It's not always the case, but more often than not people do better when they work together ([9, 23] and for empirical proof, see [5, 10, 17, 27, 28, 35]). A huge part of the reason for this is because when we come together in groups, we can actually converse with one another in order to offload some of the cognitive effort onto our conversation partners [3, 9, 11]. This perspective goes by a few names in cognitive science -- distributed cognition, embedded cognition, the extended mind, and is part of embodied cognition [9, 12, 23]. But what all of these perspectives share is the following: if you think of yourself a bit like a computer (somewhat of a dubious metaphor but bear with me here), when we're talking with people the parts of that computer aren't just the parts of that are inside your body. They also include the parts of the other computers you're connected to -- that is, your conversation partner(s) body(ies) and mind(s) too. You and them, you're working together and generating new information through that interaction (via the words and ideas -- the semantic content -- that you all contribute to the conversation). So, because you are working together and enmeshing your thoughts and language behaviors into one big hodge-podge of conversation, it makes sense to think of you and them as a single computer or information generating machine.
Now, there are a loads of studies showing that the kinds of things people accomplish when they're working together can only be explained when you lean into the distributed cognition hypothesis (again, the idea that cognition is something that you can extend to your conversation partners). But there's not as much out there demonstrating how this process works. Especially not in terms of conversation. And as a computational sociolinguist, this kinda made me sad. I want a description of how conversation supports distributed cognition (or "computation" -- i.e. the ability to "compute" solutions to various problems)! What's the mechanism here? How does it work??
I'm not going to say that complex systems science (CSS) holds the answer, but it's got a really good framework that we can use to look for a mechanism of distributed cognition. There are a few things from CSS that I want us to grab and hold onto here.
First, certain computer systems (think: ways of setting up a computer in terms of its hardware or the software being fed to it) might actually be capable of "universal computation", or solving literally any problem thrown at it if given enough time to do so. If a computer can run indefinitely without arbitrarily stopping, then it can solve any problem (if given enough time to crunch out an answer). This idea is typically referred to as the "halting problem". And it, and its implications, were first proposed by Allan Turing. Whenever you hear complex systems people talking about universal computation or universal computational principles, I'll bet you 20 bucks right now that the halting problem is lurking somewhere in what they're saying (if I'm wrong and you'd like to collect your $20 please reach out to Greg Bryant at UCLA and ask him for your payout. Tell him I sent you.).
Certain complex systems are organized in a way that naturally runs head first into the halting problem. One of those organizational schema is fractal scaling. Fractal scaling occurs when something about a system--like, say, semantic information generated by two people talking to eachother during conversation--scales according to a powerlaw distribution [4, 13, 22]. Because of the way you can infinitely zoom into a fractal shape (or on some part of a power-law distribution) and get back the exact same shape, it would take an infinite amount of time to algorithmically map every part of the shape. And because it takes infinite time to do something like that, if you create an algorithm that uses fractal scaling to do some kind of work you can't actually stop it from "running" forever. And so, fractals run into the halting problem. And by running into the halting problem, fractal scaling in a complex system means that, on a long enough timeline, a computer that is organized around fractal scaling to generate its outputs might be able to solve any problem.
A couple of researchers posited a while back that complex systems comprised of multiple interacting parts, organized around fractal scaling might be able to "spontaneously support" universal computation: vis. Wolfram [34] and Langton [20]. Both researchers were interested in how "self-organization" of complex systems might yield ones that are really good at solving problems, mathematically. If you're wondering why someone might be interested in this sort of thing consider your own brain. Your brain isn't one thing. It's millions of cells all working together. And the result of them working together is your brain. Here is a complex system, then, of multiple interacting parts, that yields something that acts a whole lot like a computer (again, bad metaphor, but useful here). Systems that can spontaneously support universal computation--ones that tend to show up in nature--tend to be organized in some way that exploits fractal scaling. Thank these two eggheads for our understanding of the math behind that statement later.
Langton in particular is of interest to me here. Langton pointed out that if a system of multiple interacting parts (1) generates outputs according to a power-law distribution and (2) shows clear influence of parts of the system on other parts of the system, then it should be capable of universal computation because such a system of parts could maintain "themselves on indefinitely extended transients," or indefinitely long strings of outputs.
Langton was pretty limited in what kinds of systems he thought could produce fractal outputs like this. For the longest time, when thinking about how multiple interacting different parts/pieces/agents could come together to form a coherent, singular, information generating machine, most complex systems scientists focused a particular class of complex systems that are formed via self-organized criticality (SOC). SOC occurs when a system is perched close to a phase transition. Think for a moment about something like water boiling into steam. There isn't a characteristic size to how much water converts to steam at the boiling point. It's random. It generates "indefinitely extended transients" of converted molecules of water switching from wet to luxurious spa facial. Technically, someone who is much smarter than me (and probably actually has grant funding) could create a computer using boiling water based on that. But I digress.
