Sharp Asymptotics for the Largest Component in the Subcritical Regime of Preferential Attachment Without Vertex Growth

What the math gives to ML

The paper gives a sharp, explicit law for the largest connected component in a degree-biased random graph before its percolation transition: with edge count m=m_c(1-epsilon), the largest component has scale epsilon^{-2} log(epsilon^3 n). This can transfer to sparse neural connectivity, where preferentially sampled edges create hub-like communication patterns without crossing into a giant connected component. The most promising use is to generate sparse attention or message-passing masks from a target component-size budget, then add a few explicit global-token links if cross-component communication is needed.