Emergent explosive synchronization in adaptive complex networks

Adaptation plays a fundamental role in shaping the structure of a complex network and improving its functional fitting. Even when increasing the level of synchronization in a biological system is considered as the main driving force for adaptation, there is evidence of negative effects induced by excessive synchronization. This indicates that coherence alone cannot be enough to explain all the structural features observed in many real-world networks. In this work, we propose an adaptive network model where the dynamical evolution of the node states toward synchronization is coupled with an evolution of the link weights based on an anti-Hebbian adaptive rule, which accounts for the presence of inhibitory effects in the system. We found that the emergent networks spontaneously develop the structural conditions to sustain explosive synchronization. Our results can enlighten the shaping mechanisms at the heart of the structural and dynamical organization of some relevant biological systems, namely, brain networks, for which the emergence of explosive synchronization has been observed.

Figure 1

Asymptotic values of (a) global synchronization, $R$, and (b) total strength, $S$, in coevolving weighted networks governed by Eqs. (1) and (2), as a function of the correlation threshold $p_c$, and for different values of the coupling strength ${\sigma }_c$. When ${\sigma }_c>{\sigma }_c^{*}\simeq 0.255$, abrupt transitions indicating the emergence of explosive synchronization are observed at $p_c\simeq 0.625$.

Figure 2

(a) Size of the giant connected component, $N_G$, in the asymptotic binarized networks and (b) average degree $\langle k\rangle$, as a function of the coupling strength ${\sigma }_c$ and of the correlation threshold $p_c$.

Figure 3

Scatter plot of the node degree $k_i$ (upper row panels) and of the node neighborhood detuning ${\omega }_i-\langle {\omega }_{im}\rangle$ (central row panels) as a function of the natural frequency ${\omega }_i$ of a node for ${\sigma }_c=0.6$. The connectivity of the resulting networks is shown as matrices where nodes are placed according to their natural frequencies $\omega$ (bottom row panels).

Figure 4

(a) Global synchronization $R$ for forward and backward synchronization schemes for the binary networks considered in Fig. 3. In panels (b) and (c) we investigate the abrupt nature of the synchronization transition by reporting the maximum synchronization jump $J$ (b) and the width of the hysteresis loop $H$ (c) as a function of the coupling strength ${\sigma }_c$ and of the correlation threshold $p_c$.

Figure 5

Stability diagram for the link weight of a system of two coupled oscillators following Eqs. (5). The link weight tends to 0 in region $A_1$ and to 1 in region $A_2$, but in both cases synchronization is impossible. Regions $B_{1,2,3}$ are characterized by phase locking: in $B_1$ the weight becomes 1, whereas in $B_{2,3}$ the weight becomes ${\mathrm{\Sigma }}_1/{\sigma }_c<1$ with ${\mathrm{\Sigma }}_1$ defined in Eq. (A3). Regions $B_2$ and $B_3$ are different as the latter has a spiral sink, resulting in a slower convergence to the asymptotic state.