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A Categorical Framework for Quantifying Emergent Effects in Network Topology

Johnny Jingze Li, Sebastian Prado Guerra, Kalyan Basu, Gabriel A. Silva

TL;DR

The paper addresses quantifying emergent effects in network topology by proposing a categorical framework that treats emergence as a loss of exactness in a functor between abelian categories. It models networks as quiver representations and uses derived functors $R^1\Phi$ and $L^1\Phi$ to produce a computable numerical measure of emergence, ultimately connecting to cohomology. It validates the approach with numerical experiments on random Boolean networks, showing correlation with an information-theoretic emergence measure and providing mechanistic insights into which components contribute to emergence. The work offers a principled bridge between algebraic topology, category theory, and network science with potential applications to large-scale ML architectures and biological systems.

Abstract

Emergent effect is crucial to understanding the properties of complex systems that do not appear in their basic units, but there has been a lack of theories to measure and understand its mechanisms. In this paper, we consider emergence as a kind of structural nonlinearity, discuss a framework based on homological algebra that encodes emergence as the mathematical structure of cohomologies, and then apply it to network models to develop a computational measure of emergence. This framework ties the potential for emergent effects of a system to its network topology and local structures, paving the way to predict and understand the cause of emergent effects. We show in our numerical experiment that our measure of emergence correlates with the existing information-theoretic measure of emergence.

A Categorical Framework for Quantifying Emergent Effects in Network Topology

TL;DR

The paper addresses quantifying emergent effects in network topology by proposing a categorical framework that treats emergence as a loss of exactness in a functor between abelian categories. It models networks as quiver representations and uses derived functors and to produce a computable numerical measure of emergence, ultimately connecting to cohomology. It validates the approach with numerical experiments on random Boolean networks, showing correlation with an information-theoretic emergence measure and providing mechanistic insights into which components contribute to emergence. The work offers a principled bridge between algebraic topology, category theory, and network science with potential applications to large-scale ML architectures and biological systems.

Abstract

Emergent effect is crucial to understanding the properties of complex systems that do not appear in their basic units, but there has been a lack of theories to measure and understand its mechanisms. In this paper, we consider emergence as a kind of structural nonlinearity, discuss a framework based on homological algebra that encodes emergence as the mathematical structure of cohomologies, and then apply it to network models to develop a computational measure of emergence. This framework ties the potential for emergent effects of a system to its network topology and local structures, paving the way to predict and understand the cause of emergent effects. We show in our numerical experiment that our measure of emergence correlates with the existing information-theoretic measure of emergence.
Paper Structure (10 sections, 7 theorems, 21 equations, 4 figures)

This paper contains 10 sections, 7 theorems, 21 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Emergence as Structural Nonlinearity
  3. Machine Learning
  4. Biology
  5. Statistical Mechanics
  6. Emergence as Loss of Exactness
  7. Quiver Representations
  8. Computing the Derived Functors
  9. A Numerical Measure of Emergence
  10. Discussion and Conclusion

Key Result

Proposition 3.5

In an Abelian category, the square \begin{tikzcd}[row sep=3em] C \arrow[r,"a"] \arrow[d,"b"] & A \arrow[d,"b'"] \\ B \arrow[r,"a'"] & P \end{tikzcd}is a pushout if and only if the sequence \begin{tikzcd}[row sep=3em] C \arrow[r,"{(a,b)}"] &A\oplus B \arrow[r,"{(\begin{smallmatrix} b' \\ -a' \end{sma

Figures (4)

  • Figure 1: A simple example of representation of quiver $\bullet \rightarrow \bullet \leftarrow \bullet$ over a field $k$. Adapted from schiffler2014quiver.
  • Figure 2: Additional examples of quivers. (a): A quiver $Q$ with vertices $V = \{a,b,c,d\}$ and oriented edges $E = \{\alpha, \beta, \gamma, \delta\}$, (b) and (c): two quiver representations over $Q$. Adapted from armenta2021representation.
  • Figure 3: Examples of random Boolean network for connectivity $K = 1,2,3$.
  • Figure 4: Numerical results showing the correlation of emergence and information loss

Theorems & Definitions (21)

  • Definition 2.1
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Proposition 3.5
  • Definition 3.6
  • Definition 3.7
  • Proposition 3.8
  • Proposition 3.9
  • ...and 11 more