Discovering hidden layers in quantum graphs
DDiscovering hidden layers in quantum graphs (cid:32)Lukasz G. Gajewski ∗ and Julian Sienkiewicz Faculty of Physics, Warsaw University of Technology, Koszykowa 75, 00-662 Warszawa, Poland
Janusz A. Ho(cid:32)lyst
Faculty of Physics, Warsaw University of Technology, Koszykowa 75, 00-662 Warszawa, Poland andITMO University, Kronverkskiy Prospekt 49, St Petersburg, Russia 197101
Finding hidden layers in complex networks is an important and a non-trivial problem in modernscience. We explore the framework of quantum graphs to determine whether concealed parts of amulti-layer system exist and if so then what is their extent, i.e. how many unknown layers there are.We assume that all information we have is the time evolution of a wave propagation on a single layerof a network and show that it is indeed possible to uncover that which is hidden by merely observingthe dynamics. We present evidence on both synthetic and real world networks that the frequencyspectrum of the wave dynamics can express distinct features in the form of additional frequencypeaks. These peaks exhibit dependence on the number of layers taking part in the propagationand thus allowing for the extraction of said number. In fact, with sufficient observation time, onecan fully reconstruct the row-normalized adjacency matrix spectrum. We compare this approachto the work of Aziz et al. in which there has been established a wave packet signature methodfor discerning between various single layer graphs and we modify it for the purposes of multi-layersystems.
I. INTRODUCTION
A plethora of contemporary dynamical systems andcollective phenomena can be expressed in terms of com-plex networks and recently often with multi-layer models.Whether it is a transportation network (e.g. buses andtrams) or a social one (e.g. Twitter and Facebook) vari-ous forms of information propagation or state dynamicscan be described with multi-layer networks. [1–7]However, it is not uncommon for certain parts of asystem to remain hidden from observers and it can becrucial to be able to discern what characteristics are un-known and then to try to obtain them. Such inverse prob-lems have been studied in various settings in both topol-ogy and dynamics parameters reconstruction in mono-and multi-layer scenarios alike [8–20]. Recently therehas been some pivotal advances in terms of determiningwhether one can detect hidden layers in non-markoviandynamics and even obtain how many of these layers arethere [21].Nevertheless, to our knowledge, there has not beenmuch done specifically for uncovering the multi-layerstructures in quantum graphs and thus we address thisissue and present two potentially viable approaches of es-tablishing if hidden layers exist and in some scenarios toascertain the exact count of these layers.Traditionally graphs are discrete, combinatorial ab-stract mathematical objects. If we supply them with ametric and topology we call such objects metric graphs .Those in turn equipped with a second order differentialoperator acting on its vertices and edges - a Hamilto-nian - and appropriate boundary conditions are called ∗ [email protected] quantum graphs [22–25]. It is worth underlining herethat with this definition we do not specify the exact na-ture of the Hamiltonian and while it is often a quantummechanical one it need not be so and thus here we fol-low the interpretation of taut strings, fused together atthe vertices that can be seen as the “limiting case” ofa “quantum wire”.[26, 27]. The (most likely) first useof this framework can be traced back to Pauling’s paperin 1936 [28], however, for the most part quantum graphshave not been widely used until more recently. Nowadaysthey see many various applications in dynamical systems,nanotechnology, photonic crystals and many others [29–34]. Most recently Aziz et al. established a method basedon a wave packet propagation on quantum graphs thatallows to distinguish between structures in complex net-works [35] thanks to many well studied properties of theLaplacian (e.g. finite speed of propagation [36] as op-posed to a discrete Laplacian [37, 38]) and its spectra inquantum graphs [39–45]. The idea of determining the ashape of an object based on observable dynamics on itgoes back to the work of Kac in 1966 [46] in which heasks whether it is possible to hear the shape of a drum.Giraud and Thas showed that the eigenvalues of differ-ent shapes can be identical and therefore answered Kac’squestion in the negative. Gutkin and Smilansky, on theother hand, showed that in quantum graphs specifically,under certain conditons, one can indeed “hear” the shapeas the Hamiltonian uniquely defines the connections andtheir lengths when the graph is finite (and simple), thebond lengths are rationally independent and the vertexscattering matrices are properly connecting. It is alsoworth noting that this inverse spectral problem can beextended onto scattering systems as also stated in thesame paper (a so called inverse scattering problem [47]),however, in this case it is not always possible [48, 49] (i.e.there is a way to construct isoscattering pairs of graphs a r X i v : . [ c ond - m a t . d i s - nn ] D ec FIG. 1. Three-layered multiplex representation of the Vickers[54] data. with identical polar structure of their scattering matri-ces) which was also showed experimentally via microwavenetworks by Hul et al. [50]. Wave packets specifically hasalso attracted some attention in recent years but not forthe purposes of the goal we aim for in this paper [51, 52].Some work has been done in context of sufficient cover-age with sensors [53], however, in this case we also willnot share all the assumptions and thus those methodsare not applicable to our problem.In this paper we tackle the problem of determiningwhether there are hidden layers in the complex systemwe are observing and if so then how many. We assume awave packet propagation dynamics on a quantum graphas our model for the dynamical system. Each edge e inthe graph G has an associated length l e = 1 and a spatialcoordinate variable x e ∈ [0 , l e ] along said edge. We use aspecial case of a Hamiltonian - an edge based Laplaciangiving us an edge-based wave equation on a graph in theform: u tt d E = − ∆ u (1)where E is a Lebesgue measure on the graph’s edges [26,55].We use Neumann boundary conditions stating that thesum of outward pointing gradients at every vertex mustvanish [26, 35]: ∀ v ∈ G, (cid:88) e (cid:51) v ( − − x e,v ∇ f ( e, x e,v ) = 0 (2)The initial condition for the wave equation is Gaussianwave packet: f ( e, x ) = exp (cid:0) − a ( x − µ ) (cid:1) (3)which is fully contained within a single edge with thehighest betweenness centrality[56] following the conven-tions of Aziz et al [35]. We simulate many layer system in a multiplex configuration, i.e. each node is connected toits reflection in a neighbouring layer (see Fig. 1 for a realworld network example and Fig 2 for wave propagationexample on a simple synthetic graph). While the propa-gation simulations are computed on full systems, for thedetection purposes we always only use information froma single layer, i.e. all but one layer are hidden from theperspective of our methods at all times.This rest of the paper consists of three main parts fol-lowed by a discussion. Firstly we briefly outline the ap-proach introduced by Aziz et al. in [35] and show itsviability for the purposes of multi-layer networks in thecontext described above. Secondly we introduce our ownapproach with the use of a Fourier transform on the timeevolution of sum of amplitudes in the visible part of thesystem. Thirdly, we show that with sufficient resources(i.e. observation time) one can fully reconstruct the spec-trum of the row-normalized adjacency matrix. II. GAUSSIAN WAVE PACKET SIGNATURE(WPS)
