Wave Localization in Complex Networks with High Clustering

We show that strong clustering of links in complex networks, i.e., a high probability of triadic closure, can induce a localization-delocalization quantum phase transition (Anderson-like transition) of coherent excitations. For example, the propagation of light wave packets between two distant nodes of an optical network (composed of fibers and beam splitters) will be absent if the fraction of closed triangles exceeds a certain threshold. We suggest that such an experiment is feasible with current optics technology. We determine the corresponding phase diagram as a function of clustering coefficient and disorder for scale-free networks of different degree distributions $P(k)\sim k^{-\lambda }$. Without disorder, we observe no phase transition for $\lambda <4$, a quantum transition for $\lambda >4$, and an additional distinct classical transition for $\lambda >4.5$. Disorder reduces the critical clustering coefficient such that phase transitions occur for smaller $\lambda$.

Figure 1

Representative pictures of the giant component of scale-free networks ($\lambda =5$) (a) without and (b) with clustering ($C_0=0.6$). Both networks have giant components of similar size ($N\sim 150$); the size of the whole network being $N=150$ for (a) and $N=200$ for (b). The logarithmically scaled coloring presents the intensity of a mode with $E\approx 0.2$, dark gray (red) indicating the highest, lighter colors for lower, and black (violet) for the lowest probability. (c) Degree distribution $P(k)$ and (d) clustering coefficients $\overline{C}(k)$ for scale-free networks with $\lambda =4$ [line in (c)], $C_0=0.65$ [line in (d) according to Eq. (1)] and $N=15000$ nodes, averaged over 120 configurations. Blue circles for distributions regarding the whole network and red squares for the giant component (with $⟨N_1⟩=11906$ nodes; shifted vertically by a factor of 2). The inset shows that a linear dependence between $C_0$ and $C$ holds also for the giant component if $C_0\le 0.9$.

Figure 2

Level spacing distribution $P(s)$ for optical modes on scale-free networks with $\lambda =5$, $N=12500$ and no disorder, $W=0$. A clear transition from Wigner (dashed red curve) to Poisson (dash-dotted blue curve) behavior is observed as a function of the clustering coefficient prefactor that is increased from $C_0=0.0$ (continuous red curve, next to the dashed curve) to $C_0=0.90$ (continuous blue curve, next to the dash-dotted curve). Inset: Localization parameter $\gamma$ [see Eq. (3)] versus $C_0$ for networks with (from top to bottom on the left) $N=5000$ (red), $N=7500$ (light green), $N=10000$ (green), $N=12500$ (blue), and $N=15000$ (purple). A transition from extended modes for small $C_0$ to localized modes for large $C_0$ is observed at $C_{0,q}\approx 0.69$. The results are based on eigenvalues around $|E|=0.2$ and 0.5.

Figure 3

(a) Phase diagram for transitions from localized optical modes (upper right) to extended modes in parts of the spectrum (lower left) for different degree distribution exponents $\lambda$, $\lambda =4$ (blue diamonds), 4.25 (magenta circles), and 5 (red squares). Data for $C_0>0.9$ are not reliable for network generation reasons, and the error bar for the point at $C_0=1$ is about 0.1. (b) Exponent $\nu$ for different $\lambda$ and $C_{0,q}$. The values are, within the error bars (not shown), consistent with the mean-field prediction $\nu =0.5$. (c) Quantum transitions without disorder (blue circles) and classical transitions (red squares) as a function of the degree exponent $\lambda$. In the regime $4<\lambda <4.5$ only quantum transitions occur. For $W>0$ the curves move downwards making quantum transitions possible for $\lambda <4$ (green circles for $W=5$ and magenta circles for $W=10$).