8.5 Cell assembly enumeration in random graphs  (Page 2/4)

Goal

We seek to understand Palm's definition of cell assembly in context. In particular, we seek to determine whether a brain-sized random graph can contain a realistic number of assemblies by Palm's definition.

In this report, we describe the process of k-assembly enumeration and explain some preliminary experimentation using that algorithm.

The report exists in two forms. The abridged version includes the material described above, while the full version goes on to include more trend examples, an extensive collection of visual examples of k-Assemblies, and implementation.

This is the abridged version. The full version is available here .

Finding k-assemblies

Finding k-assemblies takes place in two steps: k-core enumeration and k-assembly confirmation:

K-core enumeration

The process of k-Core enumeration follows the general form of a branch and bound algorithm. That is, it follows the process:

1. If possible, solve the problem, otherwise:
2. Break the problem into several smaller problems, for each of these smaller problems, go to step 1.

Applied to our problem, the general algorithm looks like this, given some input graph $G$ ( [link] ):

1. If $G$ is a minimal k-core, stop, otherwise:
2. Dissect $G$ as follows:
   1. Find the maximum k-core that is an induced subgraph of $G,$ call it $H$ ( [link] ).
   2. For each vertex $v\subset V\left(H\right),$ go to step 1, using for the new $G$ and induced subgraph of $H$ such that $V\left(G\right)=V\left(H\right)\setminus v.$

This process finds all k-cores in the original $G.$ To sketch a proof:

• An algorithm that finds all induced subgraphs in $G$ and then filters k-cores from the rest will trivially enumerate all induced k-cores in $G.$
• The k-core enumeration algorithm described is equivalent to such an algorithm. By in turn eliminating every vertex from a graph, the algorithm find all induced subgraphs except for those it skips. The skipped subgraphs will never generate k-cores not already generated:
  • Skipped subgraphs have at least one vertex of degree less than $k,$ with respect to that subgraph; call this vertex set $W_1.$ The subgraph may contain vertices that would have degrees less than $k$ if excluding the vertices in $W_1;$ call this set $W_2.$ There may also be vertices dependent on the vertices in $W_1$ and $W_2$ to maintain a degree of $k$ ; call those $W_3.$ We can continue forming these sets until $G\setminus (W_1\cup W_2\cup ...\cup W_n)$ is a k-core for some positive integer $n.$ Define $W$ as $(W_1\cup V_2\cup ...\cup W_n)$
  • Trivially, take some induced subgraph of $G$ called $g.\forall v\in V\left(g\right),D(v,g)\le D(v,G)$
  • Consequently, no induced subgraph of $G$ will contain a k-core including any vertices in $W,$ since no vertex in $W$ has gained degree upon finding an induced subgraph, and, consequently, the collective $W$ still cannot meet the degree threshold to be included in a k-core.
  • Subgraphs are skipped iff they contain vertices in $W.$