Gaussian wave packet signature (WPS) is a methodol-ogy developed by Aziz et al. [35] that allows to distin-guish between various types of graphs. The procedurestarts by initiating an edge with a Gaussian wave packetthat is completely contained on said edge (see Fig. 2a).The edge is chosen to be one with highest betweennesscentrality as to assure fastest possible spread of the waveon the graph (although this can, of course, be relaxedin general). Then on each integer time we measure theamplitude in the centre of every edge 2 | E | times in to-tal, where | E | is the number of edges (see Fig. 2b andAppendix A for a detailed description of the way theamplitude is calculated). We measure the centre of eachedge because at integer times the highest value is in thecentre. Finally create a histogram with 100 bins of thesemeasurements - this is the WPS of the graph. Aziz et al.show that particular graph types (say Barab´asi-Albert,Erd˝os–R´enyi etc.) will have similar WPSs yet different incomparison to other types (so e.g. one can differentiatean ER from BA). In order to actually do this differen-tiation one must build a classifier. In their work a K-nearest neighbours (K-NN) classifier was chosen. Fromthe perspective of the machine learning tools we use inthis paper (K-NN, PCA) each histogram bin of the WPSis a dimension in the feature space.For our purposes we will deviate slightly from this pro-cedure. Namely to us the whole graph is not known andthe graph itself is a multiplex. Additionally we assumethat the wave propagation is an actual process ongoingthrough some the network. Thus, we assume we haveaccess to a single layer on which a certain spread hashappened that can be modelled with a Gaussian wavepacket and we suspect there may be hidden layers in thenetwork. Question is - can we detect their presence?The rest of the procedure is similar, i.e. we create a FIG. 2. An exemplary picture of a wave starting at a specific edge (left, t = 0) and then propitiating through the multiplexsystem (right, t > N = 50, m = 3) graph with 5 and 1 layers(measured on one layer only). WPS projected onto a 2D space with PCA (centre) with colours distinguishing no. of layers.Each point is a different BA graph. PCA transformation matrix (right) showing the contributions of given WPS bins intoprincipal components. WPS and we train a classifier on a given type of a graph(e.g. BA) and this time the classes are the number oflayers. See Fig. 3 and 4 where we compare WPSs of a5-layer BA graph vs 1-layer and a 5-layer ER graph vs 1-layer respectively. One can clearly see that the signaturesare distinct. To further illustrate this we use PrincipalComponent Analysis (PCA, see Appendix B for details)to project the WPSs onto a 2D space (see the centrepieces of Fig. 3, 4). There one can see that each classtakes a distinct region of space and thus we should beable to discriminate between them. However, it is worthnoting that the more layers there are the more tightlypacked the observations are, i.e. discriminating betweenmono-layer and penta-layer graphs is fairly easy, betweentetra- and penta- not as much. It is also worth notingthat through PCA we can see that the centre bins carrythe most variance in the feature space (see panels onthe right in Fig. 3, 4) and that there are certain distinctstructures visible in the higher PCs for both BA and ERgraphs.As mentioned before, we follow Aziz et al. and alsochoose the K-NN for classification (see Appendix C for adescription of the K-NN method). We build the model on various graphs with between 1 to 5 layers (with valuesonly from one layer each time as explained earlier) andthen test it to see if it can recognise how many layersthere are in an unknown graph. We conducted our testsfor BA and ER graphs (see Fig. 5 for results of classifi-cation). For each type of graph we simulated 500 inde-pendent realisations (100 per each number of layers) withmean degree (cid:104) k (cid:105) = 6 and network size N = 50. We takeout a 100 randomly chosen, in a stratified manner (i.e.classes are equally represented), realisations are takenout of the training set (this will be the test set). On thetraining set we conduct 100 rounds of training and thentesting. That is in each round we do the training/testsplit, then a 10-fold cross-validation on the training setto find the best K parameter of the K-NN model andthen use that on the test set. We present the results ina form of box plots. In Fig. 5 we show the accuracy ofthe classifier for both BA and ER. We can see that whilefor ER the results are slightly lower than for BA, in bothcases the accuracy is still fairly high with medians of 92and 89 for BA and ER respectively.To additionally illustrate the point made earlier thathigher numbers of layers are more difficult to discern FIG. 4. Wave packet signatures (left) for various realisations of a Erd˝os–R´enyi ( N = 50, (cid:104) k (cid:105) = 6) graph with 5 and 1 layers(measured on one layer only). WPS projected onto a 2D space with PCA (centre) with colours distinguishing no. of layers.Each point is a different ER graph. PCA transformation matrix (right) showing the contributions of given WPS bins intoprincipal components.FIG. 5. K-nearest neighbours classification accuracy of the layer count for a BA and ER graph as box plots (left) based on singlelayer WPSs. Contingency table diagonal values as box plots for a BA (centre) and ER (right). We simulated 400 independentrealisations of a given graph type (BA, N = 50, m = 3; ER, N = 50, (cid:104) k (cid:105) = 6). For each type of graph we conducted 100rounds of 10-fold cross-validation to determine the K parameter in K-NN withdrawing 100 realisations (25%) for purposes offinal evaluation. Train/test split was random and stratified. amongst one another we show the diagonal values of con-tingency tables from all 100 rounds for BA (centre panelof Fig. 5) and ER (the right panel of Fig. 5). One canclearly see that identifying mono-layer systems is practi-cally 100% accurate and it is the more layered systemsthat cause trouble for the classifier.While the classification results seem very promising itis important to note the obvious and major disadvantageof this approach. One must build a training set for itto work. With synthetic networks it is easy to generateas many as one wants and the limitation is purely com-puting power. When dealing with real world networksone often does not simply have the ability to have simi-lar enough graphs to the one currently under observationbut with added layers. In such a circumstance perhaps acombination of many different synthetic networks couldsuffice and other, more advanced, classification methodsthan K-NN could be utilised. That, however, is beyondthe scope of this study. III. FOURIER TRANSFORM OF THEAMPLITUDE SIGNAL
