In mathematics, calculus on finite weighted graphs is a discrete calculus for functions whose domain is the vertex set of a graph with a finite number of vertices and weights associated to the edges.
This involves formulating discrete operators on graphs which are analogous to differential operators in calculus, such as graph Laplacians (or discrete Laplace operators) as discrete versions of the Laplacian, and using these operators to formulate differential equations, difference equations, or variational models on graphs which can be interpreted as discrete versions of partial differential equations or continuum variational models.
Such equations and models are important tools to mathematically model, analyze, and process discrete information in many different research fields, e.g., image processing, machine learning, and network analysis.
In applications, finite weighted graphs represent a finite number of entities by the graph's vertices, any pairwise relationships between these entities by graph edges, and the significance of a relationship by an edge weight function.
Differential equations or difference equations on such graphs can be employed to leverage the graph's structure for tasks such as image segmentation (where the vertices represent pixels and the weighted edges encode pixel similarity based on comparisons of Moore neighborhoods or larger windows), data clustering, data classification, or community detection in a social network (where the vertices represent users of the network, the edges represent links between users, and the weight function indicates the strength of interactions between users).
The main advantage of finite weighted graphs is that by not being restricted to highly regular structures such as discrete regular grids, lattice graphs, or meshes, they can be applied to represent abstract data with irregular interrelationships.
If a finite weighted graph is geometrically embedded in a Euclidean space, i.e., the graph vertices represent points of this space, then it can be interpreted as a discrete approximation of a related nonlocal operator in the continuum setting.
[1] On the remainder of this page, the graphs will be assumed to be undirected, unless specifically stated otherwise.
Many of the ideas presented on this page can be generalized to directed graphs.
For both mathematical and application specific reasons, the weight function on the edges is often required to be strictly positive and on this page it will be assumed to be so unless specifically stated otherwise.
Generalizations of many of the ideas presented on this page to include negatively weighted edges are possible.
Sometimes an extension of the domain of the edge weight function to
usually represents a single entity in the given data, e.g., elements of a finite data set, pixels in an image, or users in a social network.
A graph edge represents a relationship between two entities, e.g. pairwise interactions or similarity based on comparisons of geometric neighborhoods (for example of pixels in images) or of another feature, with the edge weight encoding the strength of this relationship.
Most commonly used weight functions are normalized to map to values between 0 and 1, i.e.,
In the following it is assumed that the considered graphs are connected without self-loops or multiple edges between vertices.
(see the section on differential graph operators below), and edge weights can encode similar information as multiple edges could.
is the weighted size of its neighborhood: Note that in the special case where
In applications vertex functions are useful to label the vertices of the nodes.
For example, in graph-based data clustering, each node represents a data point and a vertex function is used to identify cluster membership of the nodes.
The space of those extended edge functions is still denoted by
This means that each edge function can be identified with a linear matrix operator.
An important ingredient in the calculus on finite weighted graphs is the mimicking of standard differential operators from the continuum setting in the discrete setting of finite weighted graphs.
This allows one to translate well-studied tools from mathematics, such as partial differential equations and variational methods, and make them usable in applications which can best be modeled by a graph.
Based on this one can derive higher-order difference operators, e.g., the graph Laplacian.
To measure the local variation of a vertex function
of the weighted gradient operator is a linear operator defined by For undirected graphs with a symmetric weight function
of the Laplace operator in the continuum setting, the weighted graph Laplacian can be derived for any vertex
, i.e., Calculus on finite weighted graphs is used in a wide range of applications from different fields such as image processing, machine learning, and network analysis.
A non-exhaustive list of tasks in which finite weighted graphs have been employed is: