Contagion Networks: Evaluator Bias Propagation in Multi-Agent LLM Systems Liu Zewen Qilu Institute of Technology, School of Software Engineering Tai’an, Shandong, China [email protected]
arXiv:2606.20493v1 [cs.LG] 18 Jun 2026
June 2026
Abstract When large language models serve as evaluators in multi-agent systems, their systematic evaluation biases propagate through the agent network. We introduce Contagion Networks, a formal framework for measuring how evaluator biases spread across interacting LLM agents. In a controlled 3-agent experiment using DeepSeek-chat with three distinct evaluator bias profiles (structured, balanced, evidence-based), we measure the Cross-Agent Contagion Matrix Γ3 and find that evaluator biases consistently propagate between agents (γ ∈ [0.157, 0.352]), even within the same underlying model. We identify three propagation regimes governed by the spectral radius ρ(ΓN ), and demonstrate that homogeneous-model agents produce contagion coefficients 3–5× weaker than cross-model coefficients observed in prior work (MM-EPC: γ ≈ 0.85–1.3), placing them in the suppression regime. We show that increasing evaluator committee size from k = 1 to k = 3 reduces effective contagion by 72.4%, providing an actionable mitigation strategy. We release the open-source Contagion Network experimental framework.
1
Introduction
Multi-agent LLM systems represent a rapidly maturing paradigm for complex reasoning, code generation, and autonomous task completion [7, 8]. In these systems, agents evaluate each other’s outputs to guide collaboration, allocate sub-tasks, and select optimal responses [12, 7]. This peer evaluation mechanism is structurally analogous to the LLM-as-judge paradigm [2], but with a critical difference: the evaluations form a closed feedback loop where biased judgments from one agent directly shape another agent’s subsequent outputs. Our prior work on Multimodal Evaluator Preference Collapse (MM-EPC) [1] demonstrated that evaluator preferences can contaminate strategy selection across modalities. Here, we ask a fundamentally different question: Do evaluator biases propagate across agents, and if so, under what conditions do they cascade into system-wide preference collapse? Consider a concrete scenario: Agent A (GPT-4o) evaluates Agent B’s (DeepSeek) code generation output. GPT-4o, shaped by RLHF training on structured explanations, strongly prefers step-by-step reasoning. This feedback causes DeepSeek to increasingly adopt step-bystep strategies. Now Agent B evaluates Agent C’s (Claude) output—and applies the same step-by-step preference it absorbed from Agent A. Claude, in turn, shifts toward step-by-step. The bias has propagated two hops: A → B → C. If the process iterates, the entire agent network converges to a single strategy, eliminating the very cognitive diversity that multi-agent systems are designed to exploit. We formalize this phenomenon as Contagion Networks and make the following contributions:
1
1. Cross-Agent Contagion Matrix ΓN : We extend the 2 × 2 cross-modal contagion matrix to N agents, providing a quantitative framework for measuring evaluator bias propagation in arbitrary agent topologies.
2. Propagation Regime Theorem: We characterize three dynamical regimes (suppression/persistence/cascad governed by the spectral radius ρ(ΓN ), and prove that ρ(ΓN ) > 1 is the cascade condition for fully-connected networks. 3. Empirical Suppression Validation: In a 3-agent chain experiment, we observe rapid bias attenuation across hops (β3 = 0.0055), confirming the suppression regime for homogeneousmodel agents. 4. Critical Diversity Threshold: We prove that k ≥ 3 evaluators with diverse preference profiles reduce effective contagion below the cascade threshold, and verify that even in the suppression regime, increasing committee size from k = 1 to k = 3 reduces γeff by 72.4%. 5. Open Framework: We release the Contagion Network experimental framework, enabling the community to measure ΓN for arbitrary agent ensembles.
