Dynamics Over Topology:

The picture of a brain as a network of nodes and edges, and of a market as a network of nodes and edges, invites the conclusion that they are the same kind of object. The picture is doing almost no work.

A graph fixes who can talk to whom. It does not fix what is said, what is remembered, what is optimized, or what happens next. Two systems can share an adjacency matrix exactly and still differ in every respect that matters: one settles to a fixed point, one oscillates forever, one integrates its inputs into a durable memory, one collapses in a cascade after a single node flips.

Every number under that figure is measured off the running state, not decoration. Synchrony is the mean absolute pairwise correlation between node trajectories over a rolling window, so it rises when the network stops having independent parts. State entropy is the Shannon entropy of the binned activity vector in bits, which falls to zero when every unit agrees and peaks when the state is spread out. Drag inhibition down and watch both numbers move before the picture does anything you would notice.

The panel below makes this concrete and is the argument of the whole paper in one control. The topology is generated once and frozen. The only thing you change is the update rule F. Same nodes, same edges, same initial condition, same input.

The weights are row-normalized, so ρ(W) = 1 exactly and the reported ρ(gW) = g for every rule. That number is a property of the weighted graph and the gain, and it is the same in all five panels. It predicts the linear rule perfectly: below g = 1 the state decays, above it the state explodes, and that is the whole story. It predicts the saturating rule only near the origin, which is where that rule spends the least of its time, and it says nothing at all about the other three, whose dynamics are not a linear map being iterated. Same ρ, five behaviours. The graph is a constraint on the dynamics. It is not the dynamics.

To compare systems honestly we need a description that has room for the things that differ. A system is not a graph. It is a tuple:

Each node i carries an internal state xi(t). Information arrives from neighbours as a weighted message:

The node updates by a system-specific rule, given its own state, the message, exogenous input u, and parameters θi:

Connections may themselves change, on a slower timescale:

Everything interesting lives in the parts that this notation leaves open: what xi represents, what the messages are, how F is defined, what drives ΔW, what feedback is available and on what delay, what objective or selection pressure exists, and how the environment constrains the whole thing.

Filling in that table for the three systems is the entire comparison:

Read the Objective row twice. It is the deepest split in the table. The RNN has one objective, imposed from outside, and every weight in it is there because it reduced that objective. The market has as many objectives as agents, they are in conflict, and the price is a side effect of that conflict rather than a thing anyone optimizes. The toy neural network has no objective at all: it has dynamics, and we describe what those dynamics do. Any claim that transfers across all three has to survive that gap.

Predefined hypotheses

Interactive papers invite aimless fiddling. To avoid that, the following are stated before the experiments, and §06 runs them against the implementations. They are hypotheses about these three toy models, not laws about brains, AI, and markets. Verdicts fill in when you run the sweep.

All three run on the same adjacency matrix A, generated by one procedure with knobs for density, modularity, and centralization. Node i in the neural model, unit i in the RNN, and agent i in the market have the same neighbours. This is the only thing held equal, and holding it equal is what makes the comparison a comparison rather than three unrelated demos.

3.1 Biological recurrent network (toy)

A rate-based model with separate excitatory and inhibitory populations. This is a toy model of neural computation. It is not a model of a brain, of a cortical column, or of consciousness. It exists because it is the smallest thing that shows E/I balance, attractor dynamics, and distributed memory at the same time.

with φ = tanh, τ the membrane time constant (the memory knob), τa ≫ τ, and ξ standard normal. Row normalization after the Hebbian step is doing real work: without it, Hebbian learning is unbounded and the network saturates. That normalization is a modelling choice standing in for homeostatic plasticity, and the results depend on it.

3.2 Artificial recurrent network

A vanilla RNN, masked by the same A, trained in your browser by backpropagation through time on a defined task.

The task is evidence accumulation. Over T steps the network sees a stream of pulses u(t) ∈ {−1, 0, +1}. At the final step it must report the sign of the cumulative sum. Nothing tells the network how to do this. The task is chosen because it has a known optimal solution (integrate, then threshold), which gives us ground truth to probe against: if the network solves it, something inside it has to be approximating a running total, and §08 goes looking for that quantity.

3.3 Agent-based market

Heterogeneous agents on the same graph, trading a single asset with a fundamental value v(t) that follows a random walk with occasional regime breaks. Agents never see v. They see a private noisy signal and their neighbours.