The thing of it is, phase transitions are rare. Or at least rarer than you might think. And it's not clear that they even apply to all systems composed of interacting parts. What's the phase transition in two people talking with eachother? Is it the boundary between turns? Is every word a possible phase transition that could be flipped and switched for a new idea in a moment? Looking for a phase transition in conversation is daunting, and probably requires a universal computation machine because it's going to take a while to figure out what it is.
But SOC is not the only process that generates systems that produce fractal outputs. Highly Optimized Tolerance (HOT -- gotta love the acronyms here) refers to processes where the goal is to create an information generating system while managing sources of possible havoc in the system [8, 22]. Those sources of havoc could be from the external environment (maybe something is enacting some chaos inducing force on the system), or even from random noise in the interacting pieces themselves (as in making a system using mismatched parts). Whatever the source(s) of noise, systems generated via HOT processes (again, love the acronyms) tend to generate power-law (and thus fractal) outputs, the same as processes generated via SOCs. There are some differences--HOT outputs tend to be a little less dramatic in terms of the curl of the fractal than SOC outputs--but both generate fractal outputs and thus both facilitate the formation of systems out of many interacting parts that can (possibly) support universal computation.
This was, I know, a lot. So how does it relate to conversation? Remember earlier what I said that people work together to generate new information, and this makes them better than the sum of their parts? That's how. In conversation you have a system composed of two or more, very noisy, very different (think: differences in wants, needs, beliefs, personal history and even sometimes languages!) "agents" who are working together to have a conversation. That conversation consists of the ideas that each person contributes. And those ideas tend to be related to one another. People naturally align [13, 23], converge [14, 30], or otherwise come up with conceptual pacts [5, 6] about what they're talking about in the moment, and how they're thinking about the things they verbalize. But that simply serves as a bouncing off point for the spectacular way that conversations naturally end up generating a ton of new information, all because of people naturally working together, building on each others' ideas, and so on.
Conversation should act, thus, like a complex system. And one where people take the semantic content of what was said before and then builds on that content to come to new ideas. Langton posited that systems with fractal outputs are good at maintaining memory of prior states in the system (think: prior ideas contributed to a conversation), and rapidly generating new information on the basis of that history (think: conversation spiraling into new topics fluidly with people you love and like talking to). These systems, conversation potentially included, are a bit like building a new lego model out of old lego sets you built in the past. You can take pieces of the old sets to build the new one. Maybe it's only a brick here or there from the old ones, typically. But sometimes you might grab an entire structure like a whole wall or window. Pieces of the old build still exist. But they've been used to generate something new.
This project analyzed 5448 conversations. So many conversations... This meant looking at how 15133 different speakers worked together to build new semantic information together in a variety of settings, ranging from Zoom calls [26] to med school study groups [19]. And a big chunk of these conversations weren't in English, either. 1070 of those conversations were in languages other than English, and from a typologically diverse range of language families. It was, and is, a lot of data. Between you and me, the hardest part of this whole project was the data management and data engineering work. But it was and is worth it. Because what we find is that across all languages, people work together as an imperfect (HOT) complex system to rapidly generate new information relative to what was said in old turns, influencing each other and synchronizing how fast that system builds new conversational lego sets of ideas.
And, spoiler: That's what we found.
There are two tasks we needed to accomplish for this project. (1) measuring how much information is generated when you compare two utterances to one another (like comparing an older utterance to a later utterance), and (2) measuring how much the amount of information changes when comparing two utterances to eachother depending on how far apart the two are in time (i.e. how much new information gets added from one utterance to the next, when we keep comparing new utterances to the semantic content/ideas in an older utterance that other speakers are building on top of).
A good method for answering question one is Convergence-Entropy as described in [29]. I'm about to hit you with some math, but here's how it works at a high level to start. First, you take an earlier utterance ($x$) and an utterance spoken after it ($y$). You convert all the tokens/words (we label each one using $i$ for the $i^{th}$ token) in the utterance $x$ into what are called token embeddings ($E_{xi}$) using a language model. You do the same think for all the tokens ($j$ -- same logic) in the utterance $y$. In this study, we used a large language model (LLM) to generate embeddings, and research shows that these embeddings are actually really good representations of the semantic meaning of words (even when compared to actual human semantic processing, [16, 18, 24, 33]). This entire process is captured in equation 1.
