Analytical approach to network inference: Investigating degree distribution
aa r X i v : . [ phy s i c s . d a t a - a n ] J u l Analytical approach to network inference: Investigating degree distribution
Gloria Cecchini ∗ Institute for Complex Systems and Mathematical Biology, University of Aberdeen,Meston Building, Meston Walk, Aberdeen, AB24 3UE, United Kingdom andInstitute of Physics and Astronomy, University of Potsdam, Campus Golm,Karl-Liebknecht-Straße 24/25, 14476, Potsdam-Golm, Germany
Bj¨orn Schelter
Institute for Complex Systems and Mathematical Biology, University of Aberdeen,Meston Building, Meston Walk, Aberdeen, AB24 3UE, United Kingdom (Dated: July 18, 2018)When the network is reconstructed, two types of errors can occur: false positive and false negativeerrors about the presence or absence of links. In this paper, the influence of these two errors on thevertex degree distribution is analytically analysed. Moreover, an analytic formula of the density ofthe biased vertex degree distribution is found. In the inverse problem, we find a reliable procedureto reconstruct analytically the density of the vertex degree distribution of any network based onthe inferred network and estimates for the false positive and false negative errors based on, e.g.,simulation studies.
I. INTRODUCTION
Networks are one of the most frequently used modellingparadigms for dynamical systems. Investigations towardssynchronization phenomena in networks of coupled oscil-lators have attracted considerable attention, and so hasthe analysis of chaotic behaviour and corresponding phe-nomena in networks of dynamical systems to name justa few [1–4]. Understanding and characterizing networkbehaviour has triggered interest in a vast number of dis-ciplines, ranging from optimizing vaccination strategies[5] to understanding the functioning or malfunctioningof the human brain [6–8].While first principle modelling is feasible in some ar-eas, in others, networks need to be inferred, e.g. fromobserved data, see, e.g., [9–12]. This comes with cer-tain challenges ranging from selecting the appropriatenodes or even defining them, to the choice of an appropri-ate technique to infer the interaction between the nodes.These choices have a strong impact on the resulting net-work. Here, we discuss another related challenge thatoriginates from the fact that network inference in the
In-verse Problem typically relies on statistical methods andselection criteria.A typical network inference procedure, estimates theconnectivity between a-priori specified nodes in a net-work. If the connectivity measure passes a certain thresh-old, a link between the corresponding nodes is assumedto be present. The choice of this threshold is arbitrary,but it is intuitively clear that there is a strong corre-lation between number of links and choice of threshold.Selecting the threshold, not only controls how many linksare inferred correctly but also establishes the number ofincorrectly determined links. There are two types of er-rors, (i) a link may be erroneously considered present, ∗ Corresponding author; [email protected] this false positive conclusion is referred to as a type I er-ror ; (ii) a present link may remain undetected, this falsenegative conclusion is referred to as a type II error .In this manuscript, we present an analytical frame-work that on the network level links the reconstructednetwork structure contaminated by type I and type IIerrors to the true underlying one. While the frameworkis rather general, we used the vertex degree distributionto derive the functional relationship between the recon-structed and true underlying network. This enables us toobtain superior estimates for the vertex degree distribu-tion, a key property of a network [13]. It has been shownthat including the vertex degrees into stochastic block-models improves their performance for statistical infer-ence of group structure [11]. The functional relationshipdepends on the choice of type I error , type II error andthe dimension of the network. We demonstrate the per-formance of our novel approach in a simulation study.The manuscript is structured as follows. In Section IIa theoretical analysis of our method is presented. Sec-tion III shows some cases where the method presented inSection II is applied. II. MATERIALS AND METHODS
In section II A, a short introduction to networks is pre-sented; we analyse the influence of type I and type II er-rors on the network structure, i.e. false positive and falsenegative conclusions about links. In section II B differentmethods to solve the
Inverse Problem are presented. Sec-tion II C contains a brief description of the generalizationto directed networks.