2
Related Work
2.1
Multi-Agent LLM Systems
The rapid maturation of multi-agent frameworks [7, 8, 9, 10] has established agent collaboration as a core AI paradigm. AutoGen [7] introduced structured multi-agent conversations; MetaGPT [8] applied software engineering workflows to agent coordination; ChatDev [11] demonstrated end-to-end software development via agent roles. Critically, all these systems rely on agents evaluating each other’s outputs—yet none analyze whether this peer evaluation mechanism introduces systematic bias amplification. While these frameworks focus on efficiency (task completion, communication overhead), we study fidelity: does the evaluation signal remain faithful to task quality as it propagates through the agent network?
2.2
Evaluator Bias and Preference Collapse
MM-EPC [1] demonstrated that GPT-4o evaluator preferences contaminate strategy selection in cross-modal settings (text ↔ visual), with asymmetric contagion coefficients γV →T > γT →V . The broader LLM-as-judge literature documents systematic biases including position bias [2], verbosity bias [3], and self-preference amplification [4]. Reward overoptimization [5] and sycophancy [6] demonstrate how agents exploit evaluator weaknesses. However, all prior work studies single-evaluator or pairwise evaluation dynamics. Our contribution is the first analysis of bias propagation in multi-hop agent evaluation chains, revealing that evaluator biases behave like infectious agents in a contact network.
2.3
Epidemiological Models in AI Systems
The analogy between information propagation and disease spread has a rich history in network science [13]. SIS (Susceptible-Infected-Susceptible) and SIR (Susceptible-Infected-Recovered) models characterize threshold dynamics on complex networks. We draw on this tradition to model evaluator bias as an “infection” that propagates through agent interaction graphs, with the contagion matrix ΓN serving as the analog of the next-generation matrix in epidemiology. This framing enables us to derive precise cascade conditions and diversity thresholds that are empirically testable.
2
Table 1: Systematic comparison with related phenomena. Contagion Networks is the only framework simultaneously capturing multi-agent, multi-hop propagation dynamics. Framework
Agent Network
Propag. Dynamics
MultiHop
Cascade Threshold
Mitig. Theory
LLM-as-Judge [2] MM-EPC [1] Reward Overopt. [5] Multi-Agent Frameworks [7]
— — — ✓
— ✓ — —
— — — —
— — ✓ —
— — — —
Contagion Networks (ours)
✓
✓
✓
✓
✓
3
Contagion Network Model
3.1
Formal Definition
Consider a system of N LLM agents A = {A1 , A2 , . . . , AN }. Each agent Ai maintains a strategy space S = {s1 , s2 , . . . , sK } and an associated weight distribution wi ∈ ∆K−1 (the K-dimensional probability simplex). 3.1.1
Test-Time Reinforcement Learning (TTRL)
Agents update their strategy weights via Test-Time Reinforcement Learning (TTRL), a parameter-free adaptation mechanism introduced in MM-EPC [1]. At each round, the agent generates two responses ya , yb using strategies sa , sb sampled from Softmax(wi ). An evaluator compares the responses and designates a winner sw and loser sℓ . The agent updates weights multiplicatively: wsw ← wsw · (1 + αwin ),
wsℓ ← wsℓ · (1 − αlose )
(1)
with learning rates αwin = 0.08, αlose = 0.04. After each update, weights are normalized to sum to 1 and clipped to a minimum of 0.01 to prevent strategy extinction. TTRL requires no parameter updates to the underlying LLM—only multiplicative adjustments to the strategy sampling distribution—making it lightweight and compatible with API-only LLM access. When agent Aj evaluates agent Ai ’s output, the evaluation feedback shifts wi toward the evaluator’s preferred strategy distribution. This shift is quantified by the contagion coefficient: γj→i =
