Bayesian Optimization of Functions over Node Subsets in Graphs

TL;DR

This paper tackles the challenge of optimizing black-box utilities defined on $k$-node subsets of a graph, a problem plagued by combinatorial explosion, expensive evaluations, and partial observability of the graph. It introduces GraphComBO, which constructs a combo-graph where each node corresponds to a $k$-node subset and uses a local, recursive combo-subgraph sampling strategy to enable sample-efficient Bayesian optimization on this discrete space. A Graph Gaussian Process surrogate on the local combo-subgraph, together with a diffusion-kernel based acquisition (EI), guides the search within a trust-region-like neighborhood and updates centers when improvements occur. Empirical results on synthetic and real-world networks show GraphComBO consistently outperforming baselines, with ablations revealing how subset size, trust-region size, and function-graph alignment influence performance.

Abstract

We address the problem of optimizing over functions defined on node subsets in a graph. The optimization of such functions is often a non-trivial task given their combinatorial, black-box and expensive-to-evaluate nature. Although various algorithms have been introduced in the literature, most are either task-specific or computationally inefficient and only utilize information about the graph structure without considering the characteristics of the function. To address these limitations, we utilize Bayesian Optimization (BO), a sample-efficient black-box solver, and propose a novel framework for combinatorial optimization on graphs. More specifically, we map each $k$-node subset in the original graph to a node in a new combinatorial graph and adopt a local modeling approach to efficiently traverse the latter graph by progressively sampling its subgraphs using a recursive algorithm. Extensive experiments under both synthetic and real-world setups demonstrate the effectiveness of the proposed BO framework on various types of graphs and optimization tasks, where its behavior is analyzed in detail with ablation studies.

Figures (26)

• Figure 1: Demonstration of how the proposed framework traverses the combinatorial graph $\hat{\mathcal{G}}^{<k>}$ introduced in §\ref{['sec combo-graph']} with an exemplar original graph $\mathcal{G}$ of $6$ nodes and a subset size of $k=2$. At iteration t, we first construct a local combo-subgraph $\Tilde{\mathcal{G}}_t = \{\Tilde{\mathcal{V}}_t, \Tilde{\mathcal{E}}_t\}$ of size $Q$=6 using Algorithm \ref{['alg Combo-subgraph']} (§\ref{['sec combo-subgraph']}), which is centred at combo-node $\hat{v}^*_{t-1}$ from last iteration t-1 or initialization. Next, a $\mathcal{GP}$ surrogate is fitted on $\Tilde{\mathcal{G}}_t$ with queried combo-nodes inside $\Tilde{\mathcal{G}}_t$ being the training set. The next query location is then selected as the combo-node that maximizes the acquisition function $\hat{v}^*_t = \arg\max_{\hat{v} \in \Tilde{\mathcal{V}}_t} \alpha (\hat{v})$. If queried values $f(\hat{v}^*_t) \geq f(\hat{v}^*_{t-1})$, the next combo-subgraph $\Tilde{\mathcal{G}}_{t+1}$ will be re-sampled at a new center $\hat{v}^*_t$, or otherwise remain the same. Finally, we repeat the previous process to obtain a new query location for the next iteration t+1, and the search continues until stopping criteria are triggered.
• Figure 2: Illustration of a combinatorial graph $\hat{\mathcal{G}}^{<2>}$ constructed by the recursive combo-subgraph sampling (Algorithm \ref{['alg Combo-subgraph']}).
• Figure 3: Results for synthetic problems on BA, WS, SBM and 2D-Grid networks with $k=[4,8,16,32]$, where Regret indicates the difference between ground truth and the best query so far.
• Figure 4: Results for flattening the curve, patient-zero tracing and influence maximization.
• Figure 5: Results for road resilience testing and GNN attacks on molecules with edge-masking.

Theorems & Definitions (5)

• Definition 3.1
• Lemma 3.2
• Lemma 3.3
• proof
• proof