💻 computer science

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

This paper introduces a two-dimensional probabilistic framework that demonstrates how software systems can achieve stabilization (reduced uncertainty) without simplification (reduced structural burden) by distinguishing between expected change effort and its variance.

Original authors: Masaru Furukawa

Published 2026-04-09

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Masaru Furukawa

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Idea: A Busy City That Gets Calmer, Not Smaller

Imagine a massive, ancient city. Over the years, it has grown huge. New buildings are added, roads are connected, and traffic patterns have become incredibly complex. By almost any measure, this city is more complicated today than it was 50 years ago.

The Old Way of Thinking:
Most people assume that for a city to become "stable" (safe, predictable, easy to manage), it must be simplified. We think we need to tear down old buildings, close off streets, and reduce the population to make things easier. If the city stays huge and complex, we assume it will eventually collapse or become a nightmare to navigate.

The New Discovery:
This paper argues that the old way of thinking is wrong. It shows that a city (or a software system) can become highly predictable and stable without getting smaller or simpler.

The authors call this "Stabilization Without Simplification."

The Two Dimensions: "How Hard" vs. "How Surprising"

To understand this, the authors split the problem of "change" into two separate buckets. Think of it like driving a car:

Structural Burden (The "How Hard" Bucket):
- Definition: The average amount of work or effort required to make a change.
- Analogy: Imagine you need to fix a pothole. If the road is made of thick, heavy concrete, the effort to fix it is high. If the city grows and adds more heavy concrete, the effort stays high (or gets higher).
- In Software: This is the complexity of the code. If you change one file, how many other files does it touch? If the system is huge, the "burden" is high.
Uncertainty (The "How Surprising" Bucket):
- Definition: How unpredictable the result is. Will the fix take 1 hour or 100 hours? Will it break something else?
- Analogy: Imagine driving on a road.
  - High Uncertainty: You don't know if the road is icy, if there's a hidden hole, or if a cow will jump out. The trip is risky and scary, even if the road is short.
  - Low Uncertainty: You know exactly how long the trip takes. You know the traffic lights. You know the weather. The trip is boring but safe.
- In Software: This is the chaos. Even if a change is hard, if you know exactly how hard it will be and what will break, the system is "stable."

The Magic Trick: How to Get Stable Without Getting Simple

The paper proves mathematically that you can have High Burden (it's still hard work) but Low Uncertainty (it's very predictable).

Here is how it happens, using the City Analogy:

1. The City Doesn't Shrink (Non-Simplification)

The city keeps growing. New skyscrapers are built. The traffic is still heavy. The "Structural Burden" remains high. You still have to do a lot of work to fix a pothole because the road is thick concrete.

Result: The system is still complex.

2. But the City Becomes Routine (Stabilization)

Over time, the city planners and drivers learn the rules.

Regularization: The chaotic, messy neighborhoods become organized. Everyone knows the traffic patterns.
Process Stabilization: The city gets better tools. They have better trucks, better maps, and better training. The "surprise" factor disappears.
Covariance Control: They learn that the "scary" parts of the city (the complex intersections) are actually manageable if you follow the right procedure. They stop panicking when they hit a complex area.

The Result:
Even though the city is huge and complex (High Burden), the drivers know exactly what to expect. They aren't surprised by accidents or delays. The trip is predictable.

The Four Rules of the Game

The authors created a mathematical model with four rules that allow this "Stabilization Without Simplification" to happen:

The Load Stays High: The average complexity of the things you change doesn't go down. (The city keeps growing).
The Chaos Shrinks: The variety of problems decreases. You stop encountering weird, unique, one-off nightmares. Everything starts looking more like the "standard" problem.
The Process Gets Smooth: The random errors (like a worker dropping a tool or a sudden storm) become less frequent. The team gets better at their job.
The "Scary" Factor Drops: In the beginning, the most complex parts of the city were also the most unpredictable. Over time, the team learns how to handle the complex parts, so they stop being "extra scary."

Why Does This Matter?

For Software Engineers:
Stop obsessing over "simplifying" the code at all costs. Sometimes, a system needs to be complex to do its job. Instead of trying to tear it down, focus on making the process of changing it predictable.

Don't just ask: "How can we make this code smaller?"
Ask instead: "How can we make the changes to this code more predictable?"

For Managers:
If a system feels "heavy" and complex, don't panic. If the team is becoming more predictable, if the bugs are fewer, and if the estimates are more accurate, the system is stabilizing, even if the code hasn't gotten simpler.

The Takeaway

You don't need to turn a giant, complex machine into a tiny, simple toy to make it reliable. You just need to learn how to operate the giant machine so well that you know exactly what will happen every time you push a button.

Complexity is the weight of the machine. Predictability is the skill of the driver. You can have a heavy machine and a very skilled driver. That is the secret to long-term software stability.

