The “order” refers to the type of relationships that will be used to extract nodes from the underlaying database.
A first-order network (default) returns all seed genes and all nodes directly connected to them in the database.
A second-order network increases the size because it returns all seed genes and all nodes that are within two connections in the database. Drawing a comparison to social networks, first-order networks return seed genes and their “friends”, while second-order networks return the seed genes, their “friends”, and the “friends of their friends”.
A zero-order network will only introduce new edges between seed genes. In addition, it can reduce the number of seed genes because it retains only genes that are connected to each other within the underlaying database (i.e. orphan genes will be removed). This can help simplify your gene list to highlight the biological theme (i.e. protein-protein interactions, TF-gene interactions etc) among them based on the selected database.