Assume that a function on a discrete space (usually a graph) is given. A degree of connectivity (connectedness) will be defined to measure the connectedness of the space with respect to the function. It was invented to create a new method for image segmentation. The method has expanded to handle other problems related to uncertainty for incomplete information analysis.  
For a digital image and a certain value of , two pixels are called -connected if there is a path linking those two pixels and the connectedness of this path is at least . -connectedness is an equivalence relation.
Connectedness is a basic measure in many areas of mathematical science and social sciences. In graph theory, two vertices are said to be connected if there is a path between them. In topology, two points are connected if there is a continuous function that could move from one point to another continuously. In management science, for example, in an institution, two individuals are connected if one person is under the supervision of the other. Such connected relations only describe either full connection or no connection. lambda-connectedness is introduced to measure incomplete or fuzzy relations between two vertices, points, human beings, etc.
In fact, partial relations have been studied in other aspects. Random graph theory allows one to assign a probability to each edge of a graph. This method assumes, in most cases, each edge has the same probability. On the other hand, Bayesian networks are often used for inference and analysis when relationships between each pair of states/events, denoted by vertices, are known. These relationships are usually represented by conditional probabilities among these vertices and are usually obtained from outside of the system.
-connectedness is based on graph theory; however, graph theory only deals with vertices and edges with or without weights. In order to define a partial, incomplete, or fuzzy connectedness, one needs to assign a function on the vertex in the graph. Such a function is called a potential function. It can be used to represent the intensity of an image, the surface of a XY-domain, or the utility function of a management or economic network.
A generalized definition of -connectedness can be described as follows: a simple system , where is called a potential function of . If is an image, then is a 2D or 2D grid space and is an intensity function. For a color image, one can use to represent .
A neighbor connectivity will be first defined on a pair of adjacent points. Then one can define the general connectedness for any two points.
Assume is used to measure the neighbor-connectivity of x,y where x and y are adjacent. In graph G = (V, E), a finite sequence is called a path, if .
The path-connectivity of a path is defined as
Finally, the degree of connectedness (connectivity) of two vertices x,y with respect to is defined as
For a given , point and are said to be -connected if .
-connectedness is a equivalence relation. It can be used in image segmentation.
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