1 Majorizations for the Eigenvectors of Graph-Adjacency Matrices Rahul Dhal Sandip Roy Yan Wan Ali Saberi Abstract We develop majorization results that characterize changes in eigenvector components of a graph’s adjacency matrix when its topology is changed. Specifically, for general (weighted, directed) graphs, we characterize changes in dominant-eigenvector components for single-row and multi-row incrementations. We also show that topology changes can be tailored to set ratios between the components of the dominant eigenvector. For more limited graph classes (specifically, undirected and reversibly-structured ones), majorizations for components of the subdominant and other eigenvectors upon graph modifications are also obtained. Index Terms Algebraic Graph Theory, Network Design, Dynamical Networks, Adjacency Matrix, Eigenvectors, Majorization Mathematics Subject Classification: 15A18, 15A42, 05C85, 05C82 I. I NTRODUCTION During the last 20 years or so, new research at the interface of dynamical systems, graph theory, and linear algebra has provided significant understanding of the relationship between a network’s topology and its dynamics. However, there is an increasing need to not only characterize but also design the dynamics of complex networks, in several domains. For instance, in epidemic control, management actions that change a network topology of disease spread (e.g., targeted restriction on traveling, or quarantine) may need to be designed, to eliminate the spread [1], [2] with limited resources. Similarly, in sensor networking applications, tools for designing the sensor-to-sensor communication topology are needed to shorten the duration for the sensors to complete algorithmic tasks (e.g., reach consensus) and therefore to reduce energy consumption (e.g., [3]–[5]). While there are some nascent efforts on designing a network’s dynamics, much remains to be done—especially in the realistic circumstance that a network’s topology can only be partially designed or modified. While the details of network modification/design are application-specific, it is generally true that partial modifications of a network’s structure affect complex changes in its dynamics that are not well understood and require new methods for characterization. One promising route for understanding the impact of network modifications on dynamics is to identify how the topology changes impact the spectra (eigenvalues and eigenvectors) of certain matrices defined from the network’s topology (graph), such as the Laplacian or adjacency matrices: these matrices often define core network dynamics, and hence characterizing their spectra gives insight into the dynamics. Spectral properties of the adjacency and Laplacian matrices have been characterized in both the Algebraic Graph Theory and the non-negative matrices’ literature (e.g., [6]–[8]); a bulk of this literature is focused on characterizing eigenvalues rather than eigenvectors of the graph matrices, but some graph-based characterizations of eigenvectors are also available [9], [10]. While many spectral characterizations have been developed, only a few results specify the impact of topology changes on adjacency/Laplacian eigenvalues, and eigenvector components, especially for asymmetrical (directed) graphs. Characterization of such impacts forms a crucial step in the study of networked dynamical systems. In this paper, we investigate the impact of topology changes on the eigenvectors of the adjacency matrix of a network’s graph. Our study is aligned with [11], in which eigenvectors of Laplacian matrices of unweighted and undirected graphs are studied. In particular, the article [11] determined the effect of adding, deleting, or contracting edges in the graph on the subdominant eigenvector of the graph’s Laplacian. The results we develop here complement those presented in [11]. However, we consider weighted undirected as well as weighted directed graphs, and consider different classes of topology modifications as compared to [11]. Specifically, our focus here is on developing majorizations (comparisons) for the eigenvector components of the adjacency matrices of both directed and undirected graphs, upon structured modification of edge-weights in the graph. We consider several types of topology changes and graph topologies classes, and develop majorizations for extremal and/or subdominant eigenvalues and associated eigenvectors in these cases. The eigenvector majorization results that we develop rely critically on 1) classical results on non-negative matrices’ eigen- values [7], and 2) matrix perturbation concepts. From one viewpoint, our result extend existing eigenvalue majorizations for non-negative matrices [7], toward comparative characterizations of eigenvectors (see also [12]). From another viewpoint, our results can be viewed as providing tighter bounds on eigenvector components than would be obtained from perturbation arguments (see [9]), for the special class of non-negative matrices. While very few studies explicitly consider eigenvector-component majorization, we have recently become aware of two characterizations of nonnegative matrices ( [13] and [14]), that are closely aligned with our work. In [13], the author developed First, second, and fourth authors are with the Department of Electrical Engineering and Computer Science at Washington State University. The third author is with the Department of Electrical Engineering at University of North Texas. Correspondence should be sent to rdhal@eecs.wsu.edu