Complex quantum networks: From universal breakdown to optimal transport

We study the transport efficiency of excitations on complex quantum networks with loops. For this we consider sequentially growing networks with different topologies of the sequential subgraphs. This can lead either to a universal complete breakdown of transport for complete-graph-like sequential subgraphs or to optimal transport for ringlike sequential subgraphs. The transition to optimal transport can be triggered by systematically reducing the number of loops of complete-graph-like sequential subgraphs in a small-world procedure. These effects are explained on the basis of the spectral properties of the network's Hamiltonian. Our theoretical considerations are supported by numerical Monte Carlo simulations for complex quantum networks with a scale-free size distribution of sequential subgraphs and a small-world-type transition to optimal transport.

Figure 1

(a) Sketch of a sequentially growing network with $g=8$ SSGs and the treelike backbone. (b) Small-world-type transition in a SSG, where internal bonds (blue) are randomly added (removed) from the ringlike (complete-graph-like) SSGs with a certain probability $p_{\text{int}}$.

Figure 2

Spectral density of an ensemble of $R={10}^3$ realizations of sequentially growing networks with $N={10}^3$ and complete-graph-like SSGs built from a scale free probability distribution $p_s\left(n\right)\sim n^{-s}$. Left column: results for $\mathit{H}\left[d\right(G\left)\right]=\mathit{A}\left[d\right(G\left)\right]$ for two values of the parameter $s$. Right column: results for $\mathit{H}\left[d\right(G\left)\right]=\mathit{C}\left[d\right(G\left)\right]$ with the same values for $s$. Note the semilogarithmic scales and the different scales of the $x$ axis in the two columns.

Figure 3

Quantum transport properties of a complex network with loops and with SSG sizes chosen from a scale-free distribution. Numerical results (surface defined by dots) for the ensemble averaged global efficiency measures (a) $〈\underline{\chi }(s,p_{\text{int}})〉$ and (b) $〈\chi (s,p_{\text{int}})〉$ for $N=1000$ as a function of the scale-free parameter $s$ and of the small-world-type parameter $p_{\text{int}}$ with $R={10}^6/N$ realizations. Insets: Upper panels show $\chi (s,p_{\text{int}})$ and $\underline{\chi }(s,p_{\text{int}})$ as a function of $s$ for five fixed values of $p_{\text{int}}$. The lines represent the analytic results given in Ref. [15] for $p_{\text{int}}=1$ and for $N=1000$ (solid) as well as for $N\to \infty$ (dashed) of the lower bound $\underline{\chi }$. The lower panels show $\chi (s,p_{\text{int}})$ and $\underline{\chi }(s,p_{\text{int}})$ for $s=1$ as a function of $1-p_{\text{int}}$ in double logarithmic scale.