Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Graph Pencil Method: Mapping Subgraph Densities to Stochastic Block Models

Lee M Gunderson, Gecia Bravo-Hermsdorff, Peter Orbanz

TL;DR

This work determines an exact map from a finite set of subgraph densities to the parameters of a stochastic block model (SBM) matching these densities, and makes it possible to translate estimates of these subgraph densities into model parameters, and hence to use subgraph densities directly for inference.

Abstract

In this work, we describe a method that determines an exact map from a finite set of subgraph densities to the parameters of a stochastic block model (SBM) matching these densities. Given a number $K$ of blocks, the subgraph densities of a finite number of stars and bistars uniquely determines a single element of the class of all degree-separated stochastic block models with $K$ blocks. Our method makes it possible to translate estimates of these subgraph densities into model parameters, and hence to use subgraph densities directly for inference. The computational overhead is negligible; computing the translation map is polynomial in $K$, but independent of the graph size once the subgraph densities are given.

The Graph Pencil Method: Mapping Subgraph Densities to Stochastic Block Models

TL;DR

This work determines an exact map from a finite set of subgraph densities to the parameters of a stochastic block model (SBM) matching these densities, and makes it possible to translate estimates of these subgraph densities into model parameters, and hence to use subgraph densities directly for inference.

Abstract

In this work, we describe a method that determines an exact map from a finite set of subgraph densities to the parameters of a stochastic block model (SBM) matching these densities. Given a number of blocks, the subgraph densities of a finite number of stars and bistars uniquely determines a single element of the class of all degree-separated stochastic block models with blocks. Our method makes it possible to translate estimates of these subgraph densities into model parameters, and hence to use subgraph densities directly for inference. The computational overhead is negligible; computing the translation map is polynomial in , but independent of the graph size once the subgraph densities are given.
Paper Structure (19 sections, 45 equations, 1 figure, 3 tables)

This paper contains 19 sections, 45 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Notation and Definitions
  3. Mapping Star Densities to Block Degrees
  4. Classical Coin Collecting
  5. Distilling the Degree Distribution
  6. From Bistar Densities to a Stochastic Block Model
  7. Gluing Graphs: An Algebra
  8. Extracting the Edge Expectations
  9. Beyond Degree Correlations: Adding "Two-Hop" Subgraphs
  10. From an Observed Graph to a Stochastic Block Model
  11. Quickly Counting (Injective) Homomorphisms
  12. Discussion
  13. The Glyph Algebra
  14. Subgraph Densities
  15. Rooted Subgraph Densities
  16. ...and 4 more sections

Figures (1)

  • Figure 1: Adding "two-hop" subgraphs helps recover (dis)assortativity. We use $2$-by-$2$ SBMs with varying degrees of assortativity to compare our basic method using bistars from section \ref{['sec:GettingtheBmatrix']} (green squares) and the method that adds the two-hop subgraphs from section \ref{['sec:AddingMoreSubgraphs']} (purple circles). The vertical axis measures the expected squared error of the probability of an edge between two random nodes, and shading denotes $\pm 1$ standard deviation from the average value. The dashed orange line denotes the expected squared error if the latent blocks of the nodes were known, and both methods appear to converge at this optimal rate. When the SBM is particularly assortative (left) or disassortative (right), the inclusion of two-hop subgraphs results in a notable improvement.

Theorems & Definitions (2)

  • Remark 1: Degree separated assumption
  • Remark 2: Sufficiency of subgraph densities