Heterogeneity induces emergent functional networks for synchronization

We study the evolution of heterogeneous networks of oscillators subject to a state-dependent interconnection rule. We find that heterogeneity in the node dynamics is key in organizing the architecture of the functional emerging networks. We demonstrate that increasing heterogeneity among the nodes in state-dependent networks of phase oscillators causes a differentiation in the activation probabilities of the links when a distributed local network adaptation strategy is used in an evolutionary manner. This, in turn, yields the formation of hubs associated to nodes with larger distances from the average frequency of the ensemble. Our generic local evolutionary strategy can be used to solve a wide range of synchronization and control problems.

Figure 1

Schematic description of the evolutionary edge-snapping strategy. Step 1 (variation): computation of link activation probabilities by running the edge-snapping strategy from many different random initial conditions. Step 2 (selection): selection of those links whose activation probability is above some threshold value $p^{*}$.

Figure 2

(a) Standard deviation of the link activation probability $p_{ij}$ as a function of the number of trials $n_S$. (b) Order parameter and relative number of links of the minimal ES structure as a function of the number of trials $n_S$.

Figure 3

(a) Link activation probabilities $p_{ij}$ in the case of $N=6$ generated by the variation stage of the evolutionary ES strategy; (b) Selection of the threshold probability value $p$: order parameter $R$, relative number of links $\overline{M}$. The arrow on the $x$-axis indicates the critical threshold $p^{*}$ which gives the minimal ES network; (c) Minimal Edge-Snapping Network; (d) Optimal network maximizing $R$ obtained by exhaustive search and a Monte Carlo based method.

Figure 4

Heterogeneity induces functional structural properties of the network. $P$ matrix as a function of the heterogeneity parameter $\sigma$ when $N=20$.

Figure 5

Structural properties of the emergent minimal ES network with $N=100$ and $\sigma =0.2$. (a) Standard deviation of the link activation probabilities $p_{ij}$ as a function of $\sigma$. (b) Maximum (red dashed line) and minimum (black solid line) value of node degree $k_i$ as a function of $\sigma$. (c) Activation probability of each link against the value of the difference between the natural frequencies of the oscillators at the endpoints. (d) Node degree $k_i$ vs. ${\omega }_i$. (e) Order parameter $R$ (red solid line) and relative number of links $\overline{M}$ (blue solid line) of the ES network as a function of the threshold probability value $p$. For comparison, the value $R$ is depicted for an all-to-all network (purple dashed line) and for randomly generated networks (blue dot-dashed line) with the same number of links. The arrow on the $x$-axis represents the threshold $p^{*}$ to give the minimal ES network. (f) Order parameter $R$ of the phase oscillators interconnected by the minimal ES network when the overall coupling strength $K$ is increased (red solid) and decreased (blue dashed). We set $N=300$ and $\sigma =0.2$.

Figure 6

Edges' evolution according to the edge snapping mechanism.

Figure 7

Schematic illustration of NetEvo.