(1)$$E_{xi} = LLM(i \in x)$$
Next, you calculate how close the closest token $j$ in the utterance $y$ is to each token $i$ in the utterance $x$. You can then convert that distance to a probability using some fancy statistics (a Half-Gaussian, with mean 0, meaning that if the distance between the closes token $j$ in $y$ to a token $i$ in $x$ is 0, then the two tokens mean the same thing). See equation
(2)$$P_{xi}(E_y) = P_{\mathcal{N}_{[0,\infty]}} \left( \min_j \bigg(CoE(E_{xi}, E_{yj}) \bigg) \bigg| \mu=0, \sigma \right)$$
If you have a probability, you can calculate information content (thank you Claude Shannon [31]!). Information content is how much extra information you'd need in order to get the meaning of the token $i$ in $x$ in this case. See equation 3.
(3) $$I(x;y) = -\sum_{i \in x} \log P_{xi}(E_y)$$
What you're left with is how much semantic information is generated (how much extra statistical work you'd have to do) to get from one set of embeddings to another!
Quick note to the wise here: the equation above is for information, not the normalized entropy. And in other places this is reffered to as convergence-information (CI) not convergence-entropy (CE).
It's important to note that the scaling factor $\sigma$ in equation 2 is arbitrary. Especially if you're calculating information content like in equation 3. How big or small $\sigma$ is will quantitatively change how big or small the difference in probability is for different distance measurements, but it won't change that there is a difference. It monotonically scales based on $\sigma$. So $\sigma$ changes how sensitive your measurement is, but it won't change the existence of differences, and so long as you're using the same $\sigma$ for all measurement, it means that all those differences will at least be in the same scale, which is important for comparing across different examples and time.
Technically, equation 3 is not entropy. It's what's called the Shannon information. But it has a really important relationship with entropy proper.
Shannon defined entropy as the average, expected information for any measurement taken in a system. He also used the letter $H$ to symbolize entropy. Go figure. Even so, that means entropy is by definition this:
(4) $$H_x \doteq \lim_{n_{i \in x} \rightarrow \infty} \frac{I(x;y)}{n_{i \in x}}$$
But that also means that any measurement of information can be expressed as expected information/entropy plus some error ($\epsilon$)
(5) $$I(x;y) = \mathbb{E}[I(x;y) ] + \epsilon$$
or, more precisely as
(6) $$I(x;y) = (H_x * n_{i \in x}) + \epsilon$$
Hold onto that last point for a minute. Trust me.
CE (or CI, really...) gives us a solid mechanism for measuring how much information is required to recover the ideas in some utterance $x$ from the ideas expressed in a later utterance $y$. But speakers don't just dump information in a single utterance. They keep generating information. Over and over again. And over again. And over again and over... you get the picture.
Speakers and their partners, in subsequent utterances $y$ will keep generating new information. And it's interesting to ask what is the dynamical nature ("dynamical" meaning how something changes from one point to the next in time) of the way they do this? Do people slowly generate information after some prior turn $x$ and then leap off into the sunset in later utterances adding tons of new information rapidly after a period of relative little information generation? Do they linearly add new information, like gentlefolks, building up new ideas in a straight line over time? Or do people's information generation behaviors exhibit self-similarity or fractal patterns, where, because there is a kind of attractor to prior ideas, information is generated rapidly, starting from the lowest possible amount of information early on and hitting a ceiling later?
Two things are needd to answer these questions. First, we need a way to measure how much information gets added earlier versus later in subsequent uttterances. Second, we need a way of calculating how the rate of that information changes.
On its own, CI gives us a raw information measurement. But we still need a way to measure how that information compounds turn over turn. Enter integral calculus. Yes. I'm going to retraumatize you with integrals. Deal with it. If we take the integral (which is really just a fancy sum) for every utterance (up to 200 utterances into the future in this study) that a speaker make after the utterance $x$, we can ask what percentage of that integral has been "filled" after every utterance a speaker contributes. Cool. That gives us how much information has acrued, and the math looks like:
$$CDF(\Delta_t) = \frac{\int_{i>1}^{\Delta_t} I(x;y_{\Delta_t} \partial (\Delta_t))}{\int_{i>1}^{\Delta_t \in T} I(x;y_{\Delta_t} \partial (\Delta_t))}$$
Where CDF is the cumulative density function -- that's just what this kind of integral divided by the total integral is called in fancy statistics speak.
Now, we can tell which of the three behaviors I rattled off before by calculating a power-law exponent for how much the CDF changes as a function of changes in time $\Delta_t$. The power-law exponent for a whole dataset of CDFs can tell us how some value behaves over time. More on that in a second after I show y'all the equation. What's nice, too, is that we can calculate a power-law exponent via the following regression equation (which we can apply to all of our data)
$$\log CDF(\Delta_t) \sim \beta_0 + \alpha \log \Delta_t$$
Which can be "exponentiated" to look like
$$CDF(\Delta_t) \sim e^{\beta_0}\Delta_t^\alpha$$
and the value $\alpha$ is our power-law exponent describing the scaling/dynamical behavior of the CDF depending on temporal distance $\Delta_t$. Nifty!