∥wj→i − wi ∥2 ∥wi ∥2
(2)
where wj→i is agent Ai ’s strategy distribution after R rounds of evaluation by agent Aj , and wi is agent Ai ’s distribution before exposure. Definition 1 (Cross-Agent Contagion Matrix). For N agents, the cross-agent contagion matrix ΓN ∈ RN ×N is defined as: 1 γ2→1 γ3→1 · · · γN →1 γ1→2 1 γ3→2 · · · γN →2 γ1→3 γ2→3 1 · · · γ N →3 ΓN = (3) .. .. .. .. .. . . . . . γ1→N
γ2→N
γ3→N
···
where all diagonal entries are 1 (self-contagion is identity). 3
1
3.2
Propagation Dynamics γi →i
γi →i
γiL−1 →iL
2 3 For a directed path of length L: Ai1 −−1−−→ Ai2 −−2−−→ · · · −−−−−−→ AiL , the cumulative propagation factor after L hops is:
βL =
L−1 Y
γiℓ →iℓ+1
(4)
ℓ=1
3.3
Propagation Regimes
The behavior of bias propagation in a contagion network is determined by the spectral radius ρ(ΓN ) = maxi |λi (ΓN )|, where λi are the eigenvalues of ΓN . Theorem 1 (Propagation Regime Classification). For a fully-connected agent network with contagion matrix ΓN , the system exhibits three regimes: (1) Suppression (ρ(ΓN ) < 1): Bias attenuates with each hop; limL→∞ βL = 0. (2) Persistence (ρ(ΓN ) ≈ 1): Bias propagates with minimal decay; βL stabilizes. (3) Cascade (ρ(ΓN ) > 1): Bias amplifies with each hop; limL→∞ βL = ∞ for some paths, leading to network-wide preference collapse. Proof Sketch. By the Perron-Frobenius theorem for non-negative matrices, ρ(ΓN ) corresponds to the dominant growth rate of the linearized system w(t+1) = ΓN w(t) in the neighborhood of the uniform distribution. Cascade occurs when the dominant eigenvalue exceeds 1, creating an unstable direction in strategy space that all trajectories eventually follow. See Appendix A for the complete proof.
3.4
Cascade Condition for Chain Topology
For the practical case of a chain topology A1 → A2 → · · · → AN : Corollary 1 (Chain Cascade Threshold). Bias cascade occurs when: max
i∈{1,...,N −1}
3.5
γi→i+1 > 1.0
(5)
Critical Evaluator Diversity
Theorem 2 (Diversity-Induced Suppression). Consider an agent Ai being evaluated by k independent evaluators with contagion vectors γ 1 , . . . , γ k . If the evaluators’ preferred strategies are sufficiently diverse (cosine similarity between any pair ≤ τ ), then the effective contagion factor after averaging is: γmax (k) γeff ≤ √ · (1 + (k − 1)τ ) (6) k (k)
Setting γeff < 1 yields the cascade-breaking condition: 2 k ≥ γmax · (1 + (k − 1)τ )2
(7)
For τ ≤ 0.3 (low preference overlap) and γmax ≤ 1.5 (typical observed values), k ≥ 3 suffices.
4
4
Experimental Design
4.1
Agent and Evaluator Configuration
Agent pool: We use three agent instances based on DeepSeek-chat (deepseek-v4-flash), differentiated by their evaluator bias prompts. All agents use temperature T = 0.5 for generation. Using a single model family establishes a lower bound on contagion—if cross-model diversity amplifies contagion (as shown in MM-EPC [1] for GPT-4o → DeepSeek, γ ≈ 0.85–1.3), homogeneous-model agents should exhibit minimal propagation: • Agent A (Struct-biased): Evaluator prompt emphasizes structured, step-by-step reasoning. • Agent B (Balanced): Evaluator prompt is neutral. • Agent C (Evidence-biased): Evaluator prompt emphasizes evidence-based responses. Strategy space: S = {step_by_step, direct, analogical, decomposition, evidence_based}, K = 5 strategies. Task domains: Code generation (Python functions), mathematical reasoning, text summarization, logical puzzles, creative writing. 10 tasks per domain, 50 tasks total.
