1 Graph Fourier Transform based on Directed Laplacian Rahul Singh, Abhishek Chakraborty, Graduate Student Member, IEEE, and B. S. Manoj, Senior Member, IEEE Abstract—In this paper, we redefine the Graph Fourier Trans- [11] is limited to the analysis of graph signals lying on undi- form (GFT) under the DSPG framework. We consider the rected graphs with real non-negative weights. In the Laplacian Jordan eigenvectors of the directed Laplacian as graph harmonics based approach, eigendecomposition of the graph Laplacian and the corresponding eigenvalues as the graph frequencies. For this purpose, we propose a shift operator based on the L is used for frequency analysis of graph signals. Mathemat- directed Laplacian of a graph. Based on our shift operator, we ically, Graph Fourier Transform (GFT) of a graph signal f is T then define total variation of graph signals, which is used in defined as ^f = U f, where U = [u0 u1 ::: uN−1] is frequency ordering. We achieve natural frequency ordering and the matrix in which columns are the eigenvectors of L. Here, interpretation via the proposed definition of GFT. Moreover, we the frequency ordering is based on the quadratic form and show that our proposed shift operator makes the LSI filters under turns out to be “natural”. Natural frequency ordering means DSPG to become polynomial in the directed Laplacian. that the small eigenvalues correspond to low frequencies Index Terms—Graph signal processing, graph Fourier trans- and vice-versa. On the other hand, the weight matrix based form, directed Laplacian. framework [12], [14], also referred as the Discrete Signal Processing on Graphs (DSPG) framework, has been built on I. INTRODUCTION the graph shift operator. The weight matrix W of the graph acts as the shift operator and the shifted version of a graph ATA defined on network-like structures are encountered signal f can be found as ~f = Wf. Shift operator is treated D in a large number of scenarios including molecular as the elementary Linear Shift Invariant (LSI) graph filter, interaction in biological systems, computer networks, sensor which essentially is a polynomial in W. Based on the shift networks, social and citation networks, the Internet and the operator, authors define total variation (TV) on graphs that they World Wide Web, power grids, transportation networks [1], utilize for ordering of graph frequencies. In DSPG, the graph [2], and many more. Such data can be visualized as a set of Fourier transform of a graph signal f is defined as ^f = V−1f, scalar values, known as a graph signal, lying on a particular where V is the matrix with the Jordan eigenvectors of W as structure, i.e., a graph. In computer graphics, data defined on its columns (W is decomposed in Jordan canonical form as any geometrical shape described by polygon meshes can be W = VJV−1). The Jordan eigenvectors of W are used as formulated as a graph signal [3]. graph Fourier basis and the eigenvalues of W act as graph The irregular structure of the underlying graph, as opposed frequencies. to the regular structure in case of time-series and image Although DSPG is applicable to directed graphs with neg- signals dealt in classical signal processing [4], [5], imposes ative or complex edge weights as well, it does not provide a great challenge in analysis and processing of graph signals. “natural” intuition of frequency. The eigenvalue of W with Fortunately, recent work toward the development of important maximum absolute value acts as the lowest frequency and concepts and tools, extending classical signal processing the- as one moves away from this eigenvalue in the complex ory, including sampling and interpolation on graphs [6]–[8], frequency plane, the frequency increases. Thus, frequency arXiv:1601.03204v1 [cs.IT] 13 Jan 2016 graph-based transforms [9]–[13], and graph filters [14], [15] ordering is not natural and there is an overhead of frequency have enriched the field of graph signal processing. These tools ordering as well. Also, the interpretation of frequency is not have been utilized in solving a variety of problems such as intuitive — for example, in general, a constant graph signal signal recovery on graphs [16]–[18], clustering and community results in low as well as high frequency components in the detection [19], [20], graph signal denoising [21], and semi- spectral domain. supervised classification on graphs [22]. In this paper, we redefine Graph Fourier Transform (GFT) Transforms aimed at frequency analysis of