Impact of network characteristics on network reconstruction
IImpact of network characteristics on network reconstruction
Gloria Cecchini,
1, 2, ∗ Rok Cestnik,
2, 3, ∗ and Arkady Pikovsky
2, 4 CSDC, Department of Physics and Astronomy, University of Florence, Sesto Fiorentino, Florence, Italy Institute of Physics and Astronomy, University of Potsdam, Campus Golm,Karl-Liebknecht-Straße 24/25, 14476 Potsdam-Golm, Germany Faculty of Behavioural and Movement Sciences, Vrije Universiteit Amsterdam,Van der Boechorststraat 7, 1081 BT Amsterdam, Netherlands Department of Control Theory, Lobachevsky University of Nizhny Novgorod,Gagarin Av. 23, 603950, Nizhny Novgorod, Russia (Dated: August 14, 2020)When a network is inferred from data, two types of errors can occur: false positive and falsenegative conclusions about the presence of links. We focus on the influence of local network charac-teristics on the probability α - of type I false positive conclusions, and on the probability β - of typeII false negative conclusions, in the case of networks of coupled oscillators. We demonstrate thatfalse conclusion probabilities are influenced by local connectivity measures such as the shortest pathlength and the detour degree, which can also be estimated from the inferred network when the trueunderlying network is not known a priory. These measures can then be used for quantification ofthe confidence level of link conclusions, and for improving the network reconstruction via advancedconcepts of link thresholding. I. INTRODUCTION
Complex systems are of key interest in multiple sci-entific fields, ranging from medicine, physics, mathemat-ics, engineering, economics etc. [1–4]. Many complexsystems can be modeled, or represented as dynamicalnetworks, where nodes are the dynamical elements andlinks represent the interactions between them. In thiscontext, networks are widely used in studies of synchro-nization phenomena of coupled oscillators as well as inthe analysis of chaotic behavior in complex dynamicalsystems [5–8]. A deep understanding of network char-acteristics allows controlling the network dynamics [9],e.g., in case of optimizing vaccination strategies with theaim of controlling the spread of diseases [10]. Very oftenone faces an inverse problem : the underlying network isnot known, and a reliable inferring of the network struc-ture from the observation is crucial for understanding thesystem’s operation [11–20].When a network is to be inferred from observationdata, typical analysis techniques provide measures of con-nectivity strength for each link. Several methods havebeen suggested in the literature to reconstruct the net-work structure and decide whether these measures pass acertain threshold, thereby providing a mean to decide ifthe corresponding links are considered as present or not[21–27].If a non-existing link is erroneously detected, it is calleda false positive link and is referred to as a type I er-ror. Likewise, an existing link that remains undetectedis called a false negative link and is referred to as a typeII error. The probability of detecting a false positive linkis usually denoted by α , while β denotes the probability ∗ These two authors contributed equally that an existing link remains undetected. Of course, thegoal of a reliable reconstruction is to minimize both theseprobabilities simultaneously.In [28–30], the analysis of the errors of both types wasfocused on the influence of false positive and false neg-ative conclusions about links on the reconstructed net-work characteristics. It was demonstrated, that withinthe same network topology, the values for α and β , lead-ing to the least biased network characterisation, changedepending on the network property of interest. In thispaper, the analysis is reversed - the study focuses on theinfluence of network characteristics on the probabilitiesof type I and type II errors.Below, we first assume the knowledge of the true un-derlying network. In Section III we perform a simula-tion study to show the dependence of the probability offalse positive and false negative links on their shortestpath length and their detour degree (defined later in sec-tion II A). In Section IV, these results are applied to ascenario where the underlying network is unknown a pri-ori , so we evaluate the shortest path length and of the de-tour degree from the reconstruction to improve the qual-ity of the latter, i.e. to decrease the number of falselyconcluded links. II. NETWORKS AND METHODS
