Estimation of Propagated Risk in LLM-based Multi-Agent Systems

# Estimation of Propagated Risk in LLM-based Multi-Agent Systems
Source: [https://arxiv.org/html/2610.33282](https://arxiv.org/html/2610.33282)
Souradeep ChattopadhyayAffiliation:Iowa State UniversityEmail:[soura@iastate\.edu](mailto:)Qisai LiuAffiliation:Iowa State UniversityEmail:[supersai@iastate\.edu](mailto:)Soumik SarkarAffiliation:Iowa State UniversityEmail:[soumiks@iastate\.edu](mailto:)

###### Abstract

Large Language Model \(LLM\) based multi\-agent systems introduce a distinctive reliability challenge in which actions taken by upstream agents can affect the correctness of downstream agents\. While prior work has investigated error propagation and developed methods for estimating uncertainty inherited from upstream agents, existing approaches do not explicitly model how realized sender–receiver interactions contribute to downstream error risk under different sender correctness states\. In this work, we study agent\-level error\-risk estimation from multi\-agent execution traces\. We propose Interaction\-Effect Propagation \(IE\-Prop\), an interaction\-conditioned model that learns separate effects for correct and incorrect upstream agents\. For each sender\-receiver interaction, the model combines these conditional effects using the sender’s estimated error risk, and recursively propagates the resulting risk estimates through the execution graph\. We evaluate our approach across three multi\-agent architectures and three benchmarks\. Across all evaluated architectures, our approach raises the final\-node AUROC over the established baseline by approximately 0\.05 points on average\. Furthermore, tracking risk across multiple agent steps contributes a distinct \+0\.03 AUROC gain in sequential systems\. These results show that realized inter\-agent interactions provide predictive information beyond receiver\-local evidence, while the value of multi\-step risk tracking depends on the system’s information\-flow structure\.

## 1Introduction

Figure 1:Error dynamics in LLM\-based multi\-agent systems: propagation of an upstream error \(top\), correction of an upstream error \(middle\), and erroneous override of a correct upstream answer \(bottom\)\.LLM\-based multi\-agent systems decompose complex tasks across interacting agents, making intermediate outputs part of the context for subsequent generations\. This interaction structure creates a reliability problem that does not arise in isolated generation: an upstream error can influence downstream predictions, but its effect depends on how the receiving agent uses the information it receives\. Empirical analyses of multi\-agent failures have documented error transmission through inter\-agent communication\([Cemri et al\., 2025](https://arxiv.org/html/2610.33282#bib.bib11);[Zhang et al\., 2025](https://arxiv.org/html/2610.33282#bib.bib9)\), and controlled studies show that erroneous and corrective peer information can affect downstream behavior differently\([Qu et al\., 2026](https://arxiv.org/html/2610.33282#bib.bib24);[Hu, 2026](https://arxiv.org/html/2610.33282#bib.bib25)\)\. Together, these findings suggest that reliability in such systems depends not only on whether an upstream agent is likely to be wrong, but also on the realized interaction through which its output is consumed\.

Existing propagation\-aware approaches already account for the possibility that upstream errors are transmitted\. PropUQ\-MAS\([Liu et al\., 2026](https://arxiv.org/html/2610.33282#bib.bib1)\), for example, combines local uncertainty with inherited parent risk and uses an acceptance/critique mechanism to attenuate propagation when erroneous information is rejected or corrected\. Its recurrence nevertheless treats inherited risk as a predefined contamination contribution: for fixed local uncertainty and edge acceptance, increasing parent risk cannot lower the receiver’s estimated risk\. This leaves open a more general question, whether the association between an observed sender\-receiver interaction and receiver error should depend on the sender’s correctness\.

We introduce Interaction\-Effect Propagation \(IE\-Prop\), post\-hoc agent\-level error\-risk estimation from completed multi\-agent execution traces\. Such risk estimates can support selective prediction and deferral and targeted re\-runs\([El\-Yaniv and Wiener, 2010](https://arxiv.org/html/2610.33282#bib.bib34);[Geifman and El\-Yaniv, 2017](https://arxiv.org/html/2610.33282#bib.bib21)\)\. For post\-hoc risk estimation, we distinguish two increasingly rich information sets available in a completed trace\. First, receiver\-local uncertainty signals such as entropy, verbalized uncertainty, provide a conventional estimate of the receiver’s reliability\. Second, a multi\-agent trace exposes relational information unavailable in isolated generation: the messages received from upstream agents and the receiver’s response to them\. For each such interaction, we refer to the agent providing information as the sender and the agent consuming it as the receiver\. Our primary empirical question is whether upstream interaction information provides predictive signal beyond the receiver text\.

Table 1:Comparison of LLM error\-risk estimation methods by multi\-agent system \(MAS\) support, risk propagation \(Prop\.\), and underlying estimation mechanism\.MethodMASProp\.EstimatorP\(True\)\([Kadavath et al\., 2022](https://arxiv.org/html/2610.33282#bib.bib12)\)––Self\-eval\.SelfCheckGPT\([Manakul et al\., 2023](https://arxiv.org/html/2610.33282#bib.bib19)\)––Sample consistencySemantic Entropy\([Farquhar et al\., 2024](https://arxiv.org/html/2610.33282#bib.bib31)\)––Semantic entropyBTProp\([Hou et al\., 2025](https://arxiv.org/html/2610.33282#bib.bib37)\)–✓Belief treeSAUP\([Zhao et al\., 2025](https://arxiv.org/html/2610.33282#bib.bib30)\)–✓Situation weightsUProp\([Duan et al\., 2025](https://arxiv.org/html/2610.33282#bib.bib3)\)–✓Information theoryRUPA\([Ma et al\., 2026](https://arxiv.org/html/2610.33282#bib.bib4)\)–✓Relational graphMATU\([Chen et al\., 2026](https://arxiv.org/html/2610.33282#bib.bib2)\)✓–Tensor factorScore Deviation\([Kostka and Chudziak, 2026](https://arxiv.org/html/2610.33282#bib.bib38)\)✓–Conformal calibrationHalluProp\([Lin et al\., 2026](https://arxiv.org/html/2610.33282#bib.bib36)\)✓✓Noisy\-ORPropUQ\-MAS\([Liu et al\., 2026](https://arxiv.org/html/2610.33282#bib.bib1)\)✓✓Weighted Noisy\-ORInteraction\-Effect Propagation \(ours\)✓✓Signed logit mixWe address this question with an interaction\-conditioned error\-risk model\. For each realized interaction, the model learns two effects on the receiver’s error score, corresponding to a correct and an incorrect sender\. Because sender correctness is unknown at prediction time, the two effects are combined using the sender’s estimated error risk\. The resulting interaction adjustment is added to receiver\-local evidence, and node risks are computed recursively through the execution graph\. This formulation allows the same observed interaction to have different, and potentially opposite, associations with receiver error under the two sender states, while retaining a single risk estimate that can be propagated downstream\. The contributions of our work are:

Figure 2:Overview of Interaction\-Effect Propagation\. \(A\) Downstream error risk depends on realized sender–receiver interactions rather than monotone error propagation\. \(B\) Interaction effects conditioned on sender correctness are combined using estimated sender risk and receiver\-local evidence\. \(C\) Three evaluated multi\-agent topologies: sequential, fully connected, and shallow hierarchy\.\(1\) Interaction\-conditioned, sender\-state\-aware error modeling\(§[3](https://arxiv.org/html/2610.33282#S3)\)\. We introduce an agent\-level error\-risk model \(IE\-Prop\) in which each realized sender\-receiver interaction has separate signed effects under correct and incorrect sender states\. This allows the downstream association of an interaction to depend on both what occurred between the agents and the sender’s correctness\.

\(2\) Recursive risk propagation and learning\(§[3](https://arxiv.org/html/2610.33282#S3), §[4](https://arxiv.org/html/2610.33282#S4)\)\. We develop a tractable recursive procedure that combines the two interaction effects using estimated sender risk and propagates the resulting scalar risk through the execution graph\. The estimator is trained end to end from node\-level correctness labels, allowing a downstream node to incorporate information accumulated through earlier interactions without explicitly maintaining a joint distribution over upstream error states\.

\(3\) Empirical characterization of interaction and propagation effects\(§[7](https://arxiv.org/html/2610.33282#S7)\)\. Through controlled comparisons across three multi\-agent architectures, we examine whether realized upstream interactions provide predictive information beyond receiver\-local evidence and distinguish the contributions of interaction modeling and recursive risk propagation\. We find that interaction information generally improves downstream error\-risk discrimination, although the magnitude varies across architectures and datasets\.

Our results indicate downstream error risk is a property of realized inter\-agent interactions, while showing that the size and the source of the benefit depend on the graph, the data and the strength of the receiver’s own evidence\.