Here we introduce a new approach to detecting layerson quantum graphs. Similarly to the previous case weassume we can either produce or observe a wave propa-gation on the graph initiated by a Gaussian wave packet.For efficiency’s sake in the simulations used we used theedge with highest betwenness centrality as before and wealso measure the amplitudes at integer times in the cen-tres of all edges for 2 | E | times. While in WPS methodwe simply follow the advice of Aziz et al. for the count ofmeasurements in the approach discussed here it is usu-ally rather clear if enough data was collected by straightforward visual inspection.Our proposition is as follows, at each integer time com-pute the sum of all amplitudes in the visible part of thesystem and treat it as a time dependent signal S ( t ).Transform the signal into frequency domain via a fastFourier transform (see Appendix D and [57]) - ˆ S ( f ) -and look at the spectrum of the signal - | ˆ S ( f ) | .A mono-layer system will produce a “flat” signal S ( t )whilst multi-layer one will exhibit periodic behavioursdue to energy leaking in and out of the visible layer fromand into other layers. At sufficiently long measurement FIG. 6. Sum of amplitudes time evolution as measured on the only visible layer (left). Fast Fourier transform of this signal(centre) and its power spectrum (right). Each row represents a different number of layers (1 to 5 going top to bottom).Simulations conducted on 20 independent realisations of a BA graph ( N = 50, m = 3) per row, overlaid with transparency. time scale the signal should stabilise and become station-ary as long as no perturbation is introduced to the over-all network. This transfer of energy induces oscillationsin the amplitude sum on the visible layer which in turncreate clear peaks in the power spectrum, see Fig. 6 forresults from BA graphs and 7 for ER. Left column showsthe signal S ( t ), centre | ˆ S ( f ) | and right | ˆ S ( f ) | in deci-bels and log-log scale. Each row has 20 independent re-alisations plotted on top of each other with transparencyto show that these peaks are fairly consistent, and dif-ferent number of layers (1st row are mono-layer systems,2nd row di-layer etc.). It is quite apparent that there arevisible peaks and their count strictly corresponds to thenumber of layers in the system.These results already show the advantage of FFT overWPS as it is simpler and does not seem to suffer fromstruggling to differentiate as much between high-layersystems. Additionally it does not require any priorknowledge or learning of the model. It is far from flaw-less, however. As it is much easier to test on real networksthan WPS we applied it to three real world networks[54, 58, 59] - see descriptions in Appendix E. The resultsare in Fig. 9 where one can see that while we do get peaksin the spectrum and therefore can confidently state thatthere are hidden layers, it is much less clear how many ofthem there are. This is most likely due to the fact thatthese networks are not as clear cut multiplexes as thesynthetic systems we discussed earlier. These networkshere have various mean degrees in each layer and thatalso implies varied coupling strength between the layers. Additionally real world networks can have other char-acteristics different between the layers such as cluster-ing coefficients or degree distributions and the syntheticsystems tested simply do not have this property. How-ever, we show in the next section that it is possible toreconstruct the full spectrum of the row-normalized ad-jacency matrix with this method and therefore determinethe number of layers for any system. IV. SPECTRUM RECONSTRUCTION
In this section we show that with sufficiently long ob-servation it is possible to completely recover all eigenval-ues of row-normalized adjacency matrix of the full systemand thus trivially determine the number of hidden layers.We shift slightly from previous sections as we no longertake measurements on the edges but only on nodes. Thismakes the problem less computationally intensive andalso, in our opinion, makes for a more practical case asit might be sometimes easier to observe just the nodes’states. However, same analysis can be applied using edgemeasurements as previous sections could be done withnode values only - we chose otherwise as we are stemmingfrom the work of Aziz et al.Similarly as before we observe the sum of amplitudes,however, in this case it is important to have enough sam-ples of the signal to provide sufficient resolution in thepower spectrum. How long one needs to observe a sys-tem will of course depend on the intricacies (and mostly
FIG. 7. Sum of amplitudes time evolution as measured on the only visible layer (left). Fast Fourier transform of this signal(centre) and its power spectrum (right). Each row represents a different number of layers (1 to 5 going top to bottom).Simulations conducted on 20 independent realisations of an ER graph ( N = 50, (cid:104) k (cid:105) = 6) per row, overlaid with transparency. size) of the system in question. As the propagation pro-cess is not stochastic the time needed for observation isfinite and in our experience not unattainable. The goalis simply to have the complete power spectrum of thesignal. Then one takes note of the peaks present - weopted for an automated approach using a wavelet trans-form [60](see Appendix F). It turns out that the peaksin the power spectrum are the eigenvalues of the Hamil-tonian (divided by 2 π ) which in turn are directly relateto the eigenvalues of the row-normalized adjacency ma-trix - ˆ A - such that each eigenvalue λ / ∈ {− , } of ˆ A has corresponding Hamiltonian eigenvalues cos − ( λ ) and2 π − cos − ( λ ). This leads us to two important results, i)the number of frequencies present in the power spectrum f is two less than count of eigenvalues of the adjacencymatrix and since we know the layer sizes (i.e. node countper layer - N ) as a multiplex structure was assumed thenumber of layers K = ( f + 2) /N . ii) we can in factrecover almost exactly all eigenvalues of ˆ A as cos(2 πf i )for each frequency peak f i in the power spectrum.We present the result of the full spectrum reconstruc-tion on Fig. 8 for a complete, BA and real world graph.We chose the complete graph as it has a special casedue to the extreme symmetries of the multiplex adja-cency matrix and thus the number of peaks directly corre-sponds to the number of layers unlike more complex caseswhere the eigenvalues multiplicities behave differently.Of course, this does imply that if due to specific struc-tures in a given system some eigenvalues have high multi-plicity the simple formula K = ( f + 2) /N will not hold and system specific adjustments would be needed. Thereconstructed eigenvalues give an almost perfect matchwith those of the row-normalized adjacency matrix. Notethat the performance here is mostly limited by the peakdetection method and the resolution in the frequency do-main, i.e. the information is there in the spectrum, theonly challenge is to recover it efficiently. V. DISCUSSION
