TL;DR

Modularity seems like an important feature of neural networks, but there is currently no canonical way of properly defining or measuring it which is properly theoretically motivated and doesn’t break down in some cases - in other words, we haven’t yet found a True Name for it. Most modularity measures used in experiments are based on ad-hoc methods from graph-theory or network theory, and don’t seem to capture the kind of modularity we care about.

In this post, we explore these existing ways of measuring modularity in neural networks, and their limitations. We also outline our ideas for a new modularity metric, and in particular the important role we think two branches of mathematics will play:

Information theory, because neural networks are fundamentally information-processing devices, and we want to measure how this information is being used, exchanged and transmitted. In particular, we expect mutual information to be at the core of any metric we use.
Causal inference, because correlation (and indeed mutual information) does not automatically equal causation, and we need a way of distinguishing “modules $X$ and $Y$ fire with similar patterns” from the more specific claim “modules $X$ and $Y$ are directly sharing information with each other”. In particular, we expect counterfactuals to play an important role here.

Introduction

A first-pass definition for modularity might go something like this:

A system is modular to the extent that it can be described in terms of discrete modules, which perform more intra-modular communication than inter-module.

In previous posts, we’ve talked about some causes of modularity in biological systems, reasons we should care about modularity in the first place, and unanswered questions about modularity which require experimentation. In this post, we will return to a more theoretical question - how can we develop a yardstick for measuring modularity, which is built on sound mathematical principles?

First off, why is this an important thing to do? Because most current ways of measuring modularity fall pretty short. As recently discussed by John Wentworth, if all you have is a weak proxy for the concept you’re trying to measure, then your definition is unlikely to generalise robustly in all the cases you care about. As a simple example, most architectural ways of...

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