Recursive Formula for Labeled Graph Enumeration

TL;DR

This paper introduces a recursive formula to estimate the number of labeled graphs with specified properties, aiding social network analysis by quantifying the diversity and size of graph families with given statistics.

Contribution

The paper presents a novel recursive approach to count labeled graphs with fixed network properties, extending to various network features and bipartite graphs.

Findings

Validates the formula through simulation studies.

Demonstrates application to degree distribution and mixing.

Shows potential for assessing graph diversity within fibers.

Abstract

This manuscript presents a general recursive formula to estimate the size of fibers associated with algebraic maps from graphs to summary statistics of importance for social network analysis, such as number of edges (graph density), degree sequence, degree distribution, mixing by nodal covariates, and degree mixing. That is, the formula estimates the number of labeled graphs that have given values for network properties. The proposed approach can be extended to additional network properties (e.g., clustering) as well as properties of bipartite networks. For special settings in which alternative formulas exist, simulation studies demonstrate the validity of the proposed approach. We illustrate the approach for estimating the size of fibers associated with the Barab\'{a}si--Albert model for the properties of degree distribution and degree mixing. In addition, we demonstrate how the approach can be used to assess the diversity of graphs within a fiber.