graph signals under DSPG. In the new definition of GFT, the Jordan facilitate efficient handling of the data and remain at the heart eigenvectors of the directed Laplacian matrix are treated as of graph signal processing. In literature, there exist two frame- the graph Fourier basis and the corresponding eigenvalues works for frequency analysis and processing of graph signals: constitute the graph spectrum. The directed Laplacian matrix (i) Laplacian matrix based approach, and (ii) weight matrix of a graph is a simple extension of the symmetric Laplacian based approach. The existing Laplacian based approach [9], discussed in [11] to directed graphs. To redefine GFT under DSPG, we first propose a shift operator derived from the R. Singh, A. Chakraborty, and B. S. Manoj are with the Depart- directed Laplacian. Then, we utilize this shift operator to ment of Avionics, Indian Institute of Space Science and Technology, Thiruvananthapuram, Kerala 695547 India (e-mail: [email protected], ab- define total variation (TV) of a graph signal which is used for [email protected], [email protected]). frequency ordering. We observe “natural” frequency ordering as well as better intuition as compared to the existing GFT 2 2 3 1 2 3 definition under DSPG approach. Moreover, the new definition 4 0 0 0 0 3 1 0 2 0 0 of GFT links the DSP framework to the existing Laplacian 3 6 7 G 6 7 2 1 W = 60 0 0 3 07 based approach. We also show that considering our proposed 3 42 4 0 0 15 3 shift operator, the LSI filters under DSPG become polynomial 4 3 3 3 0 0 0 in the directed Laplacian. 1 5 (b) Weight matrix. Rest of this paper is organized as follows. In Section II, we discuss the directed Laplacian of a graph followed by the (a) An example directed graph. proposed shift operator and total variation on graph. Then, 23 0 0 0 03 2 3 0 0 0 −33 we redefine the Graph Fourier Transform based on directed 60 3 0 0 07 6−1 3 −2 0 0 7 6 7 6 7 Laplacian. We describe LSI graph filters in Section III and Din = 60 0 3 0 07 L = 6 0 0 3 −3 0 7 then conclude the paper in Section IV. 40 0 0 7 05 4−2 −4 0 7 −15 0 0 0 0 6 −3 −3 0 0 6 II. FREQUENCY ANALYSIS OF GRAPH SIGNALS (c) In-degree matrix. (d) Laplacian matrix. First, we present directed Laplacian matrix of a graph and Fig. 1: A directed graph and the corresponding matrices. then, derive the shift operator from it. Next, we define total variation of a graph signal which is utilized in frequency ordering. Finally, we redefine Graph Fourier Transform under C. Shift Operator the DSPG framework. We identify the shift operator from the graph structure corresponding to a discrete time periodic signal and then A. Graph Signals extend it to arbitrary graphs. A graph signal is a collection of values defined on a A finite duration periodic discrete signal can be thought of complex and irregular structure modeled as a graph. A graph as a graph signal lying on a directed cyclic graph shown in Fig. 2. Indeed, Fig. 2 is the support of a periodic time-series is represented as = ( ; W), where = v0; v1; : : : ; vN−1 is the set of verticesG (orV nodes) and WV isf the weight matrixg having a period of five samples. The directed Laplacian of the graph is of the graph in which an element wij represents the weight of the directed edge from node j to node i. Moreover, a graph signal is represented as an N-dimensional vector 1 2 3 4 5 f = [f(1); f(2); : : : ; f(N)]T CN , where f(i) is the value of the graph signal at node i and2 N = is the total number Fig. 2: A directed cyclic (ring) graph of five nodes. This graph is the of nodes in the graph. jVj underlying structure of a finite duration periodic (with period five) discrete signal. All edges have unit weights. B. Directed Laplacian 2 1 0 0 0 13 − As discussed in [11], the graph Laplacian for undirected 6 1 1 0 0 0 7 6− 7 graphs is a symmetric difference operator L = D W, where L = 6 0 1 1 0 0 7 : (2) − 6 − 7 D is the degree matrix of the graph and W is the weight 4 0 0 1 1 0 5 matrix of the graph. However, in case of directed graphs (or 0 0− 0 1 1 digraphs), the weight matrix W of a graph is not symmetric. In − addition, the degree of a vertex can be defined in two ways [1] Consider a discrete time finite duration periodic signal x = [9 7 1 0 6]T — in-degree and out-degree. In-degree of a node i is estimated defined on the graph shown in Fig. 2. N x as din = P w , whereas, out-degree of the node i can be Shifting the signal by one unit to the right results in the i j=1 ij signal ~x = [6 9 7 1 0]T .