In this section we give necessary network definitions.A network is defined as a set of nodes with links betweenthem [31]. In graph theory, a branch of mathematics thatstudies networks, a different notation is used: networksare called graphs, and nodes and links are called verticesand edges, respectively. Below, the notations from net-work theory and graph theory are used synonymously.In this paper, Erd˝os-R´enyi networks are used for thesimulation study. Erd˝os-R´enyi networks are random net- a r X i v : . [ phy s i c s . s o c - ph ] J u l works in which the set of nodes is fixed, and each pair ofnodes is connected with independent probability p . Theprobability mass function of the node degree distributionof an Erd˝os-R´enyi network is a binomial distribution P ( d v = k ) = (cid:18) n − k (cid:19) p k (1 − p ) n − − k , (1)where n is the number of nodes in the network. A. Binary networks
The adjacency matrix A of a binary network with n nodes is an n × n matrix with elements A ij = (cid:40) i to node j ,0 otherwise. (2)Networks can be directed or undirected. In an undi-rected network, connection from i to j implies the con-nection from j to i . Note that this implies that the ad-jacency matrix is symmetric. In a directed network, thissymmetry is broken, therefore if a path from i to j ex-ists, a path from j to i does not necessarily exist. We willconsider directed networks and hence non-symmetric ad-jacency matrices.For two randomly selected nodes i, j in a network of n nodes, the shortest path length (SPL) (cid:96) ij measures thenumber of links separating them if the shortest path istaken. For connected nodes i, j , when the oriented edge i → j exists, the SPL is (cid:96) ij = 1. For directed networksgenerally (cid:96) ij (cid:54) = (cid:96) ji .Inspired by the idea of a local clustering coefficient [31],a novel network characteristic, which we refer to as the detour degree (DD) ∆ ij , is defined here. Detour degree isa pairwise measure that quantifies detours between a pairof nodes. Namely, for every oriented node pair i → j , thedetour degree is the number of oriented paths of length2 from i to j . For example, in the case shown in Fig. 1,the DD is ∆ ij = 2, corresponding to two directed pathsof length 2 from i to j through k and k . Since the edgebetween i and k is oriented towards i , a path from i to j through k does not exist. Similarly to the SPL, theDD is non-symmetric for directed networks. Notice alsosome connection between the SPL and the DD: if (cid:96) ij ≥ ij = 0. B. Weighted networks
Often it is useful to define a network where the linksare not binary connections, but are instead described bycontinuous weights. The adjacency matrix elements ofweighted networks are real numbers. Definitions pro-vided in the previous section for the SPL and the DDin binary networks are here generalized for weighted net-works. jk i k k FIG. 1. Example of DD ∆ ij = 2. We consider the direct path length from node i to node j to be the inverse of the corresponding adjacency matrixelement A ij [32], or in other words, the inverse of the linkweight. Therefore, the SPL from node i to node j is theminimal sum of pairwise path lengths for all availablepaths between i and j , i.e. (cid:96) ij = min (cid:16) A − ik + · · · + A − k n j (cid:17) , (3)where nodes k through k n belong to all possible pathsfrom i to j . Note that for binary networks, this definitionis coherent with the one in the previous section. For abinary network, an existing link corresponds to weight1 and an absent link to weight 0, the latter would leadto an infinite contribution in the sum. Therefore Eq. (3)reduces to the number of links separating i and j if theshortest path is taken. As a sidenote, one can draw aparallel here with circuit theory [33], with link weightsrepresenting directed conductances, making the shortestpath correspond to the path of least resistance and theSPL quantify its effective resistance.The DD of an oriented node pair i → j measures thecontribution of all the possible 2-step paths from i to j .In weighted networks, such a contribution must considerthe weights of the edges. Namely, the DD is scaled bythe product of weights of the two edges that form the2-step path ∆ ij = (cid:88) k A ik A kj . (4)For binary networks, this definition is coherent with thedefinition in the previous section, since for A kh ∈ { , } Eq. (4) reduces to the total number of paths of length 2from node i to node j . In the circuit theory analogy [33],the DD roughly corresponds to the effective conductanceof all paths of length 2 (that would be (cid:80) k A ik A kj A ik + A kj ). Notethat, in both the binary and weighted case, Eq. (4) canbe expressed elegantly in matrix form as ∆ = A . C. Network inference examples
It is not a goal of this study to develop a novel net-work inference method; rather we take methods from theprevious literature and consider how they are affected by
FIG. 2. Inferred coupling strengths, relationship of α and β as function of the SPL and DD using two inference techniques: G (panels a,c,e) and G (panels b,d,f). Panels (a-b): Histograms of the inferred coupling strengths. Probabilities α and β asfunctions of the SPL (c-d), and as functions of the DD (e-f), for a specific value of the threshold (0 .
