I am after some of the definitions used for KEGG Pathway Edges. In particular — activation, inhibition, expression, repression, and indirect effect. Has anyone come across documentation or a definition guide on how these edges are described and used?

If we are going to use these models beyond graphical use, modeling, or programming purpose, what is the difference between activation vs. expression, and inhibition vs. repression? etc

The Help of the online interface of KEGG Pathways already tells more: https://www.kegg.jp/kegg/document/help_pathway.html If you look into the corresponding KGML, the type attribute of the interactions is either PPrel or GErel (protein-protein vs. gene expression). Each interaction element has subtype children to give further classification. These correspond more or less to the arrow types in the legend above (e.g. binding, phosphorylation, inhibition, expression). Each of these are also shown in an alternative, 2 or 3 characters notation (the value attribute in xml, or a slightly different notation used in KEGG-MEDICUS). In conclusion, I have little doubt that all interactions are direct physical or chemical interactions unless stated otherwise; activation and inhibition means one protein directly activate or inhibit the other in carrying out its primary function; expression and repression means that one transcription factor positively or negatively regulates the transcription of a gene. State change and indirect effect are a bit confusing for me, as the help page shows them in one category, in my understanding state change might be the same as activation/inhibition, but no information about the functional consequence (i.e. we only know that there is a direct and directed interaction). Indirect effect means a proven causal relationship with no evidence of direct interaction or happening by unknown mediators. I think both these two can be either directed or without direction. Binding, association means evidence of direct interaction, without knowing the causality. Phosphorylation and others only add information about the mechanism in case of activation, inhibition.