arXiv:1609.08355v2 [physics.soc-ph] 13 Apr 2017 3 2 India 453552, Lo spectra networks in degeneracies Analysing draft epl 1 h opeeadteprildpiain 1]o nodes of [14] duplications from partial resulted the be to and exam- shown For complete been has the complex. degeneracy degen- more 0 of the are origins ple, eigenvalues that out these and turns at 0 it eracy and at exhaustive degeneracy not a are show still can absence − cliques in and graphs stars However, of [11–13]. respectively and cliques 0 related have papers Few curneo eeeaya 0, is at cor- properties degeneracy these the of of of occurrence One those [8–10]. graphs from model different exhibit responding very networks are real-world inves- which Recent that properties revealed 7]. have [6, using modeled tigations statistics eigenvalue be ex- to generalized whereas shown the to [5], been shown theory have eigenvalues matrix [4]. been tremal random graph has using bipartite spec- eigenvalues modeled a symmetric of be to a portion related that bulk is Among Further, noted origin the been 3]. about [2, has trum graphs it of things, properties other spectra dynamical graph and struc- of of tural understanding analysis our in systems Further, significantly contributed real-world has networks. large of provided very terms later analyse in sys- power to small computing opportunity to in representing an Revolution restricted graphs was size. of work tem concept The a systems. introducing complex by [1] ory noe435,India 453552, Indore rde fK of Bridges ievle,rsetvl.A eut hs reasons these result, a As respectively. eigenvalues, 1 etefrBocecsadBoeia niern,Indian Engineering, Biomedical and Biosciences for FranceCentre Cedex, Orsay 91405 Universit´e Paris-Sud, ope ytm a,Dsiln fPyis ninInstitu Indian Physics, of Discipline Lab, Systems Complex h ae rte yLohr ue nthe on Euler Leonhard by written paper The ¨ cMarrec ıc Abstract odgnrc hc ih aea motn oei network a in role important an have might opportunit which an degeneracy gives i to approach observed This rarely networks. are light cancer various degeneracies shed network eigenvalues these the Since degenerate in the degeneracy. multiplicities with high associated mat of tors adjacency occurrence network’s of the of origins transformation simple a that PACS PACS onigsberg ¨ 1 02.10.Yn 89.75.-k , 2 and ayra-ol ewrsehbtahg eeeaya e e few at degeneracy high a exhibit networks real-world Many – aiaJalan Sarika arxtheory Matrix – ope systems Complex – ak einn fgahthe- graph of beginning a marks − − ievle osasand stars to eigenvalues 1 and 1 1 − , 3 ievle [2]. eigenvalues 2 Seven p-1 nietdgahi ymti ih0ad1etis the eigenvectors entries, associated 1 The and 0 real. with are symmetric eigenvalues is graph undirected ,..., λ ftecaatrsi oyoilo h daec matrix, adjacency the of χ polynomial characteristic the of e uhas such set eemndb t daec arxfrwihiselement its which for matrix A adjacency its by determined daec arxwl edntdby denoted be will matrix adjacency j de hstp fgah sepcal neetn since at interesting unique multiplicity especially a high very is by a graphs connected exhibit of is spectra their type nodes This of pair edge. every which for 1 hs esn eel xsec fcaatrsi structures networks. characteristic of in of one existence reveals cases, reasons particular explain In these to eigenvalue. emerge every reasons of two that degeneracy We following in the eigenvalues. lies in see degenerate interest will re- other an evolution of hence origins [15], in investigating duplication process gene fundamental with on lated light as shed systems biological they for interesting particularly are which ndby fined n tews.I h olwn,terows the following, the In otherwise. 