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6G Reasoning Agent Societies: Composition, Scaling, and Consistency-Time Laws

Hatim Chergui, Carolina Fernández-Martínez, Mustapha Benjillali, Mehdi Bennis, Merouane Debbah

arXiv:2610.34357v1cs.NIcs.MA

TL;DR

LLM control-plane agents can pass along conclusions that are unsupported despite error-free transport, while local audits do not capture collective dependence. The paper models these societies as cellular sheaves and establishes composition, scaling, and consistency-time laws. Its results include hidden divergence up to 23% and a 1.14 dB cognitive-SNR overstatement in complete societies, alongside topology-dependent consistency-time bounds.

  • Problem

    Local audits of agent messages against sender observation masks miss collective behavior when agents’ conclusions become statistically intertwined.

    Source: B. Gaps and Contributions · p. 2, B. A Running Example · p. 3

  • Method

    The paper models agent societies as cellular sheaves and derives composition, scaling, and consistency-time laws for arbitrary topologies.

    Source: B. Gaps and Contributions · p. 2

  • Results

    Complete societies hide 23% of joint divergence and overstate cognitive SNR by 1.14 dB, while reported consistency time is bounded by the worst-audited link’s inflation.

    Source: B. Gaps and Contributions · p. 2, B. The Composition Law · p. 6

  • Takeaways & Limitations

    The laws provide worst-case latency ceilings for near-real-time operation, and the paper identifies minimizing network diameter, using expanders or local audits, and estimating message correlations as practical strategies.

    Source: §VI · p. 7

Abstract

from arXiv · show

Emerging 6G control planes would compose Large Language Model (LLM) agents that exchange interpreted conclusions, so hallucinations can propagate even over error-free transport. Recent architectures audit each message locally against the sender's observation mask, but isolated audits miss collective behavior. Modeling the agent society as a cellular sheaf, we establish three laws valid on arbitrary topologies. First, the \emph{composition law}: network-wide divergence exceeds the sum of per-agent audit scores by exactly their hidden total correlation. This blind spot masks up to 23\% of joint divergence and, because the declared signal cancels in the cognitive signal-to-noise ratio (SNR), overstates it by $1.14$\,dB purely through topology; the resulting trust inflation preserves valid consensus states but slows convergence. Second, the \emph{scaling law}: reconciliation decouples across metric counters, so the worst-connected dimension sets the pace. Consistency time grows for a society size $n$ as $\Omega(n^{2/d})$ on polynomial-growth topologies of degree $d$, and paths breach a $1$\,s near-real-time budget at $n^\ast=13$ agents, whereas expanders ($\Theta(1)$) and radius-3 local audits ($O(1)$, $\approx300$\,ms) stay bounded. Third, the \emph{consistency-time law}: the timing error from omitted correlations is one-sided and capped by the worst-measured link. These laws, validated on simulated topologies and deployed LLM societies, give provable worst-case latency ceilings for near-real-time operation.

I. INTRODUCTION

The paper frames trust in LLM-based control planes as a problem of auditing interpreted conclusions rather than merely ensuring intact transport. It identifies gaps in composition, topology-dependent convergence time, and bounds on consistency-time estimates, then proposes three laws for those gaps.

I. INTRODUCTION

  • LLM agents interpret network reports and negotiate resource allocation, so intact delivery cannot guarantee that a conclusion is well-founded.Because their internal logic is proprietary, operators cannot audit how conclusions were reached; the paper frames the needed protection as trust rather than transport integrity.Source: §I · p. 1

A. Literature Review

  • Prior work covers agentic control planes, semantic communication, per-interaction trust scores, and consensus, but does not answer what a trust layer can detect when conclusions depend on one another.Near-real-time loops require local message scoring, while circulated conclusions introduce dependence that sum-based aggregation rules assume away.Source: A. Literature Review · p. 1

