# Isoperimetric number v.s synchronization

The isoperimetric number or Cheeger constant is an indicator of the “bottleneckedness” of a network.

This measure is an NP measure and can only be calculated partially. It gives an estimation of the existence of a big enough group of nodes with too few connections to the rest of the network, meaning that this group of nodes will act as a bottleneck.

There is a curious relationship between the Isoperimetric number $i(A)$ of a network A and the capacity to synchronize. This relationship is connected with a very interesting number that is associated with any network which is the algebraic connectivity which is the is the second-smallest eigenvalue of the Laplacian matrix of A. If this value is zero implies that the network is disconnected. The bigger it is this value the better will synchronize the network. Then, the interesting relationship is that $i(A) \geq \frac{1}{2}\lambda_2$ ; where $\lambda_2$ is the second smallest eigenvalue of the net Laplacian. Therefore a network that synchronize well will have a big $i(A)$ meaning that will not have bottlenecks.

That seems to imply that a network with bottlenecks synchronize worst that one without them.
Our brain dedicates a big part of its volume to connect different parts of the cortex (white matter) and other different regions. The white matter made of axons seems to try to give a good connection opportunity to the neurons, reducing bottlenecks.

Is this, maybe, the way the brain has to increase the Isoperimetric number of the network and reduce bottlenecks, increasing the possibility to have different parts of it in synchrony?…..uhhmmmm
An interesting point, is that the isoperimetric number is given by the group of nodes with smaller connectivity to the rest of the network, that implies that a good network can be seriously deteriorated by a significant group of nodes with a bottleneck. Even if we have only one of this nodes…..uhmmmm again
We will have to consider as well the inhibitory connections, which are an important part of synchronization in the brain. Too low inhibition produces explosion and too high inactivity.Actually inhibition is a way to control the network structure (when a neuron is inhibited all connections through it are closed) in a time dependent way. Neuron dynamics change connectivity and connectivity changes neuron dynamics, what makes the result horridly complex.

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