0 and , , 1 A ij eo ehooyIdr,KadaRa,Smo,Indore Simrol, Road, Khandwa Indore, Technology of te 2 h ievle r bandb optn h roots the computing by obtained are eigenvalues The opeegah eoe by denoted graph, complete A nti ae,w osdrfiieudrce rpsde- graphs undirected finite consider we paper, this In nttt fTcnlg noe hnw od Simrol, Road, Khandwa Indore, Technology of Institute ( .,N ..., , ≤ oe rps epeetrslsfor results present we graphs, model n pcr.W n htteeigenvec- the that find We spectra. s λ v osac o tutrscontributing structures for search to y s1we hr sa defo vertex from edge an is there when 1 is ntesrcue otiuigt the to contributing structures the on det( = ) i rvdsa nesadn othe to understanding an provides rix N λ 2 aif h eigen-equation the satisfy . G . ≤ = | ... V { A ,E V, | ≤ = − λ N } λI N gnaus eshow We igenvalues. with and ic h daec arxo an of matrix adjacency the Since . = ) | V E Q h oest and set, node the i N | =1 = ( m K λ rp scompletely is graph A . sa nietdgraph undirected an is , − A v λ R i i i n eoe by denoted and ) . = λ i v i E i i − overtex to with h edge the fevery of eigen- 1 v 1 , i v = 2 Lo¨ıcMarrec1,2 Sarika Jalan1,3 value. Specifically, a complete graph of N nodes has N −1 degeneracies for −1 eigenvalue [2]. However, it is mislead- 2 4 ing to associate this special graph structure with −1 de- generacy. Let us take as example the 5 nodes complete 1 3 5 graph in which we have removed an edge. In the result- K S ing graph, two −1 eigenvalues are retained whereas the (a) (b) globally connected structure is destroyed, which indicates that the globally connected structure is not sufficient to Fig. 1: (Color online) (a) and (b) verify the condition (ii) and explain occurrence of −1 degeneracy. We will see in the (iii) in A + I, respectively. following that only one type of particular structure con- sisting of a complete graph and its variants contribute to −1 eigenvalue. (Figure 1) contribute to −1 eigenvalue with multiplicity We consider the matrix A + I, where I is the identity n − 1. matrix, and we make a change of variables in the char- Next, we emphasize on the condition (iii), which for ex- ample may correspond to R1 +R2 = R3 +R4 in the matrix acteristic polynomial such as χA+I (λ) = χA(µ). By this way, µ is an eigenvalue of A if and only if λ is an eigen- A + I. For this particular case, the condition (iii) is satis- value of A+I. We can also prove that they have the same fied if and only if a1,j +a2,j = a3,j +a4,j for j =1, 2, ..., N. multiplicity. Hence, it is possible to reduce the computa- On this basis, we can clarify that the condition (iii) im- tion of −1 eigenvalue of A to the 0 eigenvalue of A + I. plies for the node j such as j = 5, 6, ..., N. Whether the This is especially interesting since the origin and impli- node j is adjacent (respectively non-adjacent) to the both cations of occurrence of 0 degeneracy in networks spec- nodes labelled 1 and 2, it is also adjacent (respectively tra are well characterized [14,16]. Spectrum of a matrix non-adjacent) to the both nodes 3 and 4. In the case of size N and rank r contains 0 eigenvalue with multi- where the node j is either connected to 1 or 2, the condi- plicity N − r. Three conditions lead to the lowering of tion (iii) imposes that j is either connected to 3 or 4 (see the rank of a matrix; (i) if the network has an isolated Table 1). node (R =0 ··· 0 ······ 0), (ii) At least two rows are equal i Node 1 Node 2 Node 3 Node 4 (Ri = Rj ), (iii) Two or more rows together are equal to 1 0 1 0 some other rows ( i aiRi = j bj Rj , where ai and bi take integer value included 0). In the case of A + I, it is 1 0 0 1 j obvious that the conditionP (i) isP never met. We focus now Node 0 1 1 0 on the condition (ii). Let us consider a network of size N 0 1 0 1 1 1 1 1 for which two nodes labelled 1 and 2 verify R1 = R2 in the adjacency matrix added to the identity matrix. 