In this paper we explore the paradigm of quantumgraphs as a potential tool for studying multi-layer net-works. The particular problem we are interested in isdetermining whether there are hidden layers of commu-nication in the system that we cannot observe by takingmeasurements of the ongoing dynamics in the single layerthat we can observe. We proposed and tested two meth-ods - one based upon a Gaussian wave packet signature(WPS) that was introduced prior by Aziz et al. to dis-criminate between various types of mono-layer systemsand the other on observing the power spectrum of thewave amplitudes.WPS is a method where a Gaussian wave packet, eitherobserved or purposefully produced, initiates the propaga-tion from a single edge and we take measurements of am-plitudes at every edge at integer times sufficiently long.Such data is then histogrammed to produce the signa-ture. This signature has the property of being similarwithin a category of graphs (originally e.g. ER vs BA, r e c o n s t r u c t i o n r e c o n s t r u c t i o n r e c o n s t r u c t i o n FIG. 8. Eigenvalues reconstruction from the Fourier spectrum of the nodes’ amplitudes sum signal. (Left) an example ofa complete graph multiplex with layer size N = 20 and number of layers K = 9. As it is a special case of an extremelysymmetric adjacency matrix there are only as many eigenvalues as there are layers. (Right) a Barab´asi-Albert graph with N = 50 , m = 3 , K = 4 has a much more complex spectrum and so does a real world network (right) - Vickers[54] - both ofwhich we attain an almost perfect match between the recovered and actual eigenvalues. The dashed diagonal line is a visualaid showing “ y = x ”.FIG. 9. Sum of amplitudes time evolution as measured on the only visible layer (left). Fast Fourier transform of this signal(centre) and its power spectrum (right). Each row represents a different real world network (as indicated by titles - Vickers[54],C.Elegans[59], Krackhardt[58]). Each graph is a 3-layered multiplex. here number of layers) while varied without. This in turncan be utilised by machine learning models, such as K-nearest neighbours, to build a model capable of discrim-inating between graphs with different number of layers.This approach suffers from several issues. Most promi-nently it requires a training sample. This can be verydifficult to obtain in real world scenarios and while per-haps a well varied synthetic data set could suffice at thispoint it is a mere speculation. The choice of appropri-ate machine learning scheme and its construction is alsoa non-trivial task. Additionally as the number of lay-ers grows differentiating between such networks becomes increasingly difficult since the signatures become less var-ied.We also introduce an approach that utilises a discreteFourier transform (DFT) instead of a machine learningmodel. Instead of histogramming the measurements asbefore one computes the some of amplitudes on the vis-ible layer at each integer time. This constitutes a sig-nal that at sufficient time scales should become station-ary. In a mono-layer system the signal will simply bea constant value due to energy conservation. However,should other layers be in the system from the perspec-tive of the mono-layer there will be oscillations as theenergy will flow out and back into it. We can inspectthose oscillations with the use of DFT and look at thepower spectrum. The spectrum will exhibit characteris-tic peaks absent in mono-layer networks. The numberof these peaks strictly corresponds with the number oflayers in the synthetic scenarios tested. This approach issignificantly advantageous over the WPS as it does notrequire building a learning sample and is in general muchsimpler. Furthermore it does not really suffer in terms ofdifferentiating e.g. tetra- from penta-layer systems etc.Although it shows much promise in synthetic scenarios,it does not perform as well in real world networks. It doesindeed indicate clearly that there are hidden layers butthe number of them can be rather tricky to discern. Thisis perhaps not that surprising considering that real worldnetworks are much more “messy” in some way than syn-thetic examples. Layers vary in size, degree distribution,clustering coefficients and so on and so forth while say apenta-layer BA graph shares all characteristics betweenlayers even though the exact connections are different.Those and other features of real world systems could alsoaffect the coupling amongst the layers that most certainlywill affect the nature of the amplitude signal.In such cases (i.e. where peak count after a brief obser-vation is not enough) we show that simply a longer ob-servation time is required. As the signal is not stochasticand oscillation periods is finite it does not seem unfeasi-ble to observe enough of the signal to determine its powerspectrum with sufficient resolution. Then each peak inthe spectrum correspond to the Hamiltonian eigenvaluesthat in turn are related to the eigenvalues of the row-normalized adjacency matrix via simple formula. Withthis we showed that it is indeed possible to recover allthese eigenvalues and thus trivially determine the num-ber of layers in the system.It is worth underlining here that a row-normalized ad-jacency matrix is in fact a so called right stochastic ma-trix of a given graph and while it goes beyond the scopeof this paper, there exist methods of reconstructing thewhole matrix from its spectrum [61–65] which we suspectshould be quite feasible considering we already assumeknowing part of it (one layer and inter-layer structure).That in turn could also open the door to the adjacency matrix itself. Recovering all the connections exactly maynot be possible, however, having a matrix isospectral tothe adjacency matrix is also very valuable as having thisspectrum allows for determining many important prop-erties of the system [66, 67].We find that both methods presented in this paper -the WPS and FFT - show enough success in these sim-ple scenarios we tested to merit further study, such asin noisy (or stochastic) systems, for instance. They eachhave their pros and cons that we hopefully managed tooutline clearly as well as the potential room for improve-ment of their applicability and understanding of waveson quantum graphs alike. ACKNOWLEDGMENTS
This work has been supported by National ScienceCentre, Poland Grant No. 2015/19/B/ST6/02612.
