Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Expectation and Variance of the Degree of a Node in Random Spanning Trees

Enrique Fita Sanmartín, Christoph Schnörr, Fred A. Hamprecht

Abstract

We consider a Gibbs distribution over all spanning trees of an undirected, edge weighted finite graph, where, up to normalization, the probability of each tree is given by the product of its edge weights. Defining the weighted degree of a node as the sum of the weights of its incident edges, we present analytical expressions for the expectation, variance and covariance of the weighted degree of a node across the Gibbs distribution. To generalize our approach, we distinguish between two types of weight: probability weights, which regulate the distribution of spanning trees, and degree weights, which define the weighted degree of nodes. This distinction allows us to define the weighted degree of nodes independently of the probability weights. By leveraging the Matrix Tree Theorem, we show that these degree moments ultimately depend on the inverse of a submatrix of the graph Laplacian. While our focus is on undirected graphs, we demonstrate that our results can be extended to the directed setting by considering incoming directed trees instead.

Expectation and Variance of the Degree of a Node in Random Spanning Trees

Abstract

We consider a Gibbs distribution over all spanning trees of an undirected, edge weighted finite graph, where, up to normalization, the probability of each tree is given by the product of its edge weights. Defining the weighted degree of a node as the sum of the weights of its incident edges, we present analytical expressions for the expectation, variance and covariance of the weighted degree of a node across the Gibbs distribution. To generalize our approach, we distinguish between two types of weight: probability weights, which regulate the distribution of spanning trees, and degree weights, which define the weighted degree of nodes. This distinction allows us to define the weighted degree of nodes independently of the probability weights. By leveraging the Matrix Tree Theorem, we show that these degree moments ultimately depend on the inverse of a submatrix of the graph Laplacian. While our focus is on undirected graphs, we demonstrate that our results can be extended to the directed setting by considering incoming directed trees instead.
Paper Structure (13 sections, 7 theorems, 58 equations, 1 table)

This paper contains 13 sections, 7 theorems, 58 equations, 1 table.

Table of Contents

  1. Introduction
  2. Notation
  3. Edge weight functions
  4. Weighted degree
  5. Spanning trees
  6. The Laplacian and the Matrix Tree Theorem
  7. Polynomial-Based Computation of Expectation and Variance of Node Degree in Spanning Trees
  8. Laplacian-Based Computation of Expectation and Variance of Node Degree in Spanning Trees
  9. Relation Between Edge Probability and Expected Node Degree in Spanning Trees
  10. Extension to Directed Graphs
  11. Conclusion
  12. Covariance
  13. Expectations of Spanning Trees over Additively Decomposable Functions

Key Result

Theorem 2.1

Given an edge weighted undirected graph $G=(V,E,w)$, let $\mathcal{T}_{\text{ }}$ represent the set of all spanning trees of $G$. Then, the total weight of these spanning trees, denoted as $w(\mathcal{T}_{\text{ }})$, can be expressed as: where $r$ is an arbitrary but fixed node, and $L_G^{[r]}$ represents the Laplacian matrix after removing the row and column indexed by $r$.

Theorems & Definitions (20)

  • Theorem 2.1: Matrix Tree Theorem, kirchhoff1847ueberTutte1984
  • Definition 1
  • Lemma 3.1
  • proof
  • Theorem 3.1
  • proof
  • Remark 3.1
  • Theorem 4.1
  • Remark 4.1
  • proof
  • ...and 10 more