08 for G and 45 for G ). the network properties. We perform our studies with twonetwork inference techniques. The first one takes contin-uous signals of all oscillators and assumes they follow theKuramoto model dynamics [4]:˙ φ k = ω k + (cid:15) (cid:88) j T kj sin( φ j − φ k − Θ jk ) (5)where (cid:15) is the coupling strength, φ k the phases, ω k thenatural frequencies and Θ jk phase shifts. It returnsstrictly positive values for interactions (cid:15)T kj . For detailssee Ref. [21]. A network inferred using this technique isindicated in this manuscript with G , and Fig. 2a showsan example of inferred coupling strengths.The second technique is designed for pulse-coupled os-cillators. It takes the observed spike times and assumesthat the interaction can be well represented with a net- work based on the Winfree phase equation [34]:˙ φ k = ω k + (cid:15)Z k ( φ k ) (cid:88) j T kj δ ( t − t j ) (6)where Z k ( φ ) is the phase response curve and t k arethe spike times of oscillator k . The technique returnsreal numbers (positive and negative) for interactions.For details see Ref. [5]. A network, inferred using thistechnique, is indicated in this manuscript with G , andFig. 2b shows an example of inferred coupling strengths. III. DEPENDENCE OF FALSE CONCLUSIONSON NETWORK CHARACTERISTICS
This section focuses on the dependence of false posi-tive and false negative link conclusions on the networkcharacteristics introduced in Section II. To this aim wesimulate an ensemble of oscillatory networks, and infertheir connectivity from limited observations of its timeseries. We consider two different inference techniques,both of which yield continuous values for link weights,see Sec. II C.We denote the true network’s binary adjacency matrixwith T and the inferred weighted one with W . The aimis to reconstruct the original binary network T from theinferred one W , i.e. determine on the basis of link weights W ij whether the links are present or not. This is typicallydone by thresholding the weights, i.e. if an inferred linkweight passes a certain threshold, the link is assumed tobe present.The inferred coupling strengths W ij have a certaindistribution. Consequently, depending on the chosenthreshold value, different numbers of false positive andfalse negative conclusions occur. This is commonly rep-resented with a receiver operating characteristic, com-monly referred to as a ROC curve [35]. In this manuscriptthe interest is focused on the influence of the probabilitiesof false conclusions on the local network characteristicsSPL and DD.The simulation study is performed on Erd˝os-R´enyinetworks with n = 100 nodes and probability of con-nection p = 0 .
15. In particular, for G the frequen-cies ω k are uniformly distributed within the interval(0 . , . jk are uniformly distributedin the interval (0 , π ), the original coupling strength isset to (cid:15) = 0 .
3, and 500 data points are used to per-form the network inference. For G , the frequencies ω k are uniformly distributed within the interval (1 . , . (cid:15) = 0 .
5, all oscillatorsare assigned the same phase response curve: Z ( ϕ ) = − sin( ϕ ) exp(3 cos( ϕ − . π )) / exp(3), and all spikes thatoccur within 500 observed periods of the slowest oscilla-tor are considered for network inference. For both G and G , 100 simulations are made to have enough statisticaldata. A. False conclusions with respect to local networkstructures
In this section we study how the inferred weights, andtherefore false conclusions, depend on the local charac-teristics of the true network T , namely the shortest pathlength (SPL) and the detour degree (DD). Since T isdiscrete so too are the SPL and DD. It is worth notinghere that we consider that all possible links i → j canbe falsely identified regardless whether they are presentin T or not. Their presence simply determines whetherthey are candidates for a false positive conclusion (not present in T ), or a false negative one (present in T ).The probability of a false positive conclusion α is eval-uated for subsets of links with the same SPL: (cid:96) = 2 , , ... ( (cid:96) = 1 means the corresponding link exists and thereforeno false positive conclusion can be made), see Fig. 2c-d. In the case of false negative conclusions however, thetrue link is present and the shortest path length there-fore equal to 1. Because of this, we consider the indirectshortest path length (iSPL), i.e. SPL when the direct linkis not considered - for clarity we distinguish its notationas ˜ (cid:96) . Note that if T ij = 0 then ˜ (cid:96) ij = (cid:96) ij . The proba-bility of a false negative conclusion β is then evaluatedon links with the same iSPL: ˜ (cid:96) = 2 , , ... (˜ (cid:96) of a binarynetwork can not be less than 2). What we observe isthat false conclusions happen more often for links withshorter (i)SPL. This intuitively makes sense. The smallerthe (indirect) distance between two nodes the more theyinfluence each other via indirect coupling, which can dis-rupt the inference algorithms [5, 21] into misinterpretingthe connectivity. This holds true for both α and β . Wedepict these dependencies in Fig. 2c-d.We perform a similar analysis using the DD in place ofthe SPL (Fig. 2e-f). The probabilities of false conclusions α and β are evaluated for subsets of links with the sameDD. We find that both α and β typically increase withthe DD. This again makes intuitive sense for the samereason as with the SPL. Namely, if the DD is low, theindirect interaction between the nodes is low regardless ofwhether the direct connection exists or not. This meansthat there are less interferences to be picked up by theinference algorithms. These dependencies are depicted inFig. 2e-f.Here we point out that the DD is effectively a measureof connectivity while SPL is a measure of detachment, i.e.they measure opposite things. In circuit theory analogyDD is a measure of effective conductance while SPL is ameasure of effective resistance. IV. WHEN THE TRUE GRAPH IS UNKNOWNA. Using network characteristics from thereconstruction
