About: Calculus on finite weighted graphs

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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

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rdfs:label	Calculus on finite weighted graphs (en)
rdfs:comment	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 (en)
dbp:wikiPageUsesTemplate	dbt:Note dbt:Ref
Subject	Calculus Graph theory
prov:wasDerivedFrom	http://en.wikipedia.org/wiki/Calculus_on_finite_weighted_graphs?oldid=1006121662&ns=0
Wikipage page ID	61556353 (xsd:integer)
page length (characters) of wiki page	21712 (xsd:nonNegativeInteger)
Wikipage revision ID	1006121662 (xsd:integer)
Link from a Wikipage to another Wikipage	Machine learning Noise reduction Nonlocal operator Vertex (graph theory) Moore neighborhood Statistical classification Calculus Calculus of variations Calculus Glossary of graph theory Graph theory Differential equation Digital image processing Directed graph Discrete Laplace operator Discrete calculus Domain of a function Degree (graph theory) Linear map Phase-field models on graphs Pixel Tomographic reconstruction Cluster analysis Community structure Component (graph theory) Connectivity (graph theory) Operator (mathematics) Ordered pair Partial differential equation Social network Finite set Graph Image segmentation Handwriting recognition Inpainting Inverse problem Transpose of a linear map Types of mesh Loop (graph theory) Laplace operator Laplacian matrix Lattice graph Mathematics Multigraph Neighbourhood Network theory Point cloud Recurrence relation Regular grid
has abstract	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. (en)
foaf:isPrimaryTopicOf	http://en.wikipedia.org/wiki/Calculus_on_finite_weighted_graphs
is Wikipage redirect of	Graph divergence Graph gradient Weighted divergence Weighted gradient
is Link from a Wikipage to another Wikipage of	Discrete calculus Graph divergence Graph gradient Laplacian matrix
is foaf:primaryTopic of	http://en.wikipedia.org/wiki/Calculus_on_finite_weighted_graphs

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