Idiosophy

A physicist loose among the liberal arts

Category: graphs Page 1 of 2

Look, Ma! I’m on a podcast!

I don’t talk about my job much, because getting permission to release things to the public is a gigantic pain. But this time someone else did all the work.  Here’s a podcast about one of my co-workers. She’s talking about a cool thing she’s doing with graph theory:

Episode 16

The coolest figure from “Graph Theory as a Mathematical Model in Social Science”

My role in the podcast is to be an authoritative old geezer who tells amusing stories about what graphs are good for. As it happens, I started my experiments in 21st-Century graph theory right here on the blog. I do a lot of it at my job now, because I happened to be thinking along those lines when a problem came across my desk that needed graphs. And it took off from there. There are a lot of people thinking about how the humanities can play a bigger role in engineering, as engineers make decisions they think are independent of squishy, qualitative stuff.  I’m not sure this is what they’re referring to.

Funny coincidence: Corey Olsen was saying something similar in the Mythgard Academy “Alice” class the other night, except he was talking about English and chemistry.

 

Literary Circles

Today is another expedition into Distributed Collaboration. Not the kind your boss means, but the truly internetted kind of research in which Idiosophers specialize. This time, all the work was done by Martin Paul Eve, who has assembled a database of all the review essays in the London Review of Books. He has put the raw data on line, so all I have to do is type some commands in R. In his conception, reviews form a network where a node is a writer and a link in the network is an arrow pointing from the writer of the review to its subject. The LRB has been running since 1979, so the database has tens of thousands of entries.

Eve showed some graphs in a blog post where he had fun finding closed loops in the graph: A reviews B’s book, B reviews C’s book, C reviews A’s book. I’ve loved this kind of analysis ever since I read The Devil’s Dictionary. I went another direction, though.

This blog has been graphing the Inklings for a while. The LRB is too late historically to help out with understanding their direct interactions, but its network is useful for understanding their reception. So here’s the question: It’s clear that J.R.R. Tolkien was not welcomed into the sacred grove of Literature until we barbarians smashed the gates. The other Inklings weren’t mentioned in my college catalogue, either. Is that also true of the literary world across the pond?

One great thing you can do with a network graph is extract the subgraph around any point you ask for. I asked for the Inklings. I used Appendix A of Diana Glyer’s book The Company They Keep to decide who is an Inkling. None of them ever wrote a review, naturally, but six reviews of them appear in the network:

  • Humphrey Carpenter and Christopher Tolkien
  • C.S. Lewis, edited by Walter Hooper
  • J.R.R. Tolkien, editor Christopher Tolkien
  • J.R.R. Tolkien, editor Alan Bliss
  • David Cecil
  • John Wain

That’s a respectable number of the group to be reviewed, considering how few of them were alive in 1979. I’m looking for how broad a reach they have with two degrees of separation. J.R.R. Tolkien has only a small network. Both of his entries in the database are due to reviews by Peter Godman. Peter Godman also reviewed a book by Tom Shippey, who wrote lots of things for LRB.

JRRT network in the LRB

J.R.R. Tolkien’s network

C.S. Lewis’s network is larger, with 32 points, but that’s entirely due to J.I.M. Stewart.  I’ve turned off the labels for anyone who was involved in fewer than 25 reviews (e.g. Arthur Conan Doyle) so we can read the graph. 32 connections would be kind of impressive, but I can’t help noticing that the graph they form is the same as the graph for Humphrey Carpenter’s biography of Tolkien. Out of the 33 nodes, there are seven writers here who are connected tightly enough to the LRB establishment to be labeled. (If I recall That Hideous Strength correctly, Lewis would shed no tears at being in a small backwater of the network.)

C.S. Lewis’s network

That’s it for the people we usually think of as the Inklings. No Barfield. A couple of minor members have much more connection.  Lord David Cecil (as one might suppose from his title) is extremely connected, but only because he’s reviewed by Frank Kermode:

Cecil network, including John Wain

David Cecil’s neighborhood (click to embiggen)

John Wain is in Cecil’s neighborhood; if we re-center it on him, the graph is just slightly different. Charles Williams is a well-connected name in Eve’s database, but the name is attached to the biographer, not the Inkling.

Conclusion

It’s safe to say that the Inklings are still out of the British literary mainstream by this measure. Were it not for three reviewers taking a brief interest, none of them would have appeared in the LRB.

