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In recent years, social networks have surged in popularity as one of the main applications of the Internet. One key aspect of social network research is exploring important unobserved network information which is not explicitly represented. This study first introduces a new path identification problem to identify the existences of unobserved paths between nodes. Given a partial social network structure where the indications of observed nodes about unobserved paths are assumed to exist, we propose a multiple-level classification based path identification method (MCPIM) for graph-based social networks. MCPIM presents the new multiple-level similarity to efficiently represent the structural positions of subgraph placeholders. Subsequently, a quantum mechanism based genetic classification algorithm (QGCA) is constructed to efficiently divide subgraph placeholders into different clusters. The nodes whose subgraph placeholders are in the same cluster owning large structural similarities are inferred to have unobserved paths. Results obtained by comparing with state-of-the-art methods via extensive experiments using disparate real-world social networks show that MCPIM can well identify the existences of unobserved paths between nodes in graph-based social networks. [ABSTRACT FROM AUTHOR]