# Friends-Of-Friends Algorithm

Friends-Of-Friends (FOF) is a common tool used to identify groups of particles in a simulation. This can, for instance, be used to identify haloes in a cosmological simulation.

In practice, this is done by using a *linking length* `l`

and
demanding that any particle that finds another particle within a
distance `l`

is linked to it to form a group. A particle is linked
directly to all other particles within a distance `l`

(its
*friends*) and indirectly to all particles that are linked to its
friends (its *friends-of-friends*). This creates networks of linked particles
which are called *groups*. The size (or length) of
a group is the number of particles in that group. If a particle does not
find any other particle within `l`

then it forms its own group of
size 1. For a given distribution of particles the resulting list of
groups is unique and unambiguously defined.

Small groups are typically discarded, the final catalogue only contains
objects with a length above a minimal threshold, typically of the
order of `20`

particles. Smaller groups can often be spurious.

Once the groups have been identified, properties can be computed for each of them. The total mass or the centre of mass are common examples. These are then stored in catalogues alongside a unique identifier for each group.

SWIFT implements FOF using a Union-Find approach. It also exploits the domain decomposition and tree structure that is created for the other parts of the code. The tree can be easily used to find neighbours of particles within the linking length.

Depending on the application, the choice of linking length and minimal group size can vary. For cosmological applications, bound structures (dark matter haloes) are traditionally identified using a linking length expressed as \(0.2\) of the mean inter-particle separation \(d\) in the simulation which is given by \(d = \sqrt[3]{\frac{V}{N}}\), where \(N\) is the number of particles in the simulation and \(V\) is the simulation (co-moving) volume. Usually only dark matter particles are considered for the number \(N\). Other particle types are linked but do not participate in the calculation of the linking length. Experience shows that this produces groups that are similar to the commonly adopted (but much more complex) definition of virialised haloes. A minimal group length of \(32\) is often adopted in order to get a robust catalogue of haloes and compute a good halo mass function.

For non-cosmological applications of the FOF algorithm, the choice of the linking length is more difficult and left to the user. The choice of the minimal group size to consider is also application dependent.

In SWIFT, FOF is also used to identify haloes in which to seed black holes in a cosmological run. When used in this mode, the code does not produce any outputs but uses the catalogue internally to convert some gas particles into black hole particles.