The Functional Proximity Law: What It Says and Why It Matters

The Core Claim

When you map the same set of objects as a multilayer network — meaning you draw multiple types of relationships between the same nodes — the nodes that matter most in one layer tend to matter most in a similar layer too.

This is the Functional Proximity Law.

Formally: the Pearson correlation between hub-importance scores (degree centrality) across two layers is significantly higher when those two layers encode functionally similar relationships than when they encode dissimilar ones.

A Concrete Example

Consider a CPU block design. The same 10 components (fetch, decode, execute, ALU, FPU, ...) can be modelled as three layers:
• signal_dependency — which blocks send/receive control signals
• power_domain — which blocks share power rails
• scan_chain — which blocks are chained for testability

The signal and power layers are both operational layers. They describe how the chip functions at runtime. The scan chain is a test layer — it exists only for manufacturing validation.

Running IRDME gives: r(signal ↔ power) = +0.62, r(signal ↔ scan) = −0.65.

The law predicts the first correlation should be higher. It is. The denial case (negative r across dissimilar layers) is equally informative — it tells you the scan chain deliberately inverts the operational hierarchy to maximise coverage.

Why It Holds

The mechanism is structural: functionally similar relationships are defined by the same underlying constraints. A node that routes many signals also tends to sit at a power-domain junction, because both reflect physical proximity and fan-out. Dissimilar layers, by contrast, are defined by different constraints and can invert the importance ranking.

What It Does Not Say

The law does not say all hubs are universal. It says similarity predicts preservation. A node can be a hub in biology and irrelevant in engineering — the law only makes predictions within a single system's layers.

Implications

Network design: if you want a robust hub, make it important in multiple functionally similar layers simultaneously — not just one.

Biology: hub proteins in co-expression and protein-protein interaction networks (both functional) should be more stable targets than hubs unique to one layer.

Falsifiability: the law has a named denial condition — resolution mismatch. When layers encode relationships at different abstraction levels, the law does not apply, and we can predict that in advance.

All 23 domain experiments are pre-registered with SHA-256 hashes at github.com/vladi160/the-beginning. The paper is at arXiv:2604.23639.