# 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.

In our implementation, we use three separate categories influencing their behaviour in the FOF code:

• linkable particles which behave as described above.

• attachable particles which can only form a link with the nearest linkable particle they find.

• And the others which are ignored entirely.

The category of each particle type is specified at run time in the parameter file. The classic scenario for the two categories is to run FOF on the dark matter particles (i.e. they are linkable) and then attach the gas, stars and black holes to their nearest DM (i.e. the baryons are attachable).

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. 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. Usually only dark matter particles are considered for the number $$N$$. In practice, the mean inter-particle separation is evaluated based on the cosmology adopted in the simulation. We use: $$d=\sqrt[3]{\frac{m_{\rm DM}}{\Omega_{\rm cdm} \rho_{\rm crit}}}$$ for simulations with baryonic particles and $$d=\sqrt[3]{\frac{m_{\rm DM}}{(\Omega_{\rm cdm} + \Omega_{\rm b}) \rho_{\rm crit}}}$$ for DMO simulations. In both cases, $$m_{\rm DM}$$ is the mean mass of the DM particles. Using this definition (rather than basing in on $$N$$) makes the code robust to zoom-in scenarios where the entire volume is not filled with particles.

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.