Null-eigenvalue localization of quantum walks on complex networks

First we report that the adjacency matrices of real-world complex networks systematically have null eigenspaces with much higher dimensions than that of random networks. These null eigenvalues are caused by duplication mechanisms leading to structures with local symmetries which should be more present in complex organizations. The associated eigenvectors of these states are strongly localized. We then evaluate these microstructures in the context of quantum mechanics, demonstrating the previously mentioned localization by studying the spread of continuous-time quantum walks. This null-eigenvalue localization is essentially different from the Anderson localization in the following points: first, the eigenvalues do not lie on the edges of the density of states, but at its center; second, the eigenstates do not decay exponentially and do not leak out of the symmetric structures. In this sense, it is closer to the bound state in continuum.

Figure 1

Eigenvalue distributions of the adjacency matrices computed from two networks and their random-network counterparts: (a) Zachary's karate-club network [4], (b) neural network of c. elegans [35], (c) airline connections between airports in the U.S. [36], and (d) collaboration network in preprints on the condensed-matter archive at www.arxiv.org [37]. The shaded columns indicate the eigenvalue histograms (i.e., the number of eigenvalues in each bin) of the complex network, while the open columns are those of their random-network counterparts with the same numbers of nodes and links. The counterpart did not have any null eigenvalues in both cases. We build the random networks according to the following five steps: (i) generate a random number in the range of 1 to $N$ and choose a node; (ii) generate another random number in the range of 1 to $N$ and choose a node; (iii) if the node chosen in (ii) is the same as the node chosen in (i), repeat (ii) until they are different; (iv) connect the two nodes with a link; (v) repeat the process from (i) to (iv), rejecting multiple links, until one has links of a predetermined number. The dashed curve indicates the semicircle law according to Ref. [39], which reproduces well the histogram for the random network.

Figure 2

(a) Two dangling nodes. (b)–(d) Other variations of complete duplications. The one in (d) was taken from Ref. [40], which is a study on alloy.

Figure 3

The IPR of each eigenvalue of (a) the neural network of c. elegans [35] [see Fig. 1] and (b) the U.S. airline network [36]. (Details of the two networks are given in Table 1.) The encircled crosses indicate the null eigenvalues and the horizontal broken line indicates our criterion of localization, ${\mathrm{IPR}}_{\mu }=10/N$.

Figure 4

(a) A hierarchical network and (b) Zachary's karate-club network [4]. Encircled nodes have the structures shown in Fig. 2.

Figure 5

Plots of ${\chi }_{jk}$ in Eq. (8), (a)–(c) for the hierarchical network in Fig. 4 and (d)–(f) for Zachary's karate club in Fig. 4. In each row, (a) and (d) represent the total of ${\chi }_{jk}$, (b) and (e) the contribution of the null eigenspace, ${\chi }_{jk}^{\left(0\right)}$, and (c) and (f) the remnants, ${\chi }_{jk}-{\chi }_{jk}^{\left(0\right)}$. The horizontal and vertical axes indicate $k$ and $j$, respectively. The legend should be augmented by a factor (a)–(c) $\times 3$ and (d)–(f) $\times 3/2$.