Technical Summary: GeomHerd

1. Problem Statement

Herding—where market participants align behaviors and act collectively—is a primary driver of market fragility and systemic risk. Existing approaches to quantify herding rely on downstream observables, such as realized returns (e.g., Cross-Sectional Absolute Deviation, CSAD) or disclosed trading flows (e.g., LSV institutional imbalance). These methods are inherently post-hoc and suffer from an observability lag: they detect coordination only after it has already propagated into price movements or public filings.

While recent geometric approaches using discrete Ricci curvature have been applied to market-stress analysis, they typically operate on price-correlation graphs, inheriting the same bottleneck. The paper posits a fundamental question: Can one directly quantify the evolving geometry of the agent coordination process itself (upstream structural information) to generate a forward-looking signal before herding manifests in returns?

2. Methodology: The GeomHerd Framework

GeomHerd is a forward-looking geometric framework that bypasses the observability lag by quantifying coordination directly on agent-interaction graphs.

2.1 Substrate and Graph Construction

The framework utilizes a heterogeneous LLM-driven multi-agent financial simulator (specifically the Cividino-Sornette continuous-spin or CWS model) where each trader is instantiated by a persona-conditioned LLM. This provides a fully observable action stream.

• Nodes: Individual agents ($N=66$).
• Edges: Constructed based on windowed action agreement. The weight $w_t(i, j)$ is the frequency of identical actions (e.g., buy/hold/sell) between agents $i$ and $j$ over a window $T_w=100$.
• Sparsification: Edges are retained only if $w_t(i, j) > 0.5$, ensuring meaningful agreement above the uniform baseline.
• Graph Dynamics: The graph is reconstructed every $\Delta t = 10$ steps.

2.2 Geometric Core: Discrete Ricci Curvature

The framework computes Ollivier-Ricci curvature ($\kappa_{OR}$) on the agent graph using a lazy-walk transition kernel.

• Sign Decomposition: The curvature is decomposed into two distinct signals based on community-bridge topology:
  1. Positive Curvature ($\bar{\kappa}^+_{OR}$): Captures within-clique coordination. As agents imitate one another, their neighborhoods overlap, increasing positive curvature. This signals the onset of an information cascade.
  2. Negative Curvature ($\beta^-$): Captures inter-community bridges. Strongly negative curvature identifies edges connecting distinct clusters, signaling the propagation of contagion.
• Detection Logic:
  • $\bar{\kappa}^+_{OR}$ is monitored using a one-sided CUSUM detector to identify level shifts.
  • $\beta^-$ is monitored using a combined CUSUM and Kendall-$\tau$ slope test to detect trends in bridge emergence, robust to non-Gaussian noise.

2.3 Forward-Looking Descriptors

• Singularity Time ($\tau_{sing}$): The framework evolves the graph under discrete Ricci flow and records the time to the first "neckpinch" (singularity). This serves as a dynamic, forward-looking descriptor of proximity to collapse, distinct from static clustering.
• Effective Vocabulary ($V_{eff}$): An information-theoretic signal measuring the entropy of the agents' action language. $V_{eff}$ contracts as agents homogenize their behaviors during a cascade.

2.4 Theoretical Anchor

The paper establishes a mean-field bridge (Proposition 1) linking the geometric metric to the classical CSAD statistic. It derives that under standard assumptions (mean-field concentration, linear price impact), CSAD is monotonically decreasing in $\bar{\kappa}_{OR}$. This theoretically anchors the geometric signal to established finance literature.

3. Key Contributions

1. Substrate Pivot: Shifts the locus of herding detection from downstream price-correlation graphs to upstream agent-interaction graphs, leveraging fully observable action streams in LLM-driven simulators.
2. Sign-Decomposed Geometry: Explicitly disentangles herding (positive curvature, intra-clique density) from contagion (negative curvature, inter-cluster bridges), providing complementary alarm signals.
3. Forward-Looking Flow: Introduces Ricci-flow singularity time ($\tau_{sing}$) as a dynamic proximity-to-collapse scalar rather than a static clustering operator.
4. Theoretical Link: Provides a scaling identity connecting the geometric curvature to the classical CSAD return-dispersion statistic.

4. Experimental Results

Experiments were conducted on the CWS continuous-spin substrate (400 trajectories) and validated via out-of-domain transfer to the Vicsek self-driven-particle model.

• Lead Time:
  • The primary detector ($\bar{\kappa}^+_{OR}$) fires a median of 178 steps before the order-parameter herding event ($\tau^*$) at a precision-oriented operating point.
  • At a recall-oriented operating point, the median lead extends to 272 steps.
  • The contagion detector ($\beta^-$) recalls 65% of critical trajectories with a median lead of 318 steps.
  • On co-firing trajectories, the agent-graph signal precedes price-correlation baselines by a pooled median of 40 steps.
• Consistency:
  • The geometric signal is sign-consistent with CSAD (negative correlation between curvature and return dispersion).
  • The agent-graph signal precedes the LSV trading-flow statistic on co-firing trajectories, as the action clique is detectable before buy/sell imbalances accumulate.
• Forecasting Utility:
  • A curvature-conditioned forecasting head (using the triplet $\bar{\kappa}_{OR}, \tau_{sing}, V_{eff}$) reduces the Mean Absolute Error (MAE) of cascade-window log-returns compared to price-only and detector-conditioned baselines.
• Generalization:
  • The geometric signature transfers out-of-domain to the Vicsek model, separating ordered from disordered regimes with an AUROC of 0.99, suggesting the signature reflects universal collective coordination rather than finance-specific artifacts.
• Ablation:
  • Binary edge construction is critical; cosine-similarity variants fail to detect herding.
  • The sign decomposition is essential; removing it eliminates the signal.

5. Significance and Claims

The paper claims that GeomHerd advances forward-looking herding quantification by addressing the fundamental "observability gap" in financial risk monitoring. By treating the agent interaction graph as a forecastable world, the framework captures the structural topology of emerging coordination before it manifests in realized returns.

The authors emphasize that GeomHerd is not merely a new statistic but a forward-looking warning system that:

1. Anticipates herding significantly earlier than aggregate market baselines.
2. Distinguishes between internal coordination (clique formation) and external contagion (bridge propagation).
3. Generalizes across different physical and financial substrates.
4. Improves downstream forecasting accuracy when used as a conditioning feature.

The work positions itself as a bridge between geometric deep learning (Ricci flow) and classical financial theory (CSAD/LSV), offering a mechanism to monitor systemic stability at the micro-structural level of agent interaction.