The Minimum Dominating Set (MDS) problem is a classic NP-hard combinatorial optimization problem with critical applications in social network analysis, including viral marketing, influence maximization, public health interventions, and information dissemination. Identifying a minimal set of influential individuals whose reach covers an entire social network is central to these applications, yet remains computationally challenging at scale. Graph neural networks (GNNs) have emerged as powerful tools for learning over graphs, and recent work explores their application to hard combinatorial problems. This paper presents a novel unsupervised GNN framework for the MDS problem that eliminates the need for ground-truth solutions during training. Trained on 12,000 synthetic graphs with diverse structural properties, our method achieves up to 55x faster inference than metaheuristic baselines and up to 14x faster inference than supervised learning approaches, while finding optimal or near-optimal dominating sets on real-world social network benchmarks. Our learned heuristic generalizes effectively to unseen graph distributions, demonstrating strong practical applicability for large-scale social network analysis.

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