Better Conclusion

If you want to see a truly amazing list of people, check out the archive of Tom Shippey’s LRB reviews.  Where else can you find Geoffrey of Monmouth and Nichelle Nichols next to each other?


I made one tweak to the original database: T.A. Shippey and Tom Shippey are the same person, so I consolidated those two nodes.

Graphing CSL’s Letters

A Pilgrim in Narnia has a post today that mentions, among other things, a social-network analysis of the relationships among Inklings, published in January 2019, that lines up neatly with one of the interests of this blog.

H.H. Kim goes through all the volumes of the letters of C.S. Lewis and produces a set of uni-polar graphs showing all the people with whom Lewis exchanged letters. The figures are impressive – that’s a lot of people. [1] Kim also sets up a chronological analysis, but not in the form of a graph, just a bunch of tables.  The reason, as the Conclusion says, is that ‘The biggest limitation of this exploratory paper was the inability to create a directed “complete” network between 1) Lewis and his recipients, and 2) among Lewis’s recipients.’

That’s where my efforts to look at time-evolution of the graphs hit a wall, too. Building off Diana Glyer’s book gives a much more complete picture of the connections among people who aren’t Lewis, but most interactions between people have nothing like the date precision that letters provide. Even when Lewis is involved, most of the important connections don’t involve written records. Kim points out that there are hardly any letters between Lewis and Tolkien.  Which makes sense: why write when you can just raise your voice? [2]

This is possibly the most frustrating thing about research in the humanities, to a scientist. When scientists think of a question that we don’t have enough information to answer, the next step is to figure out an experiment that will give us the other things we need to know. In humanities, going and getting more information is a lot less likely. It’s not at all possible with medieval literature (just finished the class, hurrah!).  Things are not quite so hopeless with the 20th Century. And, of course, as long as storage media can still be read, it’s not going to be an issue at all with the 21st.


[1] I would tell you which figure I mean, but for some reason the editors of VII don’t use figure numbers.  Possibly due to the shameful under-representation of scientific analysis in their journal to date.

[2] Roman historians make the same complaint: if Atticus had just stayed in Greece we’d know a lot more about the late Roman Republic than we do. Nobody writes letters to someone nearby, and letters are much of what we know. As Anthony Everitt put it in Cicero, “when Atticus is with Cicero in Rome the picture breaks up.” (page ix, but I promise I read the whole book.)

Five Views of the Inklings

Influence among artists is a complex and poorly-understood phenomenon.  Diana Pavlac Glyer took an excellent shot at understanding influence among the Inklings in The Company They Keep. This post is a graphical expression of her work; no additional scholarship has been committed. I had grandiose plans for network analysis of the Inklings and their influence on each other, but I hit an insurmountable stumbling block: the Inklings didn’t write very much.  It’s folly to apply big-data analytical techniques to a small set of things, so I’m just drawing pictures here. 

As anyone would anticipate, C.S. Lewis is at the center of things, almost any way we choose to plot the graph.  Here, the size of the dot indicates the number of connections to other people. There is one link per mention in TCTK, so when there are lots of mentions, the links look like a fat blob, not a line.

graph of five interactions

All interactions among Inklings

OB Owen Barfield JAWB J.A.W. Bennett
DC David Cecil NC Nevill Coghill
JDG James Dundas-Grant HD H.V.D. Dyson
AF Adam Fox CH Colin Hardie
RH Robert E. Havard CSL C.S. Lewis
WL Warren Lewis GM Gervase Mathew
RBM R.B. McCallum CES C.E Stevens
CRT Christopher Tolkien JRRT J.R.R. Tolkien
JW John Wain CW Charles Williams
CLW C.L. Wrenn All The ensemble

This is a complicated network, but it can be analyzed into components. DPG cites Karen Burke LeFevre’s book Invention as a Social Act, which identifies four different types of influence that authors (or any creators, actually) can have on one another: Resonator, Opponent, Editor, and Collaborator. To these four, DPG adds a writer-specific category: dedications.