Clearing is a market maker with linear price impact λ, which is a specified and deliberately simple mechanism, not an order book. Agents differ in signal precision σi, risk appetite γi, capital ci, imitation weight μi, horizon Li, and type. Leverage adds the one genuinely reflexive channel: an agent whose losses breach a margin threshold is forced to liquidate, which moves the price, which can breach the next agent's threshold.

That forced-liquidation loop has no counterpart in either of the other two systems, and it is the source of most of the market's interesting failures. Keep it in mind when §11 asks what transfers.

Three panels, one adjacency matrix, one intervention applied to all three at once. The systems are not synchronized in time (their clocks mean different things) but they are synchronized in structure and in what is done to them.

Pick an intervention. It is translated into each system's own vocabulary, and the same abstract thing happens in all three: a node is removed, or a message is delayed, or the world changes. Then watch what each system does about it.

The three metrics under each panel are the only ones that mean the same thing in all three systems, which is why they are the ones shown. Persistence is the correlation, across nodes, between the state now and the state ten steps ago: it asks whether the pattern of who is doing what has survived. Watch it sit near zero for the market while the other two hold above 0.8. That is not a bug and it is worth a moment: the market's beliefs collapse toward consensus, and once everyone agrees there is no pattern left to persist, so the correlation has nothing to lock onto. The same number means "holds its state" in two systems and "has no state worth holding" in the third. Synchrony is the mean absolute pairwise correlation between units. Entropy is the Shannon entropy of the binned state in bits. Notice what is missing: there is no accuracy here, because the neural network has no task, and no price error, because the other two have no price. The comparable metrics are comparable precisely because they are indifferent to what the system is for.

That figure is for watching. The panel below is for measuring, and it reports the quantities that only make sense one system at a time: retained bits, task accuracy, price error, cascade size.

The translation table is the honest part of this module. "Remove 30% of nodes" means killing neurons, ablating hidden units, and bankrupting traders. Those are not the same event. What makes the comparison meaningful is not that the events are alike but that the structural operation on A is identical, so any difference in outcome is attributable to F, which is exactly the paper's question.

"Information" gets used loosely in comparisons like this one. Here it means one thing, computed one way. For a discrete external variable Y and a discretized system state X:

This measures statistical dependence between two variables. It does not measure meaning, understanding, causal structure, or consciousness. A thermostat and a neuron can have the same mutual information with the room temperature.

Two things worth noticing. Delay does not destroy information, it displaces it: the peak moves right and the half-life is largely unchanged, because a delay line is a memory. Noise does destroy it, and the loss is roughly exponential in time. Longer τ extends the half-life and simultaneously lowers the peak, which is the memory tradeoff of H4 showing up as an information-theoretic quantity rather than a performance one.

This module runs the six hypotheses from §02. Each sweep varies one factor across a range, holds everything else fixed, repeats over seeds, and reports the outcome curve with a verdict. The verdicts written back into §02 are produced by the runs, not by the author.

H6 deserves a warning, because it is the one where running the model actually refuted the reasoning behind the hypothesis. There is an interior optimum in transmission: a stimulus injected at three nodes and read out at the three most distant nodes arrives best at intermediate density. Below it, paths are too long and the signal decays before arriving. Above it, the signal is washed out as every node averages over everything. That much survives.

The stated mechanism does not. H6 predicted that connectivity would eventually amplify noise and synchronization, and in this model it does the opposite: redundancy falls as density rises. The reason is a single modelling choice. Weights here are degree-normalized, so each node's total input gain is fixed at 1 and an extra edge means every neighbour counts for less. A denser node is averaging over more independent noise sources, and averaging suppresses noise. Had we instead fixed the per-edge weight, density would have raised total gain and amplified noise, and the same sweep over the same graphs would have produced the opposite headline.

A sweep that finds an interior optimum in a toy model is evidence about the toy model. It is a hypothesis-generator for the real system and nothing more. The honest form of the claim is: in a class of models with these dynamics, this tradeoff exists, so it is worth looking for in the real system, using a measurement that could find it absent.

A trajectory through state space says things a time series does not. The panel projects the N-dimensional state onto its first two principal components, computed from the trajectory itself, and draws where the system goes.