A. Networks change
A network is defined as a set of vertices (or nodes) withlinks (or edges) between them. To quantify the struc-ture of networks, different characteristics have been in-troduced [14]. Here, we consider two key network charac-teristics: vertex degree distribution and number of edges.The vertex degree describes the number of links of a node;if the vertex v has k edges attached, its vertex degree is d = k . The vertex degree distribution is an importantproperty of the entire network.Networks can be directed or undirected [13]. In anundirected network, connection of v to v implies theconnection of v to v . In a directed network, this sym-metry is broken, therefore if a path from v to v exists, apath from v to v does not necessarily exist. In this sec-tion, we consider undirected networks. Later [Sec. II C]a generalization to directed networks is presented.Consider a network G with n nodes and vertex degreedistribution defined by the probability function P , i.e., P i = P ( d = i ) is the probability that the degree d is i ,for i = 0 , · · · , n −
1. Note that the degree of a vertex isbetween 0 and n −
1, since each vertex can be connectedto at most n − type I and type II errors on the vertex degree distribution of a given network G .We call G ′ the network detected when type I and type IIerrors occur and we assume that α is the probability of a type I error and that β is the probability of a type IIerror . Therefore, α expresses the probability that a linkabsent in G is present in G ′ and β is the probability that alink present in G is no longer present in G ′ . Hence, the setof edges of G ′ is a combination of true positive links and false positive links of G . The vertex degree distributionof G ′ is characterised by the probability function P ′ .Consider a vertex and assume it has degree k , thereforethere are k links connected to it and n − − k absent links.We evaluate the probability that this vertex has vertexdegree k ′ in G ′ . The vertex degree k ′ = j + i (1)is given by the sum of true positive links j and falsepositive links i ; additionally, i and j have to satisfy j ≤ k and (2a) i ≤ n − − k. (2b)The condition described by Eq. (2a) guarantees that thenumber of false negative links is larger or equal than zero,and smaller or equal than the number of the original truepositive links, i.e., 0 ≤ k − j ≤ k . Likewise, the number offalse positive links must be non-negative and smaller orequal than the number of the original non-present links,Eq. (2b).The probability that a vertex has degree k ′ in G ′ ,knowing it has degree k in G is P ( d ′ = k ′ | d = k ) = k ′ X i =0 (cid:18) kk ′ − i (cid:19) (1 − β ) k ′ − i β k − k ′ + i (cid:18) n − − ki (cid:19) α i (1 − α ) n − − k − i if k ′ ≤ k and k ′ ≤ n − − k k X i =0 (cid:18) ki (cid:19) (1 − β ) i β k − i (cid:18) n − − kk ′ − i (cid:19) α k ′ − i (1 − α ) n − − k − k ′ + i if k < k ′ ≤ n − − k n − − k ′ X i =0 (cid:18) kk − i (cid:19) (1 − β ) k − i β i (cid:18) n − − kk ′ − k + i (cid:19) α k ′ − k + i (1 − α ) n − − k ′ − i if k ′ ≥ k and k ′ > n − − k n − − k X i =0 (cid:18) kk ′ − i (cid:19) (1 − β ) k ′ − i β k − k ′ + i (cid:18) n − − ki (cid:19) α i (1 − α ) n − − k − i if n − − k < k ′ < k . (3)The probability P ( d ′ = k ′ | d = k ) is a piecewise func-tion for all combinations of i and j satisfying Eqs. (1)and (2). To obtain Eq. (3) we consider, as an example,the first case, i.e., k ′ ≤ k and k ′ ≤ n − − k .The probability of having j true positive links , over all possible k original true positive links, is P ( j ) = (cid:18) kj (cid:19) (1 − β ) j β k − j , (4)which is the binomial distribution B ( k, − β ). Since j = k ′ − i [Eq. (1)], Eq. 4 corresponds to the first part of thefirst case of Eq. 3. Similarly, the probability of having i false positive links is P ( i ) = (cid:18) n − − ki (cid:19) α i (1 − α ) n − − k − i , (5)which is the binomial distribution B ( n − − k, α ).Combining Eqs. (4) and (5), changing variable j ac-cording to Eq. (1), and considering all possible combina-tions of i and j , we obtain the first case of Eq. (3). Allthe other cases can be derived in the same way followingthe conditions in Eq. (2).Applying the law of total probability P ( d ′ = k ′ ) = n − X k =0 P ( d ′ = k ′ | d = k ) P ( d = k ) ∀ k ′ ∈ { , · · · , n − } (6)we obtain the matrix equation " P ( d ′ =0) ... P ( d ′ = n − = " P ( d ′ =0 | d =0) ··· P ( d ′ =0 | d = n − ... ... ... P ( d ′ = n − | d =0) ··· P ( d ′ = n − | d = n − = A · " P ( d =0) ... P ( d = n − = P , i.e., P ′ = A P . (7)The matrix A = A ( n, α, β ) depends on n , α and β andhas determinantdet A = (1 − α − β ) n ( n − , (8)therefore it is invertible if and only if α = 1 − β , seeAppendix A for a proof.Assuming G is known, Eq. (7) characterises the influ-enced of type I and type II errors on the vertex degreedistribution, and it allows to find the vertex degree dis-tribution of the network G ′ . B. Inference of networks’ vertex degreedistribution