Resonators are not just cheerleaders; they can bring out the best in an author by insisting they produce nothing less.  C.S. Lewis was the champion resonator.

graph of resonator relationships

Resonators among the Inklings

Opponents are those who poke holes in the weak parts of a work before you finish it, so prospective publishers don’t do it.  These are not so common among such supportive friends as the Inklings, so the network is much smaller.  The line from HD to JRRT is a mathematical representation of the most famous quotation in all of Inklings scholarship.

opponent influences among the inklings

Opponent relationships among the Inklings

Editors are editors. Again, C.S. Lewis is the nexus around which everyone else is arranged. Christopher Tolkien only has one line to his father, because DPG considers him more of a collaborator than an editor of the History of Middle-earth.  “All” is there in the bottom-right corner because J.R.R. Tolkien gave credit to the whole group for helping edit The Lord of the Rings. It’s not clear whom exactly he meant, so I didn’t resolve it into individuals.

inklings who edited another's work

Editorial relationships among the Inklings

Collaborators are collaborators. This is a dense network because I drew a line between any two Inklings whose names appeared as authors on a single work. For example, Essays Presented to Charles Williams had five Inkling authors which yields ten lines. C.S. Lewis is not so central, because he’s only one of a group of equals in these cases. Here also is the dense blob of links between the Tolkiens, one line for each of the posthumous volumes of the Legendarium. Various Festschriften are most of the other lines in this graph, so ironically it is dominated by books that were written after the Inklings had dissolved. 

collaboration

Collaboration relationships among the Inklings

Dedications are another Lewicentric network. Each of the the three most-prolific authors dedicated a work to the Inklings as a group.  Without the node labelled “All”, this graph would almost look like a chain, mathematically trivial.

Dedication relationships between Inklings

Conclusion

The Inklings were a large and not-well-defined group. Writers’ groups tend to be much smaller.

“Collaborative circles usually consist of three to five members; only rarely do they consist of more than seven or eight.”

Michael P Farrell, Collaborative Circles (cited in TCTK)

Despite its size and fluidity, the group we know as the Inklings was among the most influential writers’ groups of the twentieth century. The graphs above give a hint how this could be. Resolving the network into LeFevre’s various types of artistic influence shows that the Inklings can profitably be thought of as a superposition of normal-sized collaborations, one for each type of influence.  The various graphs share C.S. Lewis as the most important member measured by degree centrality, and also by who furnished the meeting space. The other members of each sub-network vary according to type. Like a refracting crystal, the network representation of the Inklings presents a different shape according to the perspective from which we choose to look at it, but each shape shares the important features of the underlying form.

Caveat:  as DPG says, “The examples of encouragement conflict, editing, collaborating, and referencing described in this book are not intended to form a comprehensive or exhaustive list.” (TCTK, p.213) If she left it out, so did I. Apart from dedications, I have omitted the section about “referents”, where characters in one Inkling’s work are based on another Inkling. Referential relationships as DPG described them are so amorphous that indicating them with lines on a graph seemed incorrect.


Works Cited

Farrell, Michael P. Collaborative circles: Friendship dynamics and creative work. University of Chicago Press, 2003.
Glyer, Diana Pavlac. The Company They Keep: CS Lewis and JRR Tolkien as Writers in Community. Kent, OH: Kent State University Press, 2007.
LeFevre, Karen Burke. Invention as a social act. SIU Press, 1986.

 

Thee & Thouing

Lee Smith is full of good ideas these days. Her latest is a graph of the characters in LotR who call each other by the familiar pronoun.

Her graph has caused me to reconsider an earlier opinion.  I once wrote a post in which I complimented Faramir on a slick linguistic move to seduce Éowyn.  The graph, though, shows that Faramir never switched from the formal to the familiar in anything he said to her.  Worse, he went even more formal: “I will wed with the White Lady of Rohan, if it be her will.”

I hadn’t realized this until I looked in the French translation.  During all their conversations in the Houses of Healing, franco-Faramir addresses Éowyn as “Madame”. (N.B. He’s 36 years old; she’s 24.) Then, as he makes his move, he ratchets it upwards. The highest level of formality, when talking to a feudal ruler, was to address them in the third person.  We have only echoes of that in American English; we get the feeling when someone says to the Queen, “as Her Majesty commands”.

So I’ve changed my opinion. Faramir is just role-playing his feudal-prince fantasies again. (We discussed this over at Olga’s joint, a while back.)  Not that I can blame him; Lee’s graph shows that Denethor never called his son “thou”, either.  Poor guy had no idea how to use pronouns.

