Quantum walks on complex networks with connection instabilities and community structure

A continuous-time quantum walk is investigated on complex networks with the characteristic property of community structure, which is shared by most real-world networks. Motivated by the prospect of viable quantum networks, I focus on the effects of network instabilities in the form of broken links, and examine the response of the quantum walk to such failures. It is shown that the reconfiguration of the quantum walk is determined by the community structure of the network. In this context, quantum walks based on the adjacency and Laplacian matrices of the network are compared, and their responses to link failures is analyzed.

Figure 1

Community structure in the karate club (KC) network [6] with $N=34$ nodes. The two main communities, centered around nodes $1$ and $34$, are indicated by squares and circles, respectively. Colors correspond to the various possible communities in the network, including subcommunities. Reprinted from Ref. [7] (©2004, IOP Publishing and SISSA).

Figure 2

Degree centrality $C_j$ (line of squares) and population ${\mathrm{P̄}}_j$ (line of circles) for each node $j=\{1,2,\dots ,34\}$ in the KC network of Fig. 1. Parameters used for the numerical simulation: $T=100\pi$ (dimensionless units) and $\delta t={10}^{-3}T$.

Figure 3

Population flows after an edge is removed. Typical examples are shown: (a) edge $(1,3)$ belongs to the community centered around node $1$ (squares in Fig. 1); (b) edge $(19,33)$ belongs to the community centered around $34$ (circles in Fig. 1); and (c) edge $(3,9)$ is an intercommunity edge. The (blue) line of circles corresponds to the populations of Fig. 2 for an ideal network, while the (green and red) diamonds correspond to the re-calculated (square and circle) populations after the link failure.

Figure 4

Node affinity ${\alpha }_{\mathit{ij}}$ of Eq. (6) for the KC network of Fig. 1. Correspondence between shades (colors) and numerical values is shown in the vertical bar on the right.