## 2Related Work

#### LLM confidence and error\-risk estimation\.

Local signals include verbalized confidence\([Lin et al\., 2022](https://arxiv.org/html/2610.33282#bib.bib14);[Tian et al\., 2023](https://arxiv.org/html/2610.33282#bib.bib15);[Xiong et al\., 2024](https://arxiv.org/html/2610.33282#bib.bib16)\), self\-evaluation\([Kadavath et al\., 2022](https://arxiv.org/html/2610.33282#bib.bib12)\), semantic entropy\([Kuhn et al\., 2023](https://arxiv.org/html/2610.33282#bib.bib13);[Farquhar et al\., 2024](https://arxiv.org/html/2610.33282#bib.bib31)\), sampling consistency\([Manakul et al\., 2023](https://arxiv.org/html/2610.33282#bib.bib19)\), and trained correctness predictors\([Mielke et al\., 2020](https://arxiv.org/html/2610.33282#bib.bib20);[Azaria and Mitchell, 2023](https://arxiv.org/html/2610.33282#bib.bib18);[Kapoor et al\., 2024](https://arxiv.org/html/2610.33282#bib.bib17)\); benchmarks such as LM\-Polygraph compare them\([Fadeeva et al\., 2023](https://arxiv.org/html/2610.33282#bib.bib22);[Vashurin et al\., 2025](https://arxiv.org/html/2610.33282#bib.bib23)\)\. BTProp estimates claim factuality by propagating beliefs over a tree of logically related statements\([Hou et al\., 2025](https://arxiv.org/html/2610.33282#bib.bib37)\)\. These methods estimate the uncertainty or confidence in LLM response for an isolated generation\.

#### Uncertainty and reliability in agent systems\.

SAUP propagates uncertainty across steps in a single\-agent trajectory using situational weights\([Zhao et al\., 2025](https://arxiv.org/html/2610.33282#bib.bib30)\), UProp adds a mutual\-information term with earlier decisions\([Duan et al\., 2025](https://arxiv.org/html/2610.33282#bib.bib3)\), RUPA aggregates over a relation\-typed trajectory graph\([Ma et al\., 2026](https://arxiv.org/html/2610.33282#bib.bib4)\), and MATU derives a system\-level score from re\-sampled multi\-agent runs\([Chen et al\., 2026](https://arxiv.org/html/2610.33282#bib.bib2)\); surveys argue that agent UQ needs formulations beyond single outputs\([Kirchhof et al\., 2025](https://arxiv.org/html/2610.33282#bib.bib8);[Oh et al\., 2026](https://arxiv.org/html/2610.33282#bib.bib7)\)\. These works study uncertainty over sequential trajectories or whole\-system executions\. Information preservation during inter\-agent communication has also been investigated in decentralized settings\([Rillo et al\., 2026](https://arxiv.org/html/2610.33282#bib.bib39)\)\.

#### Error propagation in multi\-agent systems\.

PropUQ\-MAS is the closest method to ours\([Liu et al\., 2026](https://arxiv.org/html/2610.33282#bib.bib1)\)\. It treats an upstream error as a contamination event: each agent’s local uncertainty is combined with a self\-reported acceptance probability for every incoming message in a noisy\-OR product, so transmission is monotone and a parent error can only increase receiver risk\. We do not assign a single transmission probability to a parent error\. Rather we learn separate effects of the realized sender\-receiver interaction under correct and erroneous sender states, which allows both harmful propagation and correction\.\([Lin et al\., 2026](https://arxiv.org/html/2610.33282#bib.bib36)\)introduce HalluProp, which estimates agent\-level hallucination risk before execution using role–query alignment and topology\-aware Noisy\-OR propagation\. Descriptive studies measure how hallucinations survive handoffs\([Singh and Pawar, 2026](https://arxiv.org/html/2610.33282#bib.bib5);[Jamshidi et al\., 2026](https://arxiv.org/html/2610.33282#bib.bib6)\), and failure\-attribution methods locate the responsible agent after a failure\([Zhang et al\., 2025](https://arxiv.org/html/2610.33282#bib.bib9);[Hou et al\., 2026](https://arxiv.org/html/2610.33282#bib.bib10)\)\.

#### Graphical\-model connection\.

Our formulation is related to logistic graphical models in which node\-local and relational terms combine additively in logit space\([Neal, 1992](https://arxiv.org/html/2610.33282#bib.bib33);[Besag, 1974](https://arxiv.org/html/2610.33282#bib.bib32)\)\. We adopt this structure as an inductive bias, allowing each incoming interaction to contribute a separate, correctness\-conditioned adjustment to the receiver’s error log\-odds\. This provides a compositional formulation for propagating risk without explicitly maintaining a joint distribution over agents’ error states \(Appendix[C](https://arxiv.org/html/2610.33282#A3)\)\.

## 3Interaction\-Effect Error\-Risk Estimation

#### Problem setup\.

We represent an executed multi\-agent system as a directed acyclic graphG=\(V,ℰ\)G=\(V,\\mathcal\{E\}\), whereVVis the set of agents in the execution andℰ⊆V×V\\mathcal\{E\}\\subseteq V\\times Vis the set of directed edges representing realized information flow between agents\. An edge\(p,v\)∈ℰ\(p,v\)\\in\\mathcal\{E\}indicates that agentvvreceives the visible output of agentpp\. Each agentvvhas a roleρv\\rho\_\{v\}, a visible outputmvm\_\{v\}and, after execution, an error indicatorEv∈\{0,1\}E\_\{v\}\\in\\\{0,1\\\}, and its error risk isrv=P⁡\(Ev=1\)r\_\{v\}=P\(E\_\{v\}=1\)\.EvE\_\{v\}is observed for training and evaluation traces but is unknown at inference\. Letqqbe the task\. Each agent has a local risk score

bv=gϕ​\(wv\),b\_\{v\}=g\_\{\\phi\}\(w\_\{v\}\),\(1\)wherewvw\_\{v\}holds agent\-local risk signal andgϕg\_\{\\phi\}is a small network\. For an interactionp→vp\\rightarrow v, we refer toppas the sender \(or parent\) andvvas the receiver, and their visible outputs arempm\_\{p\}andmvm\_\{v\}, respectively\. Each realized interaction is embedded by a frozen text encoder

xp​v=Enc⁡\(q,ρp,mp,ρv,mv\)x\_\{pv\}=\\operatorname\{Enc\}\(q,\\rho\_\{p\},m\_\{p\},\\rho\_\{v\},m\_\{v\}\)\(2\)The embeddingxp​vx\_\{pv\}represents the realized exchange rather than only the parent message\. It can therefore capture textual evidence of how the receiver used the upstream information \- for example, whether it followed, rejected, revised, or amplified it\. Becausemvm\_\{v\}is required to constructxp​vx\_\{pv\}, the estimator is necessarily post\-hoc and can only score an interaction after receiver has produced its output\.

#### State\-conditioned interaction model\.

The association between a realized sender–receiver interaction and receiver error may differ depending on whether the sender is correct or incorrect\. To capture this distinction, an edge head \(fθf\_\{\\theta\}\) maps \(xp​vx\_\{pv\}\) to two scalars

fθ​\(xp​v\)=\(δp​v\(0\),δp​v\(1\)\)\.f\_\{\\theta\}\(x\_\{pv\}\)=\\big\(\\delta^\{\(0\)\}\_\{pv\},\\ \\delta^\{\(1\)\}\_\{pv\}\\big\)\.\(3\)δp​v\(0\)\\delta^\{\(0\)\}\_\{pv\}is the effect of the realized interaction on the receiver’s error log\-odds if the sender is correct \(Ep=0E\_\{p\}=0\), andδp​v\(1\)\\delta^\{\(1\)\}\_\{pv\}its effect if the sender is wrong \(Ep=1E\_\{p\}=1\)\. Positive values raise the receiver’s error log\-odds and negative values lower them\. The pair can therefore represent both propagation and correction:δp​v\(1\)\>0\\delta^\{\(1\)\}\_\{pv\}\>0describes a receiver that adopts an erroneous parent message, andδp​v\(1\)<0\\delta^\{\(1\)\}\_\{pv\}<0a receiver that detects and corrects it\.

We combine receiver\-local and interaction\-specific evidence additively in log\-odds space\. The local scorebvb\_\{v\}provides the receiver’s baseline error log\-odds, while each incoming interaction contributes an adjustment selected by the corresponding sender state\. For an error configuratione=\{ep:p∈Pa⁡\(v\)\}e=\\\{e\_\{p\}:p\\in\\mathrm\{Pa\}\(v\)\\\}, whereep∈\{0,1\}e\_\{p\}\\in\\\{0,1\\\}indicates whether senderppis correct \(00\) or incorrect \(11\), we define

ηv​\(e\)=bv\+∑p∈Pa⁡\(v\)δp​v\(ep\)\.\\eta\_\{v\}\(e\)=b\_\{v\}\+\\sum\_\{p\\in\\mathrm\{Pa\}\(v\)\}\\delta^\{\(e\_\{p\}\)\}\_\{pv\}\.\(4\)and model the receiver’s conditional error probability as

P⁡\(Ev=1∣EPa⁡\(v\)=e\)=σ⁡\(ηv​\(e\)\)\.P\(E\_\{v\}=1\\mid E\_\{\\mathrm\{Pa\}\(v\)\}=e\)=\\sigma\\bigl\(\\eta\_\{v\}\(e\)\\bigr\)\.\(5\)
This additive logistic form is a modeling choice: it provides a compositional inductive bias in which incoming interactions contribute separately to the receiver’s log\-odds\. Appendix[C](https://arxiv.org/html/2610.33282#A3)gives the corresponding graphical\-model interpretation\.