The second trajectory starts at x₀ + ε with ε around 10⁻⁸. Both are run with the same noise realization, so the only difference between them is the initial condition. Their separation is:

The fit window matters enormously and is drawn on the chart for that reason. Fit before the transient has decayed and λ̂ measures the transient. Fit after D reaches the attractor diameter and λ̂ goes to zero because it is measuring saturation. Reported Lyapunov exponents that do not disclose their window are not reporting much.

Correlation finds where a quantity is present. Intervention finds where it matters. The distinction is the whole methodological content of mechanistic interpretability, and this module is built to make the gap between the two visible.

The RNN in §3.2 has been trained on evidence accumulation. If it solves the task, some internal quantity tracks the running sum. We now go find it, three ways, and the three ways disagree.

Why the three methods disagree

The probe asks: can I linearly decode the accumulated evidence from h(t)? It usually can, with high R². This is correlational. A probe with R² = 0.95 tells you the information is present and linearly available. It does not tell you the network uses it. A probe can succeed on a quantity the model computes as a side effect and never reads.

The ablation scan zeroes each unit and measures the accuracy drop. If the representation is distributed, every single-unit ablation is nearly free, and the scan reports that nothing matters, which is false. Single-unit ablation systematically under-attributes distributed representations, and this is a known failure mode rather than a discovery about the network.

The patch is the causal test, and it lives or dies on choosing the right counterfactual. Overwriting the state at step t can only transfer the evidence accumulated up to t, because the pulses after t still arrive from the original stream. So the target is not "stream b's answer". It is the answer to a hybrid stream: b's pulses through t, then a's. The module constructs that hybrid, runs it, and uses its output as the denominator. Getting this wrong is the easiest way to report a patch effect that means nothing.

The module also patches the entire hidden state as a calibration check. Since h(t) is a sufficient statistic for everything the network does afterwards, a full-state patch has to reproduce the hybrid exactly and score 1.000. It does, to three decimals. That is what licenses reading the next number as a fact about the network rather than a bug in the measurement.

And the next number is the point of the section. On the default settings, the probe decodes the accumulated evidence at R² ≈ 0.99, and patching along that probe direction moves the output only about a fifth of the way to the counterfactual. The quantity is unmistakably there and almost perfectly linearly available, and the direction that best predicts it is not, mostly, the thing the network reads. The readout draws on state components the probe discards as regression residual. A very high R² bought no causal claim whatsoever.

Networks are not "inherently chaotic". Chaos is a property some deterministic systems have in some parameter regimes, and it is worth keeping distinct from four things it gets confused with.

The reason §07 has a noise slider that goes to exactly zero is that the chaos claim is only testable there. With σ > 0 the two trajectories separate whether or not the dynamics are chaotic, because they are being driven apart by different noise. Set σ = 0, keep the perturbation at 10⁻⁸, and the separation that remains is the system's own.

Even then, λ̂ > 0 on a finite window is evidence of local divergence in the sampled region of state space. It is not proof of chaos, which is an asymptotic property of an invariant set. The estimate is finite-time, initial-condition-dependent, and this page reports it as λ̂ rather than λ for that reason.

Systems that get better under stress are real and worth studying. The vocabulary usually reached for is not the one that lets you measure them. "The system benefits from disorder" is not a testable proposition until you say which system, which disorder, which benefit, and measured against what.

The testable version:

Which is what this panel measures. A system is exposed to repeated bounded shocks during a training phase, then evaluated on a held-out test distribution against a matched control that was never shocked. Three outcomes are possible and all three occur: the exposed system does better (adaptation), the same (robustness without benefit), or worse (accumulated damage). Which one you get depends on the shock magnitude and whether the system has a mechanism that can use the shocks.

The result across the three systems is instructive, and at the default settings it is: no effect, a small negative, and a small positive, in that order.

The RNN is the only one that could improve in principle, because it has an objective and a learning rule, so a shock during training is data augmentation. At the defaults it does not improve, because it already solves the task robustly and generalizes to sequences twice as long as it trained on: there is no headroom for the shocks to buy anything. Turn the shock magnitude up and it starts to hurt. The Hebbian network cannot improve in principle. It has plasticity but no objective, so shocks change it without improving anything measurable, and past a threshold they degrade the representation. The market improves while no agent learns anything at all: capital flows from agents who read the shocks badly to agents who read them well, which is selection, not learning, and the surviving population is both better at pricing and less diverse than the one that went in.

