*Mathematical Methods in the Applied Sciences*Identification of collective particle motion in a rotating drum using a
graph community detection algorithm

We present the method for detection of particle groups involved in
collective motion based on network analysis. Knowing the positions and
velocities of individual particles, a “velocity similarity graph’‘ is
built, where the graph vertices represent the particles. The vertex
pairs are connected by the edge if the distance between the respective
particles is small enough. The edge weight is calculated to be inversely
proportional to the difference in the respective particle velocities,
i.e., the vertex pairs representing nearby particles having similar
velocities are connected by edges of larger weight. If a group of
particles moves in a coordinated matter, the particles in this group
will have similar velocities, therefore, the corresponding vertices in
the graph will be connected by edges of larger weight in the
representing graph. Having produced the velocity similarity graph,
identification of particle groups becomes equivalent to the problem of
“community detection” in graph analysis. The algorithms and techniques
developed for community detection in graphs can be thereby applied for
identification of particle groups involved in coordinated motion in
granular matter. We illustrate this approach by an example of granular
media filled in a rotating cylinder.

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