Dichotomy-Based Multi-Agent Systems

• Dichotomy-based multi-agent systems are defined by a recursive interval refinement mechanism that forces convergence of bounded solutions to consensus.
• They integrate continuous-time coordination with explainable AI by combining structured case retrieval, chain-of-thought reasoning, and expert aggregation.
• The framework guarantees convergence under specific connectivity conditions, enhancing both distributed coordination and survival prediction applications.

A dichotomy-based multi-agent system is an architectural and algorithmic paradigm in distributed decision-making that unifies hierarchical, interval-based problem solving with multi-agent inference, leveraging the notion of dichotomy: the property that any bounded solution of a suitably-posed (differential, difference, or more generally monotone) system must converge, typically to consensus or aggregation around a target set. The approach has foundational relevance for both continuous-time coordination (as in consensus or containment over graphs) (Proskurnikov et al., 2016) and modern explainable AI systems integrating structured case retrieval, chain-of-thought reasoning, and ensemble expert aggregation for complex multimodal prediction tasks (Huang et al., 20 Nov 2025). At its core, dichotomy-based inference divides the decision or prediction space recursively by binary (or more generally, multiway) intervals, at each stage invoking agent-based reasoning—potentially informed by learned models, expert reports, and retrieved historical analogues—to refine the prediction within dynamically focused subspaces.

1. Mathematical Foundations: Dichotomy in Laplacian-Type Multi-Agent Systems

The classical dichotomy principle was formalized in the context of continuous- and discrete-time multi-agent coordination via Laplacian-type differential or difference inequalities (Proskurnikov et al., 2016). In a network of $N$ agents with scalar variables $x_i(t)$, consider the continuous-time differential inequality:

$$\frac{dx_i}{dt} \le -\sum_{j=1}^N a_{ij}(t)\bigl(x_i(t) - x_j(t)\bigr) + f_i(t)$$

where $a_{ij}(t) \ge 0$ are (possibly time-varying) interaction weights, $L(t)$ is the graph Laplacian, and $f_i(t)$ are disturbances or exogenous signals.

Dichotomy is defined as follows: the inequality is dichotomic if every bounded solution converges to a finite limit, and consensus dichotomic if the limit is a consensus point $c 1_N$. Discrete-time analogues are studied via systems of the form:

$x(k+1) \le W(k) x(k) + f(k)$

with $W(k)$ a row-stochastic matrix.

The theory shows that under natural connectivity and balance conditions, all bounded trajectories are forced to synchronize, with the result that the one-sided (not necessarily contractive) inequalities guarantee global convergence provided the graph—the communication or influence pattern—meets specified connectivity criteria. This property underpins both traditional coordination and contemporary dichotomy-based multi-agent inference.

2. Architectural Overview: Dichotomy-Based Multi-Agent Inference Systems

In modern practical deployments, such as SurvAgent for multimodal survival prediction (Huang et al., 20 Nov 2025), a dichotomy-based multi-agent system comprises the following components:

• Search Agent: Assembles attribute checklists by querying structured knowledge bases.
• Domain-Specific Expert Agents (e.g., PathAgent, GenAgent): These agents process data modalities (e.g., WSIs at various magnifications; genomics stratified into gene categories), generate structured reports, and refine chain-of-thought (CoT) explanations.
• Inference Agent: Orchestrates retrieval-augmented generation (RAG), integrates structured reports, retrieved reasoning paths, and predictions from pre-trained survival models.
• Expert Survival Models: An ensemble of $M$ models provides candidate predictions for the target outcome.
• Data Flow: For new data, agents produce structured reports; similar historical cases (with their reasoning chains and outcomes) are retrieved; expert model predictions are gathered; all are synthesized by the Inference Agent using dichotomy-based progressive interval refinement.

3. Dichotomy Mechanism: Progressive Interval Refinement

Central to inference is the hierarchical dichotomy mechanism, in which the prediction space is recursively partitioned into intervals. At each dichotomy level $d$:

• The agent faces an interval $[a_d, b_d]$; a pivot $m_d$ is selected (e.g., clinical quartiles).
• The agent receives the query: "Does the target outcome (e.g., survival time) fall within $[a_d, m_d]$ (lower) or $[m_d, b_d]$ (upper)?" Inputs include multimodal structured reports, retrieved case exemplars with CoT, and expert predictions.
• The process recurses: the selected half-interval becomes the focus for the next level.
• At the conclusion (after $D$ levels), the agent outputs a calibrated estimate within the terminal interval by integrating all evidence sources.

