For the network algorithms to process correctly all polylines must be single part polylines. Multi-part polylines are often generated during editing, especially if the original data capture had a purpose other than a topologically connected network. 

You can "explode" multi-part polylines into single parts manually or by using a Geo-processing tool

WarningExploding the multi-part polyline into it's individual parts will result in the attributes of the original polyline being passed onto all subsequent parts.  Whilst this is useful when the attribute is common to all parts (e.g. river name); node ID numbers would be incorrect. Exploding a multi-part polyline is altering the topology of the network and will influence the whole network.  You must re-run all RivEX attribution processes again!

Exploding Multi-part features manually

In edit mode select a multi-part polyline and then press the Sketch Properties button (Sketch properties button)on the editor tool bar. This opens the Sketch Properties panel, now click on the Edit vertices button (Edit vertices button) to display the individual parts.  The image below shows the vertices for a 3 part polyline.

Multipart vertices

Note if this polyline was a "Z aware" feature it would show Z information. Right clicking on different parts of the sketch properties panel will display different context menus and you can use these to remove whole parts of a multi-part feature or just a single vertex.

If a multi-part polyline makes a logical flow pattern (i.e. a single channel) then turn on the advance editing tool bar from Editor > More Editing Tools > Advance Editing. Ensure your multi-part polyline is selected and press the button highlighted below.  The original polyline is removed from your dataset and replaced with all subsequent parts as single features. 

Advance editing toolbar

Exploding multi-part features using the Geo-processing tool

You can run your entire dataset and a select subset through the Geo-processing tool that will search for multi-part shapes and explode them into individual parts.  You would then check the orientation of the newly created polyline to ensure they are flowing in a downstream direction. 

Geo-processing tool