Complexity measures from interaction structures

We evaluate information-theoretic quantities that quantify complexity in terms of $k\text{th}$-order statistical dependences that cannot be reduced to interactions among $k-1$ random variables. Using symbolic dynamics of coupled maps and cellular automata as model systems, we demonstrate that these measures are able to identify complex dynamical regimes.

Figure 1

The values of $I^{\left(k\right)}$ for the fully connected graph. For $ϵ>0.45$, the network is in the state of synchronized chaos. This is fully captured by pair interactions; therefore, $I^{\left(2\right)}$ is maximal, while the $I^{\left(k\right)}$ for $k>2$ vanish. On the edge to synchronized chaos, high correlations on different scales are detected.

Figure 2

The values of $I^{\left(k\right)}$ for the circle graph. The inset shows the parameter region $ϵ∊[0.46,0.49]$.

Figure 3

Behavior of the largest Lyapunov exponent ${\lambda }_1$ for the fully connected graph (top) and the circle graph (bottom). Note that the minima are attained in exactly the same parameter region, in which the $I^{\left(k\right)}$ have maxima.

Figure 4

A snapshot of the highly regular binary symbolic dynamics of the circle graph at $ϵ=0.476$. Each row shows one node over time. White spots indicate the value 0, black squares 1.

Figure 5

A snapshot of the weakly coupled chaotic dynamic of the circle graph at $ϵ=0.04$. Each row shows one node over time. White spots indicate the value 0, black squares 1.

Figure 6

A snapshot of the complex dynamic of the circle graph at $ϵ=0.482$. In the top the activity of node 4 is plotted together with the average activity over all nodes. One can observe that for short periods of time synchronization effects occur—for instance, between $t=30$ and $t=40$ node 4 is almost constant while the average also fluctuates less. Below, the symbolic dynamics, with black as 1, white as 0, of all ten nodes is shown. The synchronization is visible here too as the pattern after $t=35$ is very regular.