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Cheeger Constant (graph Theory)
In mathematics, the Cheeger constant (also Cheeger number or isoperimetric number) of a graph is a numerical measure of whether or not a graph has a "bottleneck". The Cheeger constant as a measure of "bottleneckedness" is of great interest in many areas: for example, constructing well-connected networks of computers, card shuffling. The graph theoretical notion originated after the Cheeger isoperimetric constant of a compact Riemannian manifold. The Cheeger constant is named after the mathematician Jeff Cheeger. Definition Let be an undirected finite graph with vertex set and edge set . For a collection of vertices , let denote the collection of all edges going from a vertex in to a vertex outside of (sometimes called the ''edge boundary'' of ): :\partial A := \. Note that the edges are unordered, i.e., \ = \. The Cheeger constant of , denoted , is defined by :h(G) := \min \left\. The Cheeger constant is strictly positive if and only if is a connected graph. Intui ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cheeger Constant
In Riemannian geometry, the Cheeger isoperimetric constant of a compact Riemannian manifold ''M'' is a positive real number ''h''(''M'') defined in terms of the minimal area of a hypersurface that divides ''M'' into two disjoint pieces. In 1971, Jeff Cheeger proved an inequality that related the first nontrivial eigenvalue of the Laplace–Beltrami operator on ''M'' to ''h''(''M''). In 1982, Peter Buser proved a reverse version of this inequality, and the two inequalities put together are sometimes called the ''Cheeger-Buser inequality''. These inequalities were highly influential not only in Riemannian geometry and global analysis, but also in the theory of Markov chains and in graph theory, where they have inspired the analogous Cheeger constant of a graph and the notion of conductance. Definition Let ''M'' be an ''n''-dimensional closed Riemannian manifold. Let ''V''(''A'') denote the volume of an ''n''-dimensional submanifold ''A'' and ''S''(''E'') denote the ''n''&mi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring Network
A ring network is a network topology in which each node connects to exactly two other nodes, forming a single continuous pathway for signals through each node – a ring. Data travels from node to node, with each node along the way handling every packet. Rings can be unidirectional, with all traffic travelling either clockwise or anticlockwise around the ring, or bidirectional (as in SONET/SDH). Because a unidirectional ring topology provides only one pathway between any two nodes, unidirectional ring networks may be disrupted by the failure of a single link. A node failure or cable break might isolate every node attached to the ring. In response, some ring networks add a "counter-rotating ring" (C-Ring) to form a redundant topology: in the event of a break, data are wrapped back onto the complementary ring before reaching the end of the cable, maintaining a path to every node along the resulting C-Ring. Such "dual ring" networks include the ITU-T's PSTN telephony systems networ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conductance (graph Theory)
In theoretical computer science, graph theory, and mathematics, the conductance is a parameter of a Markov chain that is closely tied to its Markov chain mixing time, mixing time, that is, how rapidly the chain converges to its Discrete-time Markov chain#Stationary distributions, stationary distribution, should it exist. Equivalently, the conductance can be viewed as a parameter of a directed Graph (discrete mathematics), graph, in which case it can be used to analyze how quickly random walk, random walks in the graph converge. The conductance of a graph is closely related to the Cheeger constant (graph theory), Cheeger constant of the graph, which is also known as the Expander graph#Edge expansion, edge expansion or the isoperimetic number. However, due to subtly different definitions, the conductance and the edge expansion do not generally coincide if the graphs are not Regular graph, regular. On the other hand, the notion of electrical conductance that appears in electrical ne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cheeger Bound
In mathematics, the Cheeger bound is a bound of the second largest eigenvalue of the transition matrix of a finite-state, discrete-time, reversible stationary Markov chain. It can be seen as a special case of Cheeger inequalities in expander graphs. Let X be a finite set and let K(x,y) be the transition probability for a reversible Markov chain on X. Assume this chain has stationary distribution \pi. Define :Q(x,y) = \pi(x) K(x,y) and for A,B \subset X define : Q(A \times B) = \sum_ Q(x,y). Define the constant \Phi as : \Phi = \min_ \frac. The operator K, acting on the space of functions from , X, to \mathbb, defined by : (K \phi)(x) = \sum_y K(x,y) \phi(y) has eigenvalues \lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_n . It is known that \lambda_1 = 1. The Cheeger bound is a bound on the second largest eigenvalue \lambda_2. Theorem (Cheeger bound): : 1 - 2 \Phi \leq \lambda_2 \leq 1 - \frac. See also * Stochastic matrix * Cheeger constant In Rie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Connectivity
