Vertex-transitive graphIn the mathematical field of graph theory, a vertex-transitive graph is a graph G in which, given any two vertices v_1 and v_2 of G, there is some automorphism such that In other words, a graph is vertex-transitive if its automorphism group acts transitively on its vertices. A graph is vertex-transitive if and only if its graph complement is, since the group actions are identical. Every symmetric graph without isolated vertices is vertex-transitive, and every vertex-transitive graph is regular.
Information theoryInformation theory is the mathematical study of the quantification, storage, and communication of information. The field was originally established by the works of Harry Nyquist and Ralph Hartley, in the 1920s, and Claude Shannon in the 1940s. The field, in applied mathematics, is at the intersection of probability theory, statistics, computer science, statistical mechanics, information engineering, and electrical engineering. A key measure in information theory is entropy.
Convergence of measuresIn mathematics, more specifically measure theory, there are various notions of the convergence of measures. For an intuitive general sense of what is meant by convergence of measures, consider a sequence of measures μn on a space, sharing a common collection of measurable sets. Such a sequence might represent an attempt to construct 'better and better' approximations to a desired measure μ that is difficult to obtain directly.
Outer measureIn the mathematical field of measure theory, an outer measure or exterior measure is a function defined on all subsets of a given set with values in the extended real numbers satisfying some additional technical conditions. The theory of outer measures was first introduced by Constantin Carathéodory to provide an abstract basis for the theory of measurable sets and countably additive measures.
Measure (mathematics)In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge.
PreorderIn mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. Preorders are more general than equivalence relations and (non-strict) partial orders, both of which are special cases of a preorder: an antisymmetric (or ) preorder is a partial order, and a symmetric preorder is an equivalence relation. The name comes from the idea that preorders (that are not partial orders) are 'almost' (partial) orders, but not quite; they are neither necessarily antisymmetric nor asymmetric.
Hamiltonian pathIn the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a cycle that visits each vertex exactly once. A Hamiltonian path that starts and ends at adjacent vertices can be completed by adding one more edge to form a Hamiltonian cycle, and removing any edge from a Hamiltonian cycle produces a Hamiltonian path.
Large-scale brain networkLarge-scale brain networks (also known as intrinsic brain networks) are collections of widespread brain regions showing functional connectivity by statistical analysis of the fMRI BOLD signal or other recording methods such as EEG, PET and MEG. An emerging paradigm in neuroscience is that cognitive tasks are performed not by individual brain regions working in isolation but by networks consisting of several discrete brain regions that are said to be "functionally connected".
Euclidean distanceIn mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore occasionally being called the Pythagorean distance. These names come from the ancient Greek mathematicians Euclid and Pythagoras, although Euclid did not represent distances as numbers, and the connection from the Pythagorean theorem to distance calculation was not made until the 18th century.
HypergraphIn mathematics, a hypergraph is a generalization of a graph in which an edge can join any number of vertices. In contrast, in an ordinary graph, an edge connects exactly two vertices. Formally, a directed hypergraph is a pair , where is a set of elements called nodes, vertices, points, or elements and is a set of pairs of subsets of . Each of these pairs is called an edge or hyperedge; the vertex subset is known as its tail or domain, and as its head or codomain. The order of a hypergraph is the number of vertices in .
