The Hypercube Graph and the Inhibitory Hypercube Network

In this paper we review the spectral properties of the hypercube graph Qn and derive the Walsh functions as eigenvectors of the adjacency matrix. We then interpret the Walsh functions as states of the hypercube, with vertices labeled +1 and -1, and elucidate the regularity of the induced vertex neighborhoods and subcubes. We characterize the neighborhoods of each vertex in terms of the number of similar (same sign) and dissimilar (opposite sign) neighbors and relate that to the eigenvalue of the Walsh state. We then interpret the Walsh states as states of the inhibitory hypercube network, a neural network created by placing a neuron at each corner of the hypercube and -1 connection strength on each edge. The Walsh states with positive, zero, and negative eigenvalues are shown to be unstable, weakly stable, and strongly stable states, respectively. Introduction The hypercube is an amazing mathematical object. It has applications in most areas of mathematics, science, and engineering. In this paper we analyze the inhibitory neural network formed by placing a neuron at each corner of the hypercube and an inhibitory connection on each edge. This creates a neural network similar to other inhibitory networks, such as K-WTA [5], inhibitory grids [6], and lateral inhibition [7], but distinguished by its hypercube architecture. To analyze this network we study the hypercube graph, Qn [1], [2], [4]. We start by summarizing the properties of Qn. We then derive the Walsh functions as eigenvectors of Qn, and show that they have interesting regularity properties when interpreted as binary functions on the vertices of the n-cube. Walsh functions are often presented as orthogonal sets of square waves, or as the rows of a Hadamard matrix, but the patterns they induce on the n-cube, and its subcubes, argue in favor of interpreting Walsh functions as natural boolean functions on the n-cube. Once the properties of the Walsh states are understood it is an easy step to interpret them in terms of stability conditions for the inhibitory hypercube network. We do not provide a complete analysis of all the stable states but establish the role of the Walsh functions as central to the analysis.