A note on Fiedler vectors interpreted as graph realizations

Helmberg, Christoph; Reiss, Susanna

doi:10.1016/j.orl.2010.01.005

Cited by 6 publications

(5 citation statements)

References 11 publications

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“…Like in [29] we obtain the following theorems exhibiting the direct relation between optimal solutions of E λn−λ2,l and the eigenvectors of λ 2 (L(G)) and λ n (L(G)) whenever G is not complete. The proofs are almost identical to those in [29] and are therefore omitted.…”

Section: Variable Edge Length Parametersmentioning

confidence: 73%

“…Given a graph, we consider best distributions of nonnegative weights on its edges that minimize this width and study their relation to structural properties of the graph as well as to the associated eigenvalues and eigenvectors of the corresponding weighted Laplacian. The problem of minimizing λ n (L w )−λ 2 (L w ) combines the problems of maximizing λ 2 (L w ) studied in [17,23,24] and minimizing λ n (L w ) investigated in [16,22,29]. Our work builds heavily on the semidefinite optimization approaches in [23,24,22] which offer an intriguing interpretation of the semidefinite dual as a (nonstandard) graph embedding or graph realization problem with the property that optimal realizations give rise to geometric visualizations of eigenvectors corresponding to the optimized eigenvalue.…”

Section: Introductionmentioning

confidence: 99%

“…The last two theorems allow us to transfer the structural results of [23,24,22] linking optimal graph realizations (and the associated eigenvectors) to the separator structure of the graph: optimal realizations V fold inward along separators (Corollary 4.3), optimal realizations U fold outward along separators (Corollary 4.8), and there exist optimal realizations where both realizations have dimension bounded by the tree-width of the graph plus one (Corollary 4.5, Theorem 4.9, and Corollary 4.10). Like in [29], for unweighted Laplacians the eigenvectors have an interpretation as a realization where one also optimizes over the edge length parameters l (Theorems 7.1 and 7.2).…”

Section: Introductionmentioning

confidence: 99%

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On Minimizing the Spectral Width of Graph Laplacians and Associated Graph Realizations

Göring¹,

Helmberg²,

Reiss³

2013

SIAM J. Optim.

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Extremal eigenvalues and eigenvectors of the Laplace matrix of a graph form the core of many bounds on graph parameters and graph optimization problems. For example, the value of a uniform sparsest cut is bounded from below and above by the second smallest and the largest eigenvalue of the weighted Laplacian divided by the number of nodes of the graph. Minimizing the difference between maximum and second smallest eigenvalue over edge weighted Laplacians of a graph reduces the size of this interval. We study this problem of minimizing the spectral width for its own sake with the goal of advancing the understanding of connections between structural properties of the graph and the corresponding eigenvectors and eigenvalues. Building on previous work where these eigenvalues were investigated separately, we show that a corresponding dual problem allows us to view eigenvectors to optimized eigenvalues as graph realizations in Euclidean space, whose structure is tightly linked to the separator structure of the graph. In particular, optimal realizations corresponding to the maximum eigenvalue fold toward the barycenter along separators, while for the second smallest eigenvalue they fold outward along separators. Furthermore, optimal realizations exist in dimension at most the tree-width of the graph plus one.

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Section: Variable Edge Length Parametersmentioning

confidence: 73%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On Minimizing the Spectral Width of Graph Laplacians and Associated Graph Realizations

Göring¹,

Helmberg²,

Reiss³

2013

SIAM J. Optim.

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“…A version of Eq. (5) is studied by Helmberg and Reiss46 who show similar connections between the optimal solution and the eigenspace of the algebraic connectivity. These connections were further examined by Göring et al47 both theoretically and computationally.…”

Section: Generating Bounds For λ2 Via Sdpmentioning

confidence: 91%

Network topology measures

Kincaid

Phillips

2011

WIREs Computational Stats

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Networks have emerged as a unifying modeling tool across a diverse set of problem domains. Evolving methods for the analysis and design of networks must be able to address network performance in the face of increasing demands, contain and control local network disturbances, and efficiently determine network topologies that accurately represent the underlying process. Central to this effort is the identification of network-invariant functions that measure relevant topological features such as connectivity and sparsity. Results concerning network topology measures and their relationship to global and local network structures are surveyed.

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“…Remark 2.5. The algebraic multiplicity of λ 2 (L) sets an upper-bound on the dimension of realization (Helmberg and Reiss, 2010). This can be understood from Proposition 2.3 by recalling that the dimension of the eigenspace corresponding to λ 2 is at most the multiplicity of λ 2 .…”

Section: Dual and Embedding Problems For λmentioning

confidence: 99%

Designing Optimal Multiplex Networks for Certain Laplacian Spectral Properties

Shakeri,

Tavassoli,

Ardjmand

et al. 2019

Preprint

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We discuss the design of interlayer edges in a multiplex network, under a limited budget, with the goal of improving its overall performance, e.g. structural robustness, shrinking the frequency band of parametric resonance, etc. We analyze the following three problems separately; first, we maximize the smallest nonzero eigenvalue, also known as the algebraic connectivity; secondly, we minimize the largest eigenvalue, also known as the spectral radius; and finally, we minimize the spectral width. Maximizing the algebraic connectivity requires identical weights on the interlayer edges for budgets less than a threshold value. However, for larger budgets, the optimal weights are generally non-uniform. The dual formulation transforms the problem into a graph realization (embedding) problem that allows us to give a fuller picture. Namely, before the threshold budget, the optimal realization is one-dimensional with nodes in the same layer embedded to a single point; while, beyond the threshold, the optimal embeddings generally unfold into spaces with dimension bounded by the multiplicity of the algebraic connectivity. Finally, for extremely large budgets the embeddings revert again to lower dimensions. Minimizing the largest eigenvalue is driven by the spectral radius of the individual networks and its corresponding eigenvector. Before a threshold, the total budget is distributed among interlayer edges

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A note on Fiedler vectors interpreted as graph realizations

Cited by 6 publications

References 11 publications

On Minimizing the Spectral Width of Graph Laplacians and Associated Graph Realizations

On Minimizing the Spectral Width of Graph Laplacians and Associated Graph Realizations

Network topology measures

Designing Optimal Multiplex Networks for Certain Laplacian Spectral Properties

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