Lines of familiar address by chapter

Network of Fools

Over at her blog, Lee Smith has found something fun to do on a rainy February day.  She’s collected every time somebody insulted somebody else in The Lord of the Rings.  To nobody’s surprise, “fool” is the most common way to insult someone.  There’s more give-and-take than I’d thought, though.  If we define “calling someone a fool” as a relationship, it makes a fairly complex network.

Lee confines her attentions to insulting people to their face. This has an elegant directness, but it misses some things that interest me, like Sam calling himself a fool. I’m going to expand on Lee’s definition for the sake of entertainment and include any time someone calls someone a fool, or a group of up to ten others.

The network looks like this:

Graph of accusations of foolishness

Whom are you calling a fool?

I have omitted three trivial subgraphs, involving Farmer Cotton/Ted Sandyman, Shagrat/Gorbag, and Wormtongue/Hàma.  I was expecting the graph to fall into two tight cliques with loose links between them, but that turns out not to be the case.  Saruman’s insults at the end of the book tie everything together neatly into a tightly-bound community of disregard.

Here’s a table of fool-counts, sorted by the fraction of their arrows that point outwards.

Character Speaker Referent Disdain
Grishnakh 4 100%
Witch-King 2 100%
Shagrat 1 100%
Rory Brandybuck 1 100%
Farmer Cotton 1 100%
Wormtongue 1 100%
Gandalf 14 3 82%
Saruman 6 2 75%
Gollum 2 1 67%
Boromir 1 1 50%
Denethor 1 1 50%
Gimli 1 1 50%
Nameless Orc 1 1 50%
Ugluk 1 1 50%
Pippin 2 5 29%
Frodo 1 4 20%
Sam 1 4 20%
Merry 3 0%
Bilbo 2 0%
Legolas 2 0%
Aragorn 1 0%
Ted Sandyman 1 0%
Butterbur 1 0%
Eowyn 1 0%
Gorbag 1 0%
Hama 1 0%
Lotho 1 0%
Radagast 1 0%
Nameless Ruffian 1 0%
Sauron 1 0%

And here’s the Queen of Soul, misapprehending the topology:

Some Networks are Simple

In which the Idiosopher gets to do Inklings stuff on the clock.

logo of the winter simulation conferenceLast summer, the blog started following Sørina Higgins’s suggestion about network analysis to see how interactions between the Inklings in real life turned into stylistic evolution in their literary styles. I haven’t mentioned it since, due to a lack of discoveries that are interesting (even to me).

This week was the 2017 Winter Simulation Conference, a world-wide hootenanny of computer-simulation experts. With all my responsibilities discharged, I got to spend the last half-day attending talks on anything that sounded interesting.  Here’s a good one from Wai Kin (Victor) Chan of Tsinghua University:

This paper studies social influence (i.e., adoption of belief) using agent-based simulation and regression models. Each agent is modeled by a linear regression model. Agents interact with neighbors by exchanging social beliefs. It is observed that if individual belief is linear in neighbors’ beliefs, system-level belief and aggregated neighbors’ beliefs can also be described by a linear regression model. Analysis is conducted on a simplified 2-node network to provide insight into the interactions and results of general models. Least squares estimates are developed. Explicit expressions are obtained to explain relationship between initial belief and current belief.

Social networks are complicated. People go in and out, they talk more or less, they form cliques, etc. If you want to measure things about them, you usually have to build a computer model with a clock and a bunch of “agents” in it. An agent is (in this case) a person with ideas (represented by a number), and as time advances, the agents pick up ideas from each other. Then you inspect the agents after a the simulation has been calculated and find out what ideas each agent has absorbed.

What Prof. Chan has discovered is that, as long as each person in a social network only picks up an idea from three others (which is comparable to the situation for the Inklings), all the complicated stuff drops out – the results of the full-powered simulation always look like a straight-line influence!  This paper is his attempt to prove that’s actually true.  If he’s right, then the spread of some ideas through a literary group will show a very simple pattern, and a literary scholar will be able to do lots of Digital Humanities research with just a spreadsheet. (Hi, Sparrow!)