Here's where this gets really useful. When $\alpha \approx 0$, this means that information accumulates chaotically. What does that mean? It means some of the jumps in the density of information, early on, are much higher than you'd expect even compared to later additions logarithmically, and the opposite is true too. Picture a completely random squiggly line on a piece of paper. That's what $\alpha \approx 0$ indicates in terms of behavior. Conversely, $|\!|\alpha|\!| \approx 1$ is a different kind of randomness -- Gaussian randomness. In this case, each turn adds about the same amount of information, and thus we can say when this is true that information adds up linearly and (seemingly) orderly over time. Neat! When $|\!|\alpha|\!| > 1$, that means information starts of by accruing slowly and then shoots to the moon as time goes on. And when $|\!|\alpha|\!| < 1$, that means we have something that at least has self-similar tendencies if not outright fractal scaling. Information starts low, jumps up quick, and hits a ceiling/constraint.
One measurement, but excellent resolution into different kinds of possible dynamical behaviors!
An important part of Langton's definition of what makes a complex system capable of distributed computation across its parts is demonstrating that the pieces of that system are influencing eachother's behavior. Otherwise, there's no proof that they're ultimately working together.
It's hard to show that power-law scaling (and thus fractal information generation) are related between the parts of a system. Langton is a bit of a cheater, here, because all the systems he studied were simulated -- he could literally write into the program itself "this part talks to this part, make it so!" But in the real world, we need actual proof that the pieces are talking. And even more specifically, we need to show in our case that power-law scaling in one part of the system affects the power-law scaling in another part of that system.
Enter the idea of complexity matching [21]. Complexity matching is the idea that entire distributions of behavior influence in part of a system influence the distributions on behavior in the other part. It's a comparison thus between two distributions. When complexity matching is 0, it means the two systems are synchronized in their behavior. Because it means there is no difference in how the two distributions themselves behave. But systematic differences are still useful and important here. Because when two systems have systematic differences in behaviors (like the systems always being some value like 2 apart), then they're actually synchronized, and one demonstrably influences the other.
This definition was important to Mahmoodi because Mahmoodi was interested in coordinated behavior in very different systems or between radically different kinds of things. Think: measuring how the behavior of waggling your fingers influences the behavior of tapping your toes. That's why he focused on differences in whole distributions (because fingers and toes don't move the same, but how they move relative to what affordances in movement they have can become influenced by one another). Other folks though have used complexity matching in ways like tracking how people manage rate of speaking and onset times [1], though, amongst a whole fleet of other things.
For us, we test whether the difference between speakers' power-law distributed rates of information generation relative to their own prior utterances and their conversation partners' rates of information generation systematically differ or are the same. This actually is easy to do in the current study, given that it's a comparison between two clearly different, but potentially related behaviors, and we measure the difference in distributions using a paired t-test for this reason.
If you want to see what our entire process looks like, using all of these steps from start to finish, check out this visualization, as well!
The data for the english, monolingual study we did concsisted of 4,378 conversations between 12,578 speakers, and comparing utterances $x$ and $y$ a maximum of 200 turns away from eachother, making a grand total of 261,254,181 datapoints we had to analyze.
If you're looking for descriptive stats, check out...
The actual numbers, including the power-law scaling exponents, statistical significance, and other descriptive statistics are all in this table:
Across all speakers and conversations, power-law scaling for how speakers generate new information relative to their own old turns exhibits power-law scaling. The same is true for their partners, though it's a little less extreme.
In terms of complexity matching, we don't find that people's power-law exponents are equivalent to their partners when dealing with the ideas they (as in the speaker of the utterance $x$) contributed to the conversation themselves. But it does look like there is some synchrony there that is significant!
All of our results are visualized below. The top part of each plot shows where the mean for power-law exponent $\alpha$ sits for all the data, while the plots below it show the actual distribution of power-law exponents.
The data for our crosslinguistic study consisted of 1,286 (with some repeated conversations in English) conversations between 3,020 speakers, and comparing utterances $x$ and $y$ a maximum of 200 turns away from eachother, making a grand total of 100,930,208 datapoints we had to analyze.
If you're looking for descriptive stats, check out...
The actual numbers, including the power-law scaling exponents, statistical significance, and other descriptive statistics are all in this table:
Across all speakers and conversations, power-law scaling for how speakers generate new information relative to their own old turns exhibits power-law scaling. The same is true for their partners, though it's a little less extreme.
In terms of complexity matching, We find much weaker evidence for synchrony. And the Wilcoxan rank sum test (which is a non-parametric test, and so a little more prone to Type II errors) shows that there is no difference, i.e. complexity matching.
All of our results are visualized below. The top part of each plot shows where the mean for power-law exponent $\alpha$ sits for all the data, while the plots below it show the actual distribution of power-law exponents.
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