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Geometric and spectral analysis on weighted digraphs

Fernando Lledó, Ignacio Sevillano

Abstract

In this article we give a geometrical description of the (in general non-selfadjoint) in/out Laplacian $\mathcal{L}^{+/-} = (d^{+/-})^* d$ and adjacency matrix on digraphs with arbitrary weights, where $(d^{+/-})^*$ is the adjoint of the evaluation map $d^{+/-}$ on the terminal/initial vertex of each arc and $d = d^+ + d^-$ denotes the discrete gradient. We prove that the multiplicity of the zero eigenvalue of $\mathcal{L}^{+/-} = (d^{+/-})^* d$ coincides with the number of sources/sinks of the digraph. We also show that for an acyclic digraph with combinatorial weights the spectrum is contained in the set of non-zero integers. The geometrical perspective allows to interpret the set of circulations $\mathcal{C}$ of a weighted digraph as coclosed forms on the arcs, i.e. as the kernel of the discrete divergence $d^*$. Moreover, $\mathcal{C}$ is perpendicular to the set of discrete gradients of functions on the vertices. We also give formulas to compute the capacity of a cut and the value of a flow in terms of $\mathcal{L}^-$ and $d$. We illustrate the results with many concrete examples.

Geometric and spectral analysis on weighted digraphs

Abstract

In this article we give a geometrical description of the (in general non-selfadjoint) in/out Laplacian and adjacency matrix on digraphs with arbitrary weights, where is the adjoint of the evaluation map on the terminal/initial vertex of each arc and denotes the discrete gradient. We prove that the multiplicity of the zero eigenvalue of coincides with the number of sources/sinks of the digraph. We also show that for an acyclic digraph with combinatorial weights the spectrum is contained in the set of non-zero integers. The geometrical perspective allows to interpret the set of circulations of a weighted digraph as coclosed forms on the arcs, i.e. as the kernel of the discrete divergence . Moreover, is perpendicular to the set of discrete gradients of functions on the vertices. We also give formulas to compute the capacity of a cut and the value of a flow in terms of and . We illustrate the results with many concrete examples.
Paper Structure (9 sections, 12 theorems, 53 equations, 8 figures)

This paper contains 9 sections, 12 theorems, 53 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Structure of the article and main results:
  3. Notation:
  4. Acknowledgements:
  5. Weighted digraphs and components of the Laplacian
  6. Digraphs and weights
  7. First and second order operators on digraphs
  8. Sources, sinks and the spectrum of the in/out Laplacians
  9. A geometrical interpretation of circulations and flows in networks

Key Result

Proposition 2.10

Let $\varphi \in \ell_2(V,m)$, then Moreover, $0$ is an eigenvalue of $\mathcal{L}^\pm$ with (nonzero) constant eigenfunction.

Figures (8)

  • Figure 1: A representation of a digraph of order $5$.
  • Figure 2: Weighted digraph.
  • Figure 3: A digraph with four maximal chains. Lower/upper bounds given by $\{v_1\}$,$\{v_2\}$ and $\{v_5\}$,$\{v_6\}$.
  • Figure 7: Example of a digraph with one $F=\{v_1\}$ and sink $S=\{v_4\}$ that is not $d$-connected.
  • Figure 10: Matrices $(\mathcal{L}^{+}),(\mathcal{L}^{-})$ for a combinatorial digraph.
  • ...and 3 more figures

Theorems & Definitions (47)

  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Example 2.4
  • Definition 2.5
  • Definition 2.6
  • Example 2.7
  • Definition 2.8
  • Definition 2.9
  • Proposition 2.10
  • ...and 37 more