Appendix A: Calculation of the wave amplitude1. Overall description
The general solution of the wave equation on graph G , provided that the initial condition is Gaussian packetfully localised on a given edge f is derived and presentedin detail by Aziz et al. in [35]. Here, we re-write it interms of integer times, i.e., t = 0 , , , ... so that it fitsthe case examined in the main text. We additionally as-sume that the graph is unweighted, undirected and non-bipartite. In such a setting, we consider an arbitrary edge e = { u, v } that connects two vertices u and v and can beassociated with a variable x e ∈ [0 ,
1] that represents co-ordinate along such an edge. Then the amplitude u ofthe wave in the middle of edge e can be expressed as u ( e, f, t ) = u ( e, f, t ) + u ( e, f, t ) + 1 | E | , (A1)where | E | is the number of edges in the graph and u and u are defined as follows u ( e, f, t ) = (cid:80) ω ∈ Ω C ( e, ω ) C ( f, ω ) cos( B ( e, ω ) + ω ) cos( B ( f, ω ) + ω ( + t )) u ( e, f, t ) = 2 cos( πt ) (cid:80) i C π ( e, i ) C π ( f, i ) (A2)In the above equations ω , C ( e, ω ) and B ( e, ω )come from the edge-based eigenvalues and eigenfunc-tions, which are, respectively ω and φ ( e, x e ) = ± C ( e, ω ) cos( B ( e, ω ) + ωx e ) of the row-normalized ad-jacency matrix ˆ A of the graph G . Assuming that weknow vertex-based eigenvector-eigenvalues pairs ( g ( v ), λ ) of matrix ˆ A we can express C ( e, ω ) and B ( e, ω ) as C ( e, ω ) = g ( v,ω ) + g ( u,ω ) − g ( u,ω ) g ( v,ω ) cos ω sin ω tan B ( e, ω ) = g ( v,ω ) cos ω − g ( u,ω ) g ( v,ω ) sin ω , (A3)while ω = arccos λ . The sign of C ( e, ω ) needs to chosenin order to match the phase, in practice it can be achivedby calculating sgn[ g ( v )] | C ( e, ω ) | It is always true that oneof the eigenvalues is equal to 1 (consequently, ω = 0):this value is responsible for the constant term 1 /E in Eq.(A2) so it is not included in further calculations, i.e., it does not belong to Ω set in function u . Although φ ( e, x e )are orthogonal, they still need to be normalised. To fulfilthis condition for each ω ∈ Ω we calculate normalisationfactor ρ ( ω ) ρ ( ω ) = (cid:115)(cid:88) e C ( e, ω ) (cid:20)
12 + sin(2 ω + 2 B ( e, ω )) − sin(2 B ( e, ω ))4 ω (cid:21) (A4)where e runs over all edges in graph G . Then, in orderto obtain properly normalised value of C ( e, ω ) one needsto divide it by ρ ( ω ).For calculations of C π one first needs to transform theoriginal undirected graph G into a directed one D ( G ) bysimply replacing each edge e = { u, v } with two arcs ( u, v )and ( v, u ). In the next step we create a structure calledoriented line graph ( OLG ), constructed by substitutingeach arc of D ( G ) by a vertex (such vertices are connectedif the head of one arc meets the tail of another arc).Using the adjacency matrix A olg of the OLG we solvethe eigenproblem A olg g olg = λ olg g olg and then restrictourselves to λ olg = − | E | − | V | linearly independentsolutions) that form C π .
2. An example
In order to make the above concise description clear,let us follow a very simple example of a graph shown inFig. 10a. In such case, knowing the adjacency matrix A where A ij = 1 if nodes i and j share a link and A ij = 0otherwise, we can write the row-normalised adjacency matrix ˆ A ij = A ij / (cid:80) k A kj asˆ A =
13 13 1312 . (A5)Solving the eigenproblem ˆ Ag = λ g one obtains in thiscase the following eigenvectors g = ω ω ω ω (cid:113) √ − − √ − √ − (cid:113) − √ − − √ √ − (A6)and related eigenvalues λ = (cid:0) − , − , , (cid:1) . Each col-umn of g corresponds to different ω and consecutiverows are node numbers. As mentioned before, λ = 1is not taken into account in further calculations, so ω = { ω , ω , ω } = { arccos( − ) , arccos( − ) , π } . Now,having calculated g and ω we are able to obtain C ( e, ω and B ( e, ω ) as described in Eq. (A3). To simplify theoutcome we show it as matrices with rows denoted bygraph edges and columns — by ω values: C = ω ω ω e − √ − √ e − √ √ − √ e − √ − √ e √ − √ e √ − √ e − √ √ √ e − √ √ e − √ √ e √ √ e √ √ B = ω ω ω e arctan √ − arctan √ π e − arctan √ √ e arctan √ − arctan √ − π e − π e π e − arctan √ √ e arctan √ − arctan √ π e arctan √ − arctan √ − π e − π e π C needs to by divided by a cor- responding value of ρ ( ω ) given by Eq. (A4), i.e., in the0 FIG. 10. (a) An example of a simple graph consisting of | V | = 4 nodes and | E | = 5 edges. (b) Oriented Line Graph obtainedfrom the graph depicted in panel (a). case of the exemplary graph ρ = { , , } . In this waywe possess full information needed to evaluate values of u .Figure 10b presents an Oriented Line Graph obtainedfrom the graph shown in Fig. 10a, its adjacency matrix A olg being simply A olg = e e e e e e e e e e e e e e e e e e e e . (A8)We deliberately refrain from showing the full matrix ð olg of eigenvectors of A olg as in our case | E | − | V | = 1 sothere is exactly one eigenvector corresponding to λ = − C π = e e e e e e e e e e (cid:0) (cid:1) √ − √ √ − √ − √ √ − √ √ . (A9)It is now easy to check that if substitute Eq. (A2) withthe calculated matrices C , B , ω and C π and assumethat the wave is initially localised on edge f = { , } and t = 0, the amplitude u = 1 for e = f = { , } and u = 0 in any other case, as expected. The first 10 steps ofpropagation can be depicted in Fig. 11 (the wave movestoward v = 2). FIG. 11. Wave propagation on a graph shown in Fig. 10a forfirst 10 time steps (initial condition: a Gaussian wave fullycontained on edge f = { , } moving toward vertex v = 2). Appendix B: Principal component analysis
