# Threshold functions for small subgraphs: an analytic approach

## TL;DR

This paper introduces an analytic combinatorics approach to count small subgraphs in random graphs, considering degree constraints and overlaps, extending previous work with detailed proofs.

## Contribution

It develops a new analytic method using patchwork techniques to accurately count subgraphs in constrained random graphs, providing rigorous proofs.

## Key findings

- Effective counting method for small subgraphs
- Handles degree constraints in random graphs
- Provides detailed proofs and extended analysis

## Abstract

We revisit the problem of counting the number of copies of a fixed graph in a random graph or multigraph, including the case of constrained degrees. Our approach relies heavily on analytic combinatorics and on the notion of patchwork to describe the possible overlapping of copies. This paper is a version, extended to include proofs, of the paper with the same title to be presented at the Eurocomb 2017 meeting.