Global structure of directed networks emerging from a category theoretical formulation of the idea "Objects as processes, interactions as interfaces"

A system of interacting elements can be represented by a directed network so that elements are nodes and interaction between two elements is an arc. Conventionally, each node is just a point, each arc represents some kind of interaction between two nodes and nothing more after the system is mapped to a directed network. However, in many real systems, each element has its own intra-node process and interaction between two elements can be seen as an interface between two intra-node processes. We can formalize this idea “objects as processes, interactions as interfaces” within the framework of category theory. We show that a new notion of connectedness called lateral connectedness emerges as a canonical structure obtained from the idea. Lateral connectedness is not defined on the set of nodes of a directed network, but on the set of arcs. By its definition, it may be associated with functional commonality between arcs emerging from shared input or output. As a first application, we examine significance of lateral connectedness in the neuronal network of the nematode Caenorhabditis elegans by comparing the partition of the set of arcs induced by the connectedness and the partitions based on neuron functions. Lateral connectedness can capture a part of functional segregation of the neuronal network above a certain interaction strength level.