Principal components are a sequence of projections ofthe set of data in R p , mutually uncorrelated and orderedin variance in in R q where q ≤ p [68]. In other wordswe transform the feature space such that it becomes or-thogonal and each consecutive feature is aligned in thedirection maximising the variance of the data and hasmore variance than the last. We do that by minimising1the reconstruction error, i.e. solving:min V q N (cid:88) i =1 || ( x i − ¯ x ) − V q V Tq ( x i − ¯ x ) || (B1)where V q is a p × q matrix with q orthogonal unit vec-tors as columns. A p × p matrix V q V Tq is the transforma-tion matrix that maps each p -dimensional observationinto its q -rank reconstruction. In our case specifically p = q = 100 and the examples of transformation ma-trices are represented as heat-maps in Fig. 3 and 4. Ingeneral PCA is know to be a quick and easy method to (i)perform dimensional reduction, (ii) help to visualise high-dimensional data and (iii) aggregate high-dimensionaldata into a possibly single measure (see, e.g., [69–71]). Appendix C: K-nearest neighbours
In K-NN classification method the class estimationˆ y ( x ) of a given sample x is taken as a majority voteamongst the member of N K ( x ) - the neighbourhood of x defined as K points closest to x [68, 72]. To determinewhich points are closest a metric must be chosen andfor the purposes of this paper a Euclidean distance wasused. In our case specifically each observation is a graphrepresented by it’s WPS, i.e. each graph is a point in a100-dimensional space. Appendix D: Fourier analysis
Fourier analysis allows us to convert a given time de-pendent signal f ( t ) onto a frequency domain into ˆ f ( ω )via a Fourier transform and thus acquire the frequencydistribution of said signal as it becomes a linear com-bination of trigonometric functions each correspondingto a particular frequency. A discrete Fourier transformis as name suggest a discrete version where integrationis replaced by summation [73]. Therefore we consider aproblem where one wants to express f ( t ) as a complexFourier series: ˆ f ( ω ) = N − (cid:88) k =0 f ( k ) e πiω/N (D1)This procedure as it stands would require N opera-tions (where each operation is a complex multiplica-tion followed by a complex addition), however, Cooleyand Tukey in [57] presented a method known as the fast Fourier transform that allows to do it in less than2 N log N . Appendix E: Real world networks
We use three real world networks to test the Fouriertransform approach.Vickers et al. [54] collected data from 29 7th grade stu-dents from Victoria, Australia. Students were asked tonominate classmates in several categories, three of whichwere used to construct this 3-layer network. These threecategories were determined by questions - Who do youget on with in the class? Who are your best friends in theclass? Who would you prefer to work with? The graphhas 29 nodes and 740 edges in total.Krackhardt [58] took a record of relationship betweenmanagers in a high-tech company. The graph has 21nodes and 312 edges in a 3-layer form. Each layer repre-sents a relationship (advice, frendship, ”reports to”).Chen et al. [59] presented a
Caenorhabditis elegans multiplex connectome network with 3 layers, 279 nodesand 5863 edges. Each layer corresponds to a differ-ent synaptic junction: electric, chemical monadic, andpolyadic.
Appendix F: Peak detection using a wavelettransform
A wavelet transform is an analogous procedure to theFourier transform in the sense that we represent a givensignal as an orthonormal series [74]. In case of Fourierthose are sine and cosine while in the wavelet those arethe eponymous wavelets. A wavelet is a particularly cho-sen function that is localised, i.e. it has a finite widththe its family can compose an orthonormal basis for thesignal - s ( t ). C ( a, b ) = (cid:90) R s ( t ) 1 √ a ψ (cid:18) t − ba (cid:19) dt, a ∈ R + , b ∈ R (F1)In our case the wavelet - ψ - was a Morlet (also known asthe mexican hat) one as per the procedure described in[60] which (simplified) is as follows: perform a continu-ous wavelet transform (CWT) on the signal, identify theridge lines by linking local maxima of CWT at each scalelevel, identify the peaks based on the ridge lines withthree rules (quoted verbatim): “(1) The scale correspond-ing to the maximum amplitude on the ridge line,which isproportional to the width of the peak, should be withinacertain range;(2) The SNR should be larger than a certainthreshold;(3) The length of ridge lines should be largerthan a certain threshold;”. Here SNR is a signal to noiseratio . [1] Manlio De Domenico, Albert Sol´e-Ribalta, EmanueleCozzo, Mikko Kivel¨a, Yamir Moreno, Mason A Porter, Sergio G´omez, and Alex Arenas. Mathematical for- mulation of multilayer networks. Physical Review X ,3(4):041022, 2013.[2] Manlio De Domenico, Clara Granell, Mason A Porter,and Alex Arenas. The physics of spreading processesin multilayer networks.
Nature Physics , 12(10):901–906,2016.[3] Guilherme Ferraz de Arruda, Francisco A Rodrigues, andYamir Moreno. Fundamentals of spreading processes insingle and multilayer complex networks.
Physics Reports ,756:1–59, 2018.[4] Mikko Kivel¨a, Alex Arenas, Marc Barthelemy, James PGleeson, Yamir Moreno, and Mason A Porter. Multilayernetworks.
Journal of complex networks , 2(3):203–271,2014.[5] Stefano Boccaletti, Ginestra Bianconi, Regino Criado,Charo I Del Genio, Jes´us G´omez-Gardenes, Miguel Ro-mance, Irene Sendina-Nadal, Zhen Wang, and Massim-iliano Zanin. The structure and dynamics of multilayernetworks.
Physics Reports , 544(1):1–122, 2014.[6] Alain Barrat, Marc Barthelemy, and Alessandro Vespig-nani.
Dynamical processes on complex networks . Cam-bridge university press, 2008.[7] Romualdo Pastor-Satorras, Claudio Castellano, PietVan Mieghem, and Alessandro Vespignani. Epidemic pro-cesses in complex networks.
Reviews of modern physics ,87(3):925, 2015.[8] Andrey Lokhov. Reconstructing parameters of spreadingmodels from partial observations. In
Advances in NeuralInformation Processing Systems , pages 3467–3475, 2016.[9] Mateusz Wilinski and Andrey Y Lokhov. Scalable learn-ing of independent cascade dynamics from partial obser-vations. arXiv preprint arXiv:2007.06557 , 2020.[10] Jure Leskovec and Andreas Krause. Inferring networksof diffusion and influence.
