The classical concept of controllability from control theory is applied to social networks subjected to exogenous input, where the nodes can be divided into two groups, i.e., leaders and followers. Some necessary and sufficient conditions for the controllability of undirected and directed networks are derived based on algebra and graph theory. These conditions are, on the one hand, translated to algebraic test on the eigenvalues/eigenvectors of the matrix related to the dynamical system or on the rank of the controllability matrix and, on the other hand, are verified by the property of leader asymmetric. Under this new framework, the controllability of social networks with various essentially different structures is generalized. It will be shown that a network is completely controllable if the network is a stem, bud, or cactus graph.
Model Description Origin network: where, state vector input signal Laplacian matrix the input signal is acted on the node Leader-follower network: Follower network:
1. Undirected network
The networked system (1) with a single leader is controllable if and only if the following conditions are satisfied:
(1) The eigenvalues of are all distinct;
(2) The eigenvectors of are not orthogonal to .
The networked system (1) with multiple leaders is controllable if and only if and do not share any common eigenvalues.
Graph-theoretic condition: The networked system (1) with a single leader is uncontrollable if it is leader symmetric.
2. Directed network Algebraic condition: The networked system (1) is controllable if and only if the controllability matrixis of full row rank, i.e., .