Section II A analyses the impact of type I and typeII errors on the vertex degree distribution of a givennetwork. Equation (7) allows to obtain P ′ from P . Inthis section, we are interested in the inverse problem,i.e., inverting Eq. (7), to infer the original vertex de-gree distribution from an observed one. When { α, β } 6 = { , } , { , } , since the convergence to zero of the deter-minant of A scales like x n ( n − for | x | < n when inverting the matrix A to find P through P = A − P ′ . The cases α, β = 0 and α, β = 1 are trivial, see Appendix A.The least squares method is a standard approach tosolve problems like Eq. (7). Although the matrix A isnot singular, for reasonable parameter values for n , A istypically ill-conditioned, therefore the pseudoinverse ofthe truncated singular value decomposition of A is used.The singular value decomposition of a matrix A isthe factorization of the matrix into the product of A = U W V T where W is a diagonal matrix and the columnsof the matrices U and V are orthonormal [15]. Theelements w , · · · , w n on the diagonal of W are called singular values of A and they are ordered such that w ≥ w ≥ · · · ≥ w r > w r +1 = · · · = w n = 0, where r is the rank of A .The singular value decomposition is a tool to computethe pseudoinverse of a matrix. If A has singular valuedecomposition A = U W V T , its pseudoinverse A + is de-fined as A + = V W + U T , where W + is obtained from W replacing all the non-zero elements with their reciprocals.The truncated singular value decomposition is amethod for regularization of ill-posed least squares prob-lems [16]. Once the singular value decomposition A = U W V T is found, the matrix W is truncated at, e.g., rank t such that only the first t singular values are considered;this matrix is usually called W t . More precisely, W t is adiagonal matrix with elements w ≥ w ≥ · · · ≥ w t >w t +1 = · · · = w n = 0, with t < r . The truncated diag-onal matrix W t is used to find an approximation of thematrix A using its decomposition, i.e., A t = U W t V T .The optimal value for t has been studied in [17, 18]. Thematrix A t is the closest approximation of A of rank t ,[16]. Using W t , we calculate the pseudoinverse of A t ,i.e., A + t = V W + t U T , and we solve Eq. (7) resulting in P = A + t P ′ . (9) C. Generalization for directed networks
For directed networks the vertex degree is charac-terised by the vertex in-degree and the vertex out-degree ,[13]. Usually, in a directed network the vertex degree isthe sum of the vertex in-degree and the vertex out-degree.Both the in-degree and the out-degree of a vertex arenumbers between 0 and n −
1, if n is the number of ver-tices of the network. Therefore, the analysis shown inSection II A and II B remains valid if either the vertexin-degree or the vertex out-degree are considered insteadof the vertex degree.An undirected network with n nodes has at most n ( n − / n nodes has at most n ( n − III. RESULTS AND DISCUSSIONS
To demonstrate the abilities as well as limitations, theanalysis presented in Section II is applied to some typ-ical simulated networks. We like to highlight that ourapproach is derived analytically; simulation studies arepredominantly needed to demonstrate its applicability inreal-world examples and to check for numerical issues,etc. There might be practical issues, e.g., due to the di-mension of the network, and with the aim to show howthese challenges can be overcome we present a simulationstudy to explore the concrete applicability of our method.We study 5 network topologies that present differentcharacteristics so to have a spectrum of networks as wideas possible to which we apply our analysis. Namely, weconsider Erd˝os-R´enyi (also called random), Small-World,Scale-Free networks, a three-dimensional grid, and a net-work of randomly connected communities, [13]. We varythe probabilities α and β of type I and type II errors inthe range 1% −
10% mimicking a typical analysis methodthat has high sensitivity and high specificity. Neverthe-less, both lower and higher values for α and β can be cho-sen and the results obtained are qualitatively the sameas the ones presented below.Consider a random network G with 100 nodes and aprobability of a connection of 0 .
2. The vertex degree hasbinomial distribution B (100 , . α = 0 .
05 and β = 0 .
03 resultsin a new network G ′ . The vertex degree distributionof G ′ is calculated empirically by counting the vertices’degrees. Applying the procedure explained above, thevertex degree distribution of the original network is esti-mated. Figure 1 shows the results using the cut-off forthe truncated singular value decomposition method of0 .
5, i.e., W t contains only singular values greater than0 .
