On the Preservation of Input/Output Directed Graph Informativeness under Crossover

TL;DR

This paper establishes a theoretical framework for understanding how crossover operations affect the informativeness of input/output directed graphs, which model networks connecting inputs to outputs, and explores conditions under which informativeness is preserved or lost.

Contribution

It introduces the concept of Input/Output Directed Graphs (IOD Graphs), defines informativeness levels, and analyzes how crossover impacts these levels in evolutionary algorithms.

Findings

Fully informative parents can produce non-informative children.

Under certain conditions, partially informative parents produce partially informative children.

Full informativeness may not always be preserved after crossover.

Abstract

There is a broad class of networks which connect inputs to outputs. We provide a strong theoretical foundation for crossover across this class and connect it to informativeness, a measure of the connectedness of inputs to outputs. We define Input/Output Directed Graphs (or IOD Graphs) as graphs with nodes $N$ and directed edges $E$, where $N$ contains (a) a set of "input nodes" $I \subset N$, where each $i \in I$ has no incoming edges and any number of outgoing edges, and (b) a set of "output nodes" $O \subset N$, where each $o \in O$ has no outgoing edges and any number of incoming edges, and $I\cap O = \emptyset$. We define informativeness, which involves the connections via directed paths from the input nodes to the output nodes: A partially informative IOD Graph has at least one path from an input to an output, a very informative IOD Graph has a path from every input to some output, and a fully informative IOD Graph has a path from every input to every output. A perceptron is an example of an IOD Graph. If it has non-zero weights and any number of layers, it is fully informative. As links are removed (assigned zero weight), the perceptron might become very, partially, or not informative. We define a crossover operation on IOD Graphs in which we find subgraphs with matching sets of forward and backward directed links to "swap." With this operation, IOD Graphs can be subject to evolutionary computation methods. We show that fully informative parents may yield a non-informative child. We also show that under conditions of contiguousness and the no dangling nodes condition, crossover compatible, partially informative parents yield partially informative children, and very informative input parents with partially informative output parents yield very informative children. However, even under these conditions, full informativeness may not be retained.