#### Inference with uncertain sender states\.

Eq\.[5](https://arxiv.org/html/2610.33282#S3.E5)specifies the receiver risk if the correctness state of every sender were known\. At inference these states are unknown, so for each incoming edge we use the estimated sender error probabilityr^p\\hat\{r\}\_\{p\}as a soft belief over the two sender states and average the corresponding interaction contributions in logit space:

δ¯p​v=\(1−r^p\)​δp​v\(0\)\+r^p​δp​v\(1\)\.\\bar\{\\delta\}\_\{pv\}=\(1\-\\hat\{r\}\_\{p\}\)\\delta^\{\(0\)\}\_\{pv\}\+\\hat\{r\}\_\{p\}\\delta^\{\(1\)\}\_\{pv\}\.\(6\)
The receiver’s predicted risk is obtained by adding these expected interaction contributions to its local score,

r^v=σ⁡\(bv\+∑p∈Pa⁡\(v\)δ¯p​v\)\.\\hat\{r\}\_\{v\}=\\sigma\\left\(b\_\{v\}\+\\sum\_\{p\\in\\mathrm\{Pa\}\(v\)\}\\bar\{\\delta\}\_\{pv\}\\right\)\.\(7\)
This averaging can be expressed in terms of expectations\. Specifically, Eq\.[6](https://arxiv.org/html/2610.33282#S3.E6)computesδ¯​p​v=𝔼Ep​\[δp​v\(Ep\)\]\\bar\{\\delta\}\{pv\}=\\mathbb\{E\}\_\{E\_\{p\}\}\[\\delta\_\{pv\}^\{\(E\_\{p\}\)\}\], where the expectation is taken over the sender’s error state using its estimated error probabilityr^p\\hat\{r\}\_\{p\}\. By linearity of expectation, Eq\.[7](https://arxiv.org/html/2610.33282#S3.E7)then applies the sigmoid to the expected receiver logit, yieldingr^v=σ⁡\(𝔼EPa⁡\(v\)​\[ηv​\(EPa⁡\(v\)\)\]\)\\hat\{r\}\_\{v\}=\\sigma\(\\mathbb\{E\}\_\{E\_\{\\mathrm\{Pa\}\(v\)\}\}\[\\eta\_\{v\}\(E\_\{\\mathrm\{Pa\}\(v\)\}\)\]\)\.

Risk estimates are computed recursively in topological order over the execution graph\. Roots receiver^v=σ⁡\(bv\)\\hat\{r\}\_\{v\}=\\sigma\(b\_\{v\}\), while each non\-root node uses its parents’ recursively estimated risks in Eq\.[6](https://arxiv.org/html/2610.33282#S3.E6)\-[7](https://arxiv.org/html/2610.33282#S3.E7)\. So in a sequential graph a node’s risk depends on interactions further upstream through its parent’s risk\. Appendix[B\.1](https://arxiv.org/html/2610.33282#A2.SS1)shows that this update also admits a reverse\-KL barycenter interpretation\.

#### Approximation to marginal inference\.

Eq\.[7](https://arxiv.org/html/2610.33282#S3.E7)is a compositional inductive bias, not an exact identity: withZv=ηv​\(EPa⁡\(v\)\)Z\_\{v\}=\\eta\_\{v\}\(E\_\{\\mathrm\{Pa\}\(v\)\}\), exact marginalization of Eq\.[5](https://arxiv.org/html/2610.33282#S3.E5)over the unknown parent states would compute𝔼⁡\[σ⁡\(Zv\)\]\\mathbb\{E\}\[\\sigma\(Z\_\{v\}\)\], whereas Eq\.[7](https://arxiv.org/html/2610.33282#S3.E7)computesσ⁡\(𝔼⁡\[Zv\]\)\\sigma\(\\mathbb\{E\}\[Z\_\{v\}\]\)from one scalar per parent, at costO⁡\(k\)O\(k\)instead of2k2^\{k\}forkkparents, and is exact when the two effects of every uncertain edge coincide\. Appendix[A](https://arxiv.org/html/2610.33282#A1)derives a variance\-based bound on the local discrepancy between exact marginalization and Eq\.[7](https://arxiv.org/html/2610.33282#S3.E7)\. Appendix[C](https://arxiv.org/html/2610.33282#A3)describes the induced probabilistic model\. Either way,r^p\\hat\{r\}\_\{p\}enters only as the weight that mixes the two state\-conditional effects, so how it is used is fixed by the model rather than learned\.

## 4Training

We train modelsfθ​\(xp​v\)f\_\{\\theta\}\(x\_\{pv\}\)andgϕ​\(wv\)g\_\{\\phi\}\(w\_\{v\}\), in an end\-to\-end setting, on execution traces generated from training questions\. Each trace contains the observed interaction within agents\. Predictions are computed recursively using Eqs\.[6](https://arxiv.org/html/2610.33282#S3.E6)–[7](https://arxiv.org/html/2610.33282#S3.E7), and the model parametersϕ\\phiandθ\\thetaare optimized jointly by minimizing binary cross\-entropy between the predicted riskr^v\\hat\{r\}\_\{v\}and the correctness labelEvE\_\{v\}, summed over labeled nodes:

ℒ=∑v∈VBCE⁡\(r^v,Ev\)\.\\mathcal\{L\}=\\sum\_\{v\\in V\}\\operatorname\{BCE\}\(\\hat\{r\}\_\{v\},E\_\{v\}\)\.\(8\)Gradients therefore propagate through the recursive risk computation, allowing both the receiver\-local modelgϕg\_\{\\phi\}and the interaction modelfθf\_\{\\theta\}to be learned jointly\. Evaluation is performed on execution traces from held\-out questions \(§[6](https://arxiv.org/html/2610.33282#S6)\)\. Algorithm[1](https://arxiv.org/html/2610.33282#alg1)summarizes the complete procedure for recursive risk estimation, end\-to\-end training, and K\-fold prediction\.

## 5Evaluation Design

We evaluate four estimators under common training and evaluation protocol discussed in details in §[6](https://arxiv.org/html/2610.33282#S6)\.

Receiver\-Only\.The local estimatorσ⁡\(bv\)\\sigma\(b\_\{v\}\)\(Eq\.[1](https://arxiv.org/html/2610.33282#S3.E1)\) predicts the receiver’s error risk using three local uncertainty signals: verbalized uncertainty \(*verb\.*\), Maximum Sequence Probability \(*MSP*\), prediction entropy\.

Receiver\-Text\.Building on the Receiver\-Only model, this baseline adds context about the receiver’s specific task, role, and final text output, but strictly ignores any incoming messages from the sender\. We use this as our main benchmark to determine whether analyzing the actual upstream conversation gives us better error predictions than just looking at the receiver in isolation\.

Free learner\.An unconstrained edge model that directly estimates the interaction adjustmentδ¯​p​v\\bar\{\\delta\}\{pv\}from the interaction representationxp​vx\_\{pv\}\. Unlike our proposed model, it does not decompose the adjustment into sender\-state\-conditioned effects or prescribe how the parent risk is used\.

GSM8KMedQAVerb\.MSPEntropyVerb\.MSPEntropyEstimator0\.6B1\.7B4B0\.6B1\.7B4B0\.6B1\.7B4B0\.6B1\.7B4B0\.6B1\.7B4B0\.6B1\.7B4B*Sequential*Receiver\-Only0\.6220\.5310\.5180\.5410\.5870\.6780\.5700\.5880\.7200\.4950\.5160\.5310\.5080\.4860\.6860\.5120\.4860\.688Receiver\-Text0\.6680\.6200\.6740\.6220\.6030\.7100\.6480\.6220\.7310\.5260\.5680\.6100\.5360\.5650\.7050\.5240\.5610\.706Free Learner0\.6870\.6820\.6710\.6330\.5840\.6880\.6570\.5600\.6910\.4980\.5970\.6570\.5110\.5720\.6960\.5010\.5710\.704PropUQ\-MAS0\.6200\.6410\.5540\.3820\.4940\.7560\.3950\.5140\.7830\.5000\.5480\.6800\.4730\.4820\.7370\.4710\.4950\.750IE\-Prop \(Ours\)0\.7110\.7080\.6840\.6600\.6220\.7160\.6710\.6100\.7230\.5210\.6130\.6900\.5330\.5840\.7280\.5180\.5840\.743*Fully connected*Receiver\-Only0\.6160\.5930\.5070\.5910\.4770\.5340\.6030\.4880\.4570\.4930\.5090\.5910\.5500\.4570\.5260\.5620\.4590\.557Receiver\-Text0\.6960\.7160\.6520\.6510\.6820\.6650\.6550\.6700\.6760\.4880\.5710\.5940\.5290\.5600\.5520\.5340\.5540\.574Free Learner0\.7180\.7800\.7190\.7040\.7450\.7650\.7230\.7600\.7820\.5120\.5570\.6650\.4850\.5770\.6690\.4950\.5830\.707PropUQ\-MAS0\.6670\.7540\.6130\.5360\.6360\.7840\.5470\.6770\.8190\.5000\.5470\.6990\.5670\.5440\.6950\.5990\.5640\.742IE\-Prop \(Ours\)0\.7390\.7990\.7330\.7270\.7630\.7990\.7430\.7790\.8120\.5290\.5730\.6920\.5090\.5950\.6900\.5190\.6020\.727*Hierarchy*Receiver\-Only0\.5600\.7120\.5640\.6360\.5680\.7490\.6520\.5270\.7830\.4980\.5910\.6720\.5100\.5990\.6820\.5110\.6000\.725Receiver\-Text0\.7090\.7760\.7620\.7310\.6640\.7740\.7400\.6590\.7850\.5020\.6280\.6940\.4780\.6090\.6830\.4910\.6150\.715Free Learner0\.7230\.7800\.7710\.7360\.7790\.8550\.7520\.8150\.8720\.4800\.6260\.7000\.4710\.6820\.7520\.5010\.6770\.769PropUQ\-MAS0\.6710\.7100\.5630\.6390\.7250\.8650\.6530\.7480\.8910\.4710\.5740\.7040\.5140\.5870\.7500\.4840\.5750\.780IE\-Prop \(Ours\)0\.7480\.8020\.7840\.7590\.7940\.8710\.7690\.8350\.8860\.5010\.6450\.7200\.4900\.7000\.7720\.5230\.6960\.788

