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キーワード:
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要旨:
Complex networks exhibit a wide range of collective dynamic phenomena, including synchronization,
diffusion, relaxation, and coordination processes. Their asymptotic dynamics is generically characterized by the
local Jacobian, graph Laplacian, or a similar linear operator. The structure of networks with regular, small-world,
and random connectivities are reasonably well understood, but their collective dynamical properties remain
largely unknown. Here we present a two-stage mean-field theory to derive analytic expressions for network
spectra. A single formula covers the spectrum from regular via small-world to strongly randomized topologies
in Watts-Strogatz networks, explaining the simultaneous dependencies on network size N, average degree k,
and topological randomness q. We present simplified analytic predictions for the second-largest and smallest
eigenvalue, and numerical checks confirm our theoretical predictions for zero, small, and moderate topological
randomness q, including the entire small-world regime. For large q of the order of one, we apply standard random
matrix theory, thereby overarching the full range from regular to randomized network topologies. These results
may contribute to our analytic and mechanistic understanding of collective relaxation phenomena of network
dynamical systems.