1. Problem Statement

The paper addresses a fundamental tension in software engineering:

Observation: Software systems are widely observed to grow in size, complexity, and interdependence over time (increasing structural burden).
Paradox: Despite this persistent or increasing structural complexity, many large-scale, mature systems do not collapse. Instead, they evolve in a stable and manageable manner.
Limitation of Current Views: Traditional software evolution models are largely one-dimensional, treating complexity growth as inherently detrimental and assuming that "stabilization" can only be achieved through "simplification" (e.g., reducing dependencies or modularity).
The Gap: Existing frameworks fail to distinguish between the expected cost of change (structural burden) and the variability of that cost (uncertainty). Consequently, they cannot explain how systems become more predictable without necessarily becoming structurally simpler.

2. Methodology

The author proposes a two-dimensional, graph-based, discrete-time probabilistic framework to model software evolution.

A. Formal Setup

Time: Evolution is modeled in discrete time steps $t = 0, 1, 2, \dots$ .
Structure: The system state $S_t$ is represented as a directed dependency graph $G_t = (V_t, E_t)$ , where $V_t$ are entities (files, modules) and $E_t$ are dependencies.
Change Events: At each step, a target entity $X_t$ is selected randomly according to a distribution $p_t(v)$ .
Effort Model: The effort $e_t$ $e_{t}$ required to change entity $X_t$ $X_{t}$ is modeled as:
$e_t = \alpha d_t(X_t) + \beta + \epsilon_t$
- $\alpha d_t(X_t)$ : Structural component based on the out-degree (dependency neighborhood) of the changed entity.
- $\beta$ : Constant base cost.
- $\epsilon_t$ : Stochastic residual term with mean zero, representing unexplained variability.

B. Key Definitions

The framework defines two core metrics:

Structural Burden ( $B_t$ ): The expected effort of change.
$B_t = E[e_t] = \alpha \mu_t + \beta$
(Where $\mu_t$ is the expected structural load of changed entities).
Uncertainty ( $U_t$ ): The variance of change effort.
$U_t = \text{Var}(e_t) = \alpha^2 \sigma^2_{d,t} + \sigma^2_{\epsilon,t} + 2\alpha c_t$
(Where $\sigma^2_{d,t}$ is the variance of structural load, $\sigma^2_{\epsilon,t}$ is residual variance, and $c_t$ is the covariance between structural load and residual effort).

C. Assumptions

To derive the main result, the paper imposes four explicit assumptions regarding the evolution of these metrics:

A1 (Non-decreasing Average Load): $\mu_{t+1} \ge \mu_t$ (Structural burden does not decrease).
A2 (Structural Regularization): $\sigma^2_{d,t+1} \le \sigma^2_{d,t}$ (Heterogeneity of structural load decreases).
A3 (Process Stabilization): $\sigma^2_{\epsilon,t+1} \le \sigma^2_{\epsilon,t}$ (Residual process variability decreases).
A4 (Covariance Control): $c_{t+1} \le c_t$ (The link between structural heaviness and residual difficulty weakens or stays constant).

3. Key Contributions

Conceptual Distinction: The paper formally separates Structural Burden (expected cost) from Uncertainty (variability of cost), creating a two-dimensional state space for software evolution.
Formal Framework: It introduces a minimal, rigorous probabilistic model using dependency graphs and stochastic variables to define these dimensions.
Theoretical Proof: It proves that "Stabilization Without Simplification" is a structurally admissible regime, not merely an empirical anomaly.

4. Results

The central theoretical result is Theorem 5.4, which establishes the conditions for Stabilization Without Simplification:

Condition: If assumptions A1–A4 hold, then:
- $\Delta B_t \ge 0$ (Structural burden remains constant or increases).
- $\Delta U_t < 0$ (Uncertainty decreases).
Implication: A software system can become more predictable (stabilized) even while its average structural complexity remains high or grows.
Mechanism: Stabilization arises not from reducing the magnitude of dependencies, but from:
1. Reducing the heterogeneity of those dependencies (making the graph more regular).
2. Reducing process noise (better testing, workflows, developer knowledge).
3. Reducing the covariance between structural complexity and unexpected difficulties (making complex changes more routine).

5. Significance and Implications

Reframing Stability: The paper redefines software stability. Stability is not the absence of complexity, but the predictability of change. A system can be highly complex yet highly stable if the variance of effort is low.
Internalization of Complexity: The model suggests that mature systems do not necessarily "simplify"; rather, they internalize complexity. Developers and tools adapt to the structural burden, turning what was once unpredictable into routine, predictable operations.
Engineering Practice: It suggests that efforts to improve stability should focus on reducing uncertainty (via better testing, standardization, and knowledge accumulation) rather than solely focusing on reducing burden (refactoring/simplification).
Theoretical Foundation: It provides a mathematical basis for empirical observations where large open-source projects become more manageable over time despite growing in size. It explains that the "burden-uncertainty decoupling" is a natural consequence of structural regularization and process stabilization.

In summary, Furukawa's work demonstrates that software evolution is a two-dimensional process. Systems can move toward stability (lower uncertainty) while simultaneously moving away from simplicity (higher or constant burden), provided that the variability and unpredictability of the change process are effectively managed.