This handout extends foundational graph concepts into more expressive network forms that capture the directional, emotional, and multi-relational nature of real-world systems. Directed and signed graphs are introduced to represent asymmetry and polarity in relationships, where influence, prestige, or antagonism are encoded through directional or signed edges. 

The computational implications of non-symmetric adjacency matrices and triadic balance calculations are discussed, highlighting how mathematical formulation reflects social tension and stability. Multigraphs and hypergraphs are then explored as models for multiplex and group-based interactions, demonstrating how multiple edge types or group memberships can be represented and analyzed using layered adjacency or incidence matrices. Techniques such as clique expansion, layer projection, and multiplex centrality computation are presented as strategies for operationalizing higher-order relationships. 

Finally, the synthesis of complex graph forms is examined in light of computational trade-offs, emphasizing the balance between model expressiveness and algorithmic tractability. The topic concludes by asserting that the power of computational social network analysis lies in the capacity to model complexity without losing interpretability.

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