In KDD10 , 2010.[11] Jiin Woo, Jungseul Ok, and Yung Yi. Iterative learn-ing of graph connectivity from partially-observed cas-cade samples. In
Proceedings of the Twenty-First In-ternational Symposium on Theory, Algorithmic Founda-tions, and Protocol Design for Mobile Networks and Mo-bile Computing , pages 141–150, 2020.[12] Jean Pouget-Abadie and Thibaut Horel. Inferring graphsfrom cascades: A sparse recovery framework. In
Proceed-ings of the 24th International Conference on World WideWeb , pages 625–626, 2015.[13] Bruno Abrahao, Flavio Chierichetti, Robert Kleinberg,and Alessandro Panconesi. Trace complexity of networkinference. In
Proceedings of the 19th ACM SIGKDD in-ternational conference on Knowledge discovery and datamining , pages 491–499, 2013.[14] Vincent Gripon and Michael Rabbat. Reconstructing agraph from path traces. In , pages 2488–2492. IEEE,2013.[15] Manuel Gomez-Rodriguez, Jure Leskovec, and AndreasKrause. Inferring networks of diffusion and influence.
ACM Transactions on Knowledge Discovery from Data(TKDD) , 5(4):1–37, 2012.[16] Praneeth Netrapalli and Sujay Sanghavi. Learning thegraph of epidemic cascades.
ACM SIGMETRICS Per-formance Evaluation Review , 40(1):211–222, 2012.[17] Alfredo Braunstein, Alessandro Ingrosso, andAnna Paola Muntoni. Network reconstruction frominfection cascades.
Journal of the Royal SocietyInterface , 16(151):20180844, 2019. [18] (cid:32)LG Gajewski, Krzysztof Suchecki, and JA Ho(cid:32)lyst. Mul-tiple propagation paths enhance locating the source ofdiffusion in complex networks.
Physica A: Statistical Me-chanics and its Applications , 519:34–41, 2019.[19] Robert Paluch, Xiaoyan Lu, Krzysztof Suchecki,Boles(cid:32)law K Szyma´nski, and Janusz A Ho(cid:32)lyst. Fast andaccurate detection of spread source in large complex net-works.
Scientific reports , 8(1):1–10, 2018.[20] Robert Paluch, (cid:32)Lukasz G Gajewski, Janusz A Ho(cid:32)lyst,and Boleslaw K Szymanski. Optimizing sensors place-ment in complex networks for localization of hidden sig-nal source: A review.
Future Generation Computer Sys-tems , 112:1070–1092, 2020.[21] Lucas Lacasa, In´es P. Mari˜no, Joaquin Miguez, VincenzoNicosia, ´Edgar Rold´an, Ana Lisica, Stephan W. Grill,and Jes´us G´omez-Garde˜nes. Multiplex decompositionof non-markovian dynamics and the hidden layer recon-struction problem.
Physical Review X , 8(3), Aug 2018.[22] Peter Kuchment. Quantum graphs: I. Some basic struc-tures.
Waves in Random media , 14(1):S107–128, 2004.Publisher: Taylor & Francis.[23] Peter Kuchment. Quantum graphs: II. Some spectralproperties of quantum and combinatorial graphs.
Journalof Physics A: Mathematical and General , 38(22):4887,2005. Publisher: IOP Publishing.[24] Peter Kuchment. Quantum graphs: an introduction anda brief survey. arXiv preprint arXiv:0802.3442 , 2008.[25] Gregory Berkolaiko and Peter Kuchment.
Introductionto quantum graphs . American Mathematical Soc., 2013.Issue: 186.[26] Joel Friedman and Jean-Pierre Tillich. Wave equationsfor graphs and the edge-based Laplacian.
Pacific Journalof Mathematics , 216(2):229–266, 2004. Publisher: Math-ematical Sciences Publishers.[27] Norman E Hurt.
Mathematical physics of quantum wiresand devices: From spectral resonances to Anderson local-ization , volume 506. Springer Science & Business Media,2013.[28] Linus Pauling. The diamagnetic anisotropy of aromaticmolecules.
The Journal of Chemical Physics , 4(10):673–677, 1936.[29] Peter Kuchment. Graph models for waves in thin struc-tures.
Waves in random media , 12(4):R1–R24, 2002.[30] Jacob Biamonte, Mauro Faccin, and ManlioDe Domenico. Complex networks from classical toquantum.
Communications Physics , 2(1):1–10, 2019.Publisher: Nature Publishing Group.[31] Pavel Exner and Olaf Post. Quantum networks modelledby graphs. In
AIP Conference Proceedings , volume 998,pages 1–17. American Institute of Physics, 2008.[32] Mauro Faccin, Piotr Migda(cid:32)l, Tomi H Johnson, VilleBergholm, and Jacob D Biamonte. Community detec-tion in quantum complex networks.
Physical Review X ,4(4):041012, 2014. Publisher: APS.[33] Mauro Faccin, Tomi Johnson, Jacob Biamonte, SabreKais, and Piotr Migda(cid:32)l. Degree distribution in quan-tum walks on complex networks.
Physical Review X ,3(4):041007, 2013. Publisher: APS.[34] Mart´ı Cuquet and John Calsamiglia. Entanglement per-colation in quantum complex networks.
Physical reviewletters , 103(24):240503, 2009. Publisher: APS.[35] Furqan Aziz, Richard C Wilson, and Edwin R Hancock.A wave packet signature for complex networks.
Journal of Complex Networks , 7(3):346–374, June 2019.[36] Vadim Kostrykin, J¨urgen Potthoff, and Robert Schrader.Finite propagation speed for solutions of the wave equa-tion on metric graphs. Journal of Functional Analysis ,263(5):1198–1223, 2012. Publisher: Elsevier.[37] Sergio Gomez, Albert Diaz-Guilera, Jesus Gomez-Gardenes, Conrad J Perez-Vicente, Yamir Moreno, andAlex Arenas. Diffusion dynamics on multiplex networks.
Physical review letters , 110(2):028701, 2013.[38] Albert Sole-Ribalta, Manlio De Domenico, Nikos E Kou-varis, Albert Diaz-Guilera, Sergio Gomez, and Alex Are-nas. Spectral properties of the laplacian of multiplexnetworks.
