Evolution of correlated multiplexity through stability maximization

Investigating the relation between various structural patterns found in real-world networks and the stability of underlying systems is crucial to understand the importance and evolutionary origin of such patterns. We evolve multiplex networks, comprising antisymmetric couplings in one layer depicting predator-prey relationship and symmetric couplings in the other depicting mutualistic (or competitive) relationship, based on stability maximization through the largest eigenvalue of the corresponding adjacency matrices. We find that there is an emergence of the correlated multiplexity between the mirror nodes as the evolution progresses. Importantly, evolved values of the correlated multiplexity exhibit a dependence on the interlayer coupling strength. Additionally, the interlayer coupling strength governs the evolution of the disassortativity property in the individual layers. We provide analytical understanding to these findings by considering starlike networks representing both the layers. The framework discussed here is useful for understanding principles governing the stability as well as the importance of various patterns in the underlying networks of real-world systems ranging from the brain to ecology which consist of multiple types of interaction behavior.

Figure 1

Schematic diagram depicting various steps of GA as well as construction of multiplex networks by combining ER networks ($G^1,G^2\cdots G^P$) and the behavior matrix $\left[b_{ij}\right]$.

Figure 2

Change in the average value of (a) $d_q$, (b) $d_p$, (c) $R_{\max }$, (d) $L_{\mathrm{corr}}$, (e) $r_m$, and (f) $r_p$ as evolution progresses for the $G$ generation. Graphs exhibit the emergence of various structural properties as a consequence of ${\overline{R}}_{\max }$ minimization. The average is taken for 1000 networks in a population. The size of ER networks and, consequently, that of the individual layer in the multiplex networks remain $N=100$. The average degree of the ER networks is 16.

Figure 3

Impact of interlayer coupling strength on the average value of various structural properties of the evolved networks. (a) $L_{\mathrm{corr}}$ increases with increase in the $D_x$ values. (b) Initial increment in $D_x$ does not lead to an evolution of $r_p$; however, large values of $D_x$ depict enhanced $r_p$. (c) Evolved values of $r_q$ do not significantly affected with $D_x$. The evolved and the initial networks are represented by ($☆$) and ($\circ$), respectively. Size and the average degree of networks remain the same as for Fig. 2.

Figure 4

Schematic diagram depicting network structure for correlated multiplexity [Case (a)] and anticorrelated multiplexity [Case (b)].

Figure 5

(a) $R_{\max }$ calculated using Eq. (4) ($\circ$) for network having correlated multiplexity [Fig. 4] and Eq. (5) ($\square$) for network having anticorrelated multiplexity [Fig. 4] as a function of $D_x$. The plots demonstrate stability of networks having correlated multiplexity. (b) $R_{\max }$ as a function of number of interlayer connections ($m$) for the anticorrelated multiplexity case [Fig. 4]. Graph is plotted for two different values of interlayer coupling strength, $D_x$ = 0.5 (stars) and $D_x=1.0$ (diamonds). The plots demonstrate that increase in number of interlinks renders the behavior of $R_{\max }$ same as of one interlayer link.