0 0 0 0 Table 1: Node j, such as j = 5, 6, ..., N, is adjacent (respec- 1 a1,2 ··· a1,N tively non-adjacent) to node i, such as i = 1, 2, 3 and 4, if the a1,2 1 ··· a2,N corresponding entry equals to 1 (respectively 0). A + I = . . . . (1) . . .. . Let us now have a closer look at constraints which nodes a1,N a2,N ··· 1 1, 2, 3 and 4 must obey. Since we have a = 1, a = a i,i i,j j,i The condition (ii) is verified for any pair of rows, say and by considering the previous constraints: st nd 1 and 2 , if and only if a1,2 = 1 and a1,i = a2,i for i =3, 4, ..., N 1+ a1,2 = a1,3 + a1,4 a +1= a + a So, the adjacency matrix A takes the following form : 1,2 2,3 2,4 (3) a + a =1+ a 1,3 2,3 3,4 a1,4 + a2,4 = a3,4 +1 0 1 ··· a1,N 1 0 ··· a1,N This set has more unknown variables than the number A = . . . . (2) . . .. . of equations. The system is underdetermined and has in- a a ··· 0 finitely many solutions. As a result, it is difficult to define 1,N 1,N a typical structure which corresponds to the condition (iii). The nodes 1 and 2 are interlinked and connected to the Here we will limit ourselves to illustrate it with the graph same set of other nodes. The rank of A + I is N − 1, and (b) in Figure 1 for which the adjacency matrix A added hence we deduce that the spectrum associated to the A to the identity matrix satisfies R1 + R4 = R2 + R5. matrix contains exactly one −1 eigenvalue. It is trivial This relation is at the origin of −1 eigenvalue observed to generalize this proof to the case R1 = R2 = ... = Rn. in the spectrum of A. More generally, each linear combi- Hence, n nodes forming a complete graph K connected to nation of rows in A+I leads to exactly one −1 eigenvalue. a same set S of other different nodes and denoted as K ∗S The power of this approach is that it can be extended to
p-2 Title
where nLC is the number of linear combinations of rows in the network. The derivation of Eq. 5 follows the same reasoning as this of Eq. 4 seeing that Rk = Rl is a partic- ular case of i aiRi = j bj Rj . It is also interesting to note that Eqs. 4 and 5 can be proved using row equiva- lent forms ofP adjacencyP matrices [17]. Since the assump- (a) (b) tions are only based on the conditions (ii) and (iii), the proofs provided in the Supplementary Material show that Fig. 2: (Color online) (a) and (b) have same number of −1 Eqs. 4 and 5 are valid for any arbitrary network. These eigenvalues but different structures. −1 degeneracy of (a) oc- curs through the condition (iii) whereas for (b) it results from properties give the opportunity to find in every network the condition (ii). the nodes which contribute to −1 degeneracy. We can go further by distinguishing the nodes which satisfy the con- dition (ii) and those which satisfy the condition (iii). In all the degenerate eigenvalues. In the case of a network order to do it, we can have a rather easy computation of which exhibits a high multiplicity at the x eigenvalue, it the rows of A + I such as Ri = Rj . Then, by consider- is wise to reduce the computation of x eigenvalue to the ing one of the eigenvectors associated to −1 eigenvalue, study of 0 eigenvalue of A−xI such as χA−xI (λ)= χA(µ). we can find all the non-zero entries. Among these non- In this manner, we are able to understand the origin zero entries, those which do not correspond to the nodes of every degenerate eigenvalue, thus enabling to focus on computed previously, satisfy the condition (iii). Besides their implications. We note that the condition (ii) is never being able to find the nodes leading to −1 degeneracy, we met for λ < −1 and 0 < λ since the entries of the adja- can associate them to (ii) or (iii). As we did in the previ- cency matrix are equal to 0 or 1. ous section, we extend this approach to all the degenerate We have seen that −1 degeneracy in networks spectra eigenvalues. Indeed, eigenvectors of x of A matrix which is related to some typical structures. However, the study satisfy (A − xI)v = λ0v are same as the eigenvectors cor- of eigenvalues and their multiplicities is not sufficient to responding to 0 eigenvalues of A − xI such as Av = λxv. determine the number and size of these structures in net- So, we can find precisely the origin of every degenerate works. For example, graphs of Figure 2 lead to the same eigenvalue, namely the condition (ii) or the condition (iii). number of −1 degeneracy but have