## Network Theory and Discrete Calculus – Graph Divergence and Graph Laplacian

This post is part of a series

### Another Note on Notation

In a previous post, I introduced a slightly generalized notation in order to deal with directed graphs with multiple directed edges between any two nodes, e.g. parallel elements in electrical networks. However, the revised notation now makes some simpler calculations look more cumbersome. This is an example of what my adviser called the **conservation of frustration**. For example, the coboundary is now given by:

Applied to a general discrete 0-form, this becomes

To re-simplify the notation while maintaining the advantages of the new generalized notation, we can define

and we’re back to

and

as before. Furthermore, we have

where is the number of directed edges from node to node .

### Trace and Inner Products

Given a discrete 0-form , we define its trace via

Similarly, given a discrete 1-form , its trace is given by

With the trace, we can define the inner product of discrete 0-forms via

and the inner product of discrete 1-forms via

where is the edge product.

### Graph Divergence

The** graph divergence** was introduced here as a boundary operator, but the relation to divergence was mentioned here.

With the inner products defined above, a simple calculation shows

so the graph divergence is the adjoint of the coboundary.

In relating discrete calculus to algebraic topology, typically, in algebraic topology you would have a coboundary operator for cochains and a boundary operator for chains. With discrete calculus, we have both and for discrete forms.

### Graph Laplacian

The **graph Laplacian** of a discrete 0-form is given by

More generally, we could define a **graph Laplace-Beltrami operator**

### Graph Dirac Operator

The **graph Dirac operator** is essentially the “square root” of the graph Laplace-Beltrami operator. Since and , we have

so the/a graph Dirac operator is given by

[…] was defined in a previous post. […]

Network Theory and Discrete Calculus – Noether’s Theorem « Phorgy PhynanceDecember 25, 2011 at 9:09 am