epl draft Analysing degeneracies in networks spectra Lo¨ıc Marrec1,2 and Sarika Jalan1,3 1 Complex Systems Lab, Discipline of Physics, Indian Institute of Technology Indore, Khandwa Road, Simrol, Indore 453552, India 2 Universit´eParis-Sud, 91405 Orsay Cedex, France 3 Centre for Biosciences and Biomedical Engineering, Indian Institute of Technology Indore, Khandwa Road, Simrol, Indore 453552, India PACS 89.75.-k – Complex systems PACS 02.10.Yn – Matrix theory Abstract – Many real-world networks exhibit a high degeneracy at few eigenvalues. We show that a simple transformation of the network’s adjacency matrix provides an understanding to the origins of occurrence of high multiplicities in the networks spectra. We find that the eigenvec- tors associated with the degenerate eigenvalues shed light on the structures contributing to the degeneracy. Since these degeneracies are rarely observed in model graphs, we present results for various cancer networks. This approach gives an opportunity to search for structures contributing to degeneracy which might have an important role in a network. The paper written by Leonhard Euler on the Seven which are particularly interesting for biological systems as Bridges of Konigsberg¨ marks a beginning of graph the- they shed light on fundamental process in evolution re- ory [1] by introducing a concept of graphs representing lated with gene duplication [15], hence an interest lies in complex systems. The work was restricted to small sys- investigating origins of other degenerate eigenvalues. We tem size. Revolution in computing power later provided will see in the following that two reasons emerge to explain an opportunity to analyse very large real-world systems degeneracy of every eigenvalue. In particular cases, one of in terms of networks. Further, analysis of graph spectra these reasons reveals existence of characteristic structures has contributed significantly in our understanding of struc- in networks. tural and dynamical properties of graphs [2, 3]. Among In this paper, we consider finite undirected graphs de- other things, it has been noted that a symmetric spec- fined by G = {V, E} with V the node set, and E the edge trum about the origin is related to a bipartite graph [4]. set such as | V |= N and | E |= m. A graph is completely Further, bulk portion of eigenvalues has been shown to determined by its adjacency matrix for which its element be modeled using random matrix theory [5], whereas ex- Aij is 1 when there is an edge from vertex i to vertex tremal eigenvalues have been shown to be modeled using j, and 0 otherwise. In the following, the rows i of every arXiv:1609.08355v2 [physics.soc-ph] 13 Apr 2017 the generalized eigenvalue statistics [6, 7]. Recent inves- adjacency matrix will be denoted by Ri. tigations have revealed that real-world networks exhibit properties which are very different from those of the cor- The eigenvalues are obtained by computing the roots responding model graphs [8–10]. One of these properties is of the characteristic polynomial of the adjacency matrix, χ (λ) = det(A − λI) = N (λ − λ ) and denoted by occurrence of degeneracy at 0, −1 and −2 eigenvalues [2]. A i=1 i λ λ λ Few papers have related 0 and −1 eigenvalues to stars and 1 ≤ 2 ≤ ... ≤ N . Since the adjacency matrix of an undirected graph is symmetricQ with 0 and 1 entries, the cliques respectively [11–13]. However, graphs in absence v v of stars and cliques can still show a degeneracy at 0 and eigenvalues are real. The associated eigenvectors 1, 2 v Av λ v i −1 eigenvalues, respectively. As a result, these reasons ,..., N satisfy the eigen-equation i = i i with = , , ..., N are not exhaustive and it turns out that origins of degen- 1 2 . eracy at these eigenvalues are more complex. For exam- A complete graph, denoted by K, is an undirected graph ple, the 0 degeneracy has been shown to be resulted from for which every pair of nodes is connected by a unique the complete and the partial duplications [14] of nodes edge. This type of graphs is especially interesting since their spectra exhibit a very high multiplicity at −1 eigen- p-1 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.