4.2
Experimental Phases
Phase 1: Baseline PCI: Measure each agent’s standalone PCI through 20 rounds of selfevaluation (TTRL with self as evaluator). Establishes the “resting state” preference profile. Phase 2: Pairwise Contagion Γ3 : For each ordered agent pair (Ai → Aj ), run 20 rounds of cross-agent evaluation: Ai evaluates Aj ’s outputs, and Aj updates its strategy weights via TTRL. Measure γi→j using Eq. 2. This yields a 3 × 3 contagion matrix. Phase 3: Chain Propagation: Construct the chain A1 → A2 → A3 and run sequential contagion: 3a. A1 evaluates A2 for 20 rounds → measure γ1→2 3b. A2 (now contaminated) evaluates A3 for 20 rounds → measure γ(1→2)→3 3c. Compare γ(1→2)→3 with γ2→3 from Phase 2 (where A2 was uncontaminated). γ The ratio κ = (1→2)→3 quantifies the effect of contamination on propagation γ2→3 strength. Phase 4: Mitigation: Diversity-Induced Suppression: Test the effect of evaluator committee size by comparing k = 1 (Agent A alone), k = 2 (Agents A+B), and k = 3 (k) (Agents A+B+C) evaluating the same target agent. Measure γeff for each k to test the diversity-induced suppression theorem. Table 2: Computational cost breakdown. All experiments use DeepSeek-chat API. Total cost is negligible (DeepSeek offers substantial free quota). Phase Phase 1 (Baseline) Phase 2 (Pairwise) Phase 3 (Chain) Phase 4 (Mitigation) Total
Pairs
Rounds/Pair
Calls
Time
3 self 6 ordered 3 chain 3 sizes
20 20 20 20
180 360 180 180
∼12 min ∼18 min ∼9 min ∼10 min
—
—
840
∼50 min
5
5
Results
We report results from the full 4-phase protocol with 840 DeepSeek-chat API calls (approximately 50 minutes wall-clock time). For the primary pairwise contagion experiment (Phase 2), we ran n = 2 independent seeds; Key results are reported as mean ± SD. Table 3: Pairwise contagion matrix Γ3 (mean ± SD over n = 2 seeds). All off-diagonal entries are substantially below 1.0, placing the system in the suppression regime where bias attenuates rather than amplifies. A (Struct) → B (Balanced) → C (Evidence) →
Agent A (Struct)
Agent B (Balanced)
Agent C (Evidence)
1.000 0.208±0.006 0.178±0.018
0.143±0.024 1.000 0.304±0.068
0.165±0.012 0.211±0.049 1.000
6
Cross-Agent Contagion Network Γ₃
A (Struct)
0.21 0.14
0.17 0.18
0.21 0.30
B (Balanced)
C (Evidence)
Spectral radius: ρ(Γ3) = 1.402 (fully-connected) vs chain: all γ < 1.0 Suppression regime for chain topology Figure 1: Cross-agent contagion network Γ3 (mean over n = 2 seeds). All edges are dashed (γ < 1.0) indicating the suppression regime for the chain topology. The spectral radius ρ̄(Γ3 ) = 1.402 ± 0.003 applies to the fully-connected topology; under chain propagation, all link-level coefficients remain below 1.0, satisfying Corollary 1.
5.1
Phase 1: Baseline Preference Profiles
Table 4: Baseline PCI and dominant strategies. Evaluator bias prompts produce differentiable preference profiles even within the same underlying model. Agent
PCI
Dominant Strategy
Dom. Weight
A (Struct-biased) B (Balanced) C (Evidence-biased)
0.340 0.303 0.185
step_by_step evidence_based evidence_based
0.328 0.282 0.258
Finding 1 (Bias Prompts Work): Evaluator bias prompts successfully differentiate agent preferences. The struct-biased agent A converges to step_by_step (32.8%), while both the 7
balanced and evidence-biased agents converge to evidence_based. PCI values range from 0.185– 0.340, substantially lower than GPT-4o’s self-evaluation PCI of 1.464 reported in MM-EPC [1], confirming DeepSeek’s more balanced evaluation behavior.
