Adjacency Functions and Their Topological Effects

Abstract

An adjacency function AF can be used to transform a given (original) network into a new network, i.e., A ^original is transformed into A = AFA ^original. We introduce several useful adjacency functions and show how they can be used to construct networks, e.g., the power adjacency function AF ^power(), which raises each adjacency to a fixed power β, can be used to construct weighted correlation networks. In general, adjacency functions depend on parameter values, e.g., AF ^power depends on the power β. The choice of these parameter values can be informed by topological criteria based on network concepts. For example, we describe the scale-free topology criterion that considers the effect of the AF parameters on the scale-free topology fitting index. The power adjacency function can also be used to calibrate one network with respect to another network, which is often a prerequisite for differential network analysis or for consensus module detection. If an adjacency function (e.g., AF ^power) is defined via a monotonically increasing real-valued function, then it is called rank-preserving since the ranking of the original adjacencies equals that of the transformed adjacencies. The terminology of “rank-equivalence” and “threshold-equivalence” between adjacency functions, networks, and network construction methods allows us (a) to provide a general definition of a correlation network and (b) to study the relationships between different network construction methods.