Dynamical complexity as a proxy for the network degree distribution

We explore the relation between the topological relevance of a node in a complex network and the individual dynamics it exhibits. When the system is weakly coupled, the effect of the coupling strength against the dynamical complexity of the nodes is found to be a function of their topological roles, with nodes of higher degree displaying lower levels of complexity. We provide several examples of theoretical models of chaotic oscillators, pulse-coupled neurons, and experimental networks of nonlinear electronic circuits evidencing such a hierarchical behavior. Importantly, our results imply that it is possible to infer the degree distribution of a network only from individual dynamical measurements.

Figure 1

(a) Phase order parameter $R$ and synchronization error $E$ vs $d$ for a star of $N=30$ identical Rössler oscillators (see the main text for the parameter values). (b) Dynamical complexity $C$ for the hub and one of the leaves vs $d$. All the measures are averaged over 10 initial conditions, storing sequences of maxima of length ${10}^4$ per node, using $D=4$ as the embedding length for the permutation patterns.

Figure 2

Dependence of the dynamical complexity at the node level and its topological role in a SF network of $N=150$ Rössler oscillators. (a) Complexity values $C_i$ vs $d$ for different values of the node degree $k_i$ in a network with $\langle k\rangle =4$. For the sake of comparison, the complete $(E$, black dashed line) and phase $(R$, black continuous line) synchronization curves are shown, rescaled for a better visualization. (b) ${\langle C\rangle }_k$ vs $k$ for the two coupling values $d$ marked in (a) with vertical dashed lines, located before $\left(d=0.02\right)$ and after $\left(d=0.29\right)$ the phase synchronization transition. (c) ${\langle C\rangle }_k$ vs the rescaled coupling $d\langle k\rangle$ for the highest (filled markers) and lowest (void markers) node degree classes for three different mean degrees $\langle k\rangle$ of the networks (see legend). Each point is the average of 10 network realizations.

Figure 3

Dependence of the dynamical complexity at the node level and its topological role in a SF network of $N=150$ chaotic Lorenz oscillators and $\langle k\rangle =4$. (a) Complexity values ${\langle C\rangle }_k$ vs $d$ for different values of the node degree $k_i$. (b) ${\langle C\rangle }_k$ vs $k$ for the two coupling values $d=0.17$ marked in (a) with a vertical dashed line.

Figure 4

Dependence of the dynamical complexity at the node level and its topological role in a SF network of $N=150$ Rössler nonidentical oscillators with frequency heterogeneity. (a) Complexity values ${\langle C\rangle }_k$ vs $d$ for different values of the node degree $k_i$ in a network with $\langle k\rangle =4$. (b) ${\langle C\rangle }_k$ vs $k$ for the two coupling values $d$ marked in (a) with vertical dashed lines, located before $\left(d=0.03\right)$ and after $\left(d=0.07\right)$ the phase synchronization transition.

Figure 5

Dependence of the dynamical complexity at the node level and its topological role in a SF network of $N=150$ Morris-Lecar neurons. (a) Complexity values ${\langle C\rangle }_k$ vs $d$ for low $\left(k=2\right)$ and high $\left(k=32\right)$ degree node values in a network with $\langle k\rangle =4$. For the sake of comparison, the phase synchronization curve $(S$, black dotted line) is shown, rescaled for a better visualization. (b) ${\langle C\rangle }_k$ vs $k$ for the two coupling values $d$ marked in (a) with vertical dashed lines. Each point is the average of 10 network realizations.

Figure 6

Schematic representation of the experimental setup of a star network with eight Rössler-type oscillators. Using ADCs, signals are measured and stored by a DAQ card and a PC, while DO ports change the value of the coupling strength $d$ through a digital potentiometer (XDCP).

Figure 7

(a) Phase order parameter $R$ and synchronization error $E$ vs $d$ for a star of $N=8$ almost identical (within the $5\%$ experimental tolerance) Rössler-type electronic circuits for the system experimental description. (b) Dynamical complexity of the hub and of one of the leaves vs $d$. Complexity measures are averaged over 30 different initial conditions and calculated over sequences of 5000 maxima with embedding length $D=3$.