An anecdote: another thing I did last summer was serve on a jury.  We had to decide on a prison sentence.  At the beginning, we went around the table and got everybody’s first impression. There was quite a bit of difference among us.  After four hours of discussion, which got pretty acrimonious at times, we agreed on a number that was within 5% of the average of everybody’s initial opinion.  That’s what a social-influence model would predict, if Prof. Chan is correct.

If the idea you’re studying can be cast as how strongly an opinion is held, on a scale from 0 to 1, then a linear regression is all you need to solve it.  That’s what I suggested Theosophy might look like, in the post from last July.  Division of territory, such as giving up Arthurian legends to one colleague, space travel to another, and focusing your own attention on time-travel, isn’t describable this way.

Prof. Chan started off his talk by saying the topic was just his own interest, not funded by any organization, and it wasn’t finished yet and he didn’t know what it meant, and the talk was still interesting.  My new scholarship goal is to be able to do that.

Irrelevant note: today is Idiosophy’s second bloggiversary.

 

What the One Ring Does

If you believe Peter Jackson, the Ring doesn’t do much of anything. It’s just a MacGuffin. With the Ring, someone who’s a hundred feet tall can knock down dozens of soldiers with one sweep of a thirty-foot mace, but I’m pretty sure I could do that without a Ring. (It’s all in the wrists.) The Ring blurs your vision and fills your ears with voices, which seems counterproductive when you’re trying to rule an evil empire. It enables you to see Ringwraiths, but who would want to?

The cost of Ringlessness

So what does the Ring actually do? We can try to figure that out by looking for things that only happened because Sauron didn’t have it. Here are a few:

The Nazgûl aren’t very effective, without their Lord

When the Black Riders split up they can scare people, but they can’t accomplish much useful. Granted, there is power working against them in the Shire, but it would have to be stronger than the military of Gondor to neutralize the Nazgûl to that extent, and I don’t think it is. Besides, the Riders seem kind of disorganized. The One Ring may be the only channel of control that works on all nine of their rings.

Saruman attacked Rohan too soon

Time to take another swipe at Peter Jackson. Saruman in his movie is a willing slave of Sauron, which makes his attack on Rohan ridiculous. If he’d held off a week,  until about March 8, the Rohirrim would have still been pinned down in Helm’s Deep as the gates of Minas Tirith were shattered. War over.

In the book, this makes perfect sense. Saruman thinks he’s an independent agent, pursuing his own ends. I can’t find any evidence that he knows Sauron’s timetable for the attack on Gondor. Sauron can’t give him a direct order without revealing his dominance, which would provoke some kind of resistance. Considering the distance and Saruman’s considerable power, resistance could have done quite a bit of damage to Sauron’s plans. All this is because Sauron doesn’t have his Ring. With the Ring, Saruman’s own ring would have bound him to Sauron’s will, with no need to keep up a pretense, and the synchronized attacks on Gondor and Rohan would have been devastating.

The fight among the orcs in the Tower of Cirith Ungol

Sam called it “lucky” that Shagrat’s and Gorbag’s forces wiped each other out, so we should keep an eye out for the hand of Providence. And everything worked out so neatly that it must have been there, but in this case it’s hardly needed. The natural centrifugal instincts of Orcs aren’t held in check by the Ring (even though it’s just a few feet away). A bit of “binding” would have kept the lines of command clear, and made sure that either Shagrat or Gorbag would have known to shut up and obey.

The chaotic orders Shagrat gets from Lugbúrz

Look at this mess:

“Any trespasser found by the guard is to be held at the tower. Prisoner is to be stripped. Full description of every article, garment, weapon, letter, ring, or trinket is to be sent to Lugbúrz at once, and to Lugbúrz only. And the prisoner is to be kept safe and intact, under pain of death for every member of the guard, until He sends or comes Himself.”

VI, i.

Shagrat’s reporting path upwards goes through the bureaucracy of Barad-dûr, but the path back down does not. This is a recipe for disaster: the middle-managers aren’t kept apprised of the actions at the top, so who knows what they’ll screw up, even if they’re following orders to the letter.

