Optimization algorithms for spectral coarse-graining of complex networks

Coarse-graining techniques of large-scale complex networks have been an important approach to reduce network size, which merge nodes who share the same or similar properties while preserving some significant functions or properties of the original networks. However, reducing network size is often at the cost of worsening network performance. Thus, there is a trade-off between the coarse-grained network sizes and network performance. To find the balance between the two sides and based on the spectral coarse-graining approach (SCG), we propose two optimization algorithms, which are called variable step size optimization algorithm (VSSOA) and variable scale optimization algorithm (VSOA). The two algorithms can calculate the optimal coarse-grained step size and the optimal scale to reduce the share size of the network. The two algorithms are applied to the coarse-graining of several typical networks. And the feasibility and validity of the proposed algorithms are further verified by phase synchronization of coupled Kuramoto oscillators on typical networks. The related investigation provides a deep insight to the coarse-graining of large-scale complex networks.