Vectorweights and dual graphs: An emphasis on connections in brain network analysis

Abstract

Graph theoretical representations of the brain as a complex network give a special emphasis to anatomical or functional units of the gray matter. These units are abstracted as the nodes of a graph and are pairwise connected by edges that embody a notion of connectivity. Graph theoretical operations in brain network analysis are typically employed to reveal organizational principles of the network nodes. At the same time, relatively little attention has been given to connection properties and the relations between them. Yet, various neuroscientific applications place an increased importance on connections and often require a characterization by multiple features per connection. It is not clear, however, how to incorporate vector edge weights in the standard graph representation. In this paper, we present a novel Dual graph formalism, in which the role of edges and vertices is inverted relative to the original (Primal) graph. This transformation shifts the emphasis of brain network analysis from gray matter units to their underlying connections in two important ways. First, it applies standard graph theoretical operations to discover the organization of connections, as opposed to that of gray matter centers. Second, it helps in removing the single scalar weight restriction and allows each connection to be characterized by a vector of several features. In this paper, we introduce the main concepts of this novel dual formalismand illustrate its potential in a population study on schizophrenia.