Let’s put these together into an Orc Chart. This graphic shows a disaster waiting to happen. Shagrat has three directions from which orders can arrive; that’s the obvious point of failure of this organization.  (JRRT’s military experience shows up again.) The Nazgûl don’t have any lines between their boxes. If I were Lord of the Nazgûl, I’d have designated two deputies, with three wraiths reporting to each one. Put one team at each Bree-gate, and LotR ends in Chapter 11. The green boxes are characters called “lieutenant”. Not very similar roles. The Ring can be thought of as a set of command and control channels among the characters that clear up that disorderly network.

organization chart of forces of Mordor

Organization of Morgul & Cirith Ungol Divisions

Looking for the Ring in our world

From the 1950s through the 1970s, people who were looking for real-world analogues of the Ring usually thought of the atomic bomb. I was never comfortable with that because of one salient feature of the Ring: when it’s destroyed, Sauron will fall. This was a very puzzling thing to a teen-aged first-time reader. What kind of tool or weapon reduces its user to nothingness when it’s taken away? Stephen Winter wrote an essay recently about Sauron’s project, which gets at that important point.

Stephen reminds us that Sauron can not create. He calls Sauron a “false maker” (which seems to be a deliberate counterpoint to the word “sub-creator”). Previously in the history of Middle-Earth, Sauron had worked via the psychology of individuals. He was effective at sowing discord, but it didn’t help him build an empire of evil because it generally led to the destruction of the place he was working in. He needed something more tangible for his attempt to conquer and hold the world. But what tangible thing is available to a false maker? Sauron found the loophole: organizing is not creating. Matter can be rearranged almost indefinitely without overstepping any bounds. In particular, people and things can be arranged into a hierarchy.

A hierarchy is a tremendous tool for multiplying the force of a leader. In the mid-twentieth century, it reached a peak of implementation in Nazi Germany, Stalin’s Soviet Union, and Mao’s China. In fact, the word “totalitarian” could be defined as a state that allows no interactions among its citizens outside the hierarchy. There’s a pattern here. I am reminded of the comment by a (probably apocryphal) German officer in World War 2: “The reason the Americans do so well in war is that war is chaos, and Americans practice chaos every day.” The forces of the West in Middle Earth are the same way. If there’s any clear path of authority across the races, it’s hard to see. This came up in the comments at Middle-Earth Reflections, a while back. Leadership among Elves and post-Numenoreans is a matter of personal relationships between a commander and the troops. (Éowyn: “They go only because they would not be parted from thee — because they love thee.” V, ii) It’s not a hierarchy, or any kind of structure you can draw on a PowerPoint chart.

To make a hierarchy work, though, the boss has to pay a price. He has to delegate both authority or responsibility. Or, in Ring-terms, “let a great part of his former power pass into it.” (I, ii). The hierarchy can be turned against the boss, or if it’s destroyed the boss finds himself sitting at a desk, with a dead telephone, commanding nothing. Powerless.

Conclusion

Organization was missing in the First Age, so it’s what Sauron added when he set up on his own. Hierarchy is the key, a thing Morgoth never had. Théoden King of Rohan had more levels of structure in a 6,000-man (ok, 5998-man) detachment than you can find mentioned in the entire Silmarillion. For us in the Information Age, the closest thing to the Ring is not a weapon, it’s a hierarchical org chart.

Won’t you be my neighbor?

I’m playing with graphs again. Here’s a picture of my net-neighborhood out to two steps, i.e., the sites on my blogroll and the sites on their blogrolls.

graph of blog links

Web Neighborhood

The funniest thing about this graph is that, despite the fact that it was designed to be my neighborhood, Idiosophy isn’t in the center.  Olga’s Middle Earth Reflections is. (Fair enough; her blog has more than a thousand followers.) Science teaches humility, along with everything else.

Nobody else is interested in economics, so Grasping Reality is ‘way over in the corner. The rest of the network is easier to read if I cut that one link.

Zooming in on the non-economic network

J.R.R. Tolkien brings together some diverse parts of the world. There are priests and theologians along the south, language-inventors up in the northwest corner, medievalists in the northeast, and a little knot of modernists on the east side.  Nobody who knows Tolkien’s curriculum vitae would be surprised to see that list (except perhaps for the economists and the physicist), but if there’s anything else in life that connects these communities, it doesn’t come immediately to mind.

Technical note

Drawing these graphs took ten minutes.  The tools you can download freely from the Web are amazing.  This was made by the “igraph” package in R.  To make these plots, I used an algorithm that simulates a simplified physical system to place the nodes. It puts an electric charge on the nodes, so they want to be separated and legible. Then it pretends the links are rubber bands, so inter-linked nodes are pulled tighter together.  I learned how to do this from an excellent tutorial by Katherine Ognyanova. (Who must be one of us; she posted the etymology of her name on her blog. I wonder if she’s related to the Vedic fire-god Agni.)

