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

TL;DR

The paper introduces Input/Output Directed Graphs (IOD Graphs) to model broad information-flow networks with disjoint input and output sets, and defines informativeness as the presence of directed paths from inputs to outputs across three levels: partial, very, and full. It then formalizes a graph-level crossover operation using IO partitions and a crossover membrane to swap subgraph connections, proving closure of the IOD Graph class under crossover and deriving bounds on how informativeness transfers from parents to offspring. A key result is that informativeness is not generally preserved—two fully informative parents can yield a non-informative child—while under contiguity and no dangling-nodes assumptions, partial and very informative inheritance can be guaranteed in certain configurations. The discussion extends the concepts to actionability, explores the competing conventions problem, discusses evo-devo considerations, and outlines future directions including generalized membranes and computational implementations, highlighting the framework's potential across domains from neural networks to engineered flow systems.

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.

Figures (11)

• Figure 1: Two Input/Output Directed Graphs of Varying Informativeness
• Figure 2: An IO Partition and corresponding membrane
• Figure 3: This IO Partition is input-contiguous but not output-contiguous
• Figure 4: The Crossover Operation on Input/Output Directed Graphs
• Figure 5: Two fully informative parents may yield a non-informative child

Theorems & Definitions (21)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Lemma 1
• Definition 6
• Definition 7
• Lemma 2
• Lemma 3