Bisection Algorithm of Increasing Algebraic Connectivity by Adding an Edge

• Published in: IEEE transactions on automatic control Vol. 55; no. 1; pp. 170 - 174
• Main Author: KIM, Yoonsoo
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.01.2010; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Africa; Algebra; Algebraic connectivity; Algorithmics. Computability. Computer arithmetics; Algorithms; Applied sciences; Automatic control; Computational complexity; Computational efficiency; Computational modeling; Computer science; control theory; systems; Computer simulation; Control theory; Eigenvalues; Eigenvalues and eigenfunctions; Energy consumption; Exact sciences and technology; Graphs; Laplace equations; Laplacian matrix; Mechatronics; Networks; Robust stability; Theoretical computing; Dichotomy; Heuristic method; Greedy algorithm; Algebraic connectivity; Eigenvalue problem; Laplacian matrix; Computational complexity; Modeling; Laplacian
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2009.2033763

For a given graph (or network) G , consider another graph G ' by adding or deleting an edge e to or from G . We propose a computationally efficient algorithm of finding e such that the second smallest eigenvalue (algebraic connectivity, ¿ 2 ( G ')) of G ' is maximized or minimized. Theoretically, the proposed algorithm runs in O (4 mn log( d /¿)), where n is the number of nodes in G , m is the number of disconnected edges in G , d is the difference between ¿ 3 ( G ) and ¿ 2 ( G ), and ¿ > 0 is a sufficiently small constant. However, extensive simulations show that the practical computational complexity of the proposed algorithm, O (5.7 mn ), is nearly comparable to that of a simple greedy-type heuristic, O (2 mn ). This algorithm can also be easily modified for finding e which affects ¿ 2 ( G ) the least.