In a previous post, rebelkingismyhero discussed a model of large scale networks called the random graph. A random graph/network is generated by connecting each pair of a number of vertices with some probability. This creates a graph with each vertex having the same connectivity or level of connectedness on average. rebelkingismyhero noted that the behavior of many networks in the real world do not seem to match this model.

I really enjoyed reviewing Barabasi and Albert’s paper “Emergence of Scaling in Random Networks” which expands the random graph model to allow for growth in the network. Previously the number of vertices in a graph was fixed at the outset of the simulation. Barabasi et al. tries to instead mimic real world systems which continuously expand. Their model starts with an initial number of vertices and then links a new vertex to some of the old vertices each time step. To determine which vertices the new vertex will attach to, Barabasi incorporates ‘preferential attachment’ i.e. a vertex is more likely to attach to a popular vertex that has many edges than an edge that has only a few. The networks that result are similar to well documented networks such as the web and citation patterns in journals in that they self-organize into a network with ‘hubs’ and have connectivity that follows a power law.

In class we have studied networks that are static. I find it interesting to think about how the networks we have used to study power imbalances would grow if we used Barabasi’s technique to expand the network. How would the network evolve if we allowed vertices with power below a certain threshold to ‘die’? Should we modify the definition of ‘preferential attachment’ to have a new vertex prefer connecting to powerful vertices rather than well connected ones? Perhaps the ‘balanced outcomes’ discussed in lecture would move away from extreme imbalances and instead move toward a more even distribution of power. I think that it is likely that individual ‘agents’ ‘ attachment preferences must be at work when you examine how social networks, trading markets, and even bone tissue grows. These personal preferences may influence the global behavior of a network. Likewise, perhaps the overall function of a network or the environment in which it grows dictates how a new vertex attaches to the rest of the network.

With the dynamic nature of the real world, studying how a network’s structure changes over time could be as important as studying the network’s structure itself.