zhaofeng-shu33 / Kuramoto_Oscillators

Exploring the dynamics towards synchronization in a system of coupled Kuramoto oscillators.

Kuramoto model for community detection

Method explanation

Let $E$ be the edge list of the undirected graph $G$. $N$ is the number of nodes.

The dynamic function is $$ \frac{\partial \theta_i}{\partial t} = \frac{1}{N}\sum_{j:(i,j) \in E} \sin(\theta_j - \theta_i) $$

Local order during simulation step

$$r_{c}=\frac{1}{|E|}\sum_{(i,j) \in E}e^{ |\theta_{i} - \theta_{j}| } $$

Solve the ODE until $r_c$ is near 1. This can be used as a termination condition for the algorithm.

Then we use $[\theta_1, \dots, \theta_N]$ to cluster the nodes into different community.

Example

For the above graph with 12 nodes. After $t = 69$ we can obtain $r_c\geq 0.99$, and the phases are

[4.85074828 4.84967319 4.84943349 4.86083052 4.846165   4.85887088
 4.91879579 4.92206347 4.90935898 4.91855667 4.90739992 4.91748298]

Pros and Cons

Pros: This method is bio-inspired, and shows potential in plausible explanation and theoretical justification.

Cons:

1. Slow as the number of nodes grows large, since we need to solve an ODE system with $N$ variables.
2. Low accuracy compared with other methods. We test this method on SBM(100,2,16,4), which shows that it has low accuracy compared with SDP-based method.

Reference

Implementation of syncnet in pyclustering.