5. The choice of t is motivated by smoothness and reg-ularity of the solution obtained.Figure 1 shows the histogram of the degrees of thevertices of the original network G , the density of the de-tected network G ′ , the reconstructed vertex degree distri-bution of the original network P resulting from Eq. (9),and the result when a non-negative constraint is appliedto the truncated singular value decomposition to avoidthat numerical issues result in negative solutions. Moreprecisely, we use lsqnonlin Matlab function with lowerbound condition lb = zeros ( n ); this function implementsthe trust region reflective algorithm [19, 20]. The densityof G ′ is estimated by P ′ i = number of nodes with vertex degree = i number of nodesits empirical distribution, and this is used to infer theoriginal network.Figure 2 shows the result when the original network G is a Small-World network. It is built from the regularnetwork of 100 nodes, vertex degree 4, and probability ofrewiring 0 .
4. The network G ′ is obtained by adding andremoving links at random with probabilities α = 0 . FIG. 1. Density histogram of the original vertex degrees,blue bars, detected density vertex degree distribution, graydotted line, the result of network reconstruction using Eq. (9)knowing A and P ′ , solid red line, and the result when a non-negative constraint is applied to the truncated singular valuedecomposition, black dashed line. The original network is arandom network with 100 nodes and probability of connection0 . and β = 0 .
05 respectively. The cut-off for the truncatedsingular value decomposition method is 0 . G , the density of the detectednetwork G ′ , the reconstructed vertex degree distribution,and the result when a non-negative constraint is appliedto the truncated singular value decomposition.Figure 3 shows the result when the original network G is a Scale-Free network. It is built using a preferentialattachment model for network growth. At each step avertex, with a link attached to it, is added. The prob-ability that the new vertex attaches to a given old oneis proportional to its vertex degree. This procedure isrepeated until the network has 100 nodes. The network G ′ is obtained by adding and removing links at randomwith probabilities α = 0 . β = 0 .
03 respectively.The cut-off for the truncated singular value decomposi-tion method is 0 . G , the density of the de-tected network G ′ , the solution of Eq. (9), and the resultwhen a non-negative constraint is applied to the trun-cated singular value decomposition.Another example we apply our method to, is when theoriginal network G is a three-dimensional grid 4 × × G has 100 nodes. The network G ′ is obtainedby adding and removing links at random with probabil-ities α = 0 . β = 0 .
05 respectively. The cut-offfor the truncated singular value decomposition methodis 0 .
38. Figure 4 shows the histogram of the degrees of
FIG. 2. Density histogram of the original vertex degrees,blue bars, detected density vertex degree distribution, graydotted line, the result of network reconstruction using Eq. (9)knowing A and P ′ , solid red line, and the result when a non-negative constraint is applied to the truncated singular valuedecomposition, black dashed line. The original network isa Small World network with 100 nodes and probability ofrewiring 0 . A and P ′ , solid red line, and the result when a non-negative constraint is applied to the truncated singular valuedecomposition, black dashed line. The original network is aScale-Free network with 100 nodes. the vertices of the original network G , the density of thedetected network G ′ , the solution of Eq. (9), and theresult when a non-negative constraint is applied to thetruncated singular value decomposition. FIG. 4. Density histogram of the original vertex degrees,blue bars, detected density vertex degree distribution, graydotted line, the result of network reconstruction using Eq.(9)knowing A and P ′ , solid red line, and the result when a non-negative constraint is applied to the truncated singular valuedecomposition, black dashed line. The original network is a4 × × Finally, we investigate the case when G is a networkof three randomly connected communities. It is builtby constructing three Erd˝os-R´enyi networks, with prob-ability of connection 0 .
3, 0 .
6, and 0 .
9, and each with 33nodes. Then, nodes from different communities are con-nected with probability 0 .
1. The network G ′ is obtainedby adding and removing links at random with probabil-ities α = 0 .
05 and β = 0 .
03 respectively. The cut-offfor the truncated singular value decomposition methodis 0 .
42. Figure 5 shows the histogram of the degrees ofthe vertices of the original network G , the density of thedetected network G ′ , the solution of Eq. (9), and theresult when a non-negative constraint is applied to thetruncated singular value decomposition.Another interesting aspect is the influence of type I and type II errors and the proposed method on the re-construction of individual nodes and not just the correctdistribution. This is particularly relevant for nodes thathave a degree much higher than average, so-called hubs.In the Scale-Free example, Fig. 3, the detected dis-tribution appears to be smoother than the original, im-plying that a hub might have been converted to a non-hub. Analysing this in more detail, there is convincingevidence that this is not the case - hubs are correctlyidentified as hubs.Consider a node d that has degree k in G that has n nodes. Due to type I and type II errors , this nodein G ′ has degree d ′ , a random variable with distributionshown in Eq. (3). Taking realisations of this random vari-able, and inverting the process using Eq. (9), allows us tocompare individual degrees for a given node of the true FIG. 5. Density histogram of the original vertex degrees,blue bars, detected density vertex degree distribution, graydotted line, the result of network reconstruction using Eq.(9)knowing A and P ′ , solid red line, and the result when a non-negative constraint is applied to the truncated singular valuedecomposition, black dashed line. A network of three ran-domly connected communities is used as original network. network with the reconstructed one. We consider a net-work with n = 100 nodes, a node d with degree k = 75,probabilities of type I and type II errors of α = 0 .