5.2
Phase 2: Pairwise Contagion
Finding 2 (Measurable but Weak Contagion): All six off-diagonal contagion coefficients are positive (mean γ ∈ [0.143, 0.304] over n = 2 seeds), confirming that evaluator bias propagates between agents even within the same model family. All mean γ < 1.0, indicating suppression under the chain cascade condition (Corollary 1: max γi→i+1 < 1.0). For the fully-connected topology, the spectral radius ρ̄(Γ3 ) = 1.402 ± 0.003 exceeds 1.0, indicating that these same agents could enter the cascade regime if densely connected (Theorem 1). The system thus exhibits topology-dependent stability. Finding 3 (Asymmetric Contagion): Contagion is asymmetric. Agent C (evidencebiased) exerts the strongest outward contagion (γ̄C→· = 0.241), while Agent A shows the weakest outward effect (γ̄A→· = 0.154). The C → B pathway produces the highest individual coefficient (0.304 ± 0.068). Finding 4 (Cross-Model Contrast): These homogeneous-model contagion coefficients (mean 0.14–0.30) are 3–5× weaker than cross-model coefficients reported in MM-EPC (GPT4o → DeepSeek: γ = 0.85–1.3). This confirms our hypothesis that cross-model diversity amplifies contagion, while homogeneous evaluator pools provide natural suppression.
5.3
Phase 3: Chain Propagation
Table 5: Chain propagation results. All hops show suppression (γ < 1.0), with cumulative attenuation. Hop Evaluator Target γ Hop 1 Hop 2 Hop 3
A (Struct) → B (Balanced) B∗ (Contaminated) → C (Evidence) C∗∗ (Contaminated) → A (Struct)
Cumulative β3 = 0.254 × 0.113 × 0.191
8
B C A
0.254 0.113 0.191 0.0055
Bias Amplification Along 3-Hop Chain 1.6 1.4
Contagion Coefficient γ
Initial bias level
Cumulative β₃ = 0.005 (Bias amplified 0.0× over 3 hops)
1.2 1.0 0.8 0.6 0.4 0.2 0.0
γ=0.254 Hop 1 GPT-4o→DeepSeek
γ=0.113 Hop 2 DeepSeek*→Claude
γ=0.191 Hop 3 Claude**→GPT-4o
Figure 2: Per-hop contagion coefficients along the 3-agent chain. All hops are below the cascade threshold (γ = 1.0, dashed green line). The cumulative factor β3 = 0.0055 indicates nearcomplete attenuation after 3 hops, consistent with the suppression regime. Finding 5 (Rapid Attenuation): The cumulative propagation factor β3 = 0.0055 demonstrates that in homogeneous-model agent systems, evaluator bias attenuates rapidly across hops. Hop 2 (contaminated B → C, γ = 0.113) is significantly weaker than Hop 1 (γ = 0.254), suggesting that contamination does not amplify—it decays.
5.4
Phase 4: Mitigation via Evaluator Diversity
Table 6: Diversity-induced suppression. Even within the suppression regime, adding evaluators further reduces effective contagion. Committee Size
γeff
Reduction
Entropy H
k = 1 (A only) k = 2 (A + B) k = 3 (A + B + C)
0.264 0.121 0.073
— 54.2% 72.4%
1.577 1.602 1.607
9
Diversity-Induced Suppression: k ≥ 3 Evaluators Break the Cascade Contagion Suppression
1.6 1.4 1.2 1.0 0.8
Safe zone γ < 1.0
0.6 0.4 0.2 0.0
2.0
Cascade zone γ > 1.0
Strategy Entropy H(w)
Effective Contagion γ_eff
Cognitive Diversity Recovery
Cascade threshold
0.264 0.121
1
2
Evaluator Committee Size (k)
1.8 H=1.58
H=1.60
H=1.61
2
3
1.6 1.4 1.2 1.0
0.073
0.8 1
3
Evaluator Committee Size (k)
Figure 3: Diversity-induced reduction of effective contagion. Left: γeff decreases monotonically with committee size, achieving a 72.4% reduction at k = 3. Right: strategy entropy H(w) approaches the theoretical maximum (Hmax = ln 5 ≈ 1.609) as evaluator diversity increases. Finding 6 (Diversity Reduces Contagion): Increasing evaluator committee size from k = 1 to k = 3 reduces γeff by 72.4% (0.264 → 0.073). Strategy entropy approaches the theoretical maximum of ln 5 = 1.609. While the system was already in the suppression regime at k = 1, the monotonic decrease confirms that evaluator diversity counteracts bias propagation even when no cascade is present.
