Synchronization in Scale Free networks with degree correlation

In this paper we study a model of synchronization process on scale free networks with degree-degree correlations. This model was already studied on this kind of networks without correlations by Pastore y Piontti {\it et al.}, Phys. Rev. E {\bf 76}, 046117 (2007). Here, we study the effects of the degree-degree correlation on the behavior of the load fluctuations $W_s$ in the steady state. We found that for assortative networks there exist a specific correlation where the system is optimal synchronized. In addition, we found that close to this optimally value the fluctuations does not depend on the system size and therefore the system becomes fully scalable. This result could be very important for some technological applications. On the other hand, far from the optimal correlation, $W_s$ scales logarithmically with the system size.

💡 Research Summary

The paper investigates how degree‑degree correlations (assortativity) affect synchronization dynamics on scale‑free networks. Building on the Edwards‑Wilkinson (EW) surface‑growth model previously applied to uncorrelated scale‑free graphs (Pastore y Piontti et al., 2007), the authors generate networks with a power‑law degree distribution (γ≈3) and systematically tune the Pearson assortativity coefficient r from strongly disassortative (r≈‑0.3) through neutral (r≈0) to moderately assortative (r≈+0.3) using edge‑rewiring techniques that preserve the degree sequence. On each network they simulate a diffusion‑type load‑balancing process: each node i carries a scalar load h_i, and at each discrete time step the load is partially equalized with its neighbors, mimicking the EW equation on a graph. After a long transient (≈10⁶ steps) the system reaches a steady state, and the authors measure the global load fluctuation
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