B. Gaps and Contributions

  • The paper identifies three open gaps: composition laws for credibility, topology-aware time-to-consensus analysis, and bounds on consistency time derived from optimistic trust weights.The cited consensus work addresses convergence and outcomes, while the paper identifies convergence time under subscription-induced topology as an unanswered near-real-time budgeting question.Source: B. Gaps and Contributions · p. 2
  • The proposed laws state that local audits miss total message correlation, reconciliation follows the worst-connected counter, and reported consistency time is bounded by the worst-audited link’s inflation.The paper says these laws apply to arbitrary topologies and are validated on simulated sheaves and LLM societies prompted with O-RAN reports.Source: B. Gaps and Contributions · p. 2

II. SYSTEM MODEL

The system models LLM agents as a graph-connected society with public observation masks and overlapping metric vocabularies. Agents send beliefs across the dimensions they score, while meaningful agreement is restricted to shared dimensions.

A. Agents, Masks, Messages, and Beliefs

  • Agents interact over graph G = (V, E), and each public mask A_v specifies the E2 counters that agent v observes.Agents score dimensions such as UP, CP, and MOB from the counters covered by their masks.Source: A. Agents, Masks, Messages, and Beliefs · p. 2, Figure 1 · p. 2
  • Each round, an agent emits a message over the shared vocabulary using its own observation and its neighbors’ previous-round messages.The agent’s belief is the portion of that message backed by its mask.Source: A. Agents, Masks, Messages, and Beliefs · p. 2
  • When agents have asymmetric views, consensus is meaningful only on mutually observable metrics, not across every dimension.The paper uses a cellular sheaf to formalize this localized agreement over overlapping vocabularies.Source: A. Agents, Masks, Messages, and Beliefs · p. 2

B. A Running Example

A two-agent signaling-storm example shows how a locally supported belief can still reflect an unsupported peer message. The dependence between agents’ messages makes the provenance of that influence invisible to a local audit.

B. A Running Example

  • Agent u monitors CP and MOB, agent v monitors UP and MOB, and both observe the shared mobility counter during a signaling storm.The scenario places the storm over a lightly loaded user plane.Source: B. A Running Example · pp. 2–3
  • Agent u reports UP = 70 without mask support, while v reports UP = 52 after reading u’s message despite its own counters indicating about 35.The example contrasts u’s unsupported assertion with v’s mask-supported but peer-influenced score.Source: B. A Running Example · p. 3
  • A local audit penalizes u’s unsupported UP claim but accepts v’s UP score, missing that the score was influenced by u’s message.The provenance of the error lies in statistical dependence between the agents, which inspection against v’s own expectation cannot detect.Source: B. A Running Example · p. 3

C. The Trust Sheaf

The trust sheaf represents each agent’s beliefs over its own metrics and compares agents only on shared dimensions. Weighted disagreement drives relaxation toward global agreement, while trust weights affect convergence speed rather than which global sections exist.

C. The Trust Sheaf

  • Each agent has a vertex stalk of beliefs over its covered dimensions, while each link’s edge stalk contains the dimensions shared by its endpoints.Restriction maps filter out dimensions the neighboring agent does not track.Source: C. The Trust Sheaf · p. 3
  • Link consensus means the agents’ beliefs match after restriction to their shared dimensions.Thus, an agent’s blindness to a dimension held by its neighbor does not itself violate agreement.Source: C. The Trust Sheaf · p. 3
  • The coboundary collects differences between restricted beliefs on links, and the sheaf Laplacian weights those residuals by link trust.Relaxation follows ẋ = −L_Fx and stops when shared-dimension disagreement vanishes.Source: C. The Trust Sheaf · p. 3
  • On a connected graph, pairwise agreements stitch into a global section, and changing positive link weights changes how fast it is reached, not which global sections exist.When calibrations conflict around loops, relaxation projects initial beliefs onto the consistent subspace, with disagreement decaying as e^−λ₂t.Source: C. The Trust Sheaf · p. 3

D. The Cognitive SNR

The cognitive SNR compares a sender’s asserted confidence with divergence from an expectation based on its public observation mask. The resulting trust weight is bottlenecked by the least credible participant on a link.