The algebraic connectivity (also known as Fiedler value or Fiedler eigenvalue after Miroslav Fiedler) of a graph ' is the second-smallest eigenvalue (counting multiple eigenvalues separately) of the Laplacian matrix of '. This eigenvalue is greater than 0 if and only if ' is a connected graph. This is a corollary to the fact that the number of times 0 appears as an eigenvalue in the Laplacian is the number of connected components in the graph. The magnitude of this value reflects how well connected the overall graph is. It has been used in analyzing the robustness and synchronizability of networks. Properties The truncated icosahedron or Buckminsterfullerene graph has a traditional connectivity (graph theory)">connectivity of 3, and an algebraic connectivity of 0.243. The algebraic connectivity of undirected graphs with nonnegative weights is a(G)\geq0, with the inequality being strict if and only if is connected. However, the algebraic connectivity can be negative for gener ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Graph Theory
In mathematics, spectral graph theory is the study of the properties of a Graph (discrete mathematics), 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 number, real symmetric matrix and is therefore Orthogonal diagonalization, orthogonally diagonalizable; its eigenvalues are real algebraic integers. While the adjacency matrix depends on the vertex labeling, its Spectrum of a matrix, spectrum is a graph invariant, although not a complete one. Spectral graph theory is also concerned with graph parameters that are defined via multiplicities of eigenvalues of matrices associated to the graph, such as the Colin de Verdière graph invariant, Colin de Verdière number. Cospectral graphs Two graphs are called cospectral or isospectral if the adjacency matrices of the graphs are isospectral, that is, if t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplacian Matrix
In 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 functional graph properties. Kirchhoff's theorem can be used to calculate the number of spanning trees for a given graph. The sparsest cut of a graph can be approximated through the Fiedler vector — the eigenvector corresponding to the second smallest eigenvalue of the graph Laplacian — as established by Cheeger's inequality. The spectral decomposition of the Laplacian matrix allows the construction of low-dimensional embeddings that appear in many machine learning applications and determines a spectral layo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Gap
In mathematics, the spectral gap is the difference between the moduli of the two largest eigenvalues of a matrix or operator; alternately, it is sometimes taken as the smallest non-zero eigenvalue. Various theorems relate this difference to other properties of the system. The spectral gap gets its name from the ''matrix spectrum'', that is, for a matrix, the list of its eigenvalues. It provides insight on diffusion within the graph: corresponding the spectral gap to the smallest non-zero eigenvalue, it is then the mode of the network state that shows the slowest exponential decay over time. See also * Cheeger constant (graph theory) * Cheeger constant (Riemannian geometry) * Eigengap * Spectral gap (physics) * Spectral radius ''Spectral'' is a 2016 Hungarian-American military science fiction action film co-written and directed by Nic Mathieu. Written with Ian Fried (screenwriter), Ian Fried & George Nolfi, the film stars James Badge Dale as DARPA research scientist Ma . ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expander Graph
In graph theory, an expander graph is a sparse graph that has strong connectivity properties, quantified using vertex, edge or spectral expansion. Expander constructions have spawned research in pure and applied mathematics, with several applications to complexity theory, design of robust computer networks, and the theory of error-correcting codes. Definitions Intuitively, an expander graph is a finite, undirected multigraph in which every subset of the vertices that is not "too large" has a "large" boundary. Different formalisations of these notions give rise to different notions of expanders: ''edge expanders'', ''vertex expanders'', and ''spectral expanders'', as defined below. A disconnected graph is not an expander, since the boundary of a connected component is empty. Every connected finite graph is an expander; however, different connected graphs have different expansion parameters. The complete graph has the best expansion property, but it has largest possible ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expander Graphs
In graph theory In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ..., an expander graph is a sparse graph that has strong connectivity (graph theory), connectivity properties, quantified using vertex (graph theory), vertex, edge (graph theory), edge or spectral graph theory, spectral expansion. Expander constructions have spawned research in pure and applied mathematics, with several applications to Computational complexity theory, complexity theory, design of robust computer networks, and the theory of error-correcting codes. Definitions Intuitively, an expander graph is a finite, undirected multigraph in which every subset of the vertices that is not "too large" has a "large" boundary (graph theory), boundary. Different formalisations of these notions give rise to different notions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
