1 Graph Equivalence Classes for Spectral Projector-Based Graph Fourier Transforms Joya A. Deri, Member, IEEE, and José M. F. Moura, Fellow, IEEE Abstract—We define and discuss the utility of two equiv- Consider a graph G = G(A) with adjacency matrix alence graph classes over which a spectral projector-based A 2 CN×N with k ≤ N distinct eigenvalues and Jordan graph Fourier transform is equivalent: isomorphic equiv- decomposition A = VJV −1. The associated Jordan alence classes and Jordan equivalence classes. Isomorphic equivalence classes show that the transform is equivalent subspaces of A are Jij, i = 1; : : : k, j = 1; : : : ; gi, up to a permutation on the node labels. Jordan equivalence where gi is the geometric multiplicity of eigenvalue 휆i, classes permit identical transforms over graphs of noniden- or the dimension of the kernel of A − 휆iI. The signal tical topologies and allow a basis-invariant characterization space S can be uniquely decomposed by the Jordan of total variation orderings of the spectral components. subspaces (see [13], [14] and Section II). For a graph Methods to exploit these classes to reduce computation time of the transform as well as limitations are discussed. signal s 2 S, the graph Fourier transform (GFT) of [12] is defined as Index Terms—Jordan decomposition, generalized k gi eigenspaces, directed graphs, graph equivalence classes, M M graph isomorphism, signal processing on graphs, networks F : S! Jij i=1 j=1 s ! (s ;:::; s ;:::; s ;:::; s ) ; (1) b11 b1g1 bk1 bkgk I. INTRODUCTION where sij is the (oblique) projection of s onto the Jordan subspace Jij parallel to SnJij. That is, the Fourier Graph signal processing [1], [2] permits applications transform of s, is the unique decomposition of digital signal processing concepts to increasingly k gi larger networks. It is based on defining a shift filter, X X for example, the adjacency matrix in [1], [3], [4] to s = sbij; sbij 2 Jij: (2) analyze undirected and directed graphs, or the graph i=1 j=1 Laplacian [2] that applies to undirected graph struc- The spectral components are the Jordan subspaces of the tures. The graph Fourier transform is defined through adjacency matrix with this formulation. the eigendecomposition of this shift operator, see these This paper presents graph equivalence classes where references. Further developments have been considered equal GFT projections by (1) are the equivalence re- in [5], [6], [7]. In particular, filter design [1], [5], [8] lation. First, the transform (1) is invariant to node and sampling [9], [10], [11] can be applied to reduce the permutations, which we formalize with the concept of computational complexity of graph Fourier transforms. isomorphic equivalence classes. Furthermore, the GFT With the objective of simplifying graph Fourier trans- permits degrees of freedom in graph topologies, which forms for large network applications, this paper explores arXiv:1701.02864v1 [cs.SI] 11 Jan 2017 we formalize by defining Jordan equivalence classes, a methods based on graph equivalence classes to reduce concept that allows graph Fourier transform computa- the computation time of the subspace projector-based tions over graphs of simpler topologies. A frequency- graph Fourier transform proposed in [12]. This transform like ordering based on total variation of the spectral extends the graph signal processing framework proposed components is also presented to motivate low-pass, high- by [1], [3], [4] to consider spectral analysis over directed pass, and pass-band graph signals. graphs with potentially non-diagonalizable (defective) adjacency matrices. The graph signal processing frame- Section II provides the graph signal processing and work of [12] allows for a unique, unambiguous signal linear algebra background for the graph Fourier trans- representation over defective adjacency matrices. form (1). Isomorphic equivalence classes are defined in Section III, and Jordan equivalence classes are defined in This work was partially supported by NSF grants CCF-1011903 and Section IV. The Jordan equivalence classes influence the CCF-1513936 and an SYS-CMU grant. definition of total variation-based orderings of the Jordan The authors are with the Department of Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh, PA 15213 USA subspaces, which is discussed in detail in Section V. (email: jderi,[email protected]) Section VI illustrates Jordan equivalence classes and total variation orderings. Limitations of the method are Jordan decomposition. Let Vij denote the N × rij discussed in Section VII. matrix whose columns form a Jordan chain of eigen- value 휆i that spans Jordan subspace Jij. Then the II. BACKGROUND eigenvector matrix V of A is This section reviews the concepts of graph signal [︀ ]︀ V = V11 ··· V1g ··· Vk1 ··· Vkg ; (9) processing and the GFT (1). Background on graphs 1 k signal processing, including definitions of graph signals where k is the number of distinct eigenvalues. The and the graph shift, is described in greater detail in [1], columns of V are a Jordan basis of CN . Then A [3], [4], [12]. For background on eigendecompositions, has block-diagonal Jordan normal form J consisting of the reader is directed to in [13], [15], [16]. Jordan blocks 2휆 1 3 A. Eigendecomposition 6 .. 