Optimized evolution of networks for principal eigenvector localization

Network science is increasingly being developed to get new insights about behavior and properties of complex systems represented in terms of nodes and interactions. One useful approach is investigating the localization properties of eigenvectors having diverse applications including disease-spreading phenomena in underlying networks. In this work, we evolve an initial random network with an edge rewiring optimization technique considering the inverse participation ratio as a fitness function. The evolution process yields a network having a localized principal eigenvector. We analyze various properties of the optimized networks and those obtained at the intermediate stage. Our investigations reveal the existence of a few special structural features of such optimized networks, for instance, the presence of a set of edges which are necessary for localization, and rewiring only one of them leads to complete delocalization of the principal eigenvector. Furthermore, we report that principal eigenvector localization is not a consequence of changes in a single network property and, preferably, requires the collective influence of various distinct structural as well as spectral features.

Figure 1

Network size $N=500$, $\langle k\rangle =10$, and we iterate the rewiring process $600000$ times and store the network after each 100 steps. Changes in various network properties: (a) IPR value of PEV, (b) maximum degree, (c) average clustering coefficient, (d) degree-degree correlation, (e) correlation between degree vector and clustering coefficient vector, and (f) correlation between PEV and clustering coefficient vector during evolution (${\tau }_{\mathrm{evolution}}$).

Figure 2

ER network as an initial network. Degree distribution of the (a) initial, (b) intermediate, and (c) optimized networks. PEV entry distribution of the (d) initial (e) intermediate, and (f) optimized networks.

Figure 3

Schematic representing the initial (left) and the most optimized (right) networks.

Figure 4

IPR as a function of edge rewiring. Networks with a large IPR value in the $r_3$ region consist of few edges, rewiring which leads to a sudden drop in the IPR value. The blue square indicates the regime where the networks attain IPR values which are very close to those of the optimized network. However, in this regime rewiring an edge does not have a significant impact on the IPR values.

Figure 5

Change in IPR value as a function of network evolution (${\tau }_{\mathrm{evolution}}$). (a) SF networks with $N=500$ and $\langle k\rangle =10$. (b) C. elegans frontal network with $N=131$ and $\langle k\rangle =11$. (c) C. elegans neural network with $N=297$ and $\langle k\rangle =14$. We consider crude approximations that edges are undirected and unweighted for C. elegans networks.

Figure 6

(a) Evolution (${\tau }_{\mathrm{evolution}}$) of IPR values for different network sizes. As the network size increases, more evolution time is required for a network to reach the optimized state. (b) IPR value of the initial ER and the final optimized networks as a function of $N$.

Figure 7

Behavior of ${\lambda }_1$ (open violet circles) and ${\lambda }_2$ (filled cyan circles) during network evolution (${\tau }_{\mathrm{revolution}}$). Initial network: (a) ER random network; (b) SF network; (c) C. elegans neural network.

Figure 8

Spreading process of the SIS model in the initial ER random network (ER-init; ${\lambda }_1^{\mathrm{init}}\approx 11.34$, $Y_{X_1}^{\mathrm{init}}\approx 0.0007$), an intermediate ER network (ER-imdt; ${\lambda }_1^{\mathrm{imdt}}\approx 11.14$, $Y_{X_1}^{\mathrm{imdt}}\approx 0.21$), and an optimized ER network (ER-opt; ${\lambda }_1^{\mathrm{opt}}\approx 10.77$, $Y_{X_1}^{\mathrm{opt}}\approx 0.22$) and in the C. elegans neural network (${\lambda }_1^{\mathrm{init}}\approx 24.36$, $Y_{X_1}^{\mathrm{init}}\approx 0.019$ and ${\lambda }_1^{\mathrm{opt}}\approx 17.18$, $Y_{X_1}^{\mathrm{opt}}\approx 0.1$). The ER network has $N=2000$ nodes with $\langle k\rangle =10$.