I think, if in order to compile a ranked or ordered list, you look at the statistics of rivers, you will need to specify some criteria for measurement.
Does it matter what your criteria are if you do not specify why you need the ordered list?
Strategies for assessment:
1. Flow Rate. Follow a river from the sea, then always take the higher flow rate branch until you get to a damp spot that does not flow. What if the flow is seasonal, a river may be dry for several years before a rain event occurs.
2. Branch Length. Map the river based on the distance between the nodes where streams meet. Search for the terminal node furthest from the estuary; by the shortest possible route?
3. Total Area of the combined watershed. Water does not need to flow all the time.
4. Nomenclature. If the name of the river changes then is it the same river? What if two different named rivers meet to form a third named river.
Where does a river start? The headwaters will probably need to be defined as a spring or a wet spot where flow downstream is first visible on the surface. If scale is ignored then it will probably come down to the hopefully convergent limit of a fractal search between mineral grains in the soil.
Can a river start at a lake? Probably not, many long rivers have deep sections that might be called lakes, so claiming a river starts at a lake is probably unreasonable. Water must somehow reach that lake.
If water rises from a cave system, then what length of the unmapped cave system should be included as the river? Chemically tracing “disappearing streams” to their outlet may lead to an extension of another river by an unknown length.
Where does a river end? Does the river end in the estuary where sea water meets the fresh water? At high or at low tide? During flood or drought?
Luckily there are few long rivers that feed a salt basin without an outflow. In Australia, Coopers Creek flows into Lake Eyre. When the lake starts to fill, how much of Lake Eyre is actually counted as Coopers Creek?
What about a delta, is it the shortest or the longest channel that should be considered? I believe the shortest should be measured as the longest will be flow variable.
Which side of an anabranch or braided channel should you follow? Should it be the shortest, or the one that flows for the greater proportion of the time.
When a river divides, say during high water, which channel should be measured? The Atchafalaya River significantly shortens the Mississippi during times of high water.
