSmain, where ak-core is defined as the “largest connectedsubgraph whose nodes all have at leastk connections” (within this subgraph), and the main core is the k-core with thehighest possible degree (kmain).24 The main core is relevant to our analysis as it consists of themost influential spreaders(of, e.g., an infection or a shock) in a network.16

• Size of the giant componentSGC at the bond percolation thresholdp∗ = k−1 (k is the mean degree of the network),where bond percolation is the process of occupying each linkof the network with probabilityp, andp∗ is the criticalvalue ofp at which a percolation cluster containing a finite fraction of all nodes first occurs.17 Note that the percolationthreshold atp∗ = k−1 (that we take as reference value) is a feature proper of homogeneous graphs in the infinite volumelimit, whereas, for scale-free networks in the same limit itis p∗ → 0. Note also that a bond percolation process can bemapped into a SIR model with infection rateβ and uniform infection timeτ. In fact, by defining the trasmissibilityT = 1− e−τβ as the probability that the infection will be transmitted from an infected node to at least a susceptibleneighbor before recovery takes place, the set of nodes reached by a SIR epidemic outbreak originated from a singlenode is statistically equivalent to the cluster of the bond percolation problem (withp≡ T) the initial node belongs to.25

We then move to properties defined for directed graphs:

• Link reciprocityr, measuring the tendency of node pairs to form mutual connections. It is defined as the ratio betweenthe number of bidirected links and the total number of network connections:r = (∑i j ai→ ja j→i)/(∑i j ai→ j). Reciprocityis considered a sensible parameter for systemic risk, giving a measure of direct mutual exposure among nodes.

• Average shortest path lengthλ ,18 where the shortest path lengthλi→ j from nodei to node j is the minimum numberof links required to connecti to j (following link directions), andλ = N(N− 1)/(∑i 6= j λ−1

i→ j) (the harmonic mean is

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commonly used to avoid problems caused by pairs of nodes thatare not reachable from one to another, and for whichλdiverges). This quantity measures the number of steps that are required, on average, for a signal or a shock to propagatebetween any two nodes of the network.

• The Group DebtRankDR,4 a measure of the total economic value in the network that is potentially affected by a distresson all nodes amounting to 0<Φ < 1, withΦ = 1 meaning default. In a nutshell,DR is based on computing the recursiveimpact (i.e., the reverberation on the network) of the initial distress, and is defined as:

DR= ∑i(h∗i −h0

i )νi (5)

whereh∗i is the final amount of distress oni (h0i = Φ) andνi is the relative economic value ofi. We refer to the original

paper4 for the details on how to computeDR.

Results

Test of FiCM modelingWhen testing our network reconstruction procedure it is important to keep in mind that the method is subject to three differentkind of errors. The first one comes from hypothesis I that the real networkG0 can be properly described by a CM, whoseLagrange multipliers are obtained by constraining the whole in-degree and out-degree sequences.12 The second one derivesinstead from hypothesis II that the node fitnessesχ ,ψi∈V are proportional to the CM’s Lagrange multipliersx,yi∈V , i.e.,from imposing a FiCM. Finally, the third one is due to the limited information available for calibrating the FiCM and obtainthe true value ofz—namely, the partial knowledge of the in-degree and out-degree sequences. Note however that the firstsource of mistakes cannot be controlled for in our context, as finding the CM that describes the data requires the knowledgeof the whole in-degree and out-degree sequences (which is not accessible for our case studies). This is exactly why we haveto make hypothesis II and impose a FiCM by assigningad hocvalues to the Lagrange multipliers. In this section we thusconcentrate on the second source of errors.

Indeed, real networks are not perfect realizations of the FiCM and can only be approximated by it.22 In order to assessqualitatively how well this FiCM describes the real networkG0, one can compare the observed in-degrees and out-degrees ofG0 with their averages〈kin

i 〉Ω and〈kouti 〉Ω computed on the FiCM ensembleΩ. Figure2 shows such comparison when the

average degrees are obtained through eq. (2) for a fully informed FiCM,i.e., with the value ofz computed via eq. (3) usingthe knowledge of in- and out-degrees for all nodes. We indeedobserve a remarkable agreement between these quantities forour empirical networks: the real degrees are scattered around the functional form of their expected values. The amount ofdeviations from perfect correlation (which would correspond to an actual realization of the FiCM) gives an indication of howwell our model describes the real network. Note that the validity of hypothesis II can be evaluated also in the case of partialinformation by performing such comparison on the subsetI of nodes whose topological properties are available.

In the following, in order to have a quantitative global assessments of the errors caused by hypothesis II, we will testour network reconstruction method both on real networks andon benchmark synthetic networks numerically generated withthe fully informed FiCM through eq. (2). In the latter case, the errors made by the method will be dueonly to the limitedinformation available about the degree sequences. It is then interesting to check whether such generated synthetic networksare equivalent to the real networks in term of systemic risk.Figure3 shows that bond percolation properties, shortest pathlength distribution and DebtRank values of synthetic networks are in excellent agreement with those of their real counterparts(the correlation coefficients between real and synthetic curves are all above 0.99). FiCM thus proves itself to be a properframework for modeling our empirical networks.