different structures. In addition, we are able to identify the nodes which con- The question we ask now is how can we identify nodes tribute to high multiplicity of eigenvalues. In brief, this which contribute to degenerate eigenvalues? In order to approach provides a quantitative measure of degeneracy address this, we consider eigenvectors associated to the in networks spectra. degenerate eigenvalues of A. First we focus on −1 degen- eracy and note that eigenvectors of −1 eigenvalue, such as What we have done so far is finding subgraphs behind Av = λ−1v, are same as the eigenvectors corresponding occurrence of degeneracy. The question we ask now is: to the 0 eigenvalues of A + I which verify (A + I)v = λ0v. do they play a significant role in real-world networks? In We find that most of the entries of such eigenvectors order to answer this, we are going to assess whether ran- (corresponding to 0 eigenvalue) are equal to zero. It turns domness enables to observe this kind of structures using out that non-null entries reveal nodes which contribute various model graphs. The 0 degeneracy being the sub- to decreasing the rank of a matrix. Moreover, nodes be- ject of a previous discussion [14], we focus in the following longing to the same structure, say K ∗ S, are linked by on degeneracy at −1 eigenvalue. First, we consider Erd¨os- the following relation (derivation is in the Supplementary Renyi model (ER) [18] in which each edge has a probability Material [17]): p of existing. Since the edges are placed randomly, most of the nodes have a degree close to the average degree hki hki vi = 0 with vi 6= 0 and p =1, 2, ..., nK∗S of the graph [18]. So, the probability p equals to . i∈Kp (4) N vj∈V \{K1∪K2∪...∪K } =0 P nK∗S We note that such networks are almost surely discon- nected if p < ln(N) [18]. In other words, if the previous where nK∗S is the number of K ∗ S structures in the N equation is verified, there are at least two nodes such that network. This relation, arising directly from Ri = Rj , enables us to distinguish each structure contributing to no path has them as endpoints. We construct ER net- −1 degeneracy through condition (ii). The same reason works with different average degrees hki and sizes N. As holds good for condition (iii). Indeed, nodes belonging to depicted by Figure 3, ER networks exhibit a low degen- a same linear combination verify: eracy for small average degrees. We can explain this by referring to n-complete graphs percolation [19]. Indeed, it has been shown that in such graphs, complete subgraphs v v p , , ..., n i∈(LC)p i = 0 with i 6= 0 and =1 2 LC occur beyond a certain probability p. The threshold of 1 vj∈V \{(LC)1∪(LC)2∪...∪(LC) } =0 P nLC − (5) this percolation is defined as pc(n) = [(n − 1)N] n − 1
p-3 Lo¨ıcMarrec1,2 Sarika Jalan1,3
120 250 10 (a) (b) (c) k 10 20 30 40 50 60 200 8 90 N=1000 N=1000 N=1000 〈k〉=20 λ0 5 10 5 20 25 5 N=2000 150 N=2000 6 0 −1 60 N=3000 −1 N=3000 λ 000000 λ −1 λ λ N=4000 100 N=4000 4 30 N=5000 N=5000 50 2 Table 2: 0 and −1 degeneracy in small-world network without 0 0 0 link rewiring such as N = 1000. Multiplicity of λ eigenvalue 2 4 6 8 10 2 4 6 8 10 0 2 4 6 8-4 (x10 ) λ 〈k〉 〈k〉 pr equals to the number of times that nm is an integer, for m = 1, 2, ...N. Fig. 3: (Color online) (a) and (b) illustrate the effect of the average degree hki on −1 degeneracy for different sizes N in ER and SF networks, respectively. All values are averaged over 20 nodes having the same neighbors. However, the preferen- random realizations of the network. (c) represents the number tial attachment makes it less probable for these nodes to be of 0 eigenvalues depending on the link rewiring probability for a interlinked. So, it is unlikely to find K ∗ S structure in SF 1000 nodes small-world network. The regular graph has degree networks. In the case of hki = 2, we find −1 degeneracy. k = 20. All values are averaged over 20 random realizations of Thanks to the eigenvectors analysis, we observe that only the