As we have seen in Section III A, false conclusion prob-ability increases with the measure of indirect distancebetween nodes, i.e. it increases with the (i)SPL and de-creases with the DD. The study presented above will benow reversed - suppose the true network T is not knownand we only have access to the inferred wights W . Inthis section we investigate the possibility of using localnetwork information of the inferred graph W to gain ad-ditional insight on the probability of link existence.We can evaluate iSPL with Eq. (3), and DD withEq. (4) on the inferred network W . If any weights arenegative we take their absolute value, the reasoning be-ing that we are interested in the estimated interactionbetween nodes and negative weights represent a kind of FIG. 3. Scatter plots of the inferred weights w versus the iSPL (panels a,c) and versus the DD (panels b,d) for both inferencemethods, G (panels a,b) and G (panels c,d). Points corresponding to true links are depicted with red and false ones withblack. interactions as well. Then we compare the relationshipsbetween the inferred link weight W ij , the iSPL ˜ (cid:96) ij andthe DD ∆ ij - all obtained from W .In Fig. 3, we show scatter plots of weights W ij versustheir corresponding links’ iSPL (panels a-b), and versusDD (panels c-d), using the two network inference meth-ods explained in Sec. II C. We color the points differentlyfor the ones that represent a true link, T ij = 1 (red), andthe ones that do not, T ij = 0 (black). This reveals thequalitative dependence of weights on indirect measures ofconnectivity: iSPL and DD. The findings are reflective ofthose in Sec. III A, namely, the probabilities of false con-clusions decrease with iSPL and increase with DD. Thismeans that these measures can be used to represent thelevel of confidence in detected links, i.e. links with lowDD and high iSPL are more likely to be accurately re-constructed by thresholding.We illustrate this with ROC curves evaluated on only aselected portion of links, according to their DD and iSPL.In particular, we consider the more confident half of linksand compute false conclusions proportionally. Thesepartial-consideration ROC curves are shown alongsidethe full-consideration curve as comparison, see Fig. 4.The DD in particular seems to be a good indicator of confidence in a link conclusion. B. Alternative thresholding
The results presented in Sec. III A show the depen-dence of the inferred coupling strengths on two net-work characteristics - the indirect shortest path length(iSPL) and the detour degree (DD). These results suggestthat network reconstructions might benefit from differ-ent strategies of determining the existence of links. Thena¨ıve choice consists of selecting a threshold value, andconsidering all links with inferred coupling larger thanthe threshold as present, while the rest as not present.In this section, two advanced thresholding strategies arediscussed.The first possibility we discuss takes into account therelationship between the link’s inferred coupling strengthand its SPL. Specifically, one of many natural choices isto only consider links as present when their inverse cou-pling strength corresponds to their SPL. In other words,consider present all links for which the inferred SPL goesthrough the direct link. This choice can be graphicallyrepresented with a curved threshold, taking the 1 /x curve FIG. 4. ROC curves corresponding to: complete networkreconstruction (thick gray line), 50% of links with the lowestDD (dashed green line) and 50% of links with the highest iSPL(dotted orange line). Best results correspond to the upperleft corner of the ROC plot. The point corresponding to the mountain-pass thresholding is depicted with a blue triangle,and the one corresponding to the
SPL-relative thresholdingwith a red circle. Both methods G and G are representedin panels a and b respectively. in the plot Fig. 3a,c. We refer to this as the SPL-relative threshold. Figure 4 shows the ROC curve correspondingto the na¨ıve choice for the threshold, and the circle redmarker corresponds to the SPL-relative threshold. Whilethis does not seem to improve the reconstruction for G ,it does significantly enhance the results for G . Fur-ther, we could consider combining SPL-relative thresholdwith the na¨ıve threshold, by simply thresholding the re-maining links. Namely, among the links whose strengthcorresponds to the reciprocal of the SPL, we performsimple thresholding. With this combined thresholdingthe reconstruction is marginally improved for G as well,i.e. within a range of threshold values both α and β aremarginally reduced.For the second thresholding, consider Fig. 3b-d. In thefigure, the na¨ıve threshold corresponds to a horizontalseparation line. We suggest to make use of the extra di-mension gained with the new DD measure and considera separation line that bends and therefore possibly sepa- D e n s i t y (a) D e n s i t y (b) FIG. 5.
