Predictive Maps of Multi-Agent Reasoning: A Successor-Representation Spectrum for LLM Communication Topologies

Abstract: Practitioners deploying multi-agent LLM systems must currently choose between communication topologies such as chain, star, mesh, and richer variants without any pre-inference diagnostic for which topology will amplify drift, converge to consensus, or remain robust under perturbation. Existing evaluation answers these questions only post hoc and only for the task measured. We introduce a structural diagnostic for multi-agent LLM communication graphs based on the successor representation $M = (I - γP)<sup>{-1}$ of the row-stochastic communication operator, and we connect three of its spectral quantities, the spectral radius $ρ(M)$, the spectral gap $Δ(M)$, and the condition number $κ(M)$, to three distinct failure modes. We derive closed-form spectra for the chain, star, and mesh under row-stochastic normalization, and validate the predictions on a 12-step structured state-tracking task with Qwen2.5-7B-Instruct over 100 independent trials. The condition number is a perfect rank-order predictor of empirical perturbation robustness ($r_s = 1.0$); the spectral gap partially predicts consensus dynamics ($r_s = 0.5$); and the spectral radius is perfectly \emph{inverted} with respect to cumulative error ($r_s = -1.0$). We trace this inversion to a regime in which linear spectra are blind to non-contracting bias drift, and we propose an affine-noise extension of the predictive map that recovers the empirical ordering. We read this as a first step toward representational, drift-aware structural diagnostics for multi-agent LLM systems, sitting alongside classical spectral and consensus theory.

• The paper introduces successor-representation diagnostics for communication graphs and finds that condition number perfectly rank-orders perturbation sensitivity, while spectral gap only partly predicts consensus.
• The experiments show that chains produce roughly twice the cumulative error of star and mesh systems because aggregation reduces stochastic agent drift, despite the chain appearing spectrally more stable.
• The paper recommends computing condition number first, spectral gap second, and a drift-aware spectral radius instead of the standard radius when selecting robust multi-agent LLM architectures.

Motivation and problem statement

Practitioners composing multi-agent LLM systems must select a communication topology—chain, star, mesh, or richer variants—without any principled, pre-inference method for anticipating which topology will amplify drift, converge to consensus, or remain robust under perturbation. Existing evaluation instruments such as AgentBench, HELM, and BIG-Bench are outcome-oriented and post hoc: they answer questions about failure modes only after the pipeline has run, and only for the task measured (2605.11453). The paper addresses this gap by asking whether the communication graph alone, analyzed before any token is generated, can predict how the system will reason.

The authors' framing borrows from the successor representation (SR) literature in reinforcement learning and computational neuroscience. Dayan introduced the SR as a compact summary of discounted future state occupancy (2605.11453), and subsequent work established that hippocampal circuits appear to encode SR-like predictive maps (2605.11453). The paper transports this object to multi-agent LLM systems: the row-stochastic adjacency of the communication graph plays the role of the environment's transition operator.

The successor-representation diagnostic

A multi-agent system is modeled as a directed graph $G = (V, E)$ whose weighted adjacency is row-normalized into a stochastic operator $P$. The SR with discount $\gamma = 0.9$ is

$$M = (I - \gamma P)^{-1} = \sum_{k=0}^{\infty} \gamma^k P^k,$$

which compresses all multi-step influence pathways into a single linear operator. Three scalar summaries of its spectrum are paired with three falsifiable empirical metrics:

• Spectral radius $\rho(M)$ predicts error-amplification tendency; the empirical counterpart is cumulative error growth $E_{\mathrm{ceg}}$, the summed per-step deviation from ground truth.
• Spectral gap $\Delta(M)$ predicts consensus dynamics; the counterpart is consensus decay rate $R_{\mathrm{cdr}}$, the mean log-rate at which pairwise disagreement contracts.
• Condition number $\kappa(M)$ predicts perturbation robustness; the counterpart is perturbation sensitivity $F_{\mathrm{ps}}$, final-state divergence under a controlled input perturbation ($P$0 applied to Value).

Closed-form spectra are derived for all three topologies. The chain's nilpotent adjacency yields $P$1 and $P$2 with $P$3; the ergodic star and mesh both saturate $P$4, with gaps of $P$5 and $P$6 and condition numbers of $P$7 and $P$8 respectively.

Experimental design

The empirical study uses a 12-step structured state-tracking task over a three-field JSON state (floating-point Value, binary Parity, bounded integer Level) governed by an order-dependent three-rule protocol coupling arithmetic precision, conditional branching, and bounded updates. The three-field state is chosen to saturate working-memory capacity limits documented in cognitive psychology. Agent contexts reset between steps so that errors propagate only through the explicit communication channel. All agents are Qwen2.5-7B-Instruct calls (temperature 0.8, top-$P$9 0.5), with 100 independent trials per (topology, condition) pair on a single A100; a full run completes within six wall-clock hours. The star uses four leaves with judge aggregation; the mesh adds a peer-critique round followed by majority vote.

Topology effects are assessed by Kruskal–Wallis $\gamma = 0.9$0 tests, and theory–experiment agreement by Spearman $\gamma = 0.9$1 over three ordered means. The authors are explicit that a Spearman coefficient over three ranks carries essentially no statistical power and report it only as rank-consistency evidence—a candid concession that shapes how strongly the headline correlations should be read.