Table 2:Final\-node AUROC on GSM8K and MedQA by base\-model size \(Qwen3 0\.6B, 1\.7B and 4B\): in\-domain settings, seed mean per model size\. Columns are grouped by dataset, then uncertainty signal, then model size\.*Bold:*best value in each column within each topology\.*Underline:*second best value in each column within each topology\.IE\-Prop\.Our model from Eq\.[7](https://arxiv.org/html/2610.33282#S3.E7), which usesr^p\\hat\{r\}\_\{p\}to mix the interaction effects for correct and incorrect sender states\.

Recursion Control\.To isolate the contribution of recursively propagated risk, we train variants of IE\-Prop in which the propagated parent riskr^p\\hat\{r\}\_\{p\}is replaced by the parent’s local riskσ⁡\(bp\)\\sigma\(b\_\{p\}\)\. This comparison isolates whether recursively accumulated upstream information improves downstream risk estimation beyond the immediate parent’s local evidence and observed sender–receiver interaction\.

As an external reference we also report PropUQ\-MAS, computed from each agent’s verbalized uncertainty, MSP, and entropy with our implementation of its propagation rule\. We report AUROC for the final node\. The description of the considered local risk estimates and metric is provided in Appendix[G](https://arxiv.org/html/2610.33282#A7)and Appendix[H](https://arxiv.org/html/2610.33282#A8)respectively\.

These comparisons address whether upstream interaction information adds predictive signal beyond what is available from the receiver’s text alone, and how our interaction\-conditioned approach performs against existing propagation\-based uncertainty methods\.

## 6Experimental Setup

#### Systems\.

We evaluate three distinct four\-agent topologies instantiated with Qwen3 models\([Yang et al\., 2025](https://arxiv.org/html/2610.33282#bib.bib29)\)\(0\.6B, 1\.7B, and 4B variants\): a*sequential*pipeline where each agent observes only its immediate predecessor; a*fully connected*architecture where agents attend to all preceding outputs; and a*specialist hierarchy*where a final summarizer aggregates independent parallel responses\. Full architectural details, including specific agent roles and prompt\-level mitigations for degenerate copying behavior in the fully connected system, are deferred to Appendix[D](https://arxiv.org/html/2610.33282#A4)\.

#### Tasks and labels\.

We evaluate across three question\-answering benchmarks: GSM8K\([Cobbe et al\., 2021](https://arxiv.org/html/2610.33282#bib.bib26)\), MedQA\([Jin et al\., 2020](https://arxiv.org/html/2610.33282#bib.bib27)\), and the text\-only subset of WebQA\([Chang et al\., 2021](https://arxiv.org/html/2610.33282#bib.bib28)\)\. Treating each dataset, graph topology, and model\-size combination as an independent setting yields 27 distinct experimental evaluations for each methods\. To train and evaluate the risk estimators, each agent is assigned a binary correctness label by extracting the final answer from its visible output and comparing it against the ground truth\. Full details regarding train\-test data splits, automated answer extraction protocols \(e\.g\., deterministic vs\. LLM\-based read\-outs\), and task\-specific matching criteria are provided in Appendix[D](https://arxiv.org/html/2610.33282#A4)\.

#### In\-Domain\.

Estimators are trained on the execution traces from a dataset’s training split and evaluated on its corresponding held\-out test split\. Detailed in\-domain results are provided in §[7](https://arxiv.org/html/2610.33282#S7)\.

#### Transfer\.

To evaluate whether the learned risk signals generalize to unseen distributions, we employ a leave\-one\-dataset\-out transfer protocol\. For each architecture and model size, estimators are trained on execution traces from two datasets and evaluated on the held\-out third dataset\. Detailed cross\-dataset transfer results are in Appendix[E](https://arxiv.org/html/2610.33282#A5)\.

## 7Results

We evaluate whether realized inter\-agent exchanges improve downstream error\-risk estimation, whether sender\-state conditioning contributes beyond an unconstrained interaction model\. We then examine downstream error discrimination with correct upstream context and isolate the contribution of recursive risk propagation\.[Table2](https://arxiv.org/html/2610.33282#S5.T2)reports seed\-averaged final\-node AUROC for Qwen3\-0\.6B, 1\.7B, and 4B across datasets\.

### 7\.1Interaction information beyond receiver text

The receiver’s visible output provides useful evidence for error\-risk estimation, but does not exhaust the predictive information available in the execution trace\. In[Table2](https://arxiv.org/html/2610.33282#S5.T2), IE\-Prop exceeds Receiver\-Text in 46 of the 54 topology–dataset–uncertainty–model\-size combinations\. Averaged over the datasets and uncertainty signals, the gains for the 0\.6B, 1\.7B, and 4B models are respectively \+0\.015, \+0\.030, and \+0\.025 in sequential systems; \+0\.036, \+0\.060, and \+0\.123 in fully connected systems; and \+0\.023, \+0\.087, and \+0\.068 in the hierarchy\. Interaction information therefore improves mean discrimination at each evaluated size within all three topologies, although magnitude of the gain depends on setting\.

[Figure3](https://arxiv.org/html/2610.33282#S7.F3)shows that IE\-Prop also improves over Receiver\-Text on WebQA in all 12 evaluated settings, achieving the highest AUROC in nine\. It leads in most hierarchical settings and at the two smaller model sizes in sequential systems\. However, PropUQ\-MAS performs best at 4B in sequential systems for both MSP and entropy, indicating that the relative advantage depends on architecture and model size\.

Figure 3:Final\-node AUROC on WebQA for sequential and hierarchical systems using MSP and entropy across Qwen3 model sizes, averaged over seeds\.
### 7\.2State\-conditioned effects

We next compare IE with the Free Learner to examine whether the structured sender\-state\-conditioned update improves on an unconstrained interaction adjustment\. In[Table2](https://arxiv.org/html/2610.33282#S5.T2), IE achieves higher seed\-averaged AUROC in all 54 displayed combinations, covering every model size, topology, dataset, and uncertainty signal\. Averaged over datasets, uncertainty signals, and model sizes, the gains are approximately \+0\.026 in the sequential system, \+0\.021 in the fully connected system, and \+0\.019 in the hierarchy\.

The consistency of this comparison across model sizes supports the state\-conditioned formulation as a useful inductive bias for the evaluated GSM8K and MedQA settings\. This predictive advantage does not, by itself, establish that the two learned heads identify distinct error\-propagation and correction mechanisms; it shows that the structured formulation improves error\-risk discrimination relative to the Free Learner\. Additional experiments with oracle\-supervised sender\-state conditioning are presented in Appendix[F](https://arxiv.org/html/2610.33282#A6)\.

Figure 4:Predicted receiver risk on sequential MedQA traces with correct upstream answers\.*Top:*Risk distributions by receiver correctness\.*Bottom:*Median risk and AUROC across Qwen3 model sizes\.
### 7\.3Downstream risk discrimination with correct upstream context

A correct upstream response does not guarantee that the receiver will remain correct\.[Figure4](https://arxiv.org/html/2610.33282#S7.F4)examines sequential MedQA interactions restricted to cases where the sender’s answer is correct\. Within this subset, IE assigns higher predicted risk to incorrect receivers than to correct receivers at each evaluated model size\.

Discrimination improves across the three sizes in this diagnostic, with AUROC values of 0\.68, 0\.72, and 0\.88 for the 0\.6B, 1\.7B, and 4B models, respectively\. These results show that the estimator can distinguish downstream errors even when the upstream answer is correct\. They do not establish independence from estimated sender risk, since the model still uses that estimate when combining its interaction effects\. Rather, they demonstrate that its predictive utility extends to interactions in which downstream failure cannot be attributed simply to an incorrect upstream answer\.

### 7\.4Contribution of recursive risk propagation

Finally, we isolate the contribution of recursive propagation by comparing IE with a matched variant that replaces recursively estimated parent risk with local parent risk\. In the sequential ablation shown in[Figure5](https://arxiv.org/html/2610.33282#S7.F5), recursion improves final\-node AUROC in all nine displayed dataset–model\-size combinations\. The overall mean increases from 0\.631 to 0\.655 \(\+0\.024\)\.