Test against limited informationIn this section we finally proceed to the key testing of the method against the third (and more relevant) source of errors: thelimitedness of the information available on the degree sequences for calibrating the FiCM. In order to obtain a quantitativeestimation of the method effectiveness in reconstructing atopological propertyX of a given a networkG0 (which can be eitherthe real one or its synthetic version), we implement a procedure consisting in the following operative steps:

• Choose a value ofn< N (the number of nodes for which the in- and out- degrees are known).

• Build a set ofM = 100 subsetsIαMα=1 of n nodes picked at random fromG0.

• For each subsetIα , use the degree sequences fromG0 to evaluatez from eq. (3), and name such valuezα .

• Build the ensembleΩ(zα) using the linking probabilities from eq. (2): generatem= 100 networks fromΩ(zα ), andcompute the average valueXα of propertyX on this ensemble.

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<k>Ω

(a) (b)

(c) (d)

Figure 2. Qualitative assessment for FiCM description of the real network G0. Scatter plots of node fitnessesχ ,ψ versusreal node in- and out-degreeskin,kout of G0 (red circles) and their ensemble averages computed via the FiCM (blueasterisks). Upper panels (a,b) refer to WTW, lower panels (c,d) to E-mid.

0,001 0,01 0,1p

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0 1 2 3 4 5 6λ

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(a) (b) (c)

(d) (e) (f)

Figure 3. Properties of real and synthetic networks. Left plots (a,d): dependence of the size of the giant componentSGC onthe occupation probabilityp (the vertical dotted line indicatesp∗). Central plots (b,e): probability distribution of the directedshortest path lengthλ . Right plots (c,f): dependence ofDR on the initial distressΦ. Top panels (a,b,c) refer to WTW, bottompanels (d,e,f) to E-mid. Correlation coefficient valuesc between real and synthetic curves: (a)c= 0.999, (b)c= 0.999, (c)c= 0.994, (d)c= 0.999, (e)c= 0.989, (f)c= 0.998.

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• Compute the relative root mean square error (rRMSE) of property X over the subsetsIαMα=1:

ρ [X] = rRMSEX ≡√

1M

M

∑α=1

[

XαX(G0)

−1

]2

(6)

whereX(G0) is the value ofX measured onG0.

We then study how the rRMSE for the various network properties we consider varies as a function of the sizen of thesubset of nodes used to calibrate the FiCM (i.e., for which in- and out- degree information is available). Results are shown inFigures4 and5. We observe that in most of the cases there is a rapid decreaseof the relative error as the number of nodesnused to reconstruct the topology increases. For instance, generally the error drops to half of the starting rRMSE (forn= 1) atn/N = 5%, and to one quarter forn/N = 10%—a value that is rather close to that of the final error madeat n≡ N. This isan indication of the goodness of the estimation provided by our method. As expected, the rRMSE is higher for real networksthan for synthetic networks, and the difference between thetwo curves gives a quantitative estimation of the error madeinmodeling real networks with the FiCM. The fact that such a difference is higher for E-mid than for WTW is directly relatedto a slightly better correlation between real and expected degrees observed in the latter case (Figure2). Note that the variousrRMSE for synthetic networks do not necessarily tend to zero, because the generated synthetic configuration might be highlyimprobable—in some cases, the synthetic network can be evenmore atypical than the real one. We thus indicate with errorbars the range of performance of our method for different choices of syntheticG0.

Generally,SGC, λ , kmain andSmain are the properties which are reconstructed better: for instance, with the knowledge ofonly 10% of the nodes, all the relative errors become smallerthan 10%, and they decrease for increasingn. The rRMSEfor r andDR show instead a behavior almost flat inn. The fact that the rRMSE forr computed for real networks remainssteadily high is probably due to the fact that reciprocity ishardly reproduced by a directed CM, and is better suited as additionalimposed constraint.26 The rRMSE forDR is instead remarkably small for real networks (with values around 0.5%), and we canthus conclude that our method is efficient in estimatingDRalso when the available information is minimal. This is particularlyrelevant to our analysis, since we are estimatingDRat its peak (i.e., at its maximum, and thus mostly fluctuating, value), wherethe details of the weighted topology play a fundamental rolein the process of risk propagation. Besides, and more importantly,the value ofDR for the real network is computed using the original weightedtopology, whereas, the computation ofDR in thereconstructed network builds on link weights obtained by the simple prescription of eq. (4).

In conclusion, the outcome of this analysis is that our network reconstruction method is able to estimate the networkproperties related to systemic risk with good approximation, by using the information on the number of connections of arelatively small fraction of nodes—as long as the fitnesses of all nodes is known.

DiscussionIn this paper we studied a novel method that allows to reconstruct a directed weighted network and estimate its topologicalproperties by using only partial information about its connection patterns, as well as two additional intrinsic properties (inter-preted as fitnesses) associated to each node. Tests on empirical networks as well as on synthetic networks generated through afitness-induced configuration model reveals that the methodis highly valuable in overcoming the lack of topological informa-tion that often hinders the estimation of systemic risk in economic and financial systems. Indeed, the information exploitedby the method is minimal but is (or should be) publicly available for these kind of systems.

