Somebody's conducting an investigation - and every little bit of information could be the break they need. Red Herrings are flying left and right, and they need to get everything organized. What better way to do it than with a pegboard (or an entire room) covered in pictures of people, maps of places, and cryptic hints? Often the items are related, and these relationships are expressed by a complex web of strings connecting pairs of items; thus the name.
An example of Law of Conservation of Detail, as almost invariably every single item will be plot relevant - although it's not always clear whether it was all planned out meticulously in advance, or whether the writer decided to use the various random items on the board as jumping off points for future episodes. Fans will naturally drive themselves crazy trying to figure out the relevance of every item. Don't stare at it too long, though.
Subtrope of Room Full of Crazy.
Not to be confused with the WebcomicString Theory or actual particle physics.
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In Batman: Year Zero, Edward Nigma's office at Wayne Industries contains a vast spiderweb of different coloured strings, as he tries to "solve" pretty much everything as though it were a riddle.
A variation occurs in the film Spider where the titular paranoid schizophrenic protagonist has a penchant for creating webs with bits of discarded string as he investigates the death of his mother during his traumatic childhood. As befits his character, the strings never connect up anything useful and his notes are complete gibberish.
In A Beautiful Mind, John Nash creates these when investigating Communist infiltrations and conspiracies. It all means nothing, however, as Nash is suffering from paranoid schizophrenia.
Control by Victor Suvorov had the protagonist doing it as a part of her job in the secretSecret Police, to track power groups within party, NKVD etc. First with photos on a stand connected by threads of relations, then she removed them all and remade as one interconnected web of small thumbnails all over several walls. It worked, though not as expected: a few bosses living in one city turned out not to interact — never met informally, nor even tried to bring each other down. Wherefore Hilarity Ensues. The author was in military intelligence, after all.
In White Night, Harry Dresden is investigating his half-brother, Thomas. During the investigation, Harry snoops around the suspect's apartment, and stumbles onto one of these.
Live Action TV
Heroes, possibly to the point of being the Trope Codifier with both Mohinder's map of specials, and Future Hiro's map of all time.
Flash Forward, blatantly following in Heroes' footsteps with Mark's Blackout wall. A case could be made for D. Gibbon's "Garden of Forking Paths" as well.
Mark's wall turns out to be the key to determining the time of the next blackout.
Star Trek: Voyager, Time Fleet and the "Year of Hell" aliens has an automated version of this to keep up with their monkeying in the timescape.
Chuck, when he is keeping data of the Intersect and Orion on the back of his Tron Poster. Granted, it's in marker, but it's the thought that counts.
The Lost Room has a couple maps of the objects, including how they supposedly relate to one another, and where they have been.
The Major Crimes unit in The Wire tends to have a pegboard like this for each of their main targets. Unlike many of these examples, it's actually realistically and sensibly organized, with strings connecting people based on their positions in the drug organization's hierarchy.
New Tricks has one. Most episodes have a few scenes with the main characters sat around and one of them explaining what they've just discovered. They once discovered that a retired fireman who was helping them was an arsonist when they realised he would have been able to find his targets after seeing their board.
Kamen Rider Double uses these in a meta sense; the second episode of each mini-arc starts off with a "corkboard" that shows the characters from the first episode and how they connect. Then Movie Wars CORE shows the origin of the corkboard in-universe.
Nick sets up one of these in Primeval, trying to track the various anomalies across time and space. Later, the characters discover a heavily upgraded holographic version of his chart brought from the future.
Supernatural presents a beautiful example of a string theory during the first season's first episode. The main character's missing father was investigating on a Woman in White, using his motel room's wall to externalize his deductive reasoning.
Sam and Dean occasionally put these up in their motel rooms, which seems like a lot of effort for something you're going to have to take down in a few days.
The episode of Castle, "Linchpin", briefly displayed a room that looked very much like the page image, as belonging to a statistics genius - the strings started at one murder, the branches were cause and effects, and they converged on World War III at the other side.
Charlie Crews on Life has an entire room dedicated to finding out who framed him for murder.
In the pilot of the 2012 spy show "Hunted", the main character Samantha owns a rather "off-the-wall version" in her Scottish hideout.
In series 1, episode 13 of Elementary, Holmes is compiling one about Moriarty after tearing down the one he made about Moran. The book-canon description of Moriarty as 'a spider at the centre of a web' makes these almost inevitable.
Dirk Gently: used for the opening credits. Dirk keeps one on the wall, on which he puts everything that has happened in the episode whether or not it's relevant it's always relevant, and which he is seen painting over in the pilot episode, for a case which references an "electric monk".
John's home computer wall in Almost Human displays an electronic version as he tries to work out how his girlfriend used him to get police information to an organised crime group.
The Loom of Fate from Exalted looks a lot like this, with strings of fate representing the lives and destinies of all the beings under its purview.
In BioShock, Andrew Ryan has one of these in his office as an aid in figuring out who Jack is and why he's survived this whole time.
There is one covering the walls of the Task Force Aurora lab in the Mass Effect 3 DLC Leviathan. Fitting, seeing how this is basically an organization of kooks who believed in aliens—before the First Contact. They have also believed in Reapers long before Shepard encountered Sovereign.
In an aversion of the Law of Conservation of Detail, some of the leads the team is following are Red Herrings that won't lead you to the objective. (In fact, if you try to cross-reference them all, you end up with no systems matching all the criteria. No wonder work was going slowly...) It's up to Shepard to either sort out which influences are genuine and which misleading, or just go gallivanting around the galaxy in your Cool Starship and narrow the set of worlds down by yourself.
The coroner in Mystery Case Files: Shadow Lake was using this method to try and figure out the mysterious deaths in her town. Cassandra Williams has a similar bulletin board in her motel room, but we never get a decent look at it so there's no telling if she was trying something similar or just trying to sort out the Ghost Patrol shooting schedule.
In Sam & Max: Freelance Police, Bosco has it in his shop starting with season 2, detailing the connections between the villains from the previous season. It's rather outdated by that point, as lampshaded by Sam.
In Watch_Dogs, Aidan has one of these in the hotel room he's living out of. Oddly, he only references it a few times and you can't interact with it in-game at all.
In one Questionable Content strip, when Faye is trying to explain the main character's relationships to one another, her therapist stops her so she can get thumbtacks and colored string and diagram everything.
Ben 10: Ultimate Alien: Jimmy has one of these going for alien encounters, specifically involving those with the Omnitrix ensignia.
In the American Dad! episode "Bush Comes to Dinner," Roger determines Osama bin Laden's location by studying a variety of popular media which he's hung all over his attic. Cue the page quote.
In Megamind, Roxanne finds Megamind's plan lain out in one of these, but can't understand it at first. When she backs up for some perspective, she sees that the strings, rather than holding information, form a picture of the plan.
The FBI had one to show Homer Simpson to demonstrate the hierarchy of Fat Tony's mob. Emphasis on had. Why, oh, why, did they have to pick that spot to keep their shredder?