network. the condition (iii) contributes to high multiplicity in this particular case. As we have already said, there is no typ- ical structure corresponding to this condition. Therefore, [19]. Higher the n, more is the increase in the threshold of it is difficult to provide an explanation to this observation. percolation. Put another way, the more n augments, the more the probability to have n-complete graphs in the ER hki graphs gets diminished. Because p = and by referring N to pc, it is very easy to show that n-complete subgraphs As we have just seen, -1 degeneracy is not observed in ER appear in a network if the average degree is more than and SF networks, leading us to believe that randomness is n−2 −1 not conducive to degeneracy in networks spectra. In order hkic ∼ N n [20]. For example, hkic ∼ 31.62 for n = 3 to convince us, we study small-world (SW) networks con- and N = 1000. In the case where hki is less than hkic, we do not observe n-complete subgraphs in ER networks. structed with the Watts-Strogatz mechanism [22]. This So, the structure K ∗ S being not obtained, the condi- graph model is interesting since it illustrates small-world phenomenon according to which distance between nodes tion (ii) is not met. Whether hkic is less than hki, it is possible to find cliques in the networks. However, ran- increases as the logarithm of the number of nodes in the domness which characterizes the ER model may not give network [23]. The generating mechanism consists of a the opportunity to organized structures such as K ∗ S to regular ring lattice where each node is connected to k emerge. Nevertheless, the previous explanations are not neighbors and for which edges are rewired with proba- exhaustive since the condition (iii) can contribute to −1 bility pr. The spectrum {λ1, λ2, ..., λN } of the Watts- degeneracy. The condition not corresponding to a defined Strogatz graph without a link rewiring, i.e. 1-d lattice structure, it is more difficult to predict its contribution. with circular boundary condition, can be computed by π(N−(m−1))(k+1) Finally, the alert reader may inquire why we observe de- sin( N ) λm = − 1 with m =1, 2, ..., N. This generacies for hki = 2. As it is specified earlier, in the case π(N−(m−1)) sin( N ) ln(N) where p< N is verified, the network is most probably relation leads to λ = −1 if and only if the following con- disconnected. More particularly, for a given N, lower the straint is fulfilled: value of p (and hence hki), higher is the number of iso- lated nodes and two connected nodes [18]. Two connected λ−1 (N−(m−1))(k+1) nodes is a K ∗S structure where K contains two nodes and nm = N λ−1 λ−1 λ−1 N−(m−1) with nm and qm integers S contains no node. That is why -1 degeneracy is existing ( qm 6= N for the low average degree. (6) λ−1 λ−1 We focus now on scale-free (SF) networks [21]. This In other terms, if nm is an integer and qm different as model is particularly interesting since its degree distribu- an integer, then λm = −1”. The same reasoning applied tion follows a power law as also observed for many real- in the case of 0 eigenvalue which leads to: world networks [18]. We generate the SF networks by k N − (m − 1) using a preferential attachment process. Each new node λ0 λ0 nm = with nm integer (7) is attached to the existing nodes with a probability which 2 N is proportional to their degrees. As a result of the power Thanks to these equations, it is possible to predict the λ law, SF networks contain a few high degree nodes and a multiplicity of 0 and −1 eigenvalues. We compute nm for λ large number of low degree nodes and −1 degeneracy is not every value of m, then the number of times where nm is observed in most of the cases (Figure 3). As explained in an integer equals to the multiplicity of λ. In the particular λ−1 λ−1 [14], any two nodes having a low degree are more suscep- case of nm , we ensure that we do not count it when qm tible to connect to the high degree nodes, leading to two is an integer.