Averaged over datasets, the gains are \+0\.024, \+0\.025, and \+0\.024 for the 0\.6B, 1\.7B, and 4B models, respectively\. Thus, the average benefit is similar across model sizes, although individual improvements vary\. The largest gain occurs on MedQA with the 4B model \(\+0\.040\), while gains averaged over model sizes are \+0\.026 on GSM8K, \+0\.026 on MedQA, and \+0\.019 on WebQA\. These results indicate that recursively accumulated parent risk provides predictive information beyond local parent uncertainty and immediate sender–receiver interaction in the evaluated sequential systems\.

Figure 5:Final\-node AUROC gains from recursive risk propagation in the sequential system using Qwen3 models\. Cells show gains and corresponding scores \(without → with recursion\), averaged over three seeds\.

## 8Discussion

Our results suggest that downstream error risk depends not only on upstream correctness but also on how receivers respond to upstream information\. As shown in[Section7\.3](https://arxiv.org/html/2610.33282#S7.SS3), IE\-Prop distinguishes downstream errors even when upstream answers are correct, illustrating why sender correctness alone is insufficient to characterize downstream risk\. Moreover, its improvement over the Free Learner \([Section7\.2](https://arxiv.org/html/2610.33282#S7.SS2)\) supports sender\-state\-conditioned interaction modeling as a useful predictive inductive bias\. The benefit of recursive propagation depends on the system’s information\-flow structure\. In sequential systems, earlier interactions reach the final agent only through intermediate agents, making recursively accumulated risk informative beyond the immediate interaction \([Section7\.4](https://arxiv.org/html/2610.33282#S7.SS4)\)\. In contrast, in shallow hierarchies, the final agent receives information directly from independent specialists, allowing interaction modeling to improve risk discrimination without multi\-hop recursive accumulation\. These findings suggest interaction modeling and recursive risk propagation provide complementary predictive benefits, whose relative importance depends on system topology\.

## 9Conclusion

In this work, we investigated post\-hoc agent\-level error\-risk estimation using execution traces from LLM\-based multi\-agent systems\. We introduced IE\-Prop, an approach that models the contribution of realized sender–receiver interactions to downstream error risk by learning separate effects conditioned on whether the sender is correct or incorrect\. By combining these effects using estimated sender risk and recursively propagating the resulting estimates through the execution graph, our method captures interaction\-dependent error dynamics beyond monotone propagation assumptions\. Across three multi\-agent architectures and three question\-answering benchmarks, our approach generally improves downstream error\-risk discrimination over receiver\-local and existing propagation\-based baselines\. Our analyses further demonstrate that interaction information remains predictive even when upstream answers are correct, while recursive propagation provides additional benefits in sequential systems\. Together, these findings underscore the value of modeling realized inter\-agent interactions for downstream error\-risk estimation\.

## Limitations

Our evaluation focuses on three four\-agent architectures, three question\-answering benchmarks, and one model family, leaving generalization to larger and more diverse multi\-agent systems for future work\. Differences in agent roles and prompting protocols also limit direct comparisons across architectures\. Although interaction modeling generally improves error\-risk discrimination, the gains vary across settings, and the distinct contributions of sender\-state conditioning and recursive propagation are not fully established in every topology\. Furthermore, our estimator operates post\-hoc and relies partly on automated correctness labels\. Finally, the additive interaction model approximates marginal inference, and its learned effects represent predictive associations rather than causal relationships\. Future work could explore broader agentic settings, alternative aggregation mechanisms, and more direct supervision of sender\-state effects\.

## Ethical Considerations

This work investigates error\-risk estimation in LLM\-based multi\-agent systems, with the goal of improving their reliability\. However, inaccurate risk estimates may create a false sense of confidence in incorrect outputs, particularly in high\-stakes domains such as medical question answering\. Our method is evaluated on established benchmarks and does not guarantee correctness or safety in real\-world deployments\. Additionally, the learned risk estimates may not generalize reliably to unseen domains, model families, or multi\-agent architectures\. Finally, the use of multiple LLM agents and additional risk\-estimation models incurs computational costs and associated environmental impacts\.

## Computational Resources\.

All experiments were conducted using a computing infrastructure equipped with four NVIDIA A100 GPUs\. These GPUs supported the complete experimental workload, including LLM hosting, multi\-agent inference, execution trace generation, and training of the risk\-estimation networks\.

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- Zhaoet al\.\(2025\)Q\. Zhao, D\. Li, Y\. Liu, W\. Cheng, Y\. Sun, M\. Oishi, T\. Osaki, K\. Matsuda, H\. Yao, C\. Zhao, H\. Chen, and X\. ZhaoUncertainty Propagation on LLM Agent\.InProceedings of the 63rd Annual Meeting of the Association for Computational Linguistics \(Volume 1: Long Papers\),pp\. 6064–6073\.External Links:[Document](https://dx.doi.org/10.18653/v1/2025.acl-long.302)Cited by:[Table 1](https://arxiv.org/html/2610.33282#S1.T1.2.6.1.1.1.2),[§2](https://arxiv.org/html/2610.33282#S2.SS0.SSS0.Px2.p1.1)\.

## Appendix AApproximation Bound

LetZv=bv\+∑pδp​v\(Ep\)Z\_\{v\}=b\_\{v\}\+\\sum\_\{p\}\\delta^\{\(E\_\{p\}\)\}\_\{pv\}, where the parent statesEPa⁡\(v\)E\_\{\\mathrm\{Pa\}\(v\)\}follow some distributionπ\\pi, and letμ=𝔼π​\[Zv\]\\mu=\\mathbb\{E\}\_\{\\pi\}\[Z\_\{v\}\]\. The exact marginal under the conditional model of Eq\.[5](https://arxiv.org/html/2610.33282#S3.E5)is𝔼π​\[σ⁡\(Zv\)\]\\mathbb\{E\}\_\{\\pi\}\[\\sigma\(Z\_\{v\}\)\], and Eq\.[7](https://arxiv.org/html/2610.33282#S3.E7)computesσ⁡\(μ\)\\sigma\(\\mu\)when given the exact parent marginals\.

#### Bound\.

Taylor’s theorem with the Lagrange remainder givesσ⁡\(Z\)=σ⁡\(μ\)\+σ′​\(μ\)​\(Z−μ\)\+12​σ′′​\(ξ\)​\(Z−μ\)2\\sigma\(Z\)=\\sigma\(\\mu\)\+\\sigma^\{\\prime\}\(\\mu\)\(Z\-\\mu\)\+\\tfrac\{1\}\{2\}\\sigma^\{\\prime\\prime\}\(\\xi\)\(Z\-\\mu\)^\{2\}for someξ\\xibetweenμ\\muandZZ\. Taking expectations, the linear term vanishes, so\|𝔼⁡\[σ⁡\(Z\)\]−σ⁡\(μ\)\|≤12​supz\|σ′′​\(z\)\|​Var​\(Z\)\|\\mathbb\{E\}\[\\sigma\(Z\)\]\-\\sigma\(\\mu\)\|\\leq\\tfrac\{1\}\{2\}\\sup\_\{z\}\|\\sigma^\{\\prime\\prime\}\(z\)\|\\,\\mathrm\{Var\}\(Z\)\. Withσ′′=σ⁡\(1−σ\)​\(1−2​σ\)\\sigma^\{\\prime\\prime\}=\\sigma\(1\-\\sigma\)\(1\-2\\sigma\), the supremum is attained atσ=12±12​3\\sigma=\\tfrac\{1\}\{2\}\\pm\\tfrac\{1\}\{2\\sqrt\{3\}\}and equals1/\(6​3\)1/\(6\\sqrt\{3\}\), so

\|𝔼⁡\[σ⁡\(Zv\)\]−σ⁡\(𝔼⁡\[Zv\]\)\|≤112​3​Var​\(Zv\)\.\\big\|\\mathbb\{E\}\[\\sigma\(Z\_\{v\}\)\]\-\\sigma\(\\mathbb\{E\}\[Z\_\{v\}\]\)\\big\|\\ \\leq\\ \\frac\{1\}\{12\\sqrt\{3\}\}\\,\\mathrm\{Var\}\(Z\_\{v\}\)\.\(9\)

#### Assumptions and interpretation\.

The bound uses only the first two moments ofZvZ\_\{v\}, so it holds for any parent\-state distribution, including correlated ones\. WritingΔp​v=δp​v\(1\)−δp​v\(0\)\\Delta\_\{pv\}=\\delta^\{\(1\)\}\_\{pv\}\-\\delta^\{\(0\)\}\_\{pv\}, we haveZv=bv\+∑pδp​v\(0\)\+∑pEp​Δp​vZ\_\{v\}=b\_\{v\}\+\\sum\_\{p\}\\delta^\{\(0\)\}\_\{pv\}\+\\sum\_\{p\}E\_\{p\}\\Delta\_\{pv\}and

Var⁡\(Zv\)=∑pΔp​v2​rp​\(1−rp\)\+2∑p<qΔp​vΔq​vCov\(Ep,Eq\)\.\\mathrm\{Var\}\(Z\_\{v\}\)=\\sum\_\{p\}\\Delta\_\{pv\}^\{2\}\\,r\_\{p\}\(1\-r\_\{p\}\)\\\\ \+2\\sum\_\{p<q\}\\Delta\_\{pv\}\\Delta\_\{qv\}\\,\\mathrm\{Cov\}\(E\_\{p\},E\_\{q\}\)\.The error therefore vanishes when the two effects coincide, and grows with strong, uncertain and positively co\-varying parent effects; correlations do not enter the update itself but change the variance that controls its error\. Exactness holds wheneverVar⁡\(Zv\)=0\\mathrm\{Var\}\(Z\_\{v\}\)=0: when the parent states are known, whenδp​v\(0\)=δp​v\(1\)\\delta^\{\(0\)\}\_\{pv\}=\\delta^\{\(1\)\}\_\{pv\}on every uncertain edge, or when every parent configuration with positive probability gives the same logit\. The bound is local: it compares the update and exact marginalization under the same parent distribution and does not account for errors in propagated parent marginals\.