Results

All three metrics show strongly significant topology effects ($\gamma = 0.9$2). The key findings:

Metric	Chain	Mesh	Star	$\gamma = 0.9$3 vs. prediction
Cumulative error $\gamma = 0.9$4	2094 ± 843	1241 ± 950	1184 ± 952	−1.00
Consensus decay $\gamma = 0.9$5	0.27	−1.66	−3.44	+0.50
Perturbation sensitivity $\gamma = 0.9$6	238 ± 191	247 ± 199	444 ± 307	+1.00

The stability paradox. The spectral radius naively predicts that the chain ($\gamma = 0.9$7) should be most stable; empirically it exhibits roughly twice the cumulative error of the aggregated topologies, an exact inversion ($\gamma = 0.9$8). The authors argue this is not a numerical artifact but a regime distinction: $\gamma = 0.9$9 governs geometric growth of deterministic perturbations to a homogeneous linear flow, whereas the observed error is dominated by accumulation of stochastic per-agent deviations, which the homogeneous spectrum cannot see. Each agent's small non-zero bias is handed forward without averaging in the chain, while star and mesh aggregation acts as implicit variance reduction.

An affine-noise model recovers the ordering. Modeling each agent as producing $$M = (I - \gamma P)^{-1} = \sum_{k=0}^{\infty} \gamma^k P^k,$$0 with iid noise of variance $$M = (I - \gamma P)^{-1} = \sum_{k=0}^{\infty} \gamma^k P^k,$$1, and assuming $$M = (I - \gamma P)^{-1} = \sum_{k=0}^{\infty} \gamma^k P^k,$$2 is Lipschitz with constant near unity, the derivation gives $$M = (I - \gamma P)^{-1} = \sum_{k=0}^{\infty} \gamma^k P^k,$$3, where $$M = (I - \gamma P)^{-1} = \sum_{k=0}^{\infty} \gamma^k P^k,$$4 is the per-step aggregation count. This predicts a chain-to-aggregated ratio of $$M = (I - \gamma P)^{-1} = \sum_{k=0}^{\infty} \gamma^k P^k,$$5; the observed ratios are 1.77 (chain/star) and 1.69 (chain/mesh), within sampling variance. A drift-corrected gain $$M = (I - \gamma P)^{-1} = \sum_{k=0}^{\infty} \gamma^k P^k,$$6 rank-orders the topologies correctly on cumulative error. The implication is that pipelines whose dominant error source is per-agent stochasticity—which the authors expect describes most LLM pipelines—are governed by drift stability rather than spectral stability, motivating diagnostics beyond classical spectra.

Consensus is only partially predicted. The spectral gap correctly places the chain last but inverts star and mesh: the star's many-to-one judge aggregation accelerates consensus beyond what the raw graph gap suggests. Explicit aggregation operators impose bottlenecks invisible to the raw spectrum, so the authors recommend treating $$M = (I - \gamma P)^{-1} = \sum_{k=0}^{\infty} \gamma^k P^k,$$7 as a lower bound on consensus speed for topologies containing judges or votes.

The condition number is the cleanest diagnostic. The ordering $$M = (I - \gamma P)^{-1} = \sum_{k=0}^{\infty} \gamma^k P^k,$$8 matches empirical perturbation sensitivity exactly ($$M = (I - \gamma P)^{-1} = \sum_{k=0}^{\infty} \gamma^k P^k,$$9), making $\rho(M)$0 the recommended first step in a practical pre-inference triage: compute $\rho(M)$1 first, $\rho(M)$2 second (with the bottleneck caveat), and $\rho(M)$3 rather than bare $\rho(M)$4 for drift. The full computation requires only the row-stochastic adjacency and in-degree profile and runs in milliseconds.

Extensions in the appendix

Three extensions sharpen the framework. Under correlated noise—suspect when agents share a base model, since identical ensembles converge to shared priors rather than truth—the chain-to-aggregated ratio generalizes from $\rho(M)$5 to $\rho(M)$6; the empirical ratio of 1.77 implies inter-agent correlation $\rho(M)$7. A reliability-weighted dynamic operator in the style of graph attention networks keeps each $\rho(M)$8 row-stochastic so the diagnostics apply pointwise, and admits a variance-reduction reading in which optimal weighting suppresses idiosyncratic variance but leaves the systemic correlation floor untouched. Finally, a malicious-leaf analysis shows that $\rho(M)$9 and $E_{\mathrm{ceg}}$0 are structurally insensitive to weight-inflation attacks while $E_{\mathrm{ceg}}$1 responds: the worst-case bound reaches approximately 98.5 versus the benign 28.6, scaling as $E_{\mathrm{ceg}}$2, so larger stars are disproportionately vulnerable; capping individual agent influence restores the benign bound structurally.

Limitations and open questions

The paper states its scope plainly. The evidence covers three topologies, one model family (Qwen2.5-7B-Instruct), one task family, and 100 trials per condition; the Spearman statistics over three ranks have essentially no statistical power. Empirical metrics reduce full reasoning trajectories to scalars, discarding trajectory-level information. The affine-noise model assumes iid noise and a Lipschitz-near-unity transition rule—both may fail under strong nonlinearity or shared systematic biases. The claim that the qualitative ordering is model-agnostic is flagged explicitly as a prediction of the framework rather than an empirical finding, and sweeps over model scale, closed-weight systems, richer topologies, and direct variation of $E_{\mathrm{ceg}}$3, $E_{\mathrm{ceg}}$4, and $E_{\mathrm{ceg}}$5 remain open. The $E_{\mathrm{ceg}}$6 prediction has been tested only at the single point implied by the chain/star/mesh triple.

Conclusion

This paper converts topology selection from a post-hoc empirical question into a closed-form linear-algebraic one. Within its controlled case study, the condition number of the successor representation perfectly rank-orders perturbation robustness, the spectral gap partially tracks consensus subject to aggregation-bottleneck caveats, and the spectral radius is exactly inverted with respect to cumulative error—an inversion explained quantitatively by an affine-noise model and corrected by the drift-aware gain $E_{\mathrm{ceg}}$7. Whether these orderings generalize across models, tasks, and richer topologies is the central question the paper leaves open.