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Conway's law, revised from a mathematical viewpoint

Shigeki Matsutani, Shousuke Ohmori, Kenji Hiranabe, Eiichi Hanyuda

TL;DR

This paper revisits Conway's law by embedding the software-organization correspondence into a three-tier mathematical framework: first, a task-graph decomposition via a surjective map from the software graph to a task graph and an injective map from the task graph to the organization; second, a relaxation to w-homomorphisms to model internal communication and hierarchical structure; and third, a topological treatment using continuous maps between collections of connected subgraphs to express hierarchy and information flow. The authors define novel notions such as t-Conway doublets/triplets and their morphisms, showing how the correspondence can be expressed as continuous relationships between graph topologies rather than strict graph homomorphisms. This provides a rigorous foundation for analyzing and designing large-scale, hierarchical software and organizations, aligning with modern concerns like security and knowledge hiding. The work also positions these mathematical tools as a formal basis for concepts like team topologies and potential extensions to collaborative artifacts, while highlighting future directions in sheaf-theoretic treatments of software-organization geometry.

Abstract

In this article, we revise Conway's Law from a mathematical point of view. By introducing a task graph, we first rigorously state Conway's Law based on the homomorphisms in graph theory for the software system and the organizations that created it. Though Conway did not mention it, the task graph shows the geometric structure of tasks, which plays a crucial role. Furthermore, due to recent requirements for high-level treatment of communication (due to security, knowledge hiding, etc.) in organizations and hierarchical treatment of organizations, we have reformulated these statements in terms of weakened homomorphisms, and the continuous maps in graph topology. In order to use graph topology and the continuous map in Conway's law, we have prepared them as mathematical tools, and then we show the natural expression of Conway's correspondences with hierarchical structures.

Conway's law, revised from a mathematical viewpoint

TL;DR

This paper revisits Conway's law by embedding the software-organization correspondence into a three-tier mathematical framework: first, a task-graph decomposition via a surjective map from the software graph to a task graph and an injective map from the task graph to the organization; second, a relaxation to w-homomorphisms to model internal communication and hierarchical structure; and third, a topological treatment using continuous maps between collections of connected subgraphs to express hierarchy and information flow. The authors define novel notions such as t-Conway doublets/triplets and their morphisms, showing how the correspondence can be expressed as continuous relationships between graph topologies rather than strict graph homomorphisms. This provides a rigorous foundation for analyzing and designing large-scale, hierarchical software and organizations, aligning with modern concerns like security and knowledge hiding. The work also positions these mathematical tools as a formal basis for concepts like team topologies and potential extensions to collaborative artifacts, while highlighting future directions in sheaf-theoretic treatments of software-organization geometry.

Abstract

In this article, we revise Conway's Law from a mathematical point of view. By introducing a task graph, we first rigorously state Conway's Law based on the homomorphisms in graph theory for the software system and the organizations that created it. Though Conway did not mention it, the task graph shows the geometric structure of tasks, which plays a crucial role. Furthermore, due to recent requirements for high-level treatment of communication (due to security, knowledge hiding, etc.) in organizations and hierarchical treatment of organizations, we have reformulated these statements in terms of weakened homomorphisms, and the continuous maps in graph topology. In order to use graph topology and the continuous map in Conway's law, we have prepared them as mathematical tools, and then we show the natural expression of Conway's correspondences with hierarchical structures.
Paper Structure (17 sections, 22 theorems, 1 equation, 6 figures, 1 table)

This paper contains 17 sections, 22 theorems, 1 equation, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Conway's law: mathematical revised as the first step
  3. Mathematical preliminary
  4. Review of topology
  5. Review of poset
  6. Examples of the continuous map
  7. Topological expression of graphs
  8. Review of graph theory
  9. Topology of graph
  10. Examples of the continuous maps
  11. Examples of the induced maps
  12. Conway's law
  13. The first step: original Conway's law with task graph
  14. The second step: Conway's law with w-homomorphism
  15. The third step: Conway's law under graph topology
  16. ...and 2 more sections

Key Result

Lemma 2.4

A finite poset $(P, \le)$ has a natural topology ${\mathcal{T}}_P$ induced from $\leq$, i.e., the set $P$ with ${\mathcal{T}}_P$ recovers the partial order $\leq$ again. The topological space $(P, {\mathcal{T}}_P)$ is referred to as the poset topology of $(P, \le)$.

Figures (6)

  • Figure 1: (a) There is a homomorphism, but (b) there is a w-homomorphism but not homomorphism.
  • Figure 2: Examples of the collection of connected subgraphs with the partial order $\subset$ are illustrated.
  • Figure 3: Examples of continuous injections are illustrated.
  • Figure 4: An example which we cannot define an embedding.
  • Figure 5: The graphs for the examples of the continuum injections.
  • ...and 1 more figures

Theorems & Definitions (60)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • Corollary 2.7
  • ...and 50 more