#### Single parent\.

Fork=1k=1, as in our sequential system,Var⁡\(Zv\)=Δp​v2​rp​\(1−rp\)≤Δp​v2/4\\mathrm\{Var\}\(Z\_\{v\}\)=\\Delta\_\{pv\}^\{2\}\\,r\_\{p\}\(1\-r\_\{p\}\)\\leq\\Delta\_\{pv\}^\{2\}/4, so the error is at mostΔp​v2/\(48​3\)\\Delta\_\{pv\}^\{2\}/\(48\\sqrt\{3\}\)and is largest for parents of maximal uncertainty\. The final agent of the fully connected system hask=3k=3parents that share the first agent as an ancestor, so the covariance term applies there\.

## Appendix BNatural\-Parameter Interpretation and Training Analysis

### B\.1Reverse\-KL barycenter

Each parent configurationeeinduces a receiver distributionpe=Bernoulli⁡\(σ⁡\(ηe\)\)p\_\{e\}=\\mathrm\{Bernoulli\}\(\\sigma\(\\eta\_\{e\}\)\)withηe=bv\+∑pδp​v\(ep\)\\eta\_\{e\}=b\_\{v\}\+\\sum\_\{p\}\\delta^\{\(e\_\{p\}\)\}\_\{pv\}\. Consider summarizing the collection\{pe\}\\\{p\_\{e\}\\\}by the single Bernoulliq=Bernoulli⁡\(r\)q=\\mathrm\{Bernoulli\}\(r\)that minimizes𝔼e∼π\[DKL\(q∥pe\)\]\\mathbb\{E\}\_\{e\\sim\\pi\}\[D\_\{\\mathrm\{KL\}\}\(q\\\|p\_\{e\}\)\]\. Since∂rDKL\(q∥pe\)=logit\(r\)−ηe\\partial\_\{r\}D\_\{\\mathrm\{KL\}\}\(q\\\|p\_\{e\}\)=\\operatorname\{logit\}\(r\)\-\\eta\_\{e\}and the objective is strictly convex, the minimizer satisfieslogit⁡\(r\)=𝔼π​\[ηe\]\\operatorname\{logit\}\(r\)=\\mathbb\{E\}\_\{\\pi\}\[\\eta\_\{e\}\], i\.e\.r=σ⁡\(𝔼⁡\[Zv\]\)r=\\sigma\(\\mathbb\{E\}\[Z\_\{v\}\]\)\. With the additive natural parameter,𝔼π​\[ηe\]=bv\+∑p\[\(1−rp\)​δp​v\(0\)\+rp​δp​v\(1\)\]\\mathbb\{E\}\_\{\\pi\}\[\\eta\_\{e\}\]=b\_\{v\}\+\\sum\_\{p\}\[\(1\-r\_\{p\}\)\\delta^\{\(0\)\}\_\{pv\}\+r\_\{p\}\\delta^\{\(1\)\}\_\{pv\}\], which depends onπ\\pionly through the individual parent marginalsrpr\_\{p\}, for any dependence between parents\. This is an interpretation of the scalar composition rule, not a justification that it yields the exact risk, and not a claim that it is empirically preferable to other operators\. The exact mixture over a binary variable is itself Bernoulli with mean𝔼π​\[σ⁡\(ηe\)\]\\mathbb\{E\}\_\{\\pi\}\[\\sigma\(\\eta\_\{e\}\)\]; the update instead aggregates the state\-conditioned distributions in reverse\-KL \(natural\-parameter\) geometry\. Eq\.[7](https://arxiv.org/html/2610.33282#S3.E7)further replaces the exact parent marginals by propagated ones, so its error combines the local term of Eq\.[9](https://arxiv.org/html/2610.33282#A1.E9)with errors accumulated upstream\.

## Appendix CInduced Probabilistic Model

Conditioned on a completed traceXX, the node\-wise conditionals of Eq\.[5](https://arxiv.org/html/2610.33282#S3.E5)can be combined into the DAG factorizationP⁡\(E∣X\)=∏vP⁡\(Ev∣EPa⁡\(v\),X\)P\(E\\mid X\)=\\prod\_\{v\}P\(E\_\{v\}\\mid E\_\{\\mathrm\{Pa\}\(v\)\},X\), a Bayesian network over error indicators whose structure is the communication graph\. This is a modeling assumption about dependencies among correctness variables given the trace, not a claim about the causal structure of failures, and it does not model how messages were generated\. The deployed estimator does not maintain this joint model; it propagates only the scalar risks of Eq\.[7](https://arxiv.org/html/2610.33282#S3.E7)\.

#### Relation to sigmoid belief networks and log\-linear models\.

Each conditional is a logistic regression on parent\-state indicators, i\.e\. a sigmoid belief network\([Neal, 1992](https://arxiv.org/html/2610.33282#bib.bib33)\)whose weightsΔp​v\\Delta\_\{pv\}and interceptbv\+∑pδp​v\(0\)b\_\{v\}\+\\sum\_\{p\}\\delta^\{\(0\)\}\_\{pv\}are computed from the trace\. Equivalently, the parent\-state indicators enter a log\-linear model of the receiver’s error with first\-order interaction terms only\([Besag, 1974](https://arxiv.org/html/2610.33282#bib.bib32)\)\. Noisy\-OR\([Pearl, 1988](https://arxiv.org/html/2610.33282#bib.bib35)\), the contamination rule, replaces the logistic link by a product of per\-parent failure terms, so a wrong parent can only increase the receiver’s risk\.

#### Exact marginalization and parent dependence\.

The exact marginal of a receiver is∑eπv​\(e\)​σ​\(ηv​\(e\)\)\\sum\_\{e\}\\pi\_\{v\}\(e\)\\,\\sigma\(\\eta\_\{v\}\(e\)\), whereπv\\pi\_\{v\}is the joint distribution of its parents’ states\. Parents that share ancestors are dependent under the induced network even when the conditionals are fixed, soπv\\pi\_\{v\}is not a product of marginals in general\. Exact inference must sum over the joint states of the latent ancestors that a receiver depends on\. Variable elimination reduces this cost to exponential in the induced width of the elimination order, but akk\-parent logistic receiver still requires the2k2^\{k\}\-entry joint of its parents, and overlapping ancestry forces joint tables over shared ancestors\. In the controlled sequential system every receiver has one parent, and in the fully connected system the final agent’s three dependent parents have only eight joint states, so exact marginalization is cheap in both; the update matters for wider graphs\.

## Appendix DControlled QA: Details and Full Results

SolverAlternativeCheckerAggregatorFigure 6:The fully connected system\. Every agent reads the visible outputs of all earlier agents \(six edges\)\. The Alternative\-Method agent and the Checker first write a private draft without the earlier outputs; the draft is never passed on\.#### Labels\.

Each agent’s label is derived from the answer stated in its visible output, after hidden reasoning and uncertainty tags are removed, and the same procedure labels training and test traces; training uses every node’s label as a target, not only the final node’s\. On GSM8K the answer is the last boxed number, or otherwise the last number in the text, and it is compared with the gold number\. On MedQA and WebQA the answer is extracted by a read\-out with the same Qwen3 model that generated the trace: given the question, the agent’s role and its visible response, but not the gold answer, the model returns a JSON object naming the option letter \(MedQA\) or the complete\-sentence answer \(WebQA\) that the response supports, or UNKNOWN, and it is instructed not to solve the question itself\. Decoding is greedy with thinking disabled and a short output budget, and an invalid JSON response is retried once\. Correctness is decided in code: exact option match for MedQA, and full recall of normalized gold\-answer tokens against any accepted answer for WebQA\. A node whose response supports no answer is unlabeled, and training traces with any unlabeled node are dropped\. In the hierarchy, the Summarizer’s label is the system evaluator’s verdict; in the fully connected system every agent is labeled as in the sequential system\. Test items whose final node is unlabeled are dropped, and oracle evaluations additionally drop items with an unlabeled parent; oracle predictions were not computed for the sequential MedQA 4B setting\.

Hyperparameters are shared by every cell: hidden width 256, one hidden layer, dropout 0\.1, AdamW with learning rate10−310^\{\-3\}and weight decay10−410^\{\-4\}, cosine schedule, batch 256, at most 30 epochs with patience 5, and five question\-level folds; each fold model is selected by validation final\-node AUROC and the five fold predictions are averaged at test time\. The oracle numbers should not be read as attainable targets, since parent and final\-node labels largely coincide\.