Physical Review E , 88(3):032807, 2013.[39] Furqan Aziz, Richard C Wilson, and Edwin R Hancock.Analysis of wave packet signature of a graph. In
Interna-tional Conference on Computer Analysis of Images andPatterns , pages 128–136. Springer, 2013.[40] Furqan Aziz, Richard C Wilson, and Edwin R Hancock.Graph characterization using gaussian wave packet sig-nature. In
International Workshop on Similarity-BasedPattern Recognition , pages 176–189. Springer, 2013.[41] Richard C. Wilson, Furqan Aziz, and Edwin R. Hancock.Eigenfunctions of the edge-based laplacian on a graph.
Linear Algebra and its Applications , 438(11):4183 – 4189,2013.[42] Isaac Pesenson. Analysis of band-limited functions onquantum graphs.
Applied and Computational HarmonicAnalysis , 21(2):230–244, 2006. Publisher: Elsevier.[43] Isaac Pesenson. Band limited functions on quantumgraphs.
Proceedings of the American Mathematical Soci-ety , 133(12):3647–3655, 2005.[44] Pavel Exner and Diana Barseghyan. Spectral analysis ofSchr¨odinger operators with unusual semiclassical behav-ior.
Acta Polytechnica , 53(3), 2013.[45] Carla Cattaneo. The spectrum of the continuousLaplacian on a graph.
Monatshefte f¨ur Mathematik ,124(3):215–235, 1997. Publisher: Springer.[46] Mark Kac. Can one hear the shape of a drum?
Theamerican mathematical monthly , 73(4P2):1–23, 1966.[47] Carolyn Gordon, Peter Perry, and Dorothee Schueth.Isospectral and isoscattering manifolds: a survey oftechniques and examples.
Contemporary Mathematics ,387:157–180, 2005.[48] Ram Band, Adam Sawicki, and Uzy Smilansky. Scatter-ing from isospectral quantum graphs.
Journal of PhysicsA: Mathematical and Theoretical , 43(41):415201, 2010.[49] R Band, A Sawicki, and U Smilansky. Note on the roleof symmetry in scattering from isospectral graphs anddrums.
Acta Physica Polonica A , 120(6A), 2011.[50] Oleh Hul, Micha(cid:32)l (cid:32)Lawniczak, Szymon Bauch, Adam Saw-icki, Marek Ku´s, and Leszek Sirko. Are scattering proper-ties of graphs uniquely connected to their shapes?
Phys.Rev. Lett. , 109:040402, Jul 2012.[51] Uzy Smilansky. Delay-time distribution in the scatteringof time-narrow wave packets.(i).
Journal of Physics A:Mathematical and Theoretical , 50(21):215301, 2017.[52] Uzy Smilansky and Holger Schanz. Delay-time distri-bution in the scattering of time-narrow wave packets(ii)—quantum graphs.
Journal of Physics A: Mathemat-ical and Theoretical , 51(7):075302, 2018.[53] Michael Robinson. Inverse problems in geometric graphsusing internal measurements, 2010.[54] M Vickers and S Chan. Representing classroom socialstructure.
Victoria Institute of Secondary Education, Melbourne , 1981.[55] Joel Friedman and Jean-Pierre Tillich. Calculus ongraphs. arXiv preprint cs/0408028 , 2004.[56] Ulrik Brandes. A faster algorithm for betweenness cen-trality.
Journal of mathematical sociology , 25(2):163–177,2001.[57] James W Cooley and John W Tukey. An algorithm forthe machine calculation of complex fourier series.
Math-ematics of computation , 19(90):297–301, 1965.[58] David Krackhardt. Cognitive social structures.
Socialnetworks , 9(2):109–134, 1987.[59] Beth L Chen, David H Hall, and Dmitri B Chklovskii.Wiring optimization can relate neuronal structure andfunction.
Proceedings of the National Academy of Sci-ences , 103(12):4723–4728, 2006.[60] Pan Du, Warren A. Kibbe, and Simon M. Lin. Improvedpeak detection in mass spectrum by incorporating contin-uous wavelet transform-based pattern matching.
Bioin-formatics , 22(17):2059–2065, 07 2006.[61] Moody Chu, Moody T Chu, Gene Golub, and Gene HGolub.
Inverse eigenvalue problems: theory, algorithms,and applications , volume 13. Oxford University Press,2005.[62] Gabriele Steidl and Maximilian Winkler. A new con-strained optimization model for solving the nonsymmet-ric stochastic inverse eigenvalue problem. arXiv preprintarXiv:2004.07330 , 2020.[63] Filippo Cacace, Alfredo Germani, and Costanzo Manes.Karpelevich theorem and the positive realization of ma-trices. In , pages 6074–6079. IEEE, 2019.[64] Zhi Zhao, Xiao-Qing Jin, and Zheng-Jian Bai. A geomet-ric nonlinear conjugate gradient method for stochastic in-verse eigenvalue problems.
SIAM Journal on NumericalAnalysis , 54(4):2015–2035, 2016.[65] Robert Orsi. Numerical methods for solving inverseeigenvalue problems for nonnegative matrices.
SIAMJournal on Matrix Analysis and Applications , 28(1):190–212, 2006.[66] Rub´en J S´anchez-Garc´ıa, Emanuele Cozzo, and YamirMoreno. Dimensionality reduction and spectral prop-erties of multilayer networks.
Physical Review E ,89(5):052815, 2014.[67] Emanuele Cozzo, Guilherme Ferraz de Arruda, Fran-cisco A Rodrigues, and Yamir Moreno. Multilayer net-works: metrics and spectral properties. In
InterconnectedNetworks , pages 17–35. Springer, 2016.[68] Trevor Hastie, Robert Tibshirani, and Jerome Friedman.
The elements of statistical learning: data mining, infer-ence, and prediction . Springer Science & Business Media,2009.[69] (cid:32)LG Gajewski, J Cho(cid:32)loniewski, and JA Ho(cid:32)lyst. Keycourses of academic curriculum uncovered by data miningof students’ grades.
Acta Physica Polonica, A. , 129(5),2016.[70] Julian Sienkiewicz, Krzysztof Soja, Janusz A Ho(cid:32)lyst, andPeter MA Sloot. Categorical and geographical separationin science.
Scientific reports , 8(1):1–12, 2018.[71] Jan Cho(cid:32)loniewski, Julian Sienkiewicz, Naum Dretnik,Gregor Leban, Mike Thelwall, and Janusz A. Ho(cid:32)lyst. Acalibrated measure to compare fluctuations of differententities across timescales.