Foundational models have advanced social robotics, enabling richer perception and communicative interaction with users. However, current systems still struggle with multi-turn engagement, social-relationship reasoning, and contextually grounded dialogue at scale. We present ARIS (Agentic and Relationship Intelligence System), an agentic AI framework that unifies multimodal reasoning, a graph-based Social World Model, and retrieval-augmented generation (RAG) within a single modular architecture for social robots. We evaluate ARIS with the Pepper robot in a robot-mediated dyadic conversational setting, comparing it against a large language model baseline. A user study (N=23) shows that ARIS yields significantly higher perceived intelligence, animacy, anthropomorphism, and likeability. Our contributions are threefold: (1)~a Social World Model that explicitly maps and updates social relationships between users through a knowledge graph, enabling social reasoning and re-identification across encounters; (2)~an efficient RAG-based conversational pipeline that maintains bounded latency as dialogue histories grow to thousands of exchanges while preserving response relevance; and (3)~system integration and empirical validation of these components within a modular agentic architecture that coordinates speech, vision, and physical action through structured APIs. The implementation of ARIS will be released as open source upon publication.