Random Graphs In A Neural Computation Model
Alexandros V. Gerbessiotis · International Journal of Computer Mathematics · 2003
We examine in this work the following graph theory problem that arises in neural computations that involve the learning of boolean expressions by studying the asymptotic connectivity properties of $G_{n\comma 1/\lpar kn\rpar ^{1/2}}$ random graphs, where k is a fixed positive integer. For an undirected graph $G = \lpar V\comma \; E\rpar $ let $N\lpar X\comma \; Y\rpar = \lcub v \in V - \lpar X \cup Y\rpar \!\mid$ $ \exists x \in X\ \hbox{with}\ \lpar v\comma \; x\rpar \in E\rcub $ . For fixed k construct an undirected graph $G = \lpar V\comma \; E\rpar $ such that for all disjoint sets $A\comma \; B \subseteq V$ such that $\vert A \vert = \vert B \vert = k$ , and $C = N\lpar A\comma \; B\rpar \cap N\lpar B\comma \; A\rpar $ , set C is such that $\vert C \vert$ is either exactly k or as close to k as possible. Asymptotic results for large values of k are also presented.