05 and β = 0 .
03, respectively, and we simulate 100 realisationsof the random variable described above. Figure 6 showsthe reconstruction of the degree of d using these reali-sations. The result does not only show an improvementfrom the detected degrees k , but also illustrates the highaccuracy of the reconstruction method.Figure 7 shows the reconstruction of various degrees,i.e., k from 10 to 90 in steps of 10, using the same param-eters n = 100, α = 0 . β = 0 .
03, and 100 realisationseach. This again demonstrates that our method reliablereconstructs the correct degree for this individual node.Further simulations, not presented here, varying α and β between 0 .
01 and 0 .
1, show qualitatively the same re-sults. In every case, the reconstruction is very robust,and this suggests that it is extremely unlikely that a hubis reconstructed as a non-hub. Moreover, the reconstruc-tion works correctly not only on the general distribution,but also when it is applied to single nodes.
A. Robustness of reconstruction
As stated above, our reconstruction method assumesthe probabilities of type I and type II errors to be knowna priori. While the type I error is controlled by statisticalmethods, the type II error must be inferred or reasonableassumptions from simulations, or prior studies, about the type II error must be available. To show the impact of vi-
FIG. 6. Reconstruction of the degree of a single node withoriginal degree k = 75.FIG. 7. Reconstructions of the degree of single nodes withoriginal degrees k from 10 to 90 in steps of 10. olations of this and thereby the robustness of our method,we analyse the performance of the reconstruction whenperturbations on α and β are introduced.Figures 8-11 demonstrate the robustness of our ap-proach for various examples. Figures 8-9 are used toshow robustness with respect to β , while Figs. 10-11show the robustness with respect to α . The perturba-tions are quantified in percentage using the parameter δ ,e.g. the perturbations of β are expressed by β + δβ . Notethat, since 0 ≤ β ≤
1, the conditions for the perturba-tions are − ≤ δ ≤ /β −
1; namely, if we call β p = β + δβ the perturbed β , then we have0 ≤ β p ≤ ≤ β + δβ ≤ − ≤ δ ≤ /β − . (10)Negative values for δ represent underestimated values for β and positive overestimated values for β . The sameargument is used for the perturbation of α .Figure 8 shows the reconstruction of a Scale-Free net-work with 100 nodes for the true value of β = 0 .
03, andalso for various values of β deviating up to 1000% fromthe true value, i.e., β p = 0 .
33. The cut-off for the trun-cated singular value decomposition method is 0 . type I error is α = 0 .
05, assumingto control the family-wise error rate at this value, i.e.,the probability of making at least one type I error ; it isbeyond the scope of this manuscript to discuss cases inwhich the technique selected to reconstruct the networkviolates this assumption - we will however estimate theresults for different deviations from the true α used togenerate the plots to investigate its robustness. Figure 8shows that our approach is robust to rather large pertur-bations of β , in both negative and positive directions. Upto δ = 500%, the bias of the reconstruction is negligible;only if δ = 1000% or more deviates the reconstructionsignificantly from the true one, although it still performsbetter than the na¨ıve approach of trusting the identifiednetwork structure.Figure 9 shows the reconstruction of a random networkwith 100 nodes and probability of a connection of 0 . β = 0 .
03, and also for various valuesof β deviating up to δ = 400% from the true value of β .The cut-off for the truncated singular value decomposi-tion method is 0 .
55 and the probability of type I error is α = 0 .
05. Also in this case, the method is robust tolarge perturbations of β , in both negative and positivedirections. A deviation of more than 400% is needed forthe method to fail and not to have an improvement overthe na¨ıve approach.Figure 10 shows the reconstruction of a random net-work with 100 nodes and probability of a connection of0 . α = 0 .
05, and α deviating upto − . typeII error is β = 0 .
03. Figure 10 shows that the method isaffected by relatively large perturbations of α . Namely,for δ < −
85% and δ > type I error .Figure 11 shows the reconstruction of a denser network,i.e., a random network with probability of a connectionof 0 .
