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Optimal k-centers of a graph: a control-theoretic approach

Karim Shahbaz, Madhu N. Belur, Chayan Bhawal, Debasattam Pal

TL;DR

This paper proposes two metrics and establishes connections to a well-studied metric from the literature (specifically for stochastic matrices) and proves the equivalence of three definitions for a path graph and extends the concept of central port linkage beyond Fiedler vectors to other eigenvectors associated with path graphs.

Abstract

In a network consisting of n nodes, our goal is to identify the most central k nodes with respect to the proposed definitions of centrality. Depending on the specific application, there exist several metrics for quantifying k-centrality, and the subset of the best k nodes naturally varies based on the chosen metric. In this paper, we propose two metrics and establish connections to a well-studied metric from the literature (specifically for stochastic matrices). We prove these three notions match for path graphs. We then list a few more control-theoretic notions and compare these various notions for a general randomly generated graph. Our first metric involves maximizing the shift in the smallest eigenvalue of the Laplacian matrix. This shift can be interpreted as an improvement in the time constant when the RC circuit experiences leakage at certain k capacitors. The second metric focuses on minimizing the Perron root of a principal sub-matrix of a stochastic matrix, an idea proposed and interpreted in the literature as manufacturing consent. The third one explores minimizing the Perron root of a perturbed (now super-stochastic) matrix, which can be seen as minimizing the impact of added stubbornness. It is important to emphasize that we consider applications (for example, facility location) when the notions of central ports are such that the set of the best k ports does not necessarily contain the set of the best k-1 ports. We apply our k-port selection metric to various network structures. Notably, we prove the equivalence of three definitions for a path graph and extend the concept of central port linkage beyond Fiedler vectors to other eigenvectors associated with path graphs.

Optimal k-centers of a graph: a control-theoretic approach

TL;DR

This paper proposes two metrics and establishes connections to a well-studied metric from the literature (specifically for stochastic matrices) and proves the equivalence of three definitions for a path graph and extends the concept of central port linkage beyond Fiedler vectors to other eigenvectors associated with path graphs.

Abstract

In a network consisting of n nodes, our goal is to identify the most central k nodes with respect to the proposed definitions of centrality. Depending on the specific application, there exist several metrics for quantifying k-centrality, and the subset of the best k nodes naturally varies based on the chosen metric. In this paper, we propose two metrics and establish connections to a well-studied metric from the literature (specifically for stochastic matrices). We prove these three notions match for path graphs. We then list a few more control-theoretic notions and compare these various notions for a general randomly generated graph. Our first metric involves maximizing the shift in the smallest eigenvalue of the Laplacian matrix. This shift can be interpreted as an improvement in the time constant when the RC circuit experiences leakage at certain k capacitors. The second metric focuses on minimizing the Perron root of a principal sub-matrix of a stochastic matrix, an idea proposed and interpreted in the literature as manufacturing consent. The third one explores minimizing the Perron root of a perturbed (now super-stochastic) matrix, which can be seen as minimizing the impact of added stubbornness. It is important to emphasize that we consider applications (for example, facility location) when the notions of central ports are such that the set of the best k ports does not necessarily contain the set of the best k-1 ports. We apply our k-port selection metric to various network structures. Notably, we prove the equivalence of three definitions for a path graph and extend the concept of central port linkage beyond Fiedler vectors to other eigenvectors associated with path graphs.
Paper Structure (13 sections, 23 theorems, 46 equations, 3 figures, 2 tables)

This paper contains 13 sections, 23 theorems, 46 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Notation
  3. Organization of this paper
  4. Definitions of $k$-centrality and possible inter-relations
  5. Main results: $k$ centrality and inter-relations
  6. "Central nodes" for path graph
  7. First order approximation of the perturbed Laplacian matrix smallest eigenvalue: general graphs
  8. Scaling with respect to number ports and number of nodes
  9. Relation between $1$-optimal port and $2$-optimal ports: path graphs
  10. Auxiliary results
  11. Comparison of proposed metrics with other control-theoretic measures: general graphs
  12. Concluding remark and future directions
  13. Appendix

Key Result

Proposition 2.7

SpielmanBook Let $L_n$ be the Laplacian matrix of the path graph $P_n$ of order $n$. Then the (ordered) eigenvalues $\lambda_{j}\in \mathbb{R}$ and corresponding eigenvectors, $v_j\in \mathbb{R}^{n}$ of the Laplacian matrix $L_n$ ($L_n v_j=\lambda_{j}v_j$ for $j=\{ 1,2, \dots,n\}$) satisfy the follo

Figures (3)

  • Figure 1: Choosing best $k$-ports in Graph G
  • Figure 2: $\lambda_{\min}$ function vs $p$ plot for $P_{11}$ showing local minima at the physical center.
  • Figure 3: Path graph with multiple (optimal) ports (say $k$): shift achieved is more than $k$ times the shift due to a single port

Theorems & Definitions (47)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Proposition 2.7
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • ...and 37 more