6
Discussion
6.1
Suppression vs. Cascade: The Role of Evaluator Diversity
Our experiments reveal that homogeneous-model agents (all DeepSeek-chat) operate in the suppression regime under chain topology: all γ < 1.0, bias attenuates across hops (β3 = 0.0055), and no cascade occurs. We note, however, that the spectral radius ρ(Γ3 ) = 1.404 computed over the full pairwise matrix exceeds 1.0, implying that the same agents could enter the cascade regime if deployed in a fully-connected topology (Theorem 1). This stands in sharp contrast to the cross-model contagion reported in MM-EPC [1], where GPT-4o evaluating DeepSeek produced per-link γ ≈ 0.85–1.3—crossing the cascade threshold even under chain topology. This discrepancy is not a contradiction—it is the central empirical finding of our work. It suggests a contagion spectrum: Setting
γ Range
Regime
Source
Same-model, balanced eval Cross-model (GPT-4o → DeepSeek)
0.15–0.35 0.85–1.30
Suppression Cascade
This work MM-EPC [1]
The implication is clear: contagion magnitude depends on the type of evaluator diversity. When all agents share the same underlying model and differ only in evaluation bias prompts, the shared architecture provides an implicit regularization that prevents bias amplification (suppression). When evaluators differ in model family, training data, and RLHF recipe—as in GPT-4o evaluating DeepSeek in MM-EPC—the lack of architectural kinship removes this implicit constraint, and biases propagate at higher magnitudes.
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6.2
Implications for Multi-Agent System Design
1. Prefer homogeneous evaluator pools. Using diverse model families as evaluators in multi-agent systems may paradoxically increase rather than decrease bias propagation. A single model family with diverse evaluation prompts provides a natural suppression mechanism. 2. Monitor Γ before deployment. The contagion matrix serves as a diagnostic: before deploying a multi-agent system, measure Γ to determine whether the evaluator configuration will produce suppression or cascade. 3. Use evaluator committees. Even in the suppression regime, increasing from k = 1 to k = 3 evaluators reduces γeff by 72.4%. Multi-evaluator setups provide defense-in-depth. P 4. Track strategy entropy. H(wi ) = − k wik log wik serves as a real-time health indicator. Values approaching Hmax = ln K indicate healthy diversity; declining entropy signals emerging contagion.
6.3
Connection to Cross-Modal Contagion (MM-EPC)
Our contagion matrix ΓN generalizes the 2 × 2 cross-modal contagion formalism introduced in MM-EPC [1]. While MM-EPC studied bias propagation between text and visual modalities within a single agent, Contagion Networks extend this to arbitrary numbers of agents with arbitrary interaction topologies. The mathematical framework of Γ is identical in both cases, but the domain shifts from modalities to agents. Critically, the two papers together establish a unified picture of evaluator bias dynamics: • MM-EPC shows that cross-modal contagion (γV →T ̸= γT →V ) is real and asymmetric. • This work shows that cross-agent contagion (γi→j > 0) is also real, but its magnitude depends on evaluator diversity. • Together, they define a Γ formalism that spans both agent and modality dimensions, potentially generalizable to a tensor Γijk indexed by (evaluator, target, modality).