8, for the same true values of α and β . In this case,a deviation of 150% or more is needed for the method tofail. Comparing Figs. 10 and 11, we can conclude thatdense networks are more robust to perturbations of typeI error than sparse networks. This is intuitively moti-vated by the fact that the type I error affects links that FIG. 8. Density histogram of the original vertex degrees, bluebars, detected density vertex degree distribution, gray dottedline, and the results when a non-negative constraint is appliedto the truncated singular value decomposition using the true β , black dashed line, and perturbations from the true β , red,blue, yellow, and green solid lines. The original network is aScale-Free network with 100 nodes. are not present in the network, and therefore it has abigger influence on a sparse network.The above results demonstrated that a rough estimatefor α and β is sufficient to get an accurate reconstruction;the method is robust to relatively large perturbations ofthese two errors. Rough estimates of these parametersare typically available from simulation studies or priorknowledge about the system. Note again that the role of α and β are different; α is often controlled and can be ob-tained from known statistics of the techniques under thenull hypothesis; β is more difficult as the true alternativewould need to be known. Given the above simulations,our algorithm is more robust with respect to β than α ,which aligns with the different role of these two errors.As mentioned at the beginning of Section III, we vary α and β in the range 1% −
10% mimicking a typical analy-sis method that has high sensitivity and high specificity.Choosing either lower or higher values for the true α and β does not affect the general qualitatively result of theanalysis, but it changes the range of perturbation thatleads to the failure of the reconstruction method. IV. CONCLUSIONS
We explore the impact of false positive and false nega-tive conclusions about the presence or absence of links onthe vertex degree distribution of a network. Using an an-alytical approach, we investigate this dependence on thedimension of the network and the probabilities of type I and type II errors . Equation (7) describes the density of
FIG. 9. Density histogram of the original vertex degrees, bluebars, detected density vertex degree distribution, gray dottedline, and the results when a non-negative constraint is appliedto the truncated singular value decomposition using the true β , black dashed line, and perturbations from the true β , red,blue, yellow, and green solid lines. The original network is arandom network with 100 nodes and probability of connection0 . α , black dashed line, and perturbations from thetrue α , red, blue, yellow, and green solid lines. The originalnetwork is a random network with 100 nodes and probabilityof connection 0 . the vertex degree distribution of the biased network andthus allows to calculate the influence of false positive andfalse negative conclusions about links on any kind of net- FIG. 11. Density histogram of the original vertex degrees,blue bars, detected density vertex degree distribution, graydotted line, and the results when a non-negative constraint isapplied to the truncated singular value decomposition usingthe true α , black dashed line, and perturbations from thetrue α , red, blue, yellow, and green solid lines. The originalnetwork is a random network with 100 nodes and probabilityof connection 0 . work, assuming the probabilities of type I and type IIerrors are known.In the inverse problem, the aim is to reconstruct theoriginal network. Equation (9) enables us to calculateanalytically the vertex degree distribution of the originalnetwork if the biased one and the probabilities of typeI and type II errors are given. When the dimension ofthe network is relatively large, numerical issues arise andconsequently the truncated singular value decompositionis used to calculate the original network vertex degreedistribution. Numerical simulations show that the ver-tex degree distribution is correctly recovered in all thecases discussed; the cases presented are designed to covera variety of network topologies and therefore degree dis-tributions.The outcomes of this manuscript are general resultsthat enable to reconstruct analytically the vertex de-gree distribution of any network. The analytic formula[Eq. (9)] that allows to find the original vertex degreedistribution depends only on the detected vertex degreedistribution and on the probabilities of type I and type IIerrors . This method is a powerful tool since the vertexdegree distribution is a key characteristic of networks.Moreover, we have actually shown that this method canbe used to reconstruct individual node degrees to a veryhigh accuracy. This should positively impact on vari-ous measures that can be derived from the networks.Our proposed method should outperform standard ap-proaches in terms of betweenness centrality, identificationof hubs, and other network characteristics. This shouldbe rigorously assessed in future research.A limitation of this work is the assumption that theprobabilities of type I and type II errors are known a pri-ori. Nevertheless, we show that the method is robust torelatively large perturbations of these two errors. There-fore, wrong estimates of type I and type II errors , withincertain bounds, do not cause the reconstruction of berendered invalid. We like to emphasise again that in ap-plication the type I error is typically controlled, whilean estimate for the type II error can only be obtainedthrough prior experiments/knowledge or simulation stud-ies. As shown in various simulations, our reconstructionmethod is robust to considerable deviations in β , whichsupports the usefulness of our technique over and aboveproviding deeper insights into the role of these errors innetwork reconstruction; our approach is promising forreal-world applications. Note though that it is alwaysadvisable to utilise simulation studies to characterise theadvantageous and limitations in a concrete application athand. Further analyses should study possible statisticalapproaches to infer these parameters employing Bayesianapproaches or simulation studies. We recommend per-forming the latter to get an estimate of the type II error in particular.Future studies should investigate the influence of typeI and type II errors on other network characteristics,e.g. the number of edges, the global clustering coefficient, and the efficiency. As a consequence, more informationabout the original network can be found and, therefore,combining them all a better reconstruction of the networkcan be achieved. ACKNOWLEDGMENTS
The authors thank Dr. Daniel Vogel for helpful com-ments and discussions. This project has received fundingfrom the European Union’s Horizon 2020 research and in-novation programme under the Marie Sklodowska-Curiegrant agreement No 642563. The authors declare no com-peting financial interests.
