Stabilization Without Simplification: A Two-Dimensional Model of Software Evolution

TL;DR

This paper presents a probabilistic framework explaining how software systems can stabilize over time without reducing their structural complexity, by separating burden from uncertainty.

Contribution

It introduces a novel graph-based model that formalizes the phenomenon of stabilization without simplification in software evolution.

Findings

Uncertainty can decrease while structural burden remains unchanged.

The framework formalizes conditions for stabilization without structural simplification.

Provides a theoretical basis for understanding software system stability over time.

Abstract

Software systems are widely observed to grow in size, complexity, and interdependence over time, yet many large-scale systems remain stable despite persistent structural burden. This apparent tension suggests a limitation in one-dimensional views of software evolution. This paper introduces a graph-based, discrete-time probabilistic framework that separates structural burden from uncertainty. Change effort is modeled as a stochastic variable determined by the dependency neighborhood of the changed entity and by residual variability. Within this framework, burden is defined as expected effort and uncertainty as variance of effort. We show that, under explicit assumptions on non-decreasing average structural load, structural regularization, process stabilization, and covariance control, there exists a regime in which uncertainty decreases while structural burden does not. This regime formalizes the phenomenon of stabilization without simplification. The proposed framework provides a minimal theoretical explanation for how software systems can become more predictable over time without necessarily becoming structurally simpler, and offers a foundation for further theoretical and empirical studies of software evolution.