p-4 Title
Network N hki p pc(3) λ+1 λ0 λ−1
Table 3: Statistical properties for all the normal and disease networks. The total number of nodes is N, the average degree hki, the probability p of two nodes to be connected by an edge and the threshold of 3-complete percolation pc(3). λ represent the number of eigenvalues in the cancer networks, < C.C > the average clustering coefficient and r the assortativity value.
As reported in Table 2, there is no −1 degeneracy in the (ii) (iii) (ii) (iii) Network λ−1 λ−1 Nλ−1 Nλ−1 network for pr = 0. Mathematically, it may be due to the BreastN 19 2 38 14 supplement constraint that λm must verify to equal −1. BreastD 12 0 21 0 However, we observe a degeneracy at 0 eigenvalue. It is ColonN 18 1 32 6 interesting to notice that the degeneracy does not increase ColonD 10 0 19 0 constantly with an increase in k. Let us now how attempt OralN 13 0 21 0 to understand the probability p affects the multiplicity r OralD 1 0 2 0 of 0 eigenvalue for a small-world network. Figure 3 re- OvarianN 23 8 40 36 veals that the number of 0 eigenvalues decreases quickly OvarianD 13 6 24 28 with pr. More particularly, 0 degeneracy is completely re- ProstateN 34 13 56 58 moved for low link rewiring probability. Simulations for ProstateD 117 18 199 91 different configurations (N, k) yield similar results. As a consequence, the introduction of randomness, even small, Table 4: Number of −1 eigenvalues and nodes by condition, have strong impacts on the multiplicity of eigenvalues. namely the condition (ii) and (iii), for all the normal and dis-
ease networks. λ−1 and Nλ−1 are the number of −1 eigenvalues and nodes respectively. We substantiate the previous results by considering ex- amples of few real-world networks. We analyse protein- protein interaction networks (PPI) of five cancers namely All these point out that there exists no obvious causal link Breast, Colon, Oral, Ovarian and Prostate [24]. The PPI between these macroscopic properties and degeneracy in networks have proteins as nodes and the interaction be- networks. Our experiments only indicates that random- tween those proteins as edges. These networks exhibit 0 ness has a strong impact on degeneracy in networks spec- and −1 degeneracies (see Table 3). In addition, we find a tra. low degeneracy at +1 in few of these networks. The num- Thanks to Eq. 4 and 5, we can go further by using what ber of 0 eigenvalues being more than the number of −1 we know about eigenvectors associated to −1 eigenvalue eigenvalues indicates that duplication structures are more in order to identify each contribution to degeneracy in the frequent than the K ∗ S structures. It may be tempting cancer networks. Table 4 reports the number of −1 eigen- to make a causal link between macroscopic properties such values, denoted by λ−1, and nodes, denoted by Nλ−1 , by as clustering coefficient or assortativity and eigenvalue de- condition for all the normal and disease networks. The generacy. However, as reported in Table 3, the normal condition (ii) is largely at the origin of −1 degeneracy and Oral cancer network has hCCi = 0.31 and Nλ0 = 60 in a few cases, condition (iii) is not met. Therefore, K ∗ S whereas for the disease one, hCCi = 0.35 and Nλ0 = 15. structure mostly contributes to −1 degeneracy in the net- What follows that one can find networks having lesser de- works spectra. As a conclusion, real-world networks con- generacy but with high value of hCCi. Furthermore, let us tain a large number of K ∗ S structures whose existence consider one more macroscopic quantity which is degree- is revealed by high multiplicity of −1 eigenvalues. As de- degree correlations and let us attempt to find a relation generacy at −1 eigenvalue is poorly observed in model between assortatvity (positive degree-degree correlations networks, this indicates that the resulting structures may and degeneracy at -1). Again, let us consider the Ovar- have a significance in real-world networks. ian cancer which for the disease case has r = 0.10 and
Nλ0 = 116, whereas for the normal case has r = −0.01 As a conclusion, many real-world networks exhibit a and Nλ0 = 129. Therefore, even if assortativities of net- very high degeneracy at few eigenvalues such as 0 and works are different, degeneracies may be quite the same. −1 as compared to their corresponding random networks.
