Graph theoryIn mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices (also called nodes or points) which are connected by edges (also called links or lines). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics.
Brain connectivity estimatorsBrain connectivity estimators represent patterns of links in the brain. Connectivity can be considered at different levels of the brain's organisation: from neurons, to neural assemblies and brain structures. Brain connectivity involves different concepts such as: neuroanatomical or structural connectivity (pattern of anatomical links), functional connectivity (usually understood as statistical dependencies) and effective connectivity (referring to causal interactions).
Graph drawingGraph drawing is an area of mathematics and computer science combining methods from geometric graph theory and information visualization to derive two-dimensional depictions of graphs arising from applications such as social network analysis, cartography, linguistics, and bioinformatics. A drawing of a graph or network diagram is a pictorial representation of the vertices and edges of a graph. This drawing should not be confused with the graph itself: very different layouts can correspond to the same graph.
Spectral graph theoryIn mathematics, spectral graph theory is the study of the properties of a graph in relationship to the characteristic polynomial, eigenvalues, and eigenvectors of matrices associated with the graph, such as its adjacency matrix or Laplacian matrix. The adjacency matrix of a simple undirected graph is a real symmetric matrix and is therefore orthogonally diagonalizable; its eigenvalues are real algebraic integers. While the adjacency matrix depends on the vertex labeling, its spectrum is a graph invariant, although not a complete one.
Adjacency matrixIn graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph. In the special case of a finite simple graph, the adjacency matrix is a (0,1)-matrix with zeros on its diagonal. If the graph is undirected (i.e. all of its edges are bidirectional), the adjacency matrix is symmetric. The relationship between a graph and the eigenvalues and eigenvectors of its adjacency matrix is studied in spectral graph theory.
ConnectomeA connectome (kəˈnɛktoʊm) is a comprehensive map of neural connections in the brain, and may be thought of as its "wiring diagram". An organism's nervous system is made up of neurons which communicate through synapses. A connectome is constructed by tracing the neuron in a nervous system and mapping where neurons are connected through synapses. The significance of the connectome stems from the realization that the structure and function of the human brain are intricately linked, through multiple levels and modes of brain connectivity.
Graph databaseA graph database (GDB) is a database that uses graph structures for semantic queries with nodes, edges, and properties to represent and store data. A key concept of the system is the graph (or edge or relationship). The graph relates the data items in the store to a collection of nodes and edges, the edges representing the relationships between the nodes. The relationships allow data in the store to be linked together directly and, in many cases, retrieved with one operation.
Laplacian matrixIn the mathematical field of graph theory, the Laplacian matrix, also called the graph Laplacian, admittance matrix, Kirchhoff matrix or discrete Laplacian, is a matrix representation of a graph. Named after Pierre-Simon Laplace, the graph Laplacian matrix can be viewed as a matrix form of the negative discrete Laplace operator on a graph approximating the negative continuous Laplacian obtained by the finite difference method. The Laplacian matrix relates to many useful properties of a graph.
Connective tissueConnective tissue is one of the four primary types of animal tissue, along with epithelial tissue, muscle tissue, and nervous tissue. It develops from the mesenchyme, derived from the mesoderm, the middle embryonic germ layer. Connective tissue is found in between other tissues everywhere in the body, including the nervous system. The three meninges, membranes that envelop the brain and spinal cord, are composed of connective tissue. Most types of connective tissue consists of three main components: elastic and collagen fibers, ground substance, and cells.
Small-world networkA small-world network is a mathematical graph in which most nodes are not neighbors of one another, but the neighbors of any given node are likely to be neighbors of each other. Due to this, most neighboring nodes can be reached from every other node by a small number of hops or steps. Specifically, a small-world network is defined to be a network where the typical distance L between two randomly chosen nodes (the number of steps required) grows proportionally to the logarithm of the number of nodes N in the network, that is: while the global clustering coefficient is not small.
