Convex Graph Invariants
Name
Chandrasekaran-2012-Convex graph invariants.pdf
Size
492.97 KB
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Author(s)
Chandrasekaran, Venkat
Parrilo, Pablo A.
Willsky, Alan S.
Date Issued
August 2012
Journal
SIAM Review
Publisher
Society for Industrial and Applied Mathematics
Citation
Chandrasekaran, Venkat, Pablo A. Parrilo, and Alan S. Willsky. “Convex Graph Invariants.” SIAM Review 54.3 (2012): 513–541. © 2012, Society for Industrial and Applied Mathematics
Version
Final published version
Abstract
The structural properties of graphs are usually characterized in terms of invariants, which are functions of graphs that do not depend on the labeling of the nodes. In this paper we study convex graph invariants, which are graph invariants that are convex functions of the adjacency matrix of a graph. Some examples include functions of a graph such as the maximum degree, the MAXCUT value (and its semidefinite relaxation), and spectral invariants such as the sum of the $k$ largest eigenvalues. Such functions can be used to construct convex sets that impose various structural constraints on graphs and thus provide a unified framework for solving a number of interesting graph problems via convex optimization. We give a representation of all convex graph invariants in terms of certain elementary invariants, and we describe methods to compute or approximate convex graph invariants tractably. We discuss the interesting subclass of spectral invariants, and also compare convex and nonconvex invariants. Finally, we use convex graph invariants to provide efficient convex programming solutions to graph problems such as the deconvolution of the composition of two graphs into the individual components, hypothesis testing between graph families, and the generation of graphs with certain desired structural properties.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology. Laboratory for Information and Decision Systems
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1137/100816900
