Spontaneous scale-free structure in adaptive networks with synchronously dynamical linking

Inspired by the anti-Hebbian learning rule in neural systems, we study how the feedback from dynamical synchronization shapes network structure by adding new links. Through extensive numerical simulations, we find that an adaptive network spontaneously forms scale-free structure, as confirmed in many real systems. Moreover, the adaptive process produces two nontrivial power-law behaviors of deviation strength from mean activity of the network and negative degree correlation, which exists widely in technological and biological networks. Importantly, these scalings are robust to variation of the adaptive network parameters, which may have meaningful implications in the scale-free formation and manipulation of dynamical networks. Our study thus suggests an alternative adaptive mechanism for the formation of scale-free structure with negative degree correlation, which means that nodes of high degree tend to connect, on average, with others of low degree and vice versa. The relevance of the results to structure formation and dynamical property in neural networks is briefly discussed as well.

Figure 1

The synchronization error $E\left(t\right)$ for different coupling strengths $c$, indicating transition to synchronization in the ANs of (a) Rössler oscillators and (b) the food web model. Depicted results are obtained for $N=1000$ and $h=0.02$. Here, for simplicity, we suppose that the new connection is always added between the nodes with the maximum difference ${\Delta }_{\max }$ and the minimum difference ${\Delta }_{\min }$ if the connection does not already exist; otherwise, there is no new connection at this time step.

Figure 2

(a) Degree distributions $P\left(k\right)$ as a function of $k$ in a log-log plot for our ANs of Rössler (upward triangles) and food web (downward triangles) models at $N=1000$ and $h=0.02$ and (b) their dependence on different network size $N$ at $h=0.2$. Results in (a) and (b) are averaged over 20 realizations of the ANs with random initial conditions. For clarity, only the results of Rössler oscillators are shown in (b). The solid lines in (a) and (b) are linear fittings, which respectively confirm their scale-free properties. Depicted results are obtained for $c=1.0$, which can lead to global synchronization of the network (see Fig. 1). For comparison, the inset in (a) gives the degree distributions $p\left(k\right)$ in a linear-log plot for our ANs of Rössler (upward triangles) and food web (downward triangles) models at $N=1000$, $h=0.02$, and $c=0.001$, which cannot lead to global synchronization of the network (see Fig. 1). The solid lines in the inset in (a) are linear fittings, which confirm their exponential properties.

Figure 3

The degree exponent $\gamma$ as a function of coupling strength $c$ under different values of the linking time step $h$. Results are averaged over 20 independent realizations. Since the results for the Rössler oscillators and the food web model are nearly the same as shown in Fig. 2a, we only show the results for the Rössler oscillators. The depicted results are obtained for $N=1000$.

Figure 4

(a) The ADS $\langle \Delta \left(k\right)\rangle$ of vertices with degree $k$ from the mean activity of the whole network and (b) the mean degree $\langle k_{nn}\rangle$ of the nearest neighbors of the vertices as a function of the degree $k$ in a log-log plot for Rössler oscillators (upward triangles) and food web model (downward triangles). Results are averaged over 20 independent realizations. The dotted line represents the thresholds $k_t=14$ in (a) and (b) for both oscillator models. The solid lines give slopes $1.91$ in (a) and $2.43$ in (b), which confirms their power-law decreases. The depicted results are obtained for $N=1000$, $c=1.0$, and $h=0.14$.

Figure 5

(a) The ADS $\langle \Delta \left(k\right)\rangle$ as a function of $\langle k_{nn}\rangle$ in a log-log plot for Rössler oscillators (upward triangles) and the food web model (downward triangles) with $N=1000$, $c=1.0$, and $h=0.14$ and (b) ADS $\langle \Delta \left(k\right)\rangle$ vs $\langle k_{nn}\rangle$ for different coevolutionary network parameters $N$, $c$, and $h$. Results in both panels are averaged over 20 realizations on the ANs with random initial conditions. The dotted line denotes the thresholds of $\langle k_{nn}\rangle$ at $k_t=14$ in (a) for both oscillator models. For clarity, only the results of the Rössler oscillators are shown for different coevolutionary network parameters in (b). The solid lines in (a) and (b) have a slope of $0.78$, confirming the power-law behaviors.