Each additional synapse is harder to form than the last.
Fukushima (arXiv:2603.19320) builds a minimal model with a single rule: as a neuron accumulates incoming connections, the probability of forming the next one decreases. The crowding is physical — limited dendritic surface, metabolic cost of maintenance, spatial constraints on synaptic machinery. One parameter controls how fast crowding kicks in.
From this single constraint, the model derives exact solutions. In-degree distributions follow broad, approximately power-law forms — a few neurons receive many connections, most receive few. Mean connectivity grows logarithmically with network size while variance stays bounded. Connection lengths follow a broad power-law distribution without any explicit distance-dependent wiring rule. The spatial structure of the network emerges from local capacity limits, not from a global blueprint.
When combined with shortcut rewiring — long-range connections added at random — the model produces small-world networks. High local clustering from the crowding constraint, short average path lengths from the shortcuts. The small-world property, usually imposed as a design feature or observed as an empirical fact, here emerges from a developmental process. The brain doesn't build a small-world network. It follows a local crowding rule, and the small-world structure is what happens.
The dynamics on these networks reveal that degree heterogeneity — the power-law in-degree distribution — determines attractor basin boundaries in threshold dynamics. Which initial states lead to which final states depends on the distribution of connection counts. Local clustering, by contrast, affects the prevalence of long-lived transient states near these boundaries but not the boundaries themselves. Structure and dynamics are linked, but through different statistics of the same network.
The model is biologically consistent with homeostatic synaptic density regulation — the observation that neurons maintain roughly stable total synaptic input despite varying connection numbers. Crowding is the constraint. Everything else — degree distribution, spatial structure, small-world topology — is consequence.