## Appendix ECross\-dataset transfer

Figure 7:Cross\-dataset transfer performance across three multi\-agent topologies, showing final\-node AUROC and differences relative to Receiver\-Only and PropUQ\-MAS baselines\.Figure[7](https://arxiv.org/html/2610.33282#A5.F7)evaluates leave\-one\-dataset\-out transfer with entropy\-based uncertainty: learned estimators are trained on traces from two datasets and evaluated on the held\-out third dataset\. It reports IE’s final\-node AUROC and its difference from Receiver\-Only and the training\-free PropUQ\-MAS\. The held\-out IE AUROCs are 0\.608, 0\.561, and 0\.575 in the sequential system; 0\.605, 0\.622, and 0\.546 in the fully connected system; and 0\.755, 0\.624, and 0\.556 in the hierarchy, for GSM8K, MedQA, and WebQA, respectively\.

IE improves over Receiver\-Only most clearly on held\-out MedQA across all three graphs\. Its advantage over PropUQ\-MAS is less consistent: the paired intervals overlap zero in seven of the nine graph–held\-out\-dataset comparisons, while sequential GSM8K favors IE and fully connected GSM8K favors PropUQ\-MAS\. Transfer therefore offers evidence that learned interaction information can remain useful outside its training datasets, but not that IE uniformly dominates a propagation\-based uncertainty reference\. Because Figure[7](https://arxiv.org/html/2610.33282#A5.F7)does not include Receiver\-Text or matched recursion controls, it does not establish that the transfer improvements arise from state\-conditioned recursion or exceed a receiver\-text predictor\.

## Appendix FOracle\-Supervised Sender\-State Conditioning

The original interaction\-effect \(IE\) objective supervises the sender\-state headsδp​v\(0\)\\delta\_\{pv\}^\{\(0\)\}andδp​v\(1\)\\delta\_\{pv\}^\{\(1\)\}only through their risk\-weighted mixture\. Consequently, good predictions need not imply that the heads are aligned with sender correctness\. To test this, we add an oracle\-conditioned training loss that selects the head corresponding to the observed sender label:

E^i​voracle\\displaystyle\\hat\{E\}\_\{iv\}^\{\\mathrm\{oracle\}\}=σ⁡\(bi​v\+∑p∈Pa⁡\(v\)δi​p​v\(Ei​p\)\),\\displaystyle=\\sigma\\left\(b\_\{iv\}\+\\sum\_\{p\\in\\mathrm\{Pa\}\(v\)\}\\delta\_\{ipv\}^\{\(E\_\{ip\}\)\}\\right\),\(10\)ℒoracle\\displaystyle\\mathcal\{L\}\_\{\\mathrm\{oracle\}\}=14​N​∑i=1N∑v∈𝒱ieligibleBCE⁡\(E^i​voracle,Ei​v\),\\displaystyle=\\frac\{1\}\{4N\}\\sum\_\{i=1\}^\{N\}\\sum\_\{v\\in\\mathcal\{V\}\_\{i\}^\{\\mathrm\{eligible\}\}\}\\operatorname\{BCE\}\\left\(\\hat\{E\}\_\{iv\}^\{\\mathrm\{oracle\}\},E\_\{iv\}\\right\),ℒtotal\\displaystyle\\mathcal\{L\}\_\{\\mathrm\{total\}\}=ℒpred\+λ​ℒoracle\.\\displaystyle=\\mathcal\{L\}\_\{\\mathrm\{pred\}\}\+\\lambda\\mathcal\{L\}\_\{\\mathrm\{oracle\}\}\.where𝒱ieligible\\mathcal\{V\}\_\{i\}^\{\\mathrm\{eligible\}\}contains non\-root nodes with known receiver and parent labels\. The originalℒpred\\mathcal\{L\}\_\{\\mathrm\{pred\}\}is unchanged and uses the same1/\(4​N\)1/\(4N\)normalization\. At test time, inference uses the original soft recursive mixture without sender labels\.

We evaluate all three architectures on GSM8K, MedQA, and WebQA, across three Qwen3 sizes and three uncertainty signals \(three seeds; five question\-level folds per seed\)\. We selectλ=0\.1\\lambda=0\.1from\{0,0\.1,0\.3,1,3\}\\\{0,0\.1,0\.3,1,3\\\}by mean out\-of\-fold final\-node AUROC on training data\. Relative to the original IE model, the mean change in deployed AUROC is\+0\.004\+0\.004\(sequential\), approximately00\(fully connected\), and−0\.001\-0\.001\(hierarchy\)\. Thus, oracle supervision leaves predictive performance largely unchanged\.

To assess head alignment, we compute the oracle\-state swap gap:Gswap=AUROC⁡\(Ep\)−AUROC⁡\(1−Ep\)G\_\{\\mathrm\{swap\}\}=\\operatorname\{AUROC\}\(E\_\{p\}\)\-\\operatorname\{AUROC\}\(1\-E\_\{p\}\), where each receiver uses either the observed or flipped sender labels to select its incoming heads\. Withλ=0\\lambda=0, the mean gaps are\+0\.271\+0\.271,−0\.050\-0\.050, and−0\.283\-0\.283for sequential, fully connected, and hierarchical systems, respectively; atλ=0\.1\\lambda=0\.1, they increase to\+0\.570\+0\.570,\+0\.740\+0\.740, and\+0\.679\+0\.679\. These oracle diagnostics use cross\-fitted training traces that also served in fold\-level validation; independent test traces corroborate the trend in the sequential system\. Swapping estimated sender risks in the deployed soft rule likewise produces a larger AUROC gap after oracle supervision\. These results support improved*sender\-state alignment*, but do not identify causal error\-transmission or correction effects: strong sender–receiver correctness correlations can also drive large swap gaps\.

## Appendix GLocal Uncertainty Signals

We consider three agent\-local uncertainty signals: verbalized uncertainty, Maximum Sequence Probability \(MSP\), and prediction entropy\. Each signal is evaluated independently as an input to the local risk estimatorgϕg\_\{\\phi\}in Eq\.[1](https://arxiv.org/html/2610.33282#S3.E1)\. Lethvh\_\{v\}denote the generation context of agentvv, and letyv=\(yv,1,…,yv,Tv\)y\_\{v\}=\(y\_\{v,1\},\\ldots,y\_\{v,T\_\{v\}\}\)denote the generated token sequence used for uncertainty estimation\.

#### Verbalized uncertainty\.

Verbalized uncertainty is obtained from the model’s self\-reported confidence in its generated answer\([Lin et al\., 2022](https://arxiv.org/html/2610.33282#bib.bib14);[Tian et al\., 2023](https://arxiv.org/html/2610.33282#bib.bib15);[Xiong et al\., 2024](https://arxiv.org/html/2610.33282#bib.bib16)\)\. Letcv∈\[0,1\]c\_\{v\}\\in\[0,1\]denote the confidence expressed by agentvv, where larger values indicate greater confidence\. We convert this to an uncertainty score as

uvverb=1−cv\.u\_\{v\}^\{\\mathrm\{verb\}\}=1\-c\_\{v\}\.\(11\)Thus, larger values indicate greater self\-reported uncertainty\.

#### Maximum Sequence Probability \(MSP\)\.

MSP uses the probability assigned by the language model to the generated token sequence\. LetpLM​\(yv,t∣hv,yv,<t\)p\_\{\\mathrm\{LM\}\}\(y\_\{v,t\}\\mid h\_\{v\},y\_\{v,<t\}\)denote the conditional probability of the generated tokenyv,ty\_\{v,t\}\. We compute the length\-normalized sequence probability as

svMSP=exp⁡\(1Tv​∑t=1Tvlog⁡pLM​\(yv,t∣hv,yv,<t\)\)\.s\_\{v\}^\{\\mathrm\{MSP\}\}=\\exp\\left\(\\frac\{1\}\{T\_\{v\}\}\\sum\_\{t=1\}^\{T\_\{v\}\}\\log p\_\{\\mathrm\{LM\}\}\(y\_\{v,t\}\\mid h\_\{v\},y\_\{v,<t\}\)\\right\)\.\(12\)The corresponding uncertainty score is

uvMSP=1−svMSP\.u\_\{v\}^\{\\mathrm\{MSP\}\}=1\-s\_\{v\}^\{\\mathrm\{MSP\}\}\.\(13\)Length normalization reduces the dependence of the score on response length, while lower sequence probability indicates greater uncertainty\.

#### Prediction entropy\.

Prediction entropy measures uncertainty in the model’s next\-token distributions\. Letpv,t​\(k\)p\_\{v,t\}\(k\)denote the predicted probability of tokenkkat generation steptt, and let𝒱\\mathcal\{V\}be the vocabulary\. The token\-level predictive entropy is

Hv,t=−∑k∈𝒱pv,t\(k\)logpv,t\(k\)\.H\_\{v,t\}=\-\\sum\_\{k\\in\\mathcal\{V\}\}p\_\{v,t\}\(k\)\\log p\_\{v,t\}\(k\)\.\(14\)We aggregate token\-level entropies over the generated sequence:

uvent=1Tv​∑t=1TvHv,t\.u\_\{v\}^\{\\mathrm\{ent\}\}=\\frac\{1\}\{T\_\{v\}\}\\sum\_\{t=1\}^\{T\_\{v\}\}H\_\{v,t\}\.\(15\)Higher entropy indicates a more dispersed predictive distribution and therefore greater uncertainty\. Unlike semantic entropy\([Kuhn et al\., 2023](https://arxiv.org/html/2610.33282#bib.bib13);[Farquhar et al\., 2024](https://arxiv.org/html/2610.33282#bib.bib31)\), this signal operates directly on token\-level distributions without grouping sampled responses by semantic equivalence\.

