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A defense of Tom Wolfe based on the impossibility of the law of small numbers in network structure

A tall thin young man came to my office today to talk about one of my current pet topics: stories and social science. I brought up Tom Wolfe and his goal of compressing an entire city into a single novel, and how this reminded me of the psychologists Kahneman and Tversky’s concept of “the law of small numbers,” the idea that we expect any small sample to replicate all the properties of the larger population that it represents. Strictly speaking, the law of small numbers is impossible—any small sample necessarily has its own unique features—but this is even more true if we consider network properties.

The average American knows about 700 people (depending on how you define “know”) and this defines a social network over the population. Now suppose you look at a few hundred people and all their connections. This mini-network will almost necessarily look much much sparser than the national network, as we’re removing the connections to the people not in the sample.

Now consider how this applies to Tom Wolfe. For novelistic reasons he can only have a handful of major characters and a few dozen or so minor characters. If he gives them a realistic level of interconnections, the resulting network will not be a reasonable small-scale replica of society at large—it will be too sparsely connected. To make his story-network realistic in a larger sense, he has to overload his characters with connections (“coincidences”) beyond what would actually arise in a group this small. Thus, it’s not fair to slam Wolfe for having too many connections or coincidences in his books—these are a necessary artifice that allows him to achieve a realistic density of connections in a small group of characters.

That was fun: statistics and literature!

14 Comments

  1. Anonymous says:

    But presumably much if the interest lies in the characters populating the network rather than the network itself. A network of 700 clones is not that interesting.

    So tradeoffs btw representative networks and interesting characters

  2. David Meyer says:

    Are you talking to your colleagues Elson, Dames and McKeown? In their ACL 2010 paper they observe “as the number of characters in a novel grows, so too do the cohesion, interconnectedness and balance of their social network”.

    Similarly, despite being mathematicians (how retro!), Mac Karron and Kenna analyze literary and mythological networks in their 2012 EPL paper and conclude, also more quantitatively, “fictional networks are, however, discernable from real social networks in that they mostly have exponential degree distributions, have relatively larger giant components, are robust under targeted attack and are dissassortative”, while “the network of characters in the Iliad has properties most similar to those of real social networks. It has a power-law degree distribution (with an exponential cut-off), is small world, assortative, vulnerable to targeted attack and is structurally balanced”.

    • David Meyer says:

      Sorry, that’s “Mac Carron”.

    • David Meyer,
      I was puzzled after reading your comment. Why were the characteristics of social networks in literature versus mythology, as determined by Mac Karron? McKeown? Mac Carron? and Kenna so different? I confess, I haven’t read the 2012 EPL paper. I probably should have scanned the abstract before commenting. Where angels fear to tread… well, I followed the invocation of Jimbo et. al., and decided to BE BOLD (I just read Professor Gelman’s 2006 era post about Wikipedia and Aaron Swarz).

      The mathematicians’ findings are intuitively plausible. Despite the intercession of assorted deities, centaurs and such, The Iliad’s network of characters, and the associated plot, is oddly contemporary. Since Professor Gelman mentioned Tom Wolfe, I’ll guess that the EPL paper analyzed networks in modern novels. A possible explanation for the Iliad’s network being small world, vulnerable to targeted attack yet structurally balanced (and true-to-life, Homeric and now): There were many familial ties in The Iliad. This was true, even after excluding the most obvious linkage of Paris, Helen and Menelaeus. In contrast, most 20th century fiction doesn’t emphasize familial ties; Bonfire of the Vanities did not. Maybe that accounts for the difference between real and fictional networks?

      * The Iliad was grim, complicated, tragic. The Odyssey was much better, an exciting adventure! Does *anyone* truly enjoy The Iliad, I wonder? I liked The Odyssey more than Bonfire of the Vanities too.

  3. Steve Sailer says:

    Judge Richard Posner admitted in the mid-1990s that he hadn’t thought much of “Bonfire of the Vanities” at first, but then all sorts of wacky events foreshadowed in “Bonfire” kept happening in the real world, such as Rev. Bacon’s/Sharpton’s Tawana Brawley hoax of a month after “Bonfire’s” publication date.

    Today, many assume that Tom Wolfe’s “Bonfire” was “ripped from the headlines” the way Dick Wolf’s subsequent Law & Order series is. But, Wolfe was actually out ahead of the headlines.

  4. Steve Sailer says:

    Much of art consists of the creation of a “lattice of coincidence” (to quote “Repo Man”) to make art more compressed and connected than reality.

    • Andrew says:

      Yah, but my point above is that, due to the statistical nature of samples of networks, the only way to make a story be representative of the larger world is to add these compressions and connections.

