The benchmark consensus protocol is used by the most notable distributed algorithms for controlling multiagent systems through local interaction rules, where it yields to the well-known Laplacian matrix whose null space is fixed and is spanned by the vector of ones provided that the underlying graph is connected. Since this protocol is the key building block for a wide range of other existing distributed algorithms, its extensions are likewise based on this Laplacian matrix. However, the fixed nature of the null space of this Laplacian matrix limits the composition of complex (i.e. reconfigurable) cooperative behaviours for multiagent systems. Motivated by this standpoint, the contribution of this paper is threefold: (i) For undirected and connected graphs, we introduce a new Laplacian matrix, whose null space is spanned by a user-assigned vector composed of nonzero elements. We give its mathematical definition and show that it inherits key properties of the aforementioned well-known Laplacian matrix. (ii) We next present distributed control architectures for convergence to the desired null space as well as for convergence to a specific vector within that null space. (iii) We finally multiplex information networks to distributively control the null space of this new Laplacian matrix, where this is the most important contribution of this paper. We specifically show that the resulting null space can vary through local interaction rules as desired between the spans of two or more user-assigned vectors to pave the way for composing complex cooperative behaviours for multiagent systems.