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|Abstract:||© 2016 Wiley Periodicals, Inc. We study how the spectral gap of the normalized Laplacian of a random graph changes when an edge is added to or removed from the graph. There are known examples of graphs where, perhaps counter-intuitively, adding an edge can decrease the spectral gap, a phenomenon that is analogous to Braess's paradox in traffic networks. We show that this is often the case in random graphs in a strong sense. More precisely, we show that for typical instances of Erdős-Rényi random graphs G(n, p) with constant edge density p ∈ (0, 1), the addition of a random edge will decrease the spectral gap with positive probability, strictly bounded away from zero. To do this, we prove a new delocalization result for eigenvectors of the Laplacian of G(n, p), which might be of independent interest. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 584–611, 2017.|
|Citation:||Eldan, R, Rácz, MZ, Schramm, T. (2017). Braess's paradox for the spectral gap in random graphs and delocalization of eigenvectors. Random Structures and Algorithms, 50 (4), 584 - 611. doi:10.1002/rsa.20696|
|Pages:||584 - 611|
|Type of Material:||Journal Article|
|Journal/Proceeding Title:||Random Structures and Algorithms|
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