7 6 휆 . 7 N×N J(휆) = : (10) Consider matrix A 2 C with k distinct eigen- 6 . 7 6 .. 17 values 휆1; : : : ; 휆k, k ≤ N. The algebraic multiplic- 4 5 휆 ity ai of 휆i represents the corresponding exponent of the characteristic polynomial of A. Denote by Ker(A) of size rij. The Jordan normal form J of A is unique the kernel or null space of matrix A. The geometric up to a permutation of the Jordan blocks. The Jordan multiplicity gi of eigenvalue 휆i equals the dimension of decomposition of A is A = VJV −1. Ker (A − 휆iI), which is the eigenspace of 휆i where I is the N × N identity matrix. The generalized eigenspaces B. Spectral Components Gi, i = 1; : : : ; k, of A are defined as The spectral components of the Fourier trans- mi form (1) are expressed in terms of the eigenvector Gi = Ker(A − 휆iI) ; (3) basis v1; : : : ; vN and its dual basis w1; : : : ; wN since where mi is the index of eigenvalue 휆i. The generalized the Jordan basis may not be orthogonal. Denote the N eigenspaces uniquely decompose C as the direct sum basis and dual basis matrices by V = [v1 ··· vN ] and k W = [w1 ··· ; wN ]. The dual basis matrix is the inverse N M −H C = Gi: (4) Hermitian W = V [14], [17]. i=1 Consider the jth spectral component of 휆i Jordan chains. Let v 2 Ker(A − 휆 I), v 6= 0, 1 i 1 Jij = span(v1; ··· vrij ): (11) be a proper eigenvector of A that generates generalized N eigenvectors by the recursion The projection matrix onto Jij parallel to C nJij is H Pij = VijW ; (12) Avp = 휆ivp + vp−1; p = 2; : : : ; r (5) ij where r is the minimal positive integer such that where r r−1 Vij = [v1 ··· vr ] (13) (A − 휆iI) vr = 0 and (A − 휆iI) vr 6= 0. A se- ij quence of vectors (v1; : : : ; vr) that satisfy (5) is a Jordan H rij ×N is the corresponding submatrix of V and Wij 2 C chain of length r [13]. The vectors in a Jordan chain are is the corresponding submatrix of W partitioned as linearly independent and generate the Jordan subspace W = [··· W H ··· W H ··· ]T : i1 igi (14) J = span (v1; v2; : : : ; vr) : (6) As shown in [12], the projection of signal s 2 CN Denote by the jth Jordan subspace of 휆 with Jij i onto Jordan subspace Jij can be written as dimension rij, i = 1; : : : ; k, j = 1; : : : ; gi. The Jordan s = s v + ··· + s v (15) spaces are disjoint and uniquely decompose the general- bij e1 1 erij rij H ized eigenspace Gi (3) of 휆i as = VijWij s: (16) gi M The next sections show that invariance of the graph Gi = Jij: (7) Fourier transform (1) is a useful equivalence relation on j=1 a set of graphs. Equivalence classes with respect to the The space CN can be expressed as the unique decom- GFT are explored in Sections III and IV. position of Jordan spaces III. ISOMORPHIC EQUIVALENCE CLASSES k gi N M M This section demonstrates that the graph Fourier trans- C = Jij: (8) i=1 j=1 form (1) is invariant up to a permutation of node labels 2 and establishes sets of isomorphic graphs as equivalence with respect to the invariance of the GFT (1) up to a classes with respect to invariance of the GFT (1). Two permutation of the graph signal and inverse permutation graphs G(A) and G(B) are isomorphic if their adja- of the graph Fourier transform. cency matrices are similar with respect to a permutation Theorem 1 establishes an invariance of the GFT over matrix T , or B = T AT −1 [18]. The graphs have graphs that only differ up to a node labeling, and the same Jordan normal form and the same spectra. Theorem 2 follows. Also, if V and V are eigenvector matrices of A A B The isomorphic equivalence of graphs is important and B, respectively, then V = TV . We prove that B A since it signifies that the rows and columns of an adja- the set GI of all graphs that are isomorphic to G(A) is A cency matrix can be permuted to accelerate the eigende- an equivalence class over which the GFT is preserved. composition. For example, permutations of highly sparse The next theorem shows that an appropriate permutation adjacency matrices can convert an arbitrary matrix to can be imposed on the graph signal and GFT to ensure nearly diagonal forms, such as with the Cuthill-McKee invariance of the GFT over all graphs G 2 GI .