Appendix A: Determinant of the matrix A
In this appendix we prove that the matrix A has de-terminant det A = (1 − α − β ) n ( n − . To achieve this we have to prove some intermediate steps.First, we write Eq. (3) in a more compact form as P ( d ′ = k ′ | d = k ) = min { k,n − − k ′ } X i =max { ,k − k ′ } (cid:18) ki (cid:19) (1 − β ) k − i β i (cid:18) n − − kk ′ − k + i (cid:19) α k ′ − k + i (1 − α ) n − − k ′ − i . (A1)Since the element A uv is defined as the probability P ( d ′ = u + 1 | d = v + 1), we can write A uv = min { v − ,n − u } X i =max { ,v − u } (cid:18) v − i (cid:19) (1 − β ) v − − i β i (cid:18) n − vu − v + i (cid:19) α u − v + i (1 − α ) n − u − i , (A2)for u, v ∈ { , · · · , n } and real numbers 0 ≤ α, β ≤ Proposition 1 (Limit cases) . Let A = A ( n, α, β ) be thematrix defined by Eq. (A2). For β = 1 − α or α, β = 0 , ,the determinant of A satisfies Eq. 8.Proof. When β = 1 − α , the Eq. (A2) becomes A uv ( n, α, − α ) = (cid:18) n − u − (cid:19) α u − (1 − α ) n − u . Note that A uv ( n, α, − α ) does not depend on v but onlyon u , therefore in each line all the elements are identical,i.e., it is a multiple of vector [1 , · · · , A isdet A ( n, α, − α ) = 0.If α, β = 0 then the matrix A is the identity and there-fore the determinant is det A ( n, ,
0) = 1. While if α, β =1 then the matrix A is anti-diagonal with all elementsequal to one, then the determinant is det A ( n, ,
1) =( − n ( n − .If α = 0 , β = 0, Eq. (A2) can be formulated as A uv ( n, , β ) = ((cid:0) v − u − (cid:1) (1 − β ) u − β v − u u ≤ v u > v (A3)Note that β = 1 since the case β = 1 − α has been already0considered. The matrix A ( n, , β ) is upper triangular andtherefore the determinant is the product of the elementson the diagonal A uu ( n, , β ) = (1 − β ) u − , i.e.,det A ( n, , β ) = (1 − β ) n ( n − . If α = 1 , β = 0 ,
1, Eq. (A2) can be formulated as A uv ( n, , β ) = ((cid:0) v − n − u (cid:1) (1 − β ) v − − n + u β n − u u ≥ n − v + 10 u < n − v + 1(A4)The matrix A ( n, , β ) has all zeros above the anti-diagonal and therefore the determinant is the productof the elements on the anti-diagonal A uu ( n, , β ) = β u − and sign given by ( − n ( n − , i.e.,det A ( n, , β ) = ( − β ) n ( n − . If β = 0 , α = 0 ,
1, Eq. (A2) can be formulated as A uv ( n, α,
0) = ((cid:0) n − vu − v (cid:1) α u − v (1 − α ) n − u u ≥ v u < v (A5)The matrix A ( n, α,
0) is lower triangular and thereforethe determinant is the product of the elements on thediagonal A uu ( n, α,
0) = (1 − α ) n − u , i.e.,det A ( n, α,
0) = (1 − α ) n ( n − . If β = 1 , α = 0 ,
1, Eq. (A2) can be formulated as A uv ( n, α,
1) = ((cid:0) n − vu (cid:1) α u (1 − α ) n − u − v +1 u ≤ n − v + 10 u > n − v + 1(A6)The matrix A ( n, α,
1) has all zeros below the anti-diagonal and therefore the determinant is the productof the elements on the anti-diagonal A uu ( n, α,
0) = α n − u and sign given by ( − n ( n − , i.e.,det A ( n, α,
1) = ( − α ) n ( n − . Future calculations result easier if the transpose A T of matrix A is considered. Considering A T instead of A does not affect the calculation of the determinant sinceit is in general true that det A T = det A . Proposition 2 (Transformations) . Given the matrix A,defined by Eq. (A2), let’s call A T n the transpose of A ofdimension n . Let be < α, β < and β = 1 − α . Wecall A T n the matrix with elements a nij = a nij (1 − α − β ) n − (1 − α ) n − i = 1 (cid:18) a nij − a ni a n j a n (cid:19) − α − α − β i = 2 (cid:18) a nij − β − α a ni − ,j (cid:19) − α − α − β i = 3 , · · · , n (A7) where a nij are the elements of the matrix A T n . Then, weprove that A T n = (1 − α + β ) n − A T n − . (A8) Proof.
