Connectivity-Driven Coherence in Complex Networks

We study the emergence of coherence in complex networks of mutually coupled nonidentical elements. We uncover the precise dependence of the dynamical coherence on the network connectivity, the isolated dynamics of the elements, and the coupling function. These findings predict that in random graphs the enhancement of coherence is proportional to the mean degree. In locally connected networks, coherence is no longer controlled by the mean degree but rather by how the mean degree scales with the network size. In these networks, even when the coherence is absent, adding a fraction $s$ of random connections leads to an enhancement of coherence proportional to $s$. Our results provide a way to control the emergent properties by the manipulation of the dynamics of the elements and the network connectivity.

Figure 1

${\gamma }_c$ as a function of the network size $n$ for $k=\lceil n^{\gamma }/2\rceil$. We fixed $\alpha =10$ and $\epsilon =0.2$, and recall $\beta =1$. The open circles represent the numerical determination of ${\gamma }_c$, and the solid line is the theoretical prediction, obtained by solving $\alpha \beta {\lambda }_2-{\alpha }_c=0$ for $\gamma$. The inset (a) shows the critical behavior of $E(t)$ for $\gamma =0.3$ as a function of the network size. Here at the critical network size $n=41$ coherence is lost, agreeing with the theoretical predictions. In inset (b), we present the same numerical simulations of the Lotka-Volterra model. As we predict, no abrupt transition is observed once ${\alpha }_c=0$.

Figure 2

Randomness enhance coherence. In (a) Log-Log plot of coherence $|1-E|$ versus the fraction of randomly added links $s$. For fixed $\alpha =10$ starting from a $2k$ nearest neighbor network of size $n=1000$ and $\gamma =0.3$, no coherence is observed. We then add a fraction of $s$ random links. This induces coherence in the network according to $|1-E|\propto s^{-1}$, as predicted theoretically. In (b) Log-Log plot of coherence $|1-E|$ versus the mean degree $m$. For $n=3000$, we simulate the Lorenz dynamics on Erdös-Rényi with $\epsilon =1$ (open square) and power law networks with $\theta =3$ and $\epsilon =1$ (open circle) and then with $\epsilon =0.2$ for $\theta =2.7$ (open diamond) and $\theta =4$ (open triangle). The scaling towards coherence $|1-E|\propto \delta /m^{-1}$ agrees with the theoretical prediction.