Unveiling the Dimensionality of Networks of Networks

TL;DR

By composing scale-invariant networks, tinkering decouples Fiedler and spectral dimensions, hitherto considered identical, providing valuable insights into mesoscopic and macroscopic collective regimes.

Abstract

"Every object that biology studies is a system of systems." (François Jacob, 1974). Most networks feature intricate architectures originating from tinkering, a repetitive use of existing components where structures are not invented but reshaped. Still, linking the properties of primitive components to the emergent behavior of composite networks remains a key open challenge. Here, by composing scale-invariant networks, we show how tinkering decouples Fiedler and spectral dimensions, hitherto considered identical, providing valuable insights into mesoscopic and macroscopic collective regimes.

Figures (4)

• Figure 1: (a) Third generation of the Sierpinski gasket. (b) Heat capacity, $C$, versus diffusion time, $\tau$, for the Sierpinski gasket of $n^\text{th}$ generation (see legend). The number of nodes is $N_\text{gasket}(n)=\frac{3}{2}(3^n+1)$. The inset shows the scaling of the Fiedler eigenvalue as a function of the total number of nodes. (c) Second generation of the Sierpinski carpet. (d) Heat capacity, $C$, versus diffusion time, $\tau$, for the Sierpinski carpet of $n^\text{th}$ generation (see legend). The number of nodes is $N_\text{carpet}(n)=2^{n+3}$. The inset shows the scaling of the Fiedler eigenvalue as a function of the total number of nodes. The blue dashed lines and the orange solid lines represent the theoretical expectation for both cases.
• Figure 2: (a) Dirac comb network. (b) Heat capacity, $C$, versus diffusion time, $\tau$, for the periodic Dirac comb. The first and the second plateaus reflect the spectral dimension of the fiber and the base, respectively. In this case, both are one-dimensional systems that correspond to the yellow dashed line in the figure. The blue dashed line represents the Fiedler exponent as predicted in Eq. \ref{['dg_gen']}. (c) Dirac brush network. For clarity, periodic boundary conditions have been omitted in the picture. (d) Heat capacity, $C$, versus diffusion time, $\tau$, for the periodic Dirac brush. The first plateau reflects the local dimension of the prongs (yellow dashed line), whereas the second one characterizes the base (blue dashed line).
• Figure 3: (a) Local slope of the Fiedler scaling minus the expected value given by Eq. \ref{['dg_gen']} versus system size for ring-lattice networks varying the dimensions of the base (see legend). (b) Comparison between the numerical analysis and Eq. \ref{['dg_gen']} for different composite network families: the squares are $d$-dimensional lattices as base and 1-dimensional rings as fiber; for the triangles, the bases are trees with given fractal and Fiedler dimension while the fibers are 1-dimensional rings (see end matter for details). Inset: Spectral dimension for a BA network with ring fibers and RT fibers. (c)-(d) Heat capacity, $C$, versus diffusion time, $\tau$, of different system sizes (see legend) for: (c) a BA-ring network, and (d) a RT-ring network. The first plateau reflects the local dimension of the fiber, whereas the second one characterizes the BA and RT base dimension, respectively. The blue dashed lines represent the Fiedler dimension in both cases.
• Figure 4: (a) Heat capacity, $C$, versus diffusion time, $\tau$, for a periodic Dirac comb where the base grows as a power $L^{\frac{1}{3}}$ of the linear size $L$. The only plateau featured in the profile reflects the local dimension of the rings, whereas no plateau emerges at later times. (b) Heat capacity, $C$, versus diffusion time, $\tau$, for a periodic Dirac comb where the base grows as a power $L^{\frac{4}{5}}$ of the linear size $L$. The first plateau is related to the rings, while a second plateau induce by the base emerges at later times.