To verify Eq. (A8), we have proved that the iden-tity a ni +1 ,j +1 = a n − ij i.e., (cid:18) a n j − a n a n j a n (cid:19) − α − α − β = a n − j and (cid:18) a ni +1 ,j +1 − β − α a ni,j +1 (cid:19) − α − α − β = a n − ij for i = 2 , · · · n − i = j, ( j > ij ≥ n − i − , ( j > ij < n − i − , ( j < ij > n − i , ( j < ij < n − i , ( j < ij = n − i . For each condition the identity a ni +1 ,j +1 = a n − ij has beenproved. Theorem.
The n × n matrix A, defined by Eq. (A2), hasdeterminant det A = (1 − α − β ) n ( n − . Proof.
For α, β = 0 , β = 1 − α , Proposition 1 provesthe theorem. Assume 0 < α, β < β = 1 − α . Sincea matrix and its transposed have the same determinant,we proceed considering the matrix A T n proving a proofby induction.The base of induction is n = 2; in this case the matrixis A T = " − α αβ − β and it has determinant det A T = 1 − α − β .The inductive step consists in assuming thatdet A T n − = (1 − α − β ) ( n − n − , i.e., the inductive hy-pothesis for dimension n −
1, and proving the statementfor dimension n .Since 0 < α, β < β = 1 − α , we can applyProposition 2. The transformations defined by Eq. (A7)guarantees that the matrix A T n has the same determi-nant as A T n , and according to Eq. (A8), we can express1the determinant as det A T n = (1 − α − β ) n − det A T n − .We can now apply the inductive hypothesis, thereforedet A = det A T n = det A T n =(1 − α − β ) n − det A T n − =(1 − α − β ) n − (1 − α − β ) ( n − n − =(1 − α − β ) n ( n − that concludes the proof of the theorem. [1] B. Kralemann, A. Pikovsky, and M. Rosenblum,New J. Phys. , 085013 (2014).[2] A. Pikovsky, Phys. Rev. E , 062313 (2016).[3] R. Cestnik and M. Rosenblum,Phys. Rev. E , 012209 (2017).[4] S. Li, F. Li, W. Liu, and M. Zhan, Physica A: StatisticalMechanics and its Applications , 118 (2014).[5] P. Clusella, P. Grassberger, F. J. P´erez-Reche, andA. Politi, Phys. Rev. Lett. , 208301 (2016).[6] E. Bullmore and O. Sporns,Nat. Rev. Neurosci. , 186 (2009).[7] L. Pessoa, Phys. Life Rev. , 400 (2014),arXiv:1403.7151.[8] S. E. Petersen and O. Sporns, Neuron , 207 (2015).[9] R. Guimer`a and M. Sales-Pardo,Proceedings of the National Academy of Sciences , 22073 (2009),arXiv:1004.4791.[10] Z. Levnaji´c and A. Pikovsky,Phys. Rev. Lett. , 034101 (2011).[11] B. Karrer and M. E. J. Newman, Phys. Rev. E , 016107 (2011).[12] F. Alderisio, G. Fiore, and M. di Bernardo,Phys. Rev. E , 042302 (2017).[13] M. Newman, Networks: An Introduction (Oxford Uni-versity Press, Inc., New York, NY, USA, 2010).[14] E. Olbrich, T. Kahle, N. Bertschinger, N. Ay, andJ. Jost, Eur. Phys. J. B , 239 (2010).[15] H. Yanai, K. Takeuchi, and Y. Takane, Projection Matrices, Generalized Inverse Matrices, and Singular Value Decomposition (Springer, 2011).[16] P. C. Hansen, BIT Numerical Mathematics , 534 (1987).[17] M. Frank and J. M. Buhmann, (2011),arXiv:arXiv:1102.3176v3.[18] M. Gavish and D. L. Donoho,IEEE Transactions on Information Theory , 5040 (2014).[19] T. F. Coleman and Y. Li,Mathematical Programming , 189 (1994).[20] M. A. Branch, T. F. Coleman, and Y. Li,SIAM J. Sci. Comput.21