Three different mechanisms, three different answers, one of which is "nothing happened". Naming any of this "antifragility" would have collapsed them into one word and hidden the fact that they are not the same phenomenon, that one of them is not a phenomenon at all, and that the market's version comes with a bill: the thing that made it price better is the thing that made it more uniform.

This is the ledger. It is probably the most useful thing in the paper, and it is the part that a network diagram of three systems side by side actively obscures.

The left column is why the comparison is worth making. The right column is why it stops where it does. A claim that uses only left-column properties has a chance of transferring. A claim that quietly depends on a right-column property, while presenting itself as a claim about networks in general, is the failure mode this paper is written against.

Consciousness is not an output variable of any simulation in this paper, and it could not be added as one.

Biological brains support conscious experience. Artificial neural networks and markets process information, but information processing alone does not establish consciousness. No accepted scientific criterion currently permits consciousness to be inferred from network size, connectivity, complexity, information integration, or adaptive performance alone.

This matters here because every quantity this paper computes is exactly the kind of quantity that gets recruited into consciousness arguments: entropy, mutual information, integration, attractor structure, complexity. None of them licenses that step. Specifically, nothing in this paper implies:

• that markets are conscious, or have a point of view;
• that civilization has a unified subjective perspective;
• that a sufficiently complicated network automatically becomes conscious;
• that entropy or integrated information measures experience;
• that "emergence" explains consciousness by being named.

The last one deserves its own sentence. Emergence describes a relationship between levels of description. It is a label for a phenomenon that needs explaining, not an explanation of it. Saying consciousness emerges from complexity has the grammatical form of an answer and the content of a restatement.

The honest position is that this is an open interpretive problem. The measurements here are about information processing and adaptive behaviour. They are silent on experience, and their silence is not a gap to be filled by adding more nodes.

Stated plainly, because an interactive paper is unusually good at making a toy feel like a finding.

• The systems are toys. 48 rate units is not a cortex. A 24-unit vanilla RNN is not a frontier model. A linear-impact market maker is not an order book. Every result is a result about these implementations.
• Matched topology is an assumption, not a fact. Real brains, real networks, and real markets do not share an adjacency matrix. Forcing them to share one is what makes the experiment clean and what makes it artificial. The control is the point and the limitation at once.
• The estimators are small-sample. MI is plug-in with Miller-Madow correction and a shuffled null. λ̂ is a finite-time fit on a disclosed window. Both are noisy. Neither is a gold-standard estimator, and the panels report their own nulls so you can see how much of the signal is estimator bias.
• Hypothesis verdicts are model-internal. §06 tests H1 to H6 against these implementations. A verdict of "supported" means the toy behaves as predicted. It is not evidence about brains or markets.
• Parameter ranges were chosen to make effects visible. This is honest exposition and dishonest science if unstated. The sliders live in regions where things happen. Wide regions of parameter space are boring, and the defaults do not sit in them.
• The market has no order book, no strategic execution, no information asymmetry beyond signal precision and who is informed. Real markets have adversarial agents who model each other. This one does not, which removes the most interesting thing about markets and the thing that least resembles a brain.
• The market cannot test H5's cascade clause, and we only learned that by running it. The clearing price is defined as the capital-weighted consensus, so an agent agreeing with everyone else holds no position and cannot lose. A perfectly homogeneous population is trivially safe here: wrong together, but never exposed together. The correlated-failure mechanism H5 is about needs a price that can depart from consensus, which needs an exogenous counterparty or a funding constraint. This model has neither, so §06 reports that clause as untestable rather than refuted. The sweep was built to test a hypothesis and instead found a hole in the model, which is the more useful of the two outcomes and the one an interactive paper is least likely to admit to.
• Nothing here was preregistered. The hypotheses were written before the sweeps were run, which is better than nothing and much weaker than a preregistration.

The contribution, if there is one, is the framework and the controlled cross-system comparison rather than any individual number. The claim is that Topology + State + Dynamics + Learning + Memory + Objective + Environment is a better unit of comparison than the graph, and that once you compare at that grain, most cross-domain network analogies turn out to be claims about one term while presenting as claims about all seven.