I’ve had clustering behavior in randomly generated networks on my mind, recently, so arXiv:0802.2508 naturally caught my eye. It’s entitled “Criticality of spreading dynamics in hierarchical cluster networks without inhibition.” Marcus Kaiser, Matthias Goerner and Claus C. Hilgetag write,
An essential requirement for the representation of functional patterns in complex neural networks, such as the mammalian cerebral cortex, is the existence of stable network activations within a limited critical range. In this range, the activity of neural populations in the network persists between the extremes of quickly dying out, or activating the whole network. The nerve fiber network of the mammalian cerebral cortex possesses a modular organization extending across several levels of organization. Using a basic spreading model without inhibition, we investigated how functional activations of nodes propagate through such a hierarchically clustered network. The simulations demonstrated that persistent and scalable activation could be produced in clustered networks, but not in random networks of the same size. Moreover, the parameter range yielding critical activations was substantially larger in hierarchical cluster networks than in small-world networks of the same size. These findings indicate that a hierarchical cluster architecture may provide the structural basis for the stable and diverse functional patterns observed in cortical networks.
In network theory, the clustering coefficient is the probability that two neighbors of a node will themselves be directly connected. One can speak of the clustering coefficient for each individual node, and then look at, for example, how it varies with the number of edges linking to that node; alternatively, one can calculate an average clustering coefficient for the entire network. The simplest and most classic way of stochastically building a network is the Erdös-Rényi model, in which pairs of nodes are randomly chosen to be linked or not linked, and all pairs have the same probability [tex]p[/tex] of being linked. In this model, the clustering coefficient is just the parameter [tex]p[/tex]; there’s no “additional information” at the three-node scale.
Here, we’re motivated to study networks with higher clustering coefficients because neurons in the cortex are arranged spatially. If neurons generally connect to other neurons in the same vicinity — in other words, if most connections are short-range — then two cells both connected to a common partner are likely to be directly linked themselves, because they’re probably close by.
(The term “random network,” used in opposition to “clustered network,” is something of a misnomer, since all types of networks studied in this paper had a random component to their growth.)