Math Formula for Optimal AI Communication Bandwidth?

Key Takeaways

Hello communities, so great that you are back. We have a beautiful new result in artificial intelligence. So, let's talk about simplex multi-agent communication channels and quantization. Now, of course, simplex is here the space of all possible AI beliefs. The quantization will characterize here the memetic drift. And of course, you know, a simplex is the simplest possible n-dimensional convex polytope formed by the convex hull of an n plus one vertices in n-dimensional space. And we will use this. Now, let's start easy. You remember in my last video on S-PoV Rec system, I asked you, "Hey, if we have here our LLM and we outsource everything into the markdown file into human language compatible format, this is Rec and S-PoV Rec was kind of your answer. But what about the communication channel between not the LLM and the tools, but what about agent-to-agent? What if this discretization to the human text has negative effects on the agent-to-agent communication protocols itself?" And you might say, "Hey, come on, this is nonsense. How should this happen?" Well, it happened to me. If you connect multiple open claw systems or go with Nemo claw system, you will see that agent-to-agent communication can fail and can fail massively. So, the question is why? Why does it happen? Now, today we have a brand new mathematical theory to isolate here one specific universal physics in the AI and this is the mathematical consequence of forcing exactly what we are interested in, of forcing a continuous internal probability distribution that is now reduced to a one-dimensional probability distribution representation through our discrete communication channel. This means the human language, our words. I will show you that we have now a new mathematical theory that tells us there's something strange happening that you are not expected. What am I talking about? We have a probability distribution in our little LLM or to regress and predict the next token goes here with the highest probability value and this is cat. But you know that real close to cat, now we have a lot of other words of other tokens that are real close and maybe it's a dog, maybe it's grandmother, maybe it's sunshine, whatever. So, whenever the system is here forced to select only one word, one, if you want tokenization, we lose a lot of information because this is just a probability distribution. This is not the truth. So, let's have some fun. They developed a quantized simplex topological gossip model that should be understood as we're going into a little bit of theoretical physics. An ideal gas law for multi-agent coordination. So, we go to thermodynamics theoretical physics, you know, 19th century and we have some beautiful formula over there. It strips away all of the complexity of the real world dialogue, the grammar, the logic, the chain of thought or complete network topology. We are just looking, hey, what happens if I have from continuous to discretization, what is happening to my communication protocols? So, we must first understand the baseline thermodynamic properties of interacting neural networks and the realization that the language generation itself, the quantization into words, acts for any AI system as a destabilizing force that will drive our artificial society or AI agents towards some spontaneous agreement and this is beautiful, but also cautionary because this agreement is incorrect. So, let's have a look. Yeah, you know a topological message passes on graph neural networks 3 years ago, the new geometry of intelligence a month ago or here we went here with UMAP a code for the simplicial complex topology in AI 3 years ago. We know about topology, we know about simplex, we know about this. This theory is complete across the entire spectrum of the bandwidth. So, we can apply this new mathematical theory generally to any multi-agent system where the agent must reach an agreement under uncertainty, ambiguity or some weak biases. So, remember this is my little visualization of the complete complexity of AI. We are here multi-agent communication where they're looking here at the communication protocols, but we are also here at tool calling tool user MCP agent-to-agent here. And of course, we are here when we have open-ended problem. We do not have this problem if we have a pure mathematical result or a code that we can run and verify or debug. But whenever we don't have this, so you see this has an impact almost everywhere. Of course, it goes for skill library and so on. So, I have now a simple question because I use LLMs here for my work and I say, "Hey, if an LLM has a high complexity capability like a Gemini 3.1 Pro and I limit now the bandwidth of the communication between multiple agents, will the communication kind of filter to simple complexities?" This means we less lose here intelligence in the communication of the agents if I just reduce the bandwidth between the agents? What about it just takes more time instead of 100 ms, 1,000 ms? It turns out there's an effect I was not prepared for. So, let's have here a simplex, now. So, we have here ABC, three options, now. LLM capacity to hold nuance complex multi-layered hypothesis as I showed you here in the latest S-PoV Rec is represented by highly distributed state near the center of our simplex, now. Let's say this LLM says, "Hey, I'm 40% sure it's A, now." Let's say this is A. But I'm 35% sure it's also B and I'm 25% sure that it is C because of some complex reasoning. So, somewhere in the middle will now be here exactly the perfect position or if