Forced condensation and anti-condensation on heavy-tailed networks

Abstract

We study a driven selection mechanism on a fixed heavy-tailed network. At each step, a power-normalization rule recomputes the direction of fresh injection from the current mass profile. A primitive mixing matrix then transports the existing stock and the new mass. The exponent $θ$ sets the feedback. Positive values give more weight to larger coordinates, whereas negative values favor smaller ones. After removing the deterministic growth of total mass, we give an explicit mixing--forcing condition under which the injection profile converges to a nonlinear Perron--Frobenius-type fixed point on the simplex. Hilbert's projective metric makes the stability mechanism transparent. The discounted network response draws positive profiles closer together, while the escort map scales their projective distance by $|θ|$. On heavy-tailed networks, the 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 favors high-response nodes and, when response follows degree, tilts the selected profile toward the hubs. Negative feedback favors low-response nodes. It usually produces a broad peripheral cloud unless the lower tail of the response field is itself thin. Computations on finite networks illustrate convergence, forcing-rate dependence, and the sign law. Monte Carlo samples of truncated heavy-tailed response profiles display the predicted participation-ratio crossover. A uniform comparison bound gives conditions under which this scaling transfers to the full fixed point. The mechanism differs from conserved-mass condensation and graph growth because feedback selects a non-equilibrium profile on a fixed, heterogeneous network.