The Fiedler dimension of networks of networks: from fractal to small-world architectures

Abstract

Is the relaxation of a modular network dictated by its modules or by the network that connects them? The answer fixes the time scales of diffusion, consensus, and synchronization, all set by the Fiedler eigenvalue. We show that, for bundled networks, the two levels act one after the other: a random walker must first escape from its module and then spread over the network that connects them. The equilibration time is the sum of these two times, the time needed to reach the base from inside a fiber and the relaxation time of the base, slowed down by the mass of the fibers. The consequences are unexpected. However large the modules are, a small-world core always imposes its own Fiedler dimension; the modules survive only as a logarithmic correction, which slows down equilibration and hides the true exponent up to sizes far beyond any real network. Effective dimensions measured on finite modular systems can therefore be systematically biased. We analytically work out all combinations of finite-dimensional and small-world bases and fibers, and compare them with numerically exact spectra of representatives of each category.