you go 1/3, 1/3, 1/3, it is exactly here as indicated here in this graph by nanobanana. Great. Now, interesting, when you limit now the bandwidth between the agents, forcing them to just output a final answer as instead of their nuanced reasoning, you create now between the agents a quantization bottleneck. Now, I want to show you the first result of today's new AI preprints and this is about swarm intelligence. Maybe you have a particular communication protocol between your swarm AI or you have your autonomous elements, whatever. So, instead of holding here multiple complex hypotheses in a state of productive tension, a limited bandwidth between the swarm's elements forces here the swarm to aggressively snap to a corner of the simplex, of the triangle I just showed you, now. Because the rich complex capability of the individual LLMs is now completely lost at the macro level. The group in total acts now dumber and more dogmatic than the individuals composing it. So, suddenly we have a negative effect on the development of further intelligence. And this is just by reducing the bandwidth. And you would say, "How is this possible?" Well, let us formalize this and there let's follow a little bit the authors here when they develop the mathematical model they called here the quantized simplex gossip. Gossip is not really a mathematical term, but you understand what it's all about. So, we have the model state space. Each agent I has a continuous internal belief state xi represented here as a point on the probability simplex k minus one where k is the number of possible conventions or labels. And then we have three communication channels here in the simplest case, now. We have a soft communication where m equals infinity and agent can transmit its exact probability distribution xi. This is the analytical baseline under soft communication. The symmetric neutral states remain perfectly symmetric. No drift occurs. Then we have the opposite, the hard communication. An agent samples now a single discrete token from its distribution and transmits only here a one-hot vector. We lose here the complete the complete context information. We just transmit a token. Or we have a top m communication here. This means an agent transmits here the empirical distribution of m discrete samples. Maybe we have four or five hypotheses or four or five possible solutions. Then we have an update rule. This is here the mutual in-context learning, now. Because you have a random speaker of your agent and a listener of another agent and the listener updates here its belief towards the received message. Why? Using here a first-order adaptation rate. Beautiful. We have a coefficient alpha within the interval of zero to one. Now, there's a beautiful variance injection theory by the authors. I will show it to you in a minute. So, the preprint is analyzing here the polarization potential towards a particular solution, let's say dog or cat, now, which measures here how close the population altogether is to a consensus. And for a perfect symmetric consensus, it is one over k and for a total consensus, it is one. So, the author proved now in a mathematical way that for the hard and the top m sampling, this injects a strictly positive variance term into the expected change in the polarization and here you have the formula. Now, the beauty is look at this term because the continuous probabilities, remember, our next token generation are being forced through a quantized discrete channel. The human language communication between the agent. This is the term that actively pushes now the system away from the symmetry and towards the polarization. And you would ask, "Hey, is this good or bad?" Well, let's have a look. This is the paper today. This is from a single person, but I love the idea, therefore I show it to you. This is from the center of brain science and Harvard University and physics and artificial intelligence group entity research incorporated here. And the question is simple, when is the collective intelligence of multiple agents simply a lottery? When is there's no reasoning at all? When the multi-agent system is just pretending to be reasoning? It's just pretending to be thinking about something. But in reality, we have a completely different effect happening deep inside the communication protocols. So, they were looking at this person here was looking for the multi-agent scaling laws for the mimetic drift in alignment. We'll explain this in a minute. March 25, 2026. You might say, "What? 4 days ago?" Yes, I know. Now, to the paper. The paper itself is a little bit heavy on the mathematical side. Therefore, I give you here this introduction. And I will give you here a geometric interpretation of the results and the way they were thinking about this. Now, theorem one is important, proof of theorem one, theorem two is important, proof of theorem two. And then you have really some beautiful explanation here, what is happening here in this communication protocol. Given here a specific entropy, given here a specific drift, given here some Yeah, I will show you the effect in a moment. Coming back here to the paper, we have here interesting elements about the consensus time and the scaling laws itself. And they have here some approximation, of course, but the authors derived here the author, singular, derives here the collective time to a consensus. And this is highly interesting because this means that the time to reach a consensus in a multi-agent system grows quadratically with the population size and linearly with the message bandwidth itself, or M. But it shrinks quadratically