The line cannot split, it cannot run in two directions at the same time. so If it is starting at the glowing dot on the far left right it is probably unsolvable and should be thrown out.
In fact if you run across any research you cannot solve, throw it out. If it looks to difficult, there are too many runes to move, or you have to connect the 4 corners of the page, throw it out. It is usually cheaper and easier to throw out research and get a new puzzle than try and fiddle with difficult notes.
also I am derp: right not left. right node as start is impossible
I just want to restate the bolded part, as it can't be stated enough. Also, If your first active node is right in the middle of the other nodes, throw it out. It's far easier and cheaper to go for another research page than to waste ink, time, and aspects fiddling around.
Another thing: Always start your research with Cognito. Keep doing Cognito until you have research mastery. Even though you can only move an active rune one square with Research Mastery, and even though it might deactivate when you do it, there's one big thing to remember... if the rune deactivates because you moved it, that deactivation doesn't cost you an aspect point. You just cut your point expenditure in half.
Here's one of the simplest ways I've seen to determine whether to go for a new page... When I get a new research page, I start by drawing a line with my finger on the screen from the starting node to all other nodes. If on the shortest path any two nodes would take more than 2 runes to connect, I usually toss it. I will sometimes make an exception if there are very few nodes, or if they form a fairly straight and easy line, or if
all other node connections would only take one rune.