This is a very good article on the physics of scale-free networks such as the Web.
Lately I have been getting increasingly interested in graph theory and also in knot theory. There is a similarity between networks and knots and it should be possible to do a mapping such that the theorems and algorithms of knot theory could be translated to apply to network topologies. I’m sure someone is already working on this, but it’s worth pointing out. For example, many networks could be viewed as planar projections of knots (the shadows that knots cast on a plane). I have also been thinking about the subject of loop quantum gravity which I learned about from my friend Bram Boroson.
All of this connects to an idea that I’ve been thinking about lately for a new kind of discrete system for evolving topologies that I call a “loop automata” — basically the idea is to use networks of interlinked loops as the fundamental building blocks for evolving spaces, and the dynamics within them. In my conception of a “loop automaton” the points at which loops intersect (“crossings,” to use knot theory terminology) are “nodes” and the segments of loops between intersections are “arcs.” So using a single construct we can have both nodes and arcs in our model. In other words we can construct graphs out of systems of interlinked loops.
Loops can have various states (a simple model might have a single valued state for the “energy” of the loop, while more complex models might deal with oscillation frequencies or even shapes of loops) The next step is to design functions on such networks of loops that modify the state of each loop based on the states of loops it intersects with (it’s neighborhood). This function should govern the creation, destruction, linking and unlinking of loops, as well as the states of loops. By specifying either that all loops are fixed diameter (regardless of what other loops they intersect with) or that loops can only intersect other loops in a single point (in other words that intersecting loops are never on the same plane) then we can interpret the resulting network of loops as a space that must have one of a set of certain dimensions and shapes. This enables such a system to represent any potential space. Information propagates along such spaces as the states of loops interact, causing feedback between the topology and the energy state of space.
In such models, every pair of directly connected nodes have two arcs connecting them — one in each direction (I assume that all loops are directed arcs that circle back on themselves endlessly). This enables information to propagate along different paths in different directions, enabling a form of “social interaction” between nodes. For example, imagine that every loop is a little clock around which a single pulse of energy is circling at some frequency. Whenever the pulse passes through an intersection point with another loop (ie. through a node) an interaction takes place between the two loops. This has the effect of modifying the state of the loop we are looking at such that as the pulse continues from that point onwards around the circumference of the loop it may have a different frequency. In other words, as the pulse goes “to” a node it has some state, and as it returns back “from” that node it may have a different state. This “back and forth message passing” takes place between directly connect loops as well as along transitive chains of loops.
Unless you spend a lot of time thinking about networks, graphs, knots, cellular automata and digital physics all of the above is probably incomprehensible. I apologize for the “rough” sketch but these are preliminary ideas at this stage. Still, from my reading on knot theory, graph theory and other related subjects I am starting to see a pattern here. Perhaps using loops as the fundamental building blocks of networks is not such a bad idea.