Signed Graphs and Interesting Stories

Of all the types of graphs, signed graphs are probably the most interesting for looking at interactions among groups of people. Definition: a signed graph is a collection of nodes and links between them, just like a regular graph, but each link is flagged with a positive or negative sign. When we’re using the graph to describe a social network, those might be “loves” and “hates”, “admires” and “sneers at”, or any other dichotomy that comes to mind.

Some graphs have closed paths in them. Mathematicians call a closed path a “cycle”. If you go around a cycle and encounter an even number of negative signs, it’s a balanced cycle. A graph that contains only balanced cycles is a balanced graph. These are the simplest cycles:
three-node signed graphsHere’s where things get interesting: signed graphs apparently figure into human sociology. If a network of relationships forms a balanced graph, it’s a stable structure. Relationships that fall into unbalanced graphs aren’t stable, and lead to drama. (That last word can be taken either literally, or in the euphemistic sense that people give it these days.) I don’t think anybody knows why such a simple mathematical condition seems to be true; that’s just the kind of thing mathematics does every now and then. Some brilliant psychologist will figure it out someday.

In the unbalanced graph “a”, I colored the vertices pink and blue because my wife watches soap operas, and nearly as I can tell, there’s a cycle like that at the heart of every one of them. It never ends well, because it’s unbalanced. The balanced graph “b”, by contrast, is “you and me against the world”, which is a stable configuration. The all-negative graph “c” can go one of two ways. If the vertices represent people, two of them just go away and the network disintegrates. If the vertices represent countries, or something else that’s forced into interaction because it can’t quit the game, the network changes when two of the nodes look for an advantage by conspiring against the third. The fourth possibility, “d”, is that all the links are positive. It is balanced and kind of boring in its dramatic implications.

There’s a perfect example of three-party instability in Book IV of LotR among Frodo, Sam, and Gollum.  From the time they meet up, it’s graph “a”: Sam loves Frodo, Sméagol loves Frodo, Sam doesn’t like Sméagol.  Type “a” is unstable, so something’s got to give. The crisis comes in  Chapter 8, when Sméagol (possibly) tries to turn the graph into a stable all-positive triangle, but Sam intrudes, the opportunity is lost, and Gollum plots with Shelob to turn the graph into type “b”.

Of course, three-person networks are easy to understand without graph theory. The real advantage of mathematics is that it becomes possible to handle any size network. There’s a theorem about graph balance that applies in general: Any balanced graph can be re-drawn in a simple form. (Can I say “isomorphic”? Sure I can. Y’all are tough enough.) All balanced graphs are isomorphic to a graph that’s split into two parts, where there are only positive links within each part, and all the links between the parts are negative.  That’s called the Cartwright-Harary Theorem. Prof. Harary says that the theorem is unexpected and counter-intuitive, which I am half in agreement. The positive interpretation is easy to accept:  if the world consists of two parties, and every member of a party agrees with each other, and every member of each party disagrees with all members of the other party, the situation is stable.  (Then Romeo meets Juliet, and the stability is history.)  The counter-intuitive part is that this is the only way for a graph to be stable.  That’s it – the one way you can build a stable social system is for everybody on your side to agree and to hate the other side, and contrariwise on the other side.  In practice, I suppose you could allow disagreement on issues that were irrelevant to the structure, and thereby outside the graph model. But on any important issue, perfect party unity and perfect hatred of the other side is your only chance.

Interesting stories, whether they’re fictional or meta-fictional, don’t have balanced graphs.  One of the most intriguing things I scribbled down during Sørina’s lecture was that we might be able to define a new subset of graphs under the rubric of “interestingly-unbalanced”.

Illiterate Coda

Maintaining stable structures without fomenting partisan warfare is critically important in a society as complex as ours.  But math is math. So how do we handle the dilemma of the Cartwright-Harary Theorem?  We go around the horns. Almost every organization chart you’ll ever see has the same basic structure:  No cycles, so the theorem doesn’t apply.  That type of graph is called a tree.  It’s useful in all sorts of contexts, but until now it had never occurred to me that it means that management never has to choose between polarization and instability.

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