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Unformatted text preview: tallation cost, and E f (y, b)
is the expected optimal network ﬂow cost
m πj f (y, b(j ) ), E f (y, b) =
j =1 where f is the function deﬁned in part 1. Is (42) a convex optimization problem?
15.6 Maximizing algebraic connectivity of a graph. Let G = (V, E ) be a weighted undirected graph with
n = V  nodes, m = E  edges, and weights w1 , . . . , wm ∈ R+ on the edges. If edge k connects
nodes i and j , then deﬁne ak ∈ Rn as (ak )i = 1, (ak )j = −1, with other entries zero. The weighted
Laplacian (matrix) of the graph is deﬁned as
m wk ak aT = A diag(w)AT ,
k L=
k=1 where A = [a1 · · · am ] ∈ Rn×m is the incidence matrix of the graph. Nonnegativity of the weights
implies L 0.
Denote the eigenvalues of the Laplacian L as
λ1 ≤ λ2 ≤ · · · ≤ λn ,
which are functions of w. The minimum eigenvalue λ1 is always zero, while the second smallest
eigenvalue λ2 is called the algebraic connectivity of G and is a measure of the connectedness of
a graph: The larger λ2 is, the better connected the graph is. It is often used, for example, in
analyzing the robustn...
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This note was uploaded on 09/10/2013 for the course C 231 taught by Professor F.borrelli during the Fall '13 term at University of California, Berkeley.
 Fall '13
 F.Borrelli
 The Aeneid

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