# Math Insight

### Connecting network structure to dynamical properties

In fields from biology to engineering, one encounters dynamical processes that take place on networks. A prototypical example is a network of interacting neurons that is performing some computation in the brain. One would like to understand how the structural properties of the network shape the desired computations.

Connecting network structure to dynamics is, naturally, a broad and difficult question. Some progress has been made by focusing on a few types of dynamics. One is large-scale coherent behavior across the network, or synchrony. Another is looking at the influence of network structure on second order statistics of activity, or correlations.

#### Synchrony

One dynamical property of networks that is relatively easy to describe and analyze is synchronization. If a network is in a synchronous or coherent state, then all nodes are doing the same thing at the same time. In the completely synchronous state, all the state variables describing the state of each node have exactly the same values as they evolve in time. One can also look for less precise forms of synchrony, where in some sense the state variables describing most of the nodes have a similar evolution that is coordinated in time.

##### Phase oscillators

To illustrate the concept of synchrony, we can model the network as a dynamical system where the state space of each node is one-dimensional. Then, we can use a single state variable $\theta_j(t)$ to capture the state of node $j$ at time $t$. We let $\theta_j(t)$ range in the interval $[0, 2\pi]$ and make this state space periodic, so we can represent it as a circle. For each node $j$, its state $\theta_j(t)$ will continually move around the circle, so we say it oscillates around the circle and call the node an oscillator. We call $\theta_j(t)$ the phase of oscillator $j$, as it represents the position of the oscillator phase along the circle. When we represent the state of an oscillator by just its phase, we call it a phase oscillator.

We can use the Kuramoto order parameter $r$ to measure the synchrony of phase oscillators. We define a complex number $z$ to be the average position of the $N$ oscillators around the unit circle embedded in the complex plane, $$z = \frac{1}{N} \sum_{j=1}^N e^{i\theta_j},$$ where $i = \sqrt{-1}$. The order parameter $r$ is the magnitude of $z$, $r=|z|$, and it indicates the level of synchrony. Since $z$ is the average of complex number of length 1, its maximum length $r=1$ occurs when all the phases are indentical. Its minimum length $r=0$ occurs when the oscillators are asynchronous, or at least balanced around the circle.

For more details on phase oscillators, including interactive applets that illustrate the Kuramoto order parameter, see the page on the idea of synchrony of phase oscillators.

##### Network effects on synchrony

The structure of a network can strongly influence whether or not a network of oscillators will synchronize and to what degree they will synchronize. The dynamics of the individual oscillators and the form of their coupling will also have a large influence on the resulting synchrony. Some oscillator models won't synchronize no matter the network topology; other oscillator models will tend to synchronize easily. Therefore, if one looks at the synchrony for a particular model of oscillators connected via a particular network, it's difficult to separate the roles the network and oscillator models are playing in determining the synchrony.

If one wants to look at the influence of network structure on synchrony without regard to oscillator model, the best one can do is develop a notion of the synchronizability of a network. One could imagine a synchronizability index that indicates how likely a given network will synchronize, ranking networks independent of oscillator model. For example, if network A had a higher synchronizability index than network B, and one used the same oscillator model on both networks, one wouldn't be able to deduce if either network would synchronize. Yet, in this idealized view, one could deduce that, if network B synchronized for a particular oscillator model, then network A must also synchronize for that model. On the other hand, one might find a oscillator models for which network A would synchronize but not network B. However, even if such a synchronizability index could be assigned to each network, it seems unlikely that it would consist of a single number and allow one to rank all networks in this way. If a synchronizability index had multiple dimensions, then it would be possible that network A could synchronize better than network B for some oscillator models but that network B could synchronize better for other oscillator models.

Synchronizing oscillators. lllustration of $N=100$ phase oscillators whose state can be represented by a point around the unit circle. The state of oscillator $j$ at time $t$ is represented by the state variable $\theta_j(t)$, or phase, which we can map to the unit circle by plotting a green point in the complex plane at the location $e^{i\theta_j(t)}$ where $i=\sqrt{-1}$. For each oscillator, $\theta_j(t)$ increases with time, so each point moves counterclockwise around the circle. At time $t=0$, the oscillators are randomly spread over state space. However, as time evolves, the oscillators tend to group together to have similar phase, i.e., they tend to synchronize. You can control how well they synchronize by changing the parameter $\sigma$, which controls the final spread of points. When $\sigma=0$, the oscillators would eventually completely synchronize so that all $\theta_j(t)$ become identical. The degree of synchrony is captured by the complex number $z=\frac{1}{N}\sum_{i=1}^N e^{i\theta_j(t)},$ plotted as a blue vector and by its magnitude $r=|z|$, which ranges from 0 to 1.

Desynchronizing oscillators. A group of $N=100$ phase oscillators represented just as in the previous applet, only this group of oscillators tends to desynchronize. At time $t=0$, all oscillators start at exactly the same phase. However, as time evolves, the oscillators tend to spread out across the whole circle of phases, i.e., the system tends toward asynchrony. You can control how they desynchronize by changing the parameter $\sigma$, which controls the spread of oscillator speeds. When $\sigma=0$, the oscillators will stay synchronized forever.

The influence of complex structure on synchrony has recently be reviewed by Arenas et al.1 Here we touch on two aspects related approaches to investigate the