Matrix-Weighted Graphs

This book presents and systematizes results in matrix-weighted graphs, a powerful tool for modeling and analysis of multi-dimensional networked systems. The authors select topics addressing fundamental issues, which they arrange in four parts:

• graphs and networks with matrix weighting, showing how the matrix-weighted Laplacian forms the foundation for further theoretical developments;


• development of algorithms for various purposes from the determination of connectivity to quantitative measurement as a key pillar in network design and analysis;


• control-theoretic integration, providing a framework with the matrix-weighted consensus algorithm playing a central role and which coordinates interacting dynamical agents from each vertex in a cooperative and distributed manner; and


• applications of matrix-weighted graphs in network synchronization, social networks, networked input-output economics, network localization and formation control.

The theoretical results provide a firm foundation for researchers wishing to pursue the study of matrix-weighted networks and related topics and are accessible to graduate students with a background in engineering mathematics.

Many of the definitions, analyses, and designs in this book are accompanied by figures, examples and numerical simulations. MATLAB® and Simulink® simulations to assist the reader in understanding and further developing such features are available for download.

Part I: Graphs and Networks with Matrix-Weights. - Matrix-Weighted Graphs. - Matrix-Weighted Laplacian. - Physical Interpretation and Motivational Examples. - Part II: Algorithms. - Connectivity. - Spanning Trees. - Quantitative Measures. - Part III: Control. - Matrix-Weighted Consensus. - Discrete-Time and Randomized Algorithms. - Accelerated Algorithms. - Robustness . - Tracking. - Time-Delays. - Part IV: Applications. - Synchronization. - Scaling Matrices. - Networked Input Output Analysis. - Bearing-Only Formation Control.