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Network Controllability
Network controllability concerns the structural controllability of a Graph (discrete mathematics), network. Controllability describes our ability to guide a dynamical system from any initial state to any desired final state in finite time, with a suitable choice of inputs. This definition agrees well with our intuitive notion of control. The controllability of general directed and weighted complex networks has recently been the subject of intense study by a number of groups in wide variety of networks, worldwide. Recent studies by Sharma et al. on multi-type biological networks (gene–gene, miRNA–gene, and protein–protein interaction networks) identified control targets in phenotypically characterized Osteosarcoma showing important role of genes and proteins responsible for maintaining tumor microenvironment. Background Consider the canonical linear time-invariant dynamics on a complex network \dot(t) = \mathbf \cdot \mathbf(t) + \mathbf\cdot \mathbf(t) where the vector \m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
YYL2
Lynn Lake Airport is an airport located adjacent to Lynn Lake, Manitoba, Canada. It features a paved runway, and is the furthest north runway of its type accessible by highway in Manitoba. The airport was set to close on 6 May 2013 due to costs, although a provincial spokesperson stated in the ''Winnipeg Free Press'', "It's important for the northern economy and we'll be looking at it in the short and long term". After entering into a public/private partnership with YYL Airport Inc. for the continued operation of the airport, long-term operational status is being achieved. See also *Lynn Lake (Eldon Lake) Water Aerodrome Lynn Lake (Eldon Lake) Water Aerodrome is located south southeast of Lynn Lake, Manitoba, Canada. See also * List of airports in Manitoba * Lynn Lake Airport Lynn Lake Airport is an airport located adjacent to Lynn Lake, Manitoba Lyn ... References External links Certified airports in Manitoba {{Manitoba-airport-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Controllability Gramian
In control theory, we may need to find out whether or not a system such as \begin \dot(t)\boldsymbol(t)+\boldsymbol(t)\\ \boldsymbol(t)=\boldsymbol(t)+\boldsymbol(t) \end is controllable, where \boldsymbol, \boldsymbol, \boldsymbol and \boldsymbol are, respectively, n\times n, n\times p, q\times n and q\times p matrices. One of the many ways one can achieve such goal is by the use of the Controllability Gramian. Controllability in LTI Systems Linear Time Invariant (LTI) Systems are those systems in which the parameters \boldsymbol, \boldsymbol, \boldsymbol and \boldsymbol are invariant with respect to time. One can observe if the LTI system is or is not controllable simply by looking at the pair (\boldsymbol,\boldsymbol). Then, we can say that the following statements are equivalent: 1. The pair (\boldsymbol,\boldsymbol) is controllable. 2. The n\times n matrix \boldsymbol(t)=\int_^e^\boldsymbole^d\tau=\int_^e^\boldsymbole^d\tau is nonsingular for any t>0. 3. The n\times n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimum Rank Of A Graph
In mathematics, the minimum rank is a graph parameter \operatorname(G) for a graph ''G''. It was motivated by the Colin de Verdière graph invariant. Definition The adjacency matrix of an undirected graph is a symmetric matrix whose rows and columns both correspond to the vertices of the graph. Its elements are all 0 or 1, and the element in row ''i'' and column ''j'' is nonzero whenever vertex ''i'' is adjacent to vertex ''j'' in the graph. More generally, a ''generalized adjacency matrix'' is any symmetric matrix of real numbers with the same pattern of nonzeros off the diagonal (the diagonal elements may be any real numbers). The minimum rank of G is defined as the smallest rank of any generalized adjacency matrix of the graph; it is denoted by \operatorname (G). Properties Here are some elementary properties. *The minimum rank of a graph is always at most equal to ''n'' − 1, where ''n'' is the number of vertices in the graph. *For every induced subgraph ''H'' of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Graph Theory
Algebraic graph theory is a branch of mathematics in which algebraic methods are applied to problems about graphs. This is in contrast to geometric, combinatoric, or algorithmic approaches. There are three main branches of algebraic graph theory, involving the use of linear algebra, the use of group theory, and the study of graph invariants. Branches of algebraic graph theory Using linear algebra The first branch of algebraic graph theory involves the study of graphs in connection with linear algebra. Especially, it studies the spectrum of the adjacency matrix, or the Laplacian matrix of a graph (this part of algebraic graph theory is also called spectral graph theory). For the Petersen graph, for example, the spectrum of the adjacency matrix is (−2, −2, −2, −2, 1, 1, 1, 1, 1, 3). Several theorems relate properties of the spectrum to other graph properties. As a simple example, a connected graph with diameter ''D'' w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lenka Zdeborová
Lenka Zdeborová (born 24 November 1980) is a Czech physicist and computer scientist who applies methods from statistical physics to machine learning and constraint satisfaction problems. She is a professor of physics and computer science and communication systems at EPFL (École Polytechnique Fédérale de Lausanne). Life Zdeborová was born in Plzeň and attended a local grammar school where she excelled in math and physics. After living in France with her family and working at the Centre National de la Recherche Scientifique (CNRS), she and her partner moved to Switzerland in 2020. They are currently raising their two children there. Education and career Zdeborová earned a master's degree in physics at Charles University in 2004, and 2008, completed an international dual doctorate ("en cotutelle") at both Charles University and University of Paris-Sud. Her doctoral advisors were Václav Janiš at Charles University, and Marc Mézard at Paris-Sud. After postdoctoral re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cavity Method
