Research Spotlights

Abstract

From social networks to machine learning to biological network analysis, graphs spring up in all sorts of applications related to either the natural sciences or systems created by humans. While individual nodes and edges express relationships between interacting entities, there is often far greater insight in understanding a graph's larger structural properties, e.g., treewidths, the maximal degree, or the second smallest eigenvalue of an associated graph Laplacian. As these fundamental structural properties are independent of a specific node labeling, it is then worthwhile to examine graph invariants, functions of graphs that do not depend on specific node labels. The paper by Chandrasekaran, Parrilo, and Willsky explores the realm of convex graph invariants, convex functions of a graph's adjacency matrix which are also graph invariants. These convex graph invariants arise in many practical situations and are advantageous in that one can leverage convex optimization ideas and tools to provide tractable algorithms. Three canonical graph structural problems are introduced: graph deconvolution, graph generation, and graph hypothesis testing. Deconvolution seeks to unravel a graph into simpler graph structures. Network analysis in areas such as biological sciences and social networks often rely on this type of decomposition. Graph generation seeks to produce graphs satisfying specified constraints (e.g., each node has a certain degree or the corresponding Laplacian matrix has certain spectral properties). This might arise in situations where one wishes to construct an economical network with certain robustness characteristics. Graph hypothesis testing aims to categorize candidate graphs, i.e., deciding whether they fit better within one graph family or another (where families are specified by graph invariants). The paper argues that the key to understanding and computing convex graph invariants lies in being able to first represent them via elementary invariants. This leads to solution methodologies associated with quadratic assignment problems and tractable approximations via optimization techniques such as linear programming and semidefinite programming. Overall, the paper provides a clear view into the world of convex graph invariants. In addition to providing a general framework for computing these invariants, numerous examples introduce readers unfamiliar with this area to the wide range of practical cases where the notions of invariant convex sets and convex graph invariants emerge.