  5. Steve Sailer says:

    I’d argue that Wolfe’s novels could benefit from more coincidence. Consider “Bonfire,” in which the characters are brought into contact with each other in almost exactly the same manner as participants in the Trayvon Martin – George Zimmerman case of 25 years later came into contact with each other. (Indeed, Al Sharpton played almost exactly the same role in the Trayvon story as he did in “Bonfire” under the name Rev. Bacon.)

    Offhand, I can’t think of many coincidences in “Bonfire” at all, once you assume the random encounter of Sherman McCoy and Henry Lamb. Maybe … some of the cops remember that Lamb’s father had died bravely years before so they push the case a little harder than they normally would. That’s about it. Most of the revelations in the epilogue — e.g., defense attorney Killian is now a rich man because he has all of McCoy’s money — aren’t coincidences, they are the working out of the logic of the situation. The book would be more conventional if, say, Herbert 92X was reintroduced at the end to cause some satisfying development, but he’s gone out of the story for good.

    In comparison, Dave Barry’s “Big Trouble” is an attempt to write a Bonfire style novel about Miami. The plot is more carefully constructed than Bonfire, and thus has all sorts of coincidences in it that make it artistically satisfying. (The movie of Big Trouble was pretty good, much better cast than the movie version of Bonfire, but this late September 2001 movie that climaxes with a terrorist attack on an airliner was wiped out at the box office by, coincidentally enough, 9/11.)

    • Steve,
      You are ubiquitous. You dominated the comments on Tyler Cowen’s Marginal Revolution post, The Myth of American Meritocracy: Ivy League Admissions Corruption, with your particular contribution pertaining to the supposed mediocrity of Jewish men in mathematics and “visual-spatial analysis” versus Asian and non-Jewish men. You should have asked Professor Gelman to have a glance at the results of your statistical analyses, prior to publication in the article based on those demographic findings in The American Conservative. Next, I find your website featured in the persona non grata blogroll of Lady Liberty Anon-something-or-other. And you’re here as well, times three!

      In response to your most recent comment, I wouldn’t attach any special significance to (Reverend?) Al Sharpton’s role, 25 years ago versus the present. My impression is that he has taken up numerous similar advocacy roles on a regular basis, and will continue to do so. He is resilient, energetic and enduring.

      Regarding Tom Wolfe and “Bonfire of the Vanities”, there were many implausible coincidences. Recall the assistant district attorney and his preening displays for the Girl with the Brown Lipstick, the “rent-controlled love nest”, and how said assistant district attorney was brought down just as Sherman McCoy was? There were other similar examples, which is why I think the title was so appropriate: It was a bonfire, of vanities small and large.

      * I like FuturePaul’s comment the most. And I say “Thank you!” to Bob Carpenter, for kindly including the URL to the Elson, Dames and McKeown paper, as PDF sans Shibboleth, no less.

  6. Mayod says:

    I think the way to regard a story with implausibly many coincidences as “realistic” is not to consider it should be at all like a typical set of events, but rather a story sufficiently unusual to have merited telling (retrospectively so to speak). That is to say, in rare cases in real life there are these incredible coincidences and interconnections, and it is to such cases one would look to tell a story.

  7. K? O'Rourke says:

    Might be fun to work this through ideas of Gaussian quadrature, smart Monte-Carlo and what I believe Ming once wrote of as the (historical?) tension of systematic and random approximation.

    For instance, if you “sample” from Gauss rule, you match the 2n – 1 moments directly but it is rather discrete rather than (approx) continuous.

    As for re-representing literature, as the there is an infinite number of ways to do that, some multiplicity correction might be suitable. Or at least my take on this approach http://en.wikipedia.org/wiki/Algirdas_Julien_Greimas was not stories work on that basis but rather could be recast in those terms (confusing a representation with what it is representing).

  8. Paul says:

    “This mini-network will almost necessarily look much much sparser than the national network”

    That ‘almost necessarily’ bit is key. While a small random selection from a large network won’t capture much of the larger structure, it’s not necessarily true that particular subgraphs don’t share interesting properties with the larger network. The author isn’t performing a random selection, but a targetted one. If 10% of my connections are likely to form a single clique, so too can the protaganist’s. If 15% of connections are familial, that could also be true for one or more characters. There are certainly properties that can’t be captured (a 700 member social network in a 50 character novel) but for any network dynamic you’d wish to analyze, you should be able to find compact subgraphs sharing that dynamic somewhere in the national network.

    And granted, the characters usually are created rather than selected. But I think the capturing network dynamics can be seen as smart selection rather than artificial insertion of edges.

  9. You should talk to Kathy McKeown in CS. She and a group of students have applied natural language processing to the problem of social network inference in fiction. They were debunking the commonly held notion that “country” novels had sparser social networks than “city” novels. Here’s a link to the paper:

    Elson, Dames and McKeown. 2010. Extracting Social Networks from Literary Fiction. ACL.

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