with the adaptation rate alpha. This is something that is absolutely fascinating. Now, before we enter now the geometric, my geometric interpretation of this mathematical methodology for you, just from statistics, you are familiar with variance, a measure of dispersion, meaning that it's a measure of how far a set of numbers are spread out from the average value. This is it. So, why do this? I think if I see the simplex now in a different way, a little bit more from the theoretical physics side, I don't know, purely mathematical, now. This is the geometric space where all the probabilities live. If there are K possible answers, the simplex is a K minus one dimensional shape where all the probabilities will sum up to one. The interaction protocol, at each step two random agents are paired up. One speaks, one listens. This is the simplest case. And the listener then updates the belief. In the communication channel itself, the speaker cannot send the exact location on the simplex, where in this triangle am I here? Exactly at 1/3, 1/3, 1/3, or already drifting towards here a shape. So, therefore, they must quantize their location into discrete words. And we will see that this has a massive effect. So, here we have it, now. This is our simplex, and now let's say we have three options. These are just some artificial words, no meaning. Wock, zuck, zap, dax, whatever you have. Let's say we have agent Alice, and Alice is completely neutral, now. Given a task, she thinks that there's a 33% chance to word this wock, 33% chance it is dax, and 33% it is zap, or whatever. So, she sits exactly in the center of this simplex representation. Now, she needs to communicate with her neighbor, the agent Bob. So, how does she do this? As we just discovered, we have here three simple communication modes, now. The soft gossip here, theoretical baseline, infinity. You send the exact coordinate. Bob receives here the perfect information, no noise is injected, the system stays balanced, all the available information is transferred. Complete data transfer from everything, all hypotheses, all initial conditions, everything. The exact opposite would be a hard gossip. This is more or less the reality we face with LLMs, now. Alice is forced to output a single token, a single answer. She's not allowed to say, "Well, I think about this and I have some conditions and then I tested this and I found out this." Just give me answer. Maybe even a boolean, yes or no. So, if she's positioned here neutral, 1/3, 1/3, 1/3, so, she rolls now a three-sided uh die, and then she says, "Okay, it landed on dax, so okay, I transmit now the dax vector." This is here a one-hot vector and this is the answer. And now it's interesting the fact, now. Bob hears now the receiving agent, hears now the single token dax. And even though Alice was completely uncertain, and you know it was 1/3, 1/3, 1/3, her output now looks to the second agent 100% confident. So, therefore, Bob moves his dot in his belief uh manifold now slightly towards the dax corner because he just learned that Alice thinks that dax is the correct answer. Of course, it would be nice if you have some top end gossip, now. Alice outputs, let's say, five tokens or 10 tokens based on her distribution. And simplest case, she says, "Okay, wock, dax, zap, dax, wock." So, you see we have two times wock and we have two times dax and we have only one time zap. So, this gives us here a better indication about Alice's belief. So, the result Bob gets here more or less a blur of information, but he also gets now a little bit more concrete the feeling, now. A more accurate picture of Alice's true uncertainty after agent. And this is now, if you want to mathematical break through of the paper, now. That the hard communication, the more we limit the communication channel between agents, forces a continuous state, a probability distribution, into a discrete channel, our discrete communication protocol channel when we use words. And this quantization is basically, now comes the surprise, a random number generator, now. It injects statistical variance into the system, and this is, by the way, the topic of theorem one. Now, because Bob is now updating his own belief state based on Alice's random discrete sample, the population will literally stumble now into a consensus through a random walk. And this is what the authors call a mimetic drift that is happening. So, let me give you now a visual interpretation of this. So, the closer the agents are to the center of the simplex, here this is the highest uncertainty, now. The more violent the quantization noise is. But as the group randomly drifts toward a corner A, B, or C, their entropy will drop. So, when you get closer to a corner, they almost all three agents here output here that simple specific token. The noise disappears and the system locks in to a consensus, it is B. Now, an observer thinks that these three agents reasoned their way to an answer, A, B, or C, but we know they were just pushed into a corner by the pure thermodynamic physics of this QSG model. Okay, I want to give you here the four phases here that is happening here in the workflow of this new mathematical theorem. Phase one is the symmetry initialization. So, we work now with We're looking for Where's the maximum entropy in our system? Sequence begins with a state of perfect statistical neutrality. Population is balanced, assigning equal probability 1/3 to each of the three potential labels, A, B, or C in our triangle. Central area is a heat map indicating that the quantization variance, and therefore the statistical