#### Integration with local risk estimation\.

For each uncertainty signals∈\{verb,MSP,ent\}s\\in\\\{\\mathrm\{verb\},\\mathrm\{MSP\},\\mathrm\{ent\}\\\}, we use the corresponding scalaruv\(s\)u\_\{v\}^\{\(s\)\}as the agent\-local inputwvw\_\{v\}to the learned local estimator:

bv=gϕ​\(uv\(s\)\)\.b\_\{v\}=g\_\{\\phi\}\(u\_\{v\}^\{\(s\)\}\)\.\(16\)The local error\-risk estimate isσ⁡\(bv\)\\sigma\(b\_\{v\}\), which serves as the Receiver\-Only prediction and provides the local component of the interaction\-effect propagation model\. All three signals are evaluated separately under the same experimental protocol\.

## Appendix HEvaluation Metric

We evaluate error\-risk estimation using the Area Under the Receiver Operating Characteristic Curve \(AUROC\)\. For each evaluated final\-node responseii, letEi∈\{0,1\}E\_\{i\}\\in\\\{0,1\\\}denote its correctness label, whereEi=1E\_\{i\}=1indicates an incorrect answer andEi=0E\_\{i\}=0indicates a correct answer\. Letr^i∈\[0,1\]\\hat\{r\}\_\{i\}\\in\[0,1\]denote the corresponding predicted error risk\.

The receiver operating characteristic \(ROC\) curve plots the true positive rate \(TPR\) against the false positive rate \(FPR\) as the decision thresholdτ\\tauvaries:

TPR⁡\(τ\)\\displaystyle\\mathrm\{TPR\}\(\\tau\)=P⁡\(r^i≥τ∣Ei=1\),\\displaystyle=P\(\\hat\{r\}\_\{i\}\\geq\\tau\\mid E\_\{i\}=1\),\(17\)FPR⁡\(τ\)\\displaystyle\\mathrm\{FPR\}\(\\tau\)=P⁡\(r^i≥τ∣Ei=0\)\.\\displaystyle=P\(\\hat\{r\}\_\{i\}\\geq\\tau\\mid E\_\{i\}=0\)\.AUROC summarizes the area under this curve and can equivalently be interpreted as

AUROC=P⁡\(r^\+\>r^−\)\+12​P​\(r^\+=r^−\),\\mathrm\{AUROC\}=P\(\\hat\{r\}^\{\+\}\>\\hat\{r\}^\{\-\}\)\+\\frac\{1\}\{2\}P\(\\hat\{r\}^\{\+\}=\\hat\{r\}^\{\-\}\),\(18\)wherer^\+\\hat\{r\}^\{\+\}andr^−\\hat\{r\}^\{\-\}are the predicted risks of randomly selected incorrect and correct responses, respectively\.

An AUROC of0\.50\.5corresponds to random discrimination, while1\.01\.0indicates perfect separation between incorrect and correct responses\. Higher AUROC therefore indicates better discrimination of downstream errors\. Unlike threshold\-dependent classification metrics, AUROC evaluates the ranking of predicted risks across all decision thresholds, without requiring a fixed operating threshold\. It measures discrimination rather than probability calibration\. We compute AUROC on held\-out final\-node responses and report seed\-averaged results where applicable\.

Algorithm 1Interaction\-effect propagation: inference, training andKK\-fold prediction\.1:Trace

G=\(V,ℰ\)G=\(V,\\mathcal\{E\}\)for task

qq; agent

vvhas role

ρv\\rho\_\{v\}, visible output

mvm\_\{v\}, local signal

wvw\_\{v\}
2:procedurePropagate\(

G;θ,ϕ,T=1G;\\,\\theta,\\phi,T\{=\}1\)

3:for

\(p,v\)∈ℰ\(p,v\)\\in\\mathcal\{E\}do

4:

xp​v←Enc⁡\(q,ρp,mp,ρv,mv\)x\_\{pv\}\\leftarrow\\operatorname\{Enc\}\(q,\\rho\_\{p\},m\_\{p\},\\rho\_\{v\},m\_\{v\}\)⊳\\trianglerightEq\.[2](https://arxiv.org/html/2610.33282#S3.E2); frozen

5:

\(δp​v\(0\),δp​v\(1\)\)←fθ​\(xp​v\)\(\\delta^\{\(0\)\}\_\{pv\},\\delta^\{\(1\)\}\_\{pv\}\)\\leftarrow f\_\{\\theta\}\(x\_\{pv\}\)
6:endfor

7:for

v∈Vv\\in Vin topological orderdo

8:

bv←gϕ​\(wv\)b\_\{v\}\\leftarrow g\_\{\\phi\}\(w\_\{v\}\)⊳\\trianglerightEq\.[1](https://arxiv.org/html/2610.33282#S3.E1)

9:for

p∈Pa⁡\(v\)p\\in\\mathrm\{Pa\}\(v\)do

10:

δ¯p​v←\(1−r^p\)​δp​v\(0\)\+r^p​δp​v\(1\)\\bar\{\\delta\}\_\{pv\}\\leftarrow\(1\-\\hat\{r\}\_\{p\}\)\\,\\delta^\{\(0\)\}\_\{pv\}\+\\hat\{r\}\_\{p\}\\,\\delta^\{\(1\)\}\_\{pv\}⊳\\trianglerightEq\.[6](https://arxiv.org/html/2610.33282#S3.E6)

11:endfor

12:

r^v←σ⁡\(bv\+∑p∈Pa⁡\(v\)δ¯p​v\)\\hat\{r\}\_\{v\}\\leftarrow\\sigma\\bigl\(b\_\{v\}\+\\sum\_\{p\\in\\mathrm\{Pa\}\(v\)\}\\bar\{\\delta\}\_\{pv\}\\bigr\)⊳\\trianglerightEq\.[7](https://arxiv.org/html/2610.33282#S3.E7)

13:endfor

14:return

\{r^v\}v∈V\\\{\\hat\{r\}\_\{v\}\\\}\_\{v\\in V\}⊳\\trianglerightroots:r^v=σ⁡\(bv\)\\hat\{r\}\_\{v\}=\\sigma\(b\_\{v\}\)

15:endprocedure

16:

17:procedureTrain\(

𝒟\\mathcal\{D\}\)⊳\\trianglerightlabeled traces; §[4](https://arxiv.org/html/2610.33282#S4)

18:split

𝒟\\mathcal\{D\}into

KKquestion\-level folds

𝒟1,…,𝒟K\\mathcal\{D\}\_\{1\},\\dots,\\mathcal\{D\}\_\{K\}
19:for

k=1,…,Kk=1,\\dots,Kdo

20:initialize

θk,ϕk\\theta\_\{k\},\\phi\_\{k\}
21:repeat⊳\\trianglerightepochs; minibatchesB⊂𝒟∖𝒟kB\\subset\\mathcal\{D\}\\setminus\\mathcal\{D\}\_\{k\}

22:

\{r^v\}←Propagate​\(G,θk,ϕk\)\\\{\\hat\{r\}\_\{v\}\\\}\\leftarrow\\textsc\{Propagate\}\(G;\\,\\theta\_\{k\},\\phi\_\{k\}\),

G∈BG\\in B
23:

ℒ←∑G∈B∑v∈VBCE⁡\(r^v,Ev\)\\mathcal\{L\}\\leftarrow\\sum\_\{G\\in B\}\\sum\_\{v\\in V\}\\operatorname\{BCE\}\(\\hat\{r\}\_\{v\},E\_\{v\}\)⊳\\trianglerightEq\.[8](https://arxiv.org/html/2610.33282#S4.E8)

24:AdamW step on

∇θk,ϕkℒ\\nabla\_\{\\theta\_\{k\},\\phi\_\{k\}\}\\mathcal\{L\}⊳\\trianglerightthrough allr^p\\hat\{r\}\_\{p\}

25:untilfinal\-node AUROC on

𝒟k\\mathcal\{D\}\_\{k\}stops improving

26:keep the best epoch; record its node logits on

𝒟k\\mathcal\{D\}\_\{k\}
27:endfor

28:return

\{\(θk,ϕk\)\}k=1K\\\{\(\\theta\_\{k\},\\phi\_\{k\}\)\\\}\_\{k=1\}^\{K\}
29:endprocedure

30:

31:functionPredict\(

GG\)

32:return

1K​∑k=1KPropagate​\(G,θk,ϕk,T\)\\frac\{1\}\{K\}\\sum\_\{k=1\}^\{K\}\\textsc\{Propagate\}\(G;\\,\\theta\_\{k\},\\phi\_\{k\},T\)
33:endfunction