I found people who are actually trying to model general swarms mathematically rather than specific swarms.  This is awesome!  They are calling it swarm calculus.  I think that what we are doing would contribute to the swarm calculus.  It seems like everyone always assumes we know that what we are looking at is a swarm and that we can automatically apply swarm modeling techniques   But what if it isn’t a swarm?  Then we will have wasted time and resources learning nothing.

What happens if we used swarm calculus on a multiagent learning system? What would it tell us?  That might just be stupid.

So what does that paper say about swarm calculus?

They assume cooperation is an essential part of swarms.  So right away my first thought is: Do they think Brownian motion is a swarm?  Interesting quote:

swarms generally face a tradeoff between beneficial cooperation and obstructive interference.

They find the two main components of swarms are cooperation and interference.  So is what they consider interference what I think of coordination?

They say mathematically that cooperation is:

[math]C(n)=a_1N^b[/math]

They got this formula from Breder 1954 to model cohesiveness.

They model interference by looking at the correlation between number of robots and their efficiency in the robot foraging domain.

[math]I(N)=a_2e^{cN}+d[/math].

c is less than 0 so this is a decreasing function.  So the efficiency of each robot decreases as the number of robots increases (since each robot will have to collect less).

They showed the existence of phases in swarms.  Like when there are x individuals then they all have the same speed.  But at a point other than x there might be 2 distinct speeds.  This was interesting as it points to how the number of individuals in the system can effect the emergent behavior.