nodecentered

Node-centered is a term used to describe a design, representation, or discretization in which quantities are associated with the nodes (vertices) of a graph, mesh, or network. This approach is often contrasted with cell-centered or edge-centered schemes and is common in computational mathematics and data analysis.

In numerical methods for partial differential equations, node-centered discretizations place unknowns at mesh nodes. For example, a scalar field value is defined at each vertex, and derivatives are approximated using values from neighboring nodes. In solving Poisson’s equation on a mesh, the resulting linear system typically has one equation per node, influencing the stencil structure and how boundary conditions are applied. Node-centered schemes can be easier to implement on irregular meshes and provide straightforward coupling to nodal data.

In graph and network contexts, node-centered modeling emphasizes attributes or state variables located at nodes, with edges representing relationships. This perspective underpins many graph algorithms, network dynamics models, and some graph neural network architectures where information is aggregated over a node’s neighborhood.

Compared with edge-centered or cell-centered schemes, node-centered methods have distinct advantages and limitations. They can simplify the enforcement of nodal constraints and work well with unstructured meshes, but may require interpolation to align with data defined on edges or cells. The choice between node-centered and other centering schemes often depends on the problem geometry, data availability, and the numerical method employed.

See also: Finite element method, finite difference method, mesh discretization, cell-centered, edge-centered, graph neural networks.