Here's an interesting follow-up thought on my suggestion of some Hypothetical Laws of Social Networks.
What if in fact there is an entirely new way to design social networks, based on the mathematics of tilings? A tiling is a method of filling a space with geometric shapes. For example, you can tile a space with squares, hexagons, quasicrystals, spheres, etc. -- depending on the dimensions and topology of the space.
We could view a social network as a tiling problem -- each "tile" represents a set of nodes and arcs. "Corners" (nodes) are members of the network (such as people or organizations), the edges (arcs) are relationships between members. For this to be meaningful we must require tiles that have at least one node in them (so no circles or spheres or other tiles with only a single arc but no nodes).
Viewed in this way, we can look at particular tilings as designs for optimal social networks. What all this leads to is an insight that if there is an optimal number of relationships per member of a social network, and an optimal number of hops between members of social networks, then given a number of users in a network there is a particular tiling that optimizes those parameters. This tiling could be used as a template for the relationship structure of the social network.
So for example, when a member of a certain network adds a "new friend" in fact the network may not add that friend directly to their node, but rather may reconfigure the tiling structure such that they are optimally connected to that friend, via some number of hops instead of directly.
In other words, if the goal of a social network is to optimize relationship networking and communications -- it may require not always directly connecting people when they add one another as friends. And furthermore, it may require continual reconfigurations to optimize the evolving relationship structure as people join the network and form new relationships.
When parties add one another as friends it is essentially placing a constraint on the tiling of the network -- it makes the network attempt to optimize the H value (number of hops connecting them) for those parties. I have a hunch that this could be computed recursively and in a distributed fashion -- in fact, I think a cellular automaton rule or a tiling rule may be just the way to do this.
Roger Penrose might be a good source of ideas for this. I have a feeling that the best tilings will be highly irregular and chaotic, and ultimately we may be dealing with a high-dimensional manifold (not a simple 2 dimensional plane). But the same principles apply in any number of dimensions. Also I would suggest looking what is going on in the field of "loop quantum gravity" as a source of ideas for this. Of course graph theory, cellular automata, and the theory of geometric tilings are all relevant as well.
The key point I am making is that it should be possible to optimize the structure of the network using local distributed rules -- or at least regional neighborhood rules -- rather than global rules -- computed recursively perhaps -- that seek to optimize local tilings around nodes so that they are optimally connected to parties they add to their social networks.
Optimal connectivity is a balance between sparsity and density. The current trend in social networks of directly connecting parties to their friends is actually the very WORST way to structure these networks. A much more intelligent, and adaptive, paradigm would be to continuously reconfigure the network to suit the priorities of the nodes while seeking to provide them with optimal connectivity to their contacts. Direct connectivity is not necessarily optimal connectivity.
We need a way to design a social network such that it self-optimizes as members join and as they form relationships, such that members are optimally connected (not underconnected, not overconnected, and within range of their friends). As the network grows it has to continually retile itself to stay optimal. What this looks like is an evolving, self-optimizing irregular chaotic tiling.
I also have a hunch that this is a hint towards a general physical law -- similar to the universe optimizing the shape of space so that light travels most efficiently, I think a discrete universe could be viewed as a big social network of sorts where the relationships are the topology, the nodes are the locations in space, and the messages they exchange are light pulses.
The goal is to optimize the shape of this network around every node such that it does not experience "information overload" and is also optimally connected to other nodes it interacts with. Whenever an interaction takes place between two nodes we should consider them to have a stronger relationship, and this should bring about a perturbation and re-shaping of the network connecting them such that they are optimally connected.
Every interaction therefore brings about topological changes to the network based on the goal of optimizing information flow of further interactions. Seems to me that this could become a general physical law -- just has to be expressed in the right mathematical language. The law should be invariant across levels of scale and domains -- all networks should basically be optimizable using the same "network physics." Is this the network-equivalent of general relativitiy? I think there are some very interesting similarities.