Mountain-pass threshold (black dashed line) and apossible choice for the na¨ıve threshold (white dotted-dashedline) on top of the density histogram for the inferred couplingstrengths as a function of the DD for both inference methods G (a) and G (b). Colour code expresses the density in thelogarithmic scale. rates true links from non-links more efficiently, i.e. withless false conclusions. To this aim, we first compute thehistogram of the inferred coupling strengths as a func-tion of the DD, see Fig. 5. Then, we calculate the curvethat follows the local density minimum between the twobulges of the histogram (black dashed line in Fig. 5). Thiscurve is then used as the new threshold and we refer toit as the mountain-pass threshold. The correspondingresult of the mountain-pass threshold in terms of falseconclusion is illustrated in Fig. 4 with a blue triangularmarker. For both G and G , this choice of the thresholdresults in a better reconstruction of the true links thanboth the SPL-relative and the na¨ıve threshold. V. CONCLUSION
In this paper, the influence of local network charac-teristics on the probability of false conclusions about thelinks inferred from typical data analysis methods, hasbeen examined.We considered binary directed networks of coupledoscillators and assumed a setup where only individualnodes can be observed. Namely, connectivity can notbe measured directly, but instead can only be estimatedfrom dynamical observations of individual oscillators.The particular methods of connectivity inference adoptedin this manuscript take signals of individual nodes andyield a real-valued connectivity matrix representing linkweights. In order to obtain binary connectivity fromweighted connections, one would typically threshold linkweights to determine their presence. A portion of linksis almost always misidentified. In this paper we investi-gate the relationship between these false conclusions andlocal network characteristics. In particular we look intotwo network characteristics: the shortest path length andthe detour degree. By performing a statistical analysis onsimulations where the ground truth is known, we foundthat these local characteristics can provide additional in-formation regarding the probability of false conclusions.The knowledge of the dependency of the inferred linkweights and these characteristics allows the links to berepresented in a higher dimensional space, where moreadvanced thresholding techniques can be used. Twonovel thresholding techniques are proposed as examples, both decreasing the proportion of false conclusions forthe tested conditions, see Sec. IV.Additionally, we demonstrated that such a posteri-ory calculated local network characteristics can providegood estimators of confidence in obtained links, see ROCcurves, Fig. 4. These results can be applied to real ex-perimental settings, where the underlying true networkis not known a priori . As such, these multidimensionalthresholding techniques show potential for use in a vari-ety of further investigation.In future studies, different reconstruction methodsshould be considered to check whether the common rulesfound in this manuscript apply to a wider range of cases.Furthermore, deliberating knowledge-based criteria fordetermining how effective a particular local character-istic serves for such purposes, could lead to conception ofoptimized network characteristics.
ACKNOWLEDGMENTS
This project has received funding from the Euro-pean Union’s Horizon 2020 research and innovation pro-gramme under the Marie Sklodowska-Curie grant agree-ment No 642563. A.P. thanks Russian Science Founda-tion (Grant Number 17-12-01534). The authors declareno competing financial interests. [1] A. Barrat, M. Barthelemy, and A. Vespignani,
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