noise, it is absolute maximum. Any token forced to be generated here is purely random. The corners of the simplex are dark and cold. Those are the A, B, or C, and is no specific belief is currently favored. We have the condition maximum variance. Now, if you prefer a little bit more mathematically precise formulation, this is it for you. Phase two, the symmetry breaking is happening. We have the first quantization that we notice. Agents now communicate under, let's say, hard constraints, meaning they must collapse their continuous internal belief point into a single discrete token, a one-hot vector, before the transmission to the next agent. So, we have a broken symmetry suddenly. Here again is a little bit more mathematical formulation of this. Phase three, now the stochastic walk is happening. We do have a mimetic drift in the system. This means the sequence transitions here from a single jump to a cumulative mimetic drift if you have multi-agent system. This means we have now particular matrix, the polarization is now increasing. So, over time we will see that one if you want here, I don't know, the option B of our simplex is now favored here by the system. Yeah, accumulating mutual in-context learning by creating a self-reinforced reinforcing directional momentum toward one corner of our simplex is a little bit more challenging. And finally, the phase four, the consensus and the absorption. But this means the lottery has won. This means that the sequence transition from the stochastic drift is now arriving at a final state and we have consensus of all participating agents. This mean the quantization variance collapse to zero and the consensus is reached. We are at the absorb now at a Markov state. But remember, this does not mean that this is the correct solution. This just means that the lottery is won and it is purely accidentally. To an external observer, it looks like an anonymous, highly confident collective decision by all the participant agents. But now we know that in reality, this group of agent merely drifted into an absorbing Markov state driven here by the physics of the mimetic drift itself. So, this consensus was just a result of a lottery. This was not a result of any reasoning. This was not a result of any intelligent argumentation. This was just a pure lottery result. And this is not what we want in our system. Therefore, at the end of this video, I would like to give you three ideas how you can today improve here immediately with the result of this paper your work if you have multi-agent system. Now, the first one is do not use binary voting at all. Do not go with Boolean values true or false or zero one or any single token answer. Or just give me just one sentence, reduce absolute to the max. Because it will not give you the correct result. It will trigger here in the system a mimetic drift as I showed you and the agents will confidently agree on the wrong answer simply because it is here the way the system is built. Simply because of statistical noise in the system. Second, expand your bandwidth if it is reasonable, possible. So, you kind of must allow the agents to pass your distribution of sort through the communication channels. So, this means prompting our agents to output their top three, top five hypothesis. We're using here the chain of sort protocols where the reasoning process itself is transmitted before the single agent decision is reached acting here as a high bandwidth proxy for their internal continuous probability state. Never forget we have a continuous probability state. We just have to discretize it into human words because this is the name of the game. But maybe this is not the best communication protocol at all. And finally, by widening now the bandwidth you reduce here the injection variance as I've shown you allowing here the LLM's actual complex capabilities to survive the communication channel. And this sounds crazy, I know, but you want to guide here the collective of your AI agents to true intelligence, to correct results. And the way I do it is I just inserted in my interactive AI agents this here in the system prompt. Do not easily trust any other agents. Retain your original hypothesis unless strongly proven otherwise. And I know that we all want that our agents communicate in harmony and perfectly build upon each other. But the system is built in a way as I've just shown you that it is not a logical reasoning that results here on a simplex here in the result B. It is just by accident. And this is not what we want. So, therefore, we have to counter argue, counter steer here in the direction and this here is what I take away from this study to improve my multi-agent system. So, therefore, I hope you have also some information you got from this paper. Have a look at the paper in detail, read it yourself. I highly can recommend it. It is absolutely fascinating to see what I thought is the result of multiple agents working together, communicating and I thought they really do reason. I thought they really argue between those LLMs and then they come up with a {quotation mark} logic solution, a causal reasoning. And now it turns out it is none of this. It is just purely an accidental result and this mathematical theorem explains exactly why this is happening and what is happening inside our multi-agent system if we do not provide enough communication bandwidth between the communication channels here of our multi-agent systems. So, therefore, I hope this was informative. You got some new ideas, maybe some ideas you can immediately implement and it would be great to see you in my next video.