Evolving enhanced topologies for the synchronization of dynamical complex networks

Enhancing the synchronization of dynamical networks is of great interest to those designing and analyzing many man-made and natural systems. In this work, we investigate how network topology can be evolved to improve this property through the rewiring of edges. A computational tool called NETEVO performs this task using a simulated annealing metaheuristic. In contrast to other work which considers topological attributes when assessing current performance, we instead take a dynamical approach using simulated output from the system to direct the evolution of the network. Resultant topologies are analyzed using standard network measures, $B$ matrices, and motif distributions. These uncover the convergence of many similar features for all our networks, highlighting also significant differences between those evolved using topological rather than dynamical performance measures.

Figure 1

Flow diagram of a supervised network.

Figure 2

Master stability function for the chosen Rössler system with ${\alpha }_1\approx 0.2$ and ${\alpha }_2\approx 6.1$. A method presented in [34] was used to calculate the maximal Lyapunov exponent ${\lambda }_{\text{max}}$.

Figure 3

Synchronization response curves showing the percentage of synchronization as the simulation coupling strength $\sigma$ is varied. (a) shows averages for the initial topologies and eigenratio-evolved networks. (b) displays average results for order-parameter-evolved networks with fixed coupling strengths $\widehat{\sigma }∊\left[0.1,0.7\right]$. (c) shows average results for a break down of each type of topology found at bistable point where $\widehat{\sigma }=0.7$.

Figure 4

Final topologies for a selection of performance measures: (a) eigenratio, (b) order parameter $\widehat{\sigma }=0.1$, (c) 0.3, (d) 0.6, (e) 0.7 (type 1), (f) 0.7 (type 2). Visualization was performed using a forced based layout to help minimize edge crossing.

Figure 5

Degree distributions for a selection of the evolved networks. (a) Eigenratio. (b) Order parameter $\widehat{\sigma }=0.7$ (type 1). (c) Order parameter $\widehat{\sigma }=0.7$ (type 2).

Figure 6

Eigenspectra for (a) eigenratio, (b) order parameter $\widehat{\sigma }=0.4$, and (c) order parameter $\widehat{\sigma }=0.7$ evolved networks. Each line represents a single final topology for different initial conditions.

Figure 7

Alternative stable topologies seen when evolving networks to optimize the order-parameter performance measure with fixed coupling strength $\widehat{\sigma }=0.7$.

Figure 8

Average number of single degree nodes and standard deviation during the evolution of type 1 and 2 topologies.

Figure 9

Topological bifurcation seen when varying the fixed coupling strength $\widehat{\sigma }$ used by the order-parameter performance measure during network evolution. The change is driven by factors influencing the underlying realizable dynamics that can be exhibited at a particular coupling strength.

Figure 10

(Color online) $B$ matrix portraits for (a) 24 and (b) 25 nodes connected in a periodic ring.

Figure 11

(Color online) Average $B$ matrix portraits for a selection of performance measures and coupling strengths: (a) eigenratio, order parameter $\widehat{\sigma }=\left(\text{b}\right)$ 0.1, (c) 0.5, (d) 0.6, (e) 0.7 (type 1), (f) 0.7 (type 2). Colors represent the number of nodes at a given index within the $B$ matrix and are plotted on a log scale $\left[\text{log}\left(b_{lk}\right)\right]$.

Figure 12

(Color online) Evolution of a $B$ matrix portrait when using the order-parameter performance measure with $\widehat{\sigma }=0.6$. Initial topology was a periodic ring lattice with nearest and next-nearest-neighbor coupling. $B$ matrices were taken at iterations (a) 1, (b) 4000, (c) 8000, (d) $14000$, (e) $20000$, and (f) $180000$. Colors represent the number of nodes at a given index within the $B$ matrix and are plotted on a log scale $\left[\text{log}\left(b_{lk}\right)\right]$.

Figure 13

Master stability function for the chosen Rössler system coupled on all internal states with ${\alpha }_1\approx 0.0831$.

Figure 14

Change in bounded synchronization during the evolution of networks with differing variation in underlying model parameters (1%–50%).