Network feedback model predicts condensation and anti-condensation on heavy-tailed networks

A new arXiv paper by Ashwin Bhattathiripad and Vipin P. Veetil explores forced condensation and anti-condensation on heavy-tailed networks. The researchers model a system where fresh mass is injected at each step, its direction recomputed from the current mass profile via a power-normalization rule, and then transported by a primitive mixing matrix. The exponent θ controls feedback: positive values give more weight to larger coordinates (condensation), while negative values favor smaller ones (anti-condensation). At θ=0, injection is uniform. After scaling out deterministic growth, the long-run injection profile converges to a nonlinear Perron-Frobenius fixed point on the simplex. Hilbert's projective metric reveals that the discounted network response brings profiles closer, while the escort map scales their projective distance by |θ|.

On heavy-tailed networks, this fixed point separates three often conflated effects: 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 findings, showing convergence, the irrelevance of forcing rate when mixing is fast, and confirming both the sign law and the crossover in the participation ratio. This mechanism differs from conserved-mass condensation and graph growth, instead selecting a non-equilibrium profile on a fixed, heterogeneous network.