p-5 Lo¨ıcMarrec1,2 Sarika Jalan1,3
This suggests that the nodes contributing to high multi- plicity may play a central role in these networks. This Let- ter has numerically and analytically demonstrated the ori- gin as well as structures contributing to degeneracy, giving the opportunity to study their impact in real-world net- works. In the case of cancer networks, if such structures turn out to have a biological significance, the proposed approach will provide a new and different way to search for drug targets and biomarkers [25].
∗∗∗
SJ is grateful to Department of Science and Technology, Government of India grant EMR/2014/000368 for finan- cial support. LM acknowledges Sanjiv K. Dwivedi, Alok Yadav and Aparna Rai for help with the spectral analysis and cancer data, respectively.
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Figure 2: Complete graphs for different sizes. (a), (b) and (c) have two, three and four −1 eigenvalues, respectively.
Figure 3: Complete graph of five nodes for which we cut one edge. The resulting graph exhibits two −1 eigenvalues.
In the following, we take R1 = R2. It will be easy to extend to R1 = R2 = ... = Rn.
1 1 ··· a1,N v1 1 1 ··· a1,N v2 . . . . . =0 (4) ...... a a ··· 1 v 1,N 1,N N Thanks to the two first rows, we can write:
N
v1 + v2 + a1ivi =0 (5) i=3 X By extending to the general case, we obtain:
vi + vj =0 (6) i K j S X∈ X∈ We can note that if v is an eigenvector associated to 0 eigenvalue of A + I matrix, v will also be an eigenvector associated to 0 eigenvalue of (A + I)2, (A + I)3, etc. So:
(A + I)2v =0 (7)
N 2 N 2 N−1 2+ i=3 a1i 2+ i=3 a1i ··· 3a1N + i=3 a1iaiN v1 N 2 N 2 N−1 2+ i a1i 2+ i a1i ··· 3a1N + i a1iaiN v2 P =3 P =3 P =3 = 0 (8) ...... P. P. . P. . N−1 N−1 N−1 a a a a a a ··· a2 a2 vN 3 1N + i=3 3i iN 3 1N + i=3 1i iN 1+ 1N + i=3 iN The two first rows enableP to write: P P 3
N N N 2 (2 + a1i)(v1 + v2)+ ((3a1i + a1j aji)vi) = 0 (9) i=3 i=3 j ,j i X X =3X6=
It is obvious that the trivial case a13 = a14 = ... = a1N = 0 lead to the equation (1). The trivial case corresponds to a isolated complete subgraph. By using the equations (5) and (9), we obtain the following relations:
v + v =(2+ N a2 )(v + v ) 1 2 i=3 1i 1 2 (10) N a v = N ((3a + N a a )v ) ( i=3 1i i Pi=3 1i j=3,j6=i 1j ji i P P P 1+ N a2 =0 i=3 1i (11) 2a + N a a =0 ( 1iP j=3,j6=i 1j ji
N 2 P N However, in the non-trivial case, 1 + i=3 a1i =06 and 2a1i + j=3,j6=i a1j aji =6 0. So, the system (11) is absurd. That leads to: P P
v + v =0 1 2 (12) v = v = ... = vN =0 3 4 As a result, the equation (1) is verified.