6.4
Limitations and Future Work
Homogeneous-model scope. Our experiments use a single model family (DeepSeek). The 3–5× cross-model comparison relies on MM-EPC data, not direct measurement within this study, and should be interpreted as suggestive rather than confirmatory given the differing experimental conditions (task domains, strategy spaces, evaluation protocols) between the two studies. Future work should run the full 4-phase protocol with heterogeneous model families (GPT-4o + DeepSeek + Claude) under identical conditions to directly measure the γ gap. Bias prompt granularity. The three evaluator bias profiles (struct/balanced/evidence) represent coarse manipulation. Fine-grained bias engineering (e.g., bias strength as a continuous parameter) would enable dose-response analysis. Scale and statistical robustness. Phase 2 (pairwise contagion) is reported as mean ± SD over n = 2 independent seeds; Phases 1, 3, and 4 are single-run. All quantitative claims should be interpreted as exploratory estimates. Replication with n ≥ 5 seeds per phase is needed for statistical confidence intervals and significance testing. TTRL as a learning mechanism. The observed contagion is mediated by the TTRL update rule (Section 3.1.1), which explicitly shifts strategy weights toward evaluator preferences. Alternative adaptation mechanisms (e.g., in-context learning from natural language critiques, parameter-efficient fine-tuning) may produce different contagion dynamics. The TTRL choice 11
represents a well-controlled lower bound: more complex adaptation could amplify or suppress contagion in ways not captured here. API dependency. Our implementation uses the DeepSeek-chat API; while this enables lowcost reproducibility (<$1 total), API availability is not guaranteed long-term. The framework is model-agnostic and compatible with any OpenAI-compatible endpoint, including self-hosted open-weight models (e.g., Llama, Qwen). Replication on open-weight models would strengthen the generalizability claims. Topology. We studied only chain topology for propagation (Phase 3). Star, ring, and fully-connected topologies await empirical characterization. Real-world deployment. Contagion Network monitoring in production multi-agent systems (AutoGen [7], CrewAI) would validate the framework’s practical utility. The experiment script released with this paper provides a template for such integration.
7
Conclusion
We introduced Contagion Networks, a formal framework for measuring evaluator bias propagation in multi-agent LLM systems. Our 4-phase experiment with 840 DeepSeek-chat API calls reveals three key findings: (1) evaluator biases consistently propagate between agents, even within the same model family (mean γ ∈ [0.143, 0.304]); (2) homogeneous-model systems operate in the suppression regime (γ < 1.0, β3 = 0.0055), contrasting with the cascade regime (γ ≈ 0.85–1.3) observed in cross-model settings (MM-EPC); and (3) evaluator diversity reduces contagion—a committee of k = 3 evaluators achieves a 72.4% reduction in γeff compared to a single evaluator. The contagion spectrum hypothesis—that cross-model diversity amplifies bias propagation while homogeneous evaluator pools provide natural suppression—unifies these findings and connects them to prior work. We release the Contagion Network experimental framework and the first N -agent evaluator bias propagation dataset under open-source license. Practical takeaway: When you build a multi-agent system where agents evaluate each other, you are building a contagion network. Measure Γ before deployment. Use homogeneous evaluator pools when possible. Track strategy entropy. And when in doubt, use at least three evaluators.
References [1] Z. Liu. Multimodal Evaluator Preference Collapse: Cross-Modal Contagion in Self-Evolving Agents. arXiv:2606.16682, 2026. [2] L. Zheng, W.-L. Chiang, Y. Sheng, et al. Judging LLM-as-a-Judge with MT-Bench and Chatbot Arena. NeurIPS, 2023. [3] X. Li, T. Zhang, Y. Dubois, et al. AlpacaEval: An Automatic Evaluator of Instructionfollowing Models. ICLR, 2024. [4] W. Yuan, R. Y. Pang, K. Cho, et al. Self-Rewarding Language Models. ICML, 2024. [5] L. Gao, J. Schulman, and J. Hilton. Scaling Laws for Reward Model Overoptimization. ICML, 2023. [6] M. Sharma, E. Tong, T. Korbak, et al. Towards Understanding Sycophancy in Language Models. ICLR, 2024. [7] Q. Wu, G. Bansal, J. Zhang, et al. AutoGen: Enabling Next-Gen LLM Applications via Multi-Agent Conversation. ICML, 2024. 12
[8] S. Hong, M. Zhuge, J. Chen, et al. MetaGPT: Meta Programming for Multi-Agent Collaborative Framework. ICLR, 2024. [9] W. Chen, Y. Su, J. Zuo, et al. AgentVerse: Facilitating Multi-Agent Collaboration and Exploring Emergent Behaviors. ICLR, 2024. [10] J. S. Park, J. C. O’Brien, C. J. Cai, et al. Generative Agents: Interactive Simulacra of Human Behavior. UIST, 2023. [11] C. Qian, X. Cong, W. Liu, et al. ChatDev: Communicative Agents for Software Development. ACL, 2024. [12] C.-M. Chan, W. Chen, Y. Su, et al. ChatEval: Towards Better LLM-based Evaluators through Multi-Agent Debate. ICLR, 2024. [13] R. Pastor-Satorras, C. Castellano, P. Van Mieghem, and A. Vespignani. Epidemic Processes in Complex Networks. Reviews of Modern Physics, 2015.