Formally, representative steps and formulas include:

• Retrieval:

$$\mathcal{B}_\text{retrieved} = \mathrm{RAG}\left(\mathcal{R}_\text{test}^\text{WSI}, \mathcal{R}_\text{test}^\text{gene}; \mathcal{B}_\text{WSI}, \mathcal{B}_\text{gene}, K\right)$$

• Expert model ensemble:

$$\{\hat t_m\}_{m=1}^M = \{\mathcal{M}_m(\mathcal{W}_\text{test}, \mathcal{G}_\text{test})\}$$

• Interval splits (illustrative for survival prediction):
  • Level 1: $m_1 = 24$ months; intervals $[0, 24]$ vs. $[24, \infty)$
  • Level 2: If in $[0, 24]$, $m_2 = 12$ months; else $m_2 = 36$ months

At each step, the dichotomy decision $y_d$ is made by an Inference Agent $\mathcal{A}_\text{infer}$, leveraging all multimodal and retrieved evidence.

4. Integration of Retrieved Evidence and Expert Models

The dichotomy-based multi-agent flow tightly integrates retrieval-augmented case banks and expert predictions:

• Each instance is embedded (e.g., using joint text-vision encoders), with top-$K$ similar cases retrieved based on cosine similarity.
• Retrieved cases include summarized reports, refined CoT reasoning, and ground-truth outcomes.
• All retrieved content and expert predictions (including risk scores pre-binned into quartiles) are concatenated into a structured prompt for each dichotomy decision.
• No fixed rule (e.g., weighted averaging) aggregates expert outputs; the Inference Agent performs soft, context-sensitive aggregation, potentially overruling any individual model based on the totality of structured and historical evidence.

5. Theoretical Guarantees: Connectivity and Convergence

The classical dichotomy theory guarantees convergence under specific structural conditions (Proskurnikov et al., 2016):

Condition	Consequence for Dichotomy	Consensus Property
Static: Strongly Connected Graph	Consensus dichotomic	All bounded solutions $\to$ consensus
Time-Varying: Uniform Strong Connectivity	Consensus dichotomic	See Theorem 2: USC sufficient for consensus dichotomy
Cut-Balanced + Infinite Strong Connectivity	Consensus dichotomic iff ISC	Theorem 3: necessary & sufficient, $L^1$-integrable coupling
Isolated Strongly Connected Components	Dichotomic, not necessarily consensus	Converges within components, may differ between components

Key proof arguments leverage ordering and spread-contraction, exploiting that the maximum (or minimum) value is monotone, and agents outside consensus are progressively “dragged” toward the common limiting value.

6. Representative Applications

A. Distributed Coordination and Opinion Dynamics

In classical systems, dichotomy-based differential inequalities yield unifying proofs for:

• Consensus with disturbance: Even under perturbations $f_i(t)$ with $f \in L^1$, bounded solutions converge to consensus.
• Containment/Aggregation: Agents are driven toward target sets or sets defined by leader-followers; Lyapunov functions as distances to sets obey dichotomy-based inequalities.
• Opinion polarization (Altafini model): The absolute values of agent states converge, yielding consensus or bipartite outcomes depending on network signs.

B. Multimodal Survival Prediction

SurvAgent implements a dichotomy-based multi-agent system for survival analysis (Huang et al., 20 Nov 2025), establishing new empirical benchmarks:

• Structured reports and chain-of-thought case banks are leveraged to inform dichotomy-based inference.
• The dichotomy-based interval refinement algorithm increases the Concordance Index (C-index) from 0.461 (no inference) to 0.689 (inference only), with further gains to 0.713 when combined with multimodal CoT bank retrievals.
• Kaplan–Meier analysis confirms discriminative patient stratification.

7. Limitations and Open Problems

Known limitations and theoretical challenges include:

• The dependence of explicit convergence rates on graph structure; only in certain regimes (e.g., leader-follower) are decay rates available.
• The sufficiency gap between Uniform Strong Connectivity (USC) and Infinite Strong Connectivity (ISC) in time-varying graphs remains unresolved.
• Extensions to nonlinear interaction weights, delays, higher-order agent dynamics, and strongly heterogeneous agents require further advances in nonlinear dichotomy analysis.
• In discrete time, careful handling of order switching is needed for the spread-contraction argument.
• In practical systems such as SurvAgent, operator intervention is required to select interval cut-points, and the interpretability of multi-agent aggregation relies on the transparency of embedded case banks and CoT traces.

• "Differential Inequalities in Multi-Agent Coordination and Opinion Dynamics Modeling" (Proskurnikov et al., 2016).
• "SurvAgent: Hierarchical CoT-Enhanced Case Banking and Dichotomy-Based Multi-Agent System for Multimodal Survival Prediction" (Huang et al., 20 Nov 2025).