Our work originates from the study of Musmeciet al.,10 that represented a first attempt in tackling the problem of networkreconstruction from partial information within the framework of fitness-induced configuration models. Here however wemake fundamental improvements to the method, the key advance being that of extending it to directed weighted networks (themost general class of networks). In the present form, the method is then suited to reconstruct high-order network propertiesrelated to systemic risk, a task of primary practical importance the method was conceived to address—that was however farbeyond the reach of its original version. Besides, the validation of the fitness-induced configuration model approach tomodelreal networks, as well as the reconstruction of benchmark synthetic networks generated as fitness-based counterparts of theempirical networks, are both novel ingredients that allow to assess quantitatively the accuracy of the method. Last butnotleast, the extensive analysis of different temporal snapshots of the real networks we provide in the Supplementary Informationallows to strengthens considerably the effectiveness and robustness of our method.

We remark that the method we are proposing here, by reproducing both the binary and weighted topology of the network,represent a substantial step forward in the field of network reconstruction. In fact, most of the previous works6–9,21 focusedon reproducing the strengths of the real network to the detriment of connection patters, whereas, only recently it has beenrealized that a successful reconstruction procedure must resort also on topological constrains.2,10 Here we are proposing amethod that allows to always reproduce the strengths, but also to tune the network topology through appropriate connection

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GC] (

p=p* )

0,01 0,1 1n / N

0

0,05

0,1

0,15

0,2

0,25

ρ[r]

0,01 0,1 1n / N

0

0,02

0,04

0,06

0,08

ρ[λ]

0,01 0,1 1n / N

10-4

10-3

10-2

10-1

ρ[D

Rm

ax]

(d) (f)(e)

(a) (b) (c)

Figure 4. rRMSE of various topological properties versusn for the reconstructedG0 of the WTW (both the real networkand its synthetic version). rRMSE for: (a) degree of the maincorekmain, (b) size of the main coreSmain, (c) size of the giantcomponentSGC at the bond percolation thresholdp∗ = k−1, (d) link reciprocityr, (e) mean shortest path lengthλ , (f)maximum value of the group DebtRankDR.

0,01 0,1 1n / N

0

0,1

0,2

0,3

ρ[km

ain ]

real G0

0,01 0,1 1n / N

0

0,05

0,1

0,15

0,2

ρ[S

mai

n ]

synth. G0

0,01 0,1 1n / N

0

0,05

0,1

0,15

0,2

ρ[S

GC] (

p=p* )

0,01 0,1 1n / N

0

0,05

0,1

0,15

0,2

0,25

ρ[r]

0,01 0,1 1n / N

0

0,05

0,1

0,15

0,2

ρ[λ]

0,01 0,1 1n / N

10-3

10-2

10-1

ρ[D

Rm

ax]

(d)

(c)(a) (b)

(e) (f)

Figure 5. rRMSE of various topological properties versusn for the reconstructedG0 of the E-mid (both the real networkand its synthetic version). rRMSE for: (a) degree of the maincorekmain, (b) size of the main coreSmain, (c) size of the giantcomponentSGC at the bond percolation thresholdp∗ = k−1, (d) link reciprocityr, (e) mean shortest path lengthλ , (f)maximum value of the group DebtRankDR.

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probabilities. In this respect, the use of probabilities derived from degree constraints represent the most general case, whichinclude as specific instances both the dense reconstructiontechniques6–9 (obtained forpi→ j = 1 ∀i 6= j ∈ V) and the sparsereconstruction method2 (roughly obtained forpi→ j = 1−λ ∀i 6= j ∈V).

Note that one should not be much surprised that the knowledgeof a small number of nodes allows to precisely estimate awide range of network properties, because the method assumes the additional knowledge of the fitness parameters for all thenodes. Besides, the effectiveness of the method strongly depends on the accuracy of the fitness model used to calibrate the CMin order to fit the empirical dataset. In the case of WTW and E-mid, the fitness model well describes how links are establishedacross nodes, and our method is thus effective in reconstructing the network properties. Finally, we remark that the issue ofhaving limited information on the system under investigation, while being typical for social, economic and financial systems(that are privacy-protected), is very relevant also for biological systems such as ecological networks, metabolic networks andfunctional brain networks—where, due to observational limitations and high experimental costs for collecting data, detailedtopological information about connections is often missing. Notably, our method can be used to reconstruct any networkrepresenting a set of (directed and weighted) dependenciesamong the constituents of a complex system, and we thus believeit will find wide applicability in the field of complex networks and statistical physics of networks.

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AcknowledgementsThis work was supported by the EU project GROWTHCOM (611272), the Italian PNR project CRISIS-Lab, the EU projectMULTIPLEX (317532) and the Netherlands Organization for Scientific Research (NWO/OCW). DG acknowledges supportfrom the Dutch Econophysics Foundation (Stichting Econophysics, Leiden, the Netherlands) with funds from beneficiaries ofDuyfken Trading Knowledge BV (Amsterdam, the Netherlands). We thank Guido Caldarelli and Stefano Battiston for usefuldiscussion.

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