Geometric Criticality in Scale-Invariant Networks

Abstract

Dimension in physical systems determines universal properties at criticality. Yet, the impact of structural perturbations on dimensionality remains largely unexplored. Here, we characterize the attraction basins of structural fixed points in scale-invariant networks from a renormalization group perspective, demonstrating that basin stability connects to a structural phase transition. This topology-dependent effect, which we term geometric criticality, triggers a geometric breakdown hitherto unknown, which induces non-trivial fractal dimensions and unveils hidden LRG flows toward unstable structural fixed points. Our systematic study of how networks and lattices respond to disorder paves the way for future analysis of non-ergodic behavior induced by quenched disorder.

Figures (14)

• Figure 1: Shortcuts in 2D square lattices. (a) Heat capacity, $C$, versus diffusion time, $\tau$, for different rewiring probabilities (see colorbar, $L=64$). The black dashed line corresponds to the unperturbed lattice. Inset: Zoom of the $\Lambda$-region. (b) First derivative of the heat capacity at the first peak versus rewiring probability, $p_r$, for different lattice sizes (see legend). (c) Finite-size scaling analysis of the vanishing peaks in (b). The distance of the size-dependent peak locations $p_{r,c}(N)$ from their asymptotic value for $N\rightarrow\infty$, $p_{r,c}^\infty$, scales as a power law of the system size, only at $p_{r,c}^\infty=0.10(1)$, revealing the existence of true scaling at criticality. (d) Low-dimensional representation using the three smallest low-frequency normalized network modes as the coordinate axes for a 2D square lattice of size $L=256$. Parameters: $p_{r}=2.3\times10^{-4}$, $p_{r}=0$ (left inset), and $p_{r}=0.1$ (right inset). All curves have been averaged over $10^{3}$--$10^{5}$ realizations.
• Figure 2: Diluted 2D square lattices. (a) Low-dimensional representation using the three smallest low-frequency normalized network modes as the coordinate axes, for a lattice with $p_{d}=0.35$, $p_{d}=0.2$ (left inset), and $p_{d}=0.5$ (right inset). (b) Heat capacity, $C$, versus diffusion time, $\tau$, for different dilute probabilities (see colorbar). Gray lines represent the expected plateau for a 2D lattice and a RT. (c) Second derivative of the heat capacity at the first peak versus dilute probability, $p_d$, for different lattice sizes (see legend). The vertical gray line represents the point where the 2D ultraviolet cut-off is expected to disappear. (d) Estimated correlation dimension, $D$, versus dilution probability, $p_d$ for different lattice sizes. Inset: Correlation integral $C(\ell)$ vs distance $\ell$ for different dilute probabilities. The gray area represents dilute probabilities between $p_{d,c}$ and the percolation threshold. All curves have been averaged over $10^{3}$--$10^{4}$ realizations.
• Figure 3: Diluted scale-invariant networks. (a, c) Estimated correlation dimension, $D$, versus dilution probability, $p_d$, for networks of different generation levels, $s$ (see legend), for: (a) DGM ($N_s=\frac{3+3^s}{2}$) and (c) HMN with $m_0=3$ and $\alpha=4$ ($N_s=2^s m_0$). (b) Degree distribution, $P(\kappa)$, versus node degree, $\kappa$, for a DGM network with $p_d=0$ (blue points) and $p_d=0.75$ (orange points). The black dashed line indicates the power-law scaling of a BA network, $P(\kappa)\propto\kappa^{-3}$. Inset: Specific heat for different dilution probabilities; for $p_d=0.75$, the plateau at $C_0=1$ confirms the BA limit. (d)Lacunarity index, $\lambda$, calculated at a fixed scale $\epsilon=10^{-2}$ for different network topologies (x-axis). Colors distinguish different dilution probabilities (see legend). The horizontal dashed line marks the theoretical reference $\lambda=1$ (where variance equals mean). All curves have been averaged over $10^{3}$--$10^{4}$ realizations.
• Figure 4: Criticality in 2D triangular lattices under rewiring. (a) Finite-size scaling of the heat capacity peak locations. The convergence of the size-dependent critical points $p_{r,c}(N)$ toward the asymptotic limit $p_{r,c}^\infty$ follows a power law, uniquely identifying the critical threshold at $p_{r,c} = 0.17(4)$ in the thermodynamic limit. (b) First derivative of the heat capacity with respect to the rewiring probability $p_r$, shown for different lattice sizes (see legend).
• Figure 5: Geometric breakdown of 2D triangular lattices under dilution. (a) Heat capacity $C$ as a function of diffusion time $\tau$ for varying dilution probabilities (see colorbar). The gray line marks the Alexander-Orbach (RT) plateau. (b) Second derivative (curvature) of the heat capacity at the first peak versus dilution probability $p_d$ for different lattice sizes. The vertical gray line indicates the critical point where the 2D ultraviolet cut-off vanishes. (c) Correlation integral $C_{r}(\ell)$ versus distance $\ell$ for different dilution probabilities. (d) Estimated correlation dimension $D$ versus $p_d$ for different lattice sizes. The gray shaded region marks the regime between $p_{d,c}$ and the percolation threshold. All curves represent averages over $10^{3}$--$10^{4}$ realizations.