Adaptive Frontier Exploration on Graphs with Applications to Network-Based Disease Testing

TL;DR

The paper addresses sequential frontier-constrained exploration on graphs with label-bearing nodes drawn from a Markov random field, targeting early and efficient identification of positive labels (e.g., infections) under discounting. It introduces Adaptive Frontier Exploration on Graphs (AFEG) and a Gittins-index policy, proving optimality on forest graphs and delivering a polynomial-time implementation with $O(n^2|\mathbf{\Omega}|^2)$ time and $O(n|\mathbf{\Omega}|^2)$ oracle calls. It further formalizes network-based disease testing as AFEG via pairwise MRFs and outlines learning of distribution parameters from data using pseudo-likelihood. Empirical results on synthetic and real-world networks show that Gittins outperforms natural baselines, including in tree-like, budget-limited, and noisy settings, indicating strong practical potential for network-guided disease testing and related frontier-driven exploration tasks.

Abstract

We study a sequential decision-making problem on a $n$-node graph $\mathcal{G}$ where each node has an unknown label from a finite set $\mathbfΩ$, drawn from a joint distribution $\mathcal{P}$ that is Markov with respect to $\mathcal{G}$. At each step, selecting a node reveals its label and yields a label-dependent reward. The goal is to adaptively choose nodes to maximize expected accumulated discounted rewards. We impose a frontier exploration constraint, where actions are limited to neighbors of previously selected nodes, reflecting practical constraints in settings such as contact tracing and robotic exploration. We design a Gittins index-based policy that applies to general graphs and is provably optimal when $\mathcal{G}$ is a forest. Our implementation runs in $\mathcal{O}(n^2 \cdot |\mathbfΩ|^2)$ time while using $\mathcal{O}(n \cdot |\mathbfΩ|^2)$ oracle calls to $\mathcal{P}$ and $\mathcal{O}(n^2 \cdot |\mathbfΩ|)$ space. Experiments on synthetic and real-world graphs show that our method consistently outperforms natural baselines, including in non-tree, budget-limited, and undiscounted settings. For example, in HIV testing simulations on real-world sexual interaction networks, our policy detects nearly all positive cases with only half the population tested, substantially outperforming other baselines.

Figures (14)

• Figure 1: Illustration of how a real-world transmission graph (left) can be framed as an AFEG instance (right). Here, the joint distribution $\mathcal{P}$ over the labels $X_A, X_B, X_C, X_D \in \{+, -\}$ may depend on the covariates $\mathbf{c}_A, \mathbf{c}_B, \mathbf{c}_C, \mathbf{c}_D \in \mathbb{R}^d$ and underlying interaction graph structure.
• Figure 2: Reduction to a branching bandit on 8 nodes with root $X_1$. After acting on $\{X_1, X_3\}$, the frontier is $\{X_2, X_4, X_6\}$. Note that we have $\mathcal{P}(x_2 \mid x_1, x_3) = \mathcal{P}(x_2 \mid x_1)$ by the Markov property.
• Figure 3: Subset of synthetic tree input results. Gittins consistently beats other baselines at every fixed budget, e.g., vertical line indicates performance when only half the nodes can be acted upon.
• Figure 4: Synthetic experiment: the initial performance gains of Gittins over Greedy and DQN diminishes as we progressively add edges to $10$ random $50$-node trees with discount factor $\beta = 0.9$.
• Figure 5: Experimental results for Gonorrhea, Chlamydia, Syphilis, HIV, and Hepatitis from subsampling connected components till we have at least 300 nodes. The vertical dashed line indicates performance when only half the individuals can be tested.

Theorems & Definitions (19)

• Definition 1: The Adaptive Frontier Exploration on Graphs (AFEG) problem
• Definition 2: The Gittins policy; \ref{['alg:computing-gittins']}
• Theorem 3
• Lemma 3
• Proposition 3
• Theorem 4
• Lemma 6
• proof : Proof of \ref{['lem:number-of-changepoints']}
• Lemma 6
• Lemma 6