Extracellular matrixIn biology, the extracellular matrix (ECM), also called intercellular matrix, is a network consisting of extracellular macromolecules and minerals, such as collagen, enzymes, glycoproteins and hydroxyapatite that provide structural and biochemical support to surrounding cells. Because multicellularity evolved independently in different multicellular lineages, the composition of ECM varies between multicellular structures; however, cell adhesion, cell-to-cell communication and differentiation are common functions of the ECM.
Component (graph theory)In graph theory, a component of an undirected graph is a connected subgraph that is not part of any larger connected subgraph. The components of any graph partition its vertices into disjoint sets, and are the induced subgraphs of those sets. A graph that is itself connected has exactly one component, consisting of the whole graph. Components are sometimes called connected components. The number of components in a given graph is an important graph invariant, and is closely related to invariants of matroids, topological spaces, and matrices.
Loose connective tissueLoose connective tissue, also known as areolar tissue, is a cellular connective tissue with thin and relatively sparse collagen fibers. Its ground substance occupies more volume than the fibers do. It has a viscous to gel-like consistency and plays an important role in the diffusion of oxygen and nutrients from the capillaries that course through this connective tissue as well as in the diffusion of carbon dioxide and metabolic wastes back to the vessels.
Clustering coefficientIn graph theory, a clustering coefficient is a measure of the degree to which nodes in a graph tend to cluster together. Evidence suggests that in most real-world networks, and in particular social networks, nodes tend to create tightly knit groups characterised by a relatively high density of ties; this likelihood tends to be greater than the average probability of a tie randomly established between two nodes (Holland and Leinhardt, 1971; Watts and Strogatz, 1998). Two versions of this measure exist: the global and the local.
Matrix (biology)In biology, matrix (: matrices) is the material (or tissue) in between a eukaryotic organism's cells. The structure of connective tissues is an extracellular matrix. Fingernails and toenails grow from matrices. It is found in various connective tissues. It serves as a jelly-like structure instead of cytoplasm in connective tissue. Extracellular matrix The main ingredients of the extracellular matrix are glycoproteins secreted by the cells. The most abundant glycoprotein in the ECM of most animal cells is collagen, which forms strong fibers outside the cells.
Uncertainty coefficientIn statistics, the uncertainty coefficient, also called proficiency, entropy coefficient or Theil's U, is a measure of nominal association. It was first introduced by Henri Theil and is based on the concept of information entropy. Suppose we have samples of two discrete random variables, X and Y. By constructing the joint distribution, PX,Y(x, y), from which we can calculate the conditional distributions, PXY(xy) = PX,Y(x, y)/PY(y) and PYX(yx) = PX,Y(x, y)/PX(x), and calculating the various entropies, we can determine the degree of association between the two variables.
Cluster analysisCluster analysis or clustering is the task of grouping a set of objects in such a way that objects in the same group (called a cluster) are more similar (in some sense) to each other than to those in other groups (clusters). It is a main task of exploratory data analysis, and a common technique for statistical data analysis, used in many fields, including pattern recognition, , information retrieval, bioinformatics, data compression, computer graphics and machine learning.
Closed graph propertyIn mathematics, particularly in functional analysis and topology, closed graph is a property of functions. A function f : X → Y between topological spaces has a closed graph if its graph is a closed subset of the product space X × Y. A related property is open graph. This property is studied because there are many theorems, known as closed graph theorems, giving conditions under which a function with a closed graph is necessarily continuous. One particularly well-known class of closed graph theorems are the closed graph theorems in functional analysis.
Functional analysisFunctional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, inner product, norm, or topology) and the linear functions defined on these spaces and suitably respecting these structures. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining, for example, continuous or unitary operators between function spaces.
Fractional anisotropyFractional anisotropy (FA) is a scalar value between zero and one that describes the degree of anisotropy of a diffusion process. A value of zero means that diffusion is isotropic, i.e. it is unrestricted (or equally restricted) in all directions. A value of one means that diffusion occurs only along one axis and is fully restricted along all other directions. FA is a measure often used in diffusion imaging where it is thought to reflect fiber density, axonal diameter, and myelination in white matter.