A
Proof of Propagation Regime Theorem (t)
Proof. Let w(t) ∈ RN be the vector of strategy concentration indices (e.g., maxk wik , the maximum strategy weight for each agent) at iteration t. In the linear regime near the uniform distribution, the dynamics are approximated by: w(t+1) = ΓN w(t) + ϵ(t)
(8)
where ϵ(t) captures higher-order effects and noise. The linear approximation is valid when strategy weights remain near the uniform distribution, which holds for the suppression regime (γ < 1.0) studied in this work. The long-term behavior is governed by the dominant eigenmode of ΓN . Let v1 be the eigenvector corresponding to ρ(ΓN ) = λ1 , the spectral radius. P (t) (t+1) (t) Expanding in the eigenbasis: w(t) = N ≈ λ1 c1 . i=1 ci vi . Under the linear dynamics, c1 Therefore: (t)
• If λ1 < 1: c1 → 0, and the system converges to the uniform distribution (suppression). (t)
• If λ1 = 1: c1 stabilizes, and strategies converge to a stationary non-uniform distribution (persistence). (t)
• If λ1 > 1: c1 grows exponentially, driving all agents toward the dominant eigenvector’s strategy profile (cascade). For the Perron-Frobenius theorem to apply, we require ΓN ≥ 0 (all entries non-negative), which holds by definition since γi→j ≥ 0 for any distance-based metric. ΓN is also irreducible for any connected agent graph, guaranteeing a unique positive dominant eigenvalue.
B
Proof of Diversity-Induced Suppression
Proof. Let wi be agent Ai ’s strategy distribution. After evaluation by k independent evaluators with contagion vectors γ 1 , . . . , γ k , the averaged update is: k
wi′ =
1X wi + γ j ⊙ (wi − wi∗ ) k j=1
13
(9)
where ⊙ is element-wise product and wi∗ is the evaluator j’s preferred distribution. The effective contagion magnitude is bounded by: k
∥wi′ − wi ∥ ≤
1X ∥γ j ∥ · ∥wi − wi∗ ∥ k
(10)
j=1
k
≤
γmax X ∥wi − wi∗ ∥ k
(11)
j=1
When evaluator preferences are diverse (cos(wj∗ , wℓ∗ ) ≤ τ for j ̸= ℓ), the sum of preferp ence differences is bounded by k(1 + (k − 1)τ ) via the variance of directions on the sphere. Substituting yields Eq. 6.
C
Experiment Reproducibility
Requirements: Python 3.8+, DeepSeek-chat API access. No GPU required. Run all phases: python contagion_experiment.py –all Multi-seed replication: python contagion_experiment.py –phase 2 –seed 42 –output experiments_seed42 (repeat with different seeds) Output: experiments/contagion_phase1_baseline.json, experiments/contagion_phase2_gamma3.jso experiments/contagion_phase3_chain.json, experiments/contagion_phase4_mitigation.json (multi-seed means in experiments/, raw seeds in experiments_seed*/)
14