Lambda-connectedness

In applied mathematics, lambda-connectedness (or λ-connectedness) deals with partial connectivity for a discrete space.
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 formula_1, two pixels are called formula_1-connected if there is a path linking those two pixels and the connectedness of this path is at least formula_1. formula_1-connectedness is an equivalence relation.
Background.
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.
formula_1-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.
Basic concepts.
A generalized definition of formula_1-connectedness can be described as follows: a simple system formula_7, where formula_8 is called a potential function of formula_9. If formula_7 is an image, then formula_9 is a 2D or 2D grid space and formula_8 is an intensity function. For a color image, one can use formula_13 to represent formula_8.
Neighbor connectivity will be first defined on a pair of adjacent points. Then one can define the general connectedness for any two points.
Assume formula_15 is used to measure the neighbor-connectivity of x,y where x and y are adjacent.
In graph "G" = ("V", "E"), a finite sequence formula_16 is called a path, if formula_17.
The path-connectivity formula_18 of a path formula_19
is defined as
Finally, the degree of connectedness (connectivity) of two vertices x,y with respect to formula_8 is defined as
For a given formula_23, point formula_24 and formula_25 are said to be formula_1-connected if formula_27.
formula_1-connectedness is a equivalence relation. It can be used in image segmentation.
Relations to Image Segmentation.
The lambda-connected segmentation is a region-growing segmentation method in general. It can also be made for split-and-merge segmentation. Its time complexity also reaches the optimum at formula_29 where formula_30 is the number of pixels in the image. Also see 
The lambda-connectedness has close relationships to Data Science that can be found in the Book entitled Mathematical Problems in Data Science.
New Developments.
Researchers applied related techniques to smooth 3D data processing and transportation network management recently. 