The controllability of multiagent systems governed by node and edge dynamics on signed graphs is investigated under distinct partition rules and control algorithms. Motivated by a social network case study, dynamic models for nodes and edges are established. A novel concept “edge bidirectional mapping” is introduced, along with an associated matrix that rigorously characterizes the relationship between node and edge Laplacian matrices. Necessary and sufficient conditions for controllability equivalence between node and edge dynamics are derived across three topological regimes, determined by the cardinality relationships between the number of nodes n and edges m, namely n > m, n = m, and n < m. Furthermore, more explicit and comprehensive criteria for controllability properties of node and edge dynamics are formulated under distinct control algorithms and partition rules. Specifically, the proposed algorithms are considered in both absolute and relative forms, while the partition rules, distance partition (DP), generalized almost equitable partition (GAEP), and generalized relaxed almost equitable partition (GRAEP), are distinguished based on the number of positive, negative, and total connections between vertices (nodes or edges) within a cell and their neighboring connections to other cells. Notably, this study systematically analyzes their hierarchical inclusion relationships for the first time. Upper and lower bounds on the dimensions of the node and edge controllable subspaces are presented. The lower bound reflects DP with edge weight considerations, while the upper bound is obtained via partition algorithms that yield the best LGRAEP. Additionally, the computational complexity and convergence properties of the proposed partition rules (DP, GEP, GREP), Algorithms 1-2, and the derived bounds for the controllable subspace dimensions are analyzed. Theoretical results are comprehensively validated through multi-level simulations, including verification of theoretical bounds, analysis of clustering effects, and comparative model evaluation.