# An Exact No Free Lunch Theorem for Community Detection

## TL;DR

This paper establishes an exact No Free Lunch theorem for community detection by using an appropriate loss function and random model, extending the result to other set-partitioning tasks and reviewing evaluation functions.

## Contribution

It provides a stronger, exact No Free Lunch theorem for community detection using the correct loss function and model, generalizing to related set-partitioning problems.

## Key findings

- Provides an exact No Free Lunch theorem for community detection.
- Identifies evaluation functions compatible with the theorem.
- Generalizes the theorem to other set-partitioning tasks.

## Abstract

A precondition for a No Free Lunch theorem is evaluation with a loss function which does not assume a priori superiority of some outputs over others. A previous result for community detection by Peel et al. (2017) relies on a mismatch between the loss function and the problem domain. The loss function computes an expectation over only a subset of the universe of possible outputs; thus, it is only asymptotically appropriate with respect to the problem size. By using the correct random model for the problem domain, we provide a stronger, exact No Free Lunch theorem for community detection. The claim generalizes to other set-partitioning tasks including core/periphery separation, $k$-clustering, and graph partitioning. Finally, we review the literature of proposed evaluation functions and identify functions which (perhaps with slight modifications) are compatible with an exact No Free Lunch theorem.