EQ (1) AND (2) THROUGH ROW EQUIVALENCE AND GAUSSIAN ELIMINATION
As we have written in the manuscript, the conditions (ii) and (iii) make the rank of A − λI decrease. More particularly, if r is the number of linear combinations of rows and N the size of the matrix, one obtains rank(A − λI)=N − r. Mathematically, it results from the rank-nullity theorem according to which:
rank(A − λI) + null(A − λI)= N (13)
with null the dimension of null space of A−λI. Knowing that, it seems likely that we can take advantage of the row echelon form using Gaussian elimination. Indeed, we will see that it sheds light on Eq. 1 and 2. Before starting, we should recall that the three elementary row operations (swap the positions of two rows, multiply a row by a nonzero scalar and add to one row a scalar multiple of another) do not affect the solution set if the matrix is associated to a system of linear equations.
First, we consider the condition (ii) through a simple but typical example. We take a duplicate node structure such as two nodes have the same neighbor. This kind of graph makes appear one 0 eigenvalue. In this case, we work on A − λI = A − 0.I = A where R1 = R2.
0 0 1 A = 0 0 1 (14) 1 1 0
′ ′ Let us apply L1 ↔ L3 and L3 ← L3 − L2:
1 1 0 A ∼ 0 0 0 (15) 0 0 1 The previous matrix is a row echelon form. Since one row has all its entries equal to 0, it clearly shows that rank(A) = 2 whereas N = 3. Let us come back to the eigen-equation with this equivalent form: 4
1 1 0 v1 0 0 0 0 v = 0 (16) 2 0 0 1 v3 0 It follows that: v1 + v2 = 0 and v3 = 0. So Eq. 1 is verified.
Now, we focus on the condition (iii) using the example reported by Figure 1 (b) of the manuscript for which R1 + R4 = R2 + R5. It results one 1 eigenvalue and that is why we are going to work on A + I.
11000 11100 A + I = 01110 (17) 00111 00011 Let us apply once more the Gaussian elimination. One obtains:
1000 1 0100 −1 A + I ∼ 0010 0 (18) 0001 1 0000 0 As it was the case in the previous section, this row equivalent matrix enables to find that rank(A + I) = 4 whereas N = 5. Let us consider the eigen-equation with this equivalent form:
1000 1 v1 0 0100 −1 v 0 2 0010 0 v3 = 0 (19) 0001 1 v4 0 0000 0 v5 0 It follows that v1 + v2 + v4 + v5 = 0 and v3 = 0. We recover Eq. 2.
APPLICATION EXAMPLES OF EQS. (1) AND (2)
Since we have understood why Eq. (1) and (2) occur, let us apply them to other examples.
First, let us take an example to illustrate the relation (1): only two rows of an adjacency matrix verify the condition (ii) such as R1 = R2 and all the other ones are linearly independent. Then, Eq. 1 tells us that v1,2 =6 0, v1 + v2 = 0 and vi =0 for i =16 , 2 in every eigenvector associated to the studied degenerate eigenvalue.
Then, let us take an example for the relation (2): let us assume that an adjacency matrix satisfies the relation R1 + R2 = R3 + R4 and all the other rows are linearly independent. Then, Eq. 2 shows that v1,2,3,4 =6 0, v1 + v2 + v3 + v4 = 0 and vi = 0 for i =16 , 2, 3, 4 in every eigenvector associated to the studied degenerate eigenvalue. In other words, this means that the rows i with vi =6 0 are linearly dependent if and only if they belong to a same structure, namely a same linear combination of rows.
In the case where there are two different linear combinations of rows as follows R1 + R2 = R3 and R4 + R5 = R6, so v1,2,3,4,5,6 =6 0, v1 + v2 + v3 = 0, v4 + v5 + v6 = 0 and vi = 0 for i =16 , 2, ..., 6.