bv_cvxbook_extra_exercises

Bv_cvxbook_extra_exercises

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Unformatted text preview: ess of computer networks. Though not relevant for the rest of the problem, we mention a few other examples of how the algebraic connectivity can be used. These results, which relate graph-theoretic properties of G to properties of the spectrum of L, belong to a field called spectral graph theory. For example, λ2 > 0 if and only if the graph is connected. The eigenvector v2 associated with λ2 is often called the Fiedler vector and is widely used in a graph partitioning technique called spectral partitioning, which assigns nodes to one of two groups based on the sign of the relevant component in v2 . Finally, λ2 is also closely related to a quantity called the isoperimetric number or Cheeger constant of G, which measures the degree to which a graph has a bottleneck. The problem is to choose the edge weights w ∈ Rm , subject to some linear inequalities (and the + nonnegativity constraint) so as to maximize the algebraic connectivity: maximize λ2 subject to w 0, Fw g, with variable w ∈ Rm . The problem data are A (which gives the graph topology), and F and g (which describe the constraints on the weights). 133 (a) Describe how to solve this problem using conve...
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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.

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