Forced condensation and anti-condensation on heavy-tailed networks

We study a driven selection mechanism on a fixed heavy-tailed network.

At each step fresh mass is injected, its direction is recomputed from the current mass profile by a power-normalization rule, and the combined mass is transported by a primitive mixing matrix.

The exponent $\theta$ controls the feedback.

Positive values give more weight to larger coordinates, while negative values favor smaller ones.

When $\theta=0$, the injected mass is distributed uniformly.

After deterministic growth of the total mass is scaled out, the long-run injection profile is characterized by a nonlinear Perron-Frobenius fixed point on the simplex.

Hilbert's projective metric gives a simple way to understand the stability of the system.

The discounted network response brings positive profiles closer together, while the escort map scales their projective distance by $|\theta|$.

On heavy-tailed networks, this fixed point separates three effects that are often conflated: response or degree tilt, anomalous inverse-participation-ratio scaling, and genuine few-node localization.

Positive feedback selects high-response nodes and, when response follows degree, a hub-directed branch.

Negative feedback selects low-response nodes and typically produces a broad peripheral cloud unless the lower tail of the response field is itself thin.

Numerical experiments on finite power-law networks support these results.

They show convergence, illustrate when the forcing rate becomes unimportant because mixing is sufficiently fast, and confirm both the sign law and the crossover in the participation ratio.

This mechanism is different from both conserved-mass condensation and graph growth.

Instead, feedback selects a non-equilibrium profile on a fixed, heterogeneous network.