Say there was some situation where you have a lot of subatomic particles interacting with each other and decided to draw (say, by joining Feynmann diagrams) those interactions- so that you got some sort of (?directed) graph... what kind of network would you obtain? Would it be scale-free? Would it be a complex network similar to a social network? What properties would it have?
a difference between a social network and a Feynman diagram is that a Feynman diagram describes a possible history - something that occurs in the whole spacetime - while a social network describes a particular object that exists at one instant of time.
So one can't talk about the "Feynman diagram for particles that exist now". Instead, Feynman diagrams correspond to histories. And in fact, Feynman's main rule is to sum over all histories - all Feynman diagrams with the same external lines - before you get the result, and the result is interpreted as a probability amplitude when you're finished.
Moreover, even if you tried to neglect the fact that the Feynman diagrams have to be summed up, and that Feynman diagrams are expressing histories in spacetime rather than objects in space, there would still be another serious problem with your question because the relationship in the graph - whether two particles interact i.e. whether they are "Facebook friends" of each other - isn't really well-defined.
In principle, all two particles may interact with each other except that the interaction may be weak and fail to influence any measurable quantities.
So one can't really construct Feynman diagrams analogous to social networks. But your question has many other aspects, including self-similarity. Yes, there exists a lot of scale-free, self-similar behavior in quantum field theories that describe elementary particles. So if one properly draws some "other functions" that describe the behavior of the system with massless particles only, she will get self-similar pictures.
In particular, the "typical" trajectory of a particle in spacetime resembles the Brownian motion, or a random walk, at least in a certain parametrization. It is self-similar: if the timescale in a random walk is increased $k^2$ times, then the typical distances where the particle arrives get $k$ times longer, but the qualitative shape of the picture is unchanged.
Quantum field theory - in the scale-invariant massless limit - is full of similar chaotic, self-similar patterns of various dimensions. But they're much more like continuous functions of some real variables than combinatorial graphs.
To address another intuition that is not quite adequate, quantum field theory implies that the "number of elementary particles" in a region isn't really a finite well-defined number. In some sense, the number of quarks in a proton, when properly counted, is infinite, and the more accurate "resolution" how to look inside the proton one chooses, the more quarks she sees.