In any accidental construction you will have modes connected by accidentals.  The nature of the connection determines these three types.  These are mainly for naming purposes and don’t at the moment betray any overarching organization.

A network is open.  It is not periodic and there is no path that will lead back to where you stared.  The only native network is the Major Accidental Network.  However, any song that is not meant to loop can also be considered a network as can any sequence.

A wheel is closed but is not completely cyclic.  While there may exist “A” route back to where you started, not ALL routes do that.  In The LIttle Wheel for example, you can start on any mode on the circle and traverse the circle and return to that mode.  But if you step into Lydian Dominant (LDA) or the Altered Mode (ALT) then the only way back our out is the way you came and thus there is a non-periodic path.

A loop is completely periodic…. any path out of a mode will eventually return you to that mode.  The first four examples below are loops, circles… topologically S1.  The final two examples actually consist of two loops intertwined, forming a torus, topologically S2 or T2.