D. The Cognitive SNR

  • The cognitive SNR measures asserted precision per bit of unexplained divergence, using the sender’s mask-derived expectation as the reference.The signal is the trace of the belief’s precision matrix; the noise measures how far the sender’s restricted message departs from the expectation supported by its mask.Source: D. The Cognitive SNR · pp. 3–4
  • The dBcog reference is γ0 = 1 bit^-1, but it does not affect the three laws because the signal and reference cancel in their ratios.An x dBcog value means 10 log10(γv→e/γ0) = x.Source: D. The Cognitive SNR · p. 4
  • A high-confidence claim without observational support increases divergence and collapses the SNR.The link weight is based on the least credible participant’s SNR.Source: D. The Cognitive SNR · p. 4

E. Message Laws and the Two Divergences

The paper models messages across a society with a joint law and audits each agent against an expectation derived from that agent’s observations alone. The composition law shows that these local audits omit exactly the messages’ total correlation, regardless of topology or per-agent expectation.

E. Message Laws and the Two Divergences

  • Messages form a joint law pV with marginals pv; dependence arises from shared operating conditions and agents receiving neighbors’ messages.Each agent’s reference qv is based on its own observation alone, and its local divergence Nv measures departure from that reference.Source: E. Message Laws and the Two Divergences · p. 4
  • The local audit can clear a claim supported by an agent’s mask even when peer influence has shaped that claim, because marginal audits miss dependence.In the example, the joint divergence also charges for v’s UP score moving with u’s, which no marginal represents.Source: E. Message Laws and the Two Divergences · p. 4, §III · p. 4
  • The product reference Qv∈V qv defines the per-agent expectations against which relayed conclusions are scored.The trust layer builds each factor from that agent’s own observation alone.Source: E. Message Laws and the Two Divergences · p. 4

III. THE COMPOSITION LAW

  • Joint divergence Njoint exceeds summed local divergence Nlocal by exactly the messages’ total correlation Φ, for any finite society and interaction topology.Φ is the sum of marginal differential entropies minus the joint differential entropy; independent messages give Φ = 0.Source: §III · p. 4
  • Because declared signal S is shared and cancels, total correlation becomes an SNR over-reporting factor independent of confidence calibration.The resulting penalty is a topology- and correlation-driven decibel measure.Source: §III · p. 4

IV. SCALING AND CONSISTENCY TIME

Consistency depends on topology and on the slowest-connected metric dimension, while trust weights affect convergence speed rather than which agreements are valid. Local checks have bounded scaling, and the reported consistency time is optimistic but can be corrected using the worst-audited link.

IV. SCALING AND CONSISTENCY TIME

  • Growth and trust inflation change how quickly agents agree, not which agreements are valid.The consensus space is independent of the weights.Source: §IV · p. 5
  • Reconciliation decouples by dimension, so the least-connected shared-counter graph determines algebraic connectivity and consistency time.The Laplacian decomposes into independent processes; dimensions do not mix.Source: §IV · p. 5
  • On polynomial-growth graphs of degree d, consistency time grows at least as Ω(n^2/d) log(1/ε), while expanders maintain Θ(1) log(1/ε) time.The bounds assume link weights w ≥ wmin > 0; the slowest dimension sets the overall speed.Source: §IV · p. 5
  • With bounded maximum degree, every fixed-radius ball reconciles in O(1) time with respect to society size n.The local bound depends on the ball’s radius and maximum degree, not on the network’s global topology.Source: §IV · p. 5
  • Table I compares consistency-time scaling across named topology families as n grows, with tight orders and a fitted exponent of n.The supplied caption identifies the table’s scope but does not provide its cell values.Source: Table I · p. 5
  • Because each link inflation βe is at least 1, reported consistency time τ rep is optimistic: τ rep ≤ τ joint ≤ τ rep × maxe βe, so the true delay is capped by the single worst-measured link.Source: §IV · p. 5
  • Restricting consistency checks to a radius-r ball makes the error bound depend only on inflation inside that ball, improving fidelity when distant links are poorly audited.Local neighborhoods also reconcile faster, supporting reliable near-real-time local synchronization despite a degrading periphery.Source: §IV · p. 6

V. NUMERICAL RESULTS

The evaluation uses elicited LLM agent societies to test the trust shortfall and simulated sheaves across topologies and sizes to test scaling and consistency-time bounds.

V. NUMERICAL RESULTS

  • The evaluation tests the trust shortfall in elicited LLM agent societies and the bounds across simulated topologies and sizes.The two regimes target Theorem 1 and Proposition 1 and Theorem 2, respectively.Source: §V · p. 6
  • Elicited LLM societies are used to assess whether the trust shortfall of Theorem 1 appears in practice.The passage does not report the resulting measurements.Source: §V · p. 6
  • Simulated sheaves are swept across topologies and society sizes to test the scaling and consistency-time bounds.The passage does not provide numerical outcomes for these tests.Source: §V · p. 6

A. Experimental Setup

The experiment uses five locally hosted LLM xApps with overlapping observation masks and compares isolated, path, and complete communication topologies. Audits use same-model expectations from each agent’s observation alone, making the measured trust shortfall optimistic.

A. Experimental Setup

  • Five xApps share an MOB counter and run under isolated, path, and complete topologies to separate shared-counter correlation from communication effects.Each is a locally hosted Mistral 7B model; scenarios and seeds are held identical across topologies.Source: A. Experimental Setup · p. 6
  • Auditor and generator share a model prior, so the measured trust shortfall understates what an independent reference would measure.Expectations use each agent’s observation alone, without neighbor messages.Source: A. Experimental Setup · p. 6
  • Consistency-time measurements use 12 random sheaves per society size and 10 ms per diffusion round; the figure marks the 1 s budget.Solid lines are global times and dashed lines are radius-3 local audits.Source: Figure 3 · p. 6
  • Response variance is estimated from two elicited answers, with a two-point floor; the resulting signal cancels from reported ratios.The elicitation sets absolute SNRs, not their separation.Source: A. Experimental Setup · p. 6

B. The Composition Law

Per-link audits miss dependence among agents’ messages, so dense communication can increase reported confidence while leaving a larger share of joint divergence unseen.

B. The Composition Law

  • At n = 5, hidden divergence reaches 10%, 13%, and 23% in isolated, path, and complete societies, respectively.The complete topology hides nearly double the path’s share, with 13 of its 23 points attributed to peer persuasion rather than shared data.Source: B. The Composition Law · p. 6
  • Communication expands the audit gap: peer persuasion accounts for 13 of the 23 hidden-divergence points in complete societies.The isolated 10% baseline is a structural floor from the shared counter.Source: B. The Composition Law · p. 6
  • Reported confidence grows faster than joint cognitive SNR, widening their separation from 0.48 dB in isolation to 1.14 dB in a complete society.The path separation is 0.60 dB; dense societies report the highest confidence where the audit blind spot is largest.Source: B. The Composition Law · p. 6

C. The Scaling Law

Consistency time depends on topology and the slowest-connected metric dimension, while a worst-link inflation factor provides a bound for operational timing. The conclusion identifies topology design and correlation estimation as practical strategies.

C. The Scaling Law

  • Path consistency exceeds the 1 s budget at n*=13, grids at n*=172, while expanders remain within budget through n = 196.Paths scale quadratically, grids linearly, and expanders remain Θ(1); radius-3 audits take about 300 ms on paths regardless of size.Source: C. The Scaling Law · p. 6
  • A radius-3 audit converges in about 300 ms on paths regardless of network size, while local shortcuts are unnecessary on expanders.On grids, local times plateau near 200 ms.Source: C. The Scaling Law · pp. 6–7

D. The Consistency-Time Law

  • Across 120 random sheaves, the needed timing multiplier never exceeds the Theorem 2 bound; slack from unknown inflation location is under 1.5×.Radius-3 balls obey the same rule using the maximum inflation among links inside the ball.Source: D. The Consistency-Time Law · p. 7, Figure 4 · p. 7

VI. CONCLUSION

  • The conclusion recommends minimizing network diameter, deploying expanders, or using local audits, and estimating message correlations from relayed traffic.The informational strategy does not require access to proprietary agent internals.Source: §VI · p. 7

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