The cavity method is a mathematical method presented by Marc Mézard, Giorgio Parisi and Miguel Angel Virasoro in 1987 to solve some mean field type models in statistical physics, specially adapted to disordered systems. The method has been used to compute properties of ground states in many condensed matter and optimization problems. Initially invented to deal with the Sherrington–Kirkpatrick model of spin glasses, the cavity method has shown wider applicability. It can be regarded as a generalization of the Bethe— Peierls iterative method in tree-like graphs, to the case of a graph with loops that are not too short. The different approximations that can be done with the cavity method are usually named after their equivalent with the different steps of the replica method which is mathematically more subtle and less intuitive than the cavity approach. The cavity method has proved useful in the solution of optimization problems such as k-satisfiability and graph coloring. It ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hopcroft–Karp Algorithm
In computer science, the Hopcroft–Karp algorithm (sometimes more accurately called the Hopcroft–Karp–Karzanov algorithm) is an algorithm that takes a bipartite graph as input and produces a maximum cardinality matching as output – a set of as many edges as possible with the property that no two edges share an endpoint. It runs in O(, E, \sqrt) time in the worst case, where E is set of edges in the graph, V is set of vertices of the graph, and it is assumed that , E, =\Omega(, V, ). In the case of dense graphs the time bound becomes O(, V, ^), and for sparse random graphs it runs in time O(, E, \log , V, ) with high probability. The algorithm was discovered by and independently by . As in previous methods for matching such as the Hungarian algorithm and the work of , the Hopcroft–Karp algorithm repeatedly increases the size of a partial matching by finding ''augmenting paths''. These paths are sequences of edges of the graph, which alternate between edges in the matching ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matching (graph Theory)
In the mathematical discipline of graph theory, a matching or independent edge set in an undirected graph is a set of edges without common vertices. Finding a matching in a bipartite graph can be treated as a network flow problem. Definitions Given a graph a matching ''M'' in ''G'' is a set of pairwise non-adjacent edges, none of which are loops; that is, no two edges share common vertices. A vertex is matched (or saturated) if it is an endpoint of one of the edges in the matching. Otherwise the vertex is unmatched (or unsaturated). A maximal matching is a matching ''M'' of a graph ''G'' that is not a subset of any other matching. A matching ''M'' of a graph ''G'' is maximal if every edge in ''G'' has a non-empty intersection with at least one edge in ''M''. The following figure shows examples of maximal matchings (red) in three graphs. : A maximum matching (also known as maximum-cardinality matching) is a matching that contains the largest possible number of edges. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degree (graph Theory)
In graph theory, the degree (or valency) of a vertex of a graph is the number of edges that are incident to the vertex; in a multigraph, a loop contributes 2 to a vertex's degree, for the two ends of the edge. The degree of a vertex v is denoted \deg(v) or \deg v. The maximum degree of a graph G, denoted by \Delta(G), and the minimum degree of a graph, denoted by \delta(G), are the maximum and minimum of its vertices' degrees. In the multigraph shown on the right, the maximum degree is 5 and the minimum degree is 0. In a regular graph, every vertex has the same degree, and so we can speak of ''the'' degree of the graph. A complete graph (denoted K_n, where n is the number of vertices in the graph) is a special kind of regular graph where all vertices have the maximum possible degree, n-1. In a signed graph, the number of positive edges connected to the vertex v is called positive deg(v) and the number of connected negative edges is entitled negative deg(v). Handshaking lemma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Controllability
Controllability is an important property of a control system, and the controllability property plays a crucial role in many control problems, such as stabilization of unstable systems by feedback, or optimal control. Controllability and observability are dual aspects of the same problem. Roughly, the concept of controllability denotes the ability to move a system around in its entire configuration space using only certain admissible manipulations. The exact definition varies slightly within the framework or the type of models applied. The following are examples of variations of controllability notions which have been introduced in the systems and control literature: * State controllability * Output controllability * Controllability in the behavioural framework State controllability The state of a deterministic system, which is the set of values of all the system's state variables (those variables characterized by dynamic equations), completely describes the system at any give ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closeness Centrality
In a connected graph, closeness centrality (or closeness) of a node is a measure of centrality in a network, calculated as the reciprocal of the sum of the length of the shortest paths between the node and all other nodes in the graph. Thus, the more central a node is, the ''closer'' it is to all other nodes. Closeness was defined by Bavelas (1950) as the reciprocal of the farness, that is: : C_B(x)= \frac. where d(y,x) is the distance (length of the shortest path) between vertices x and y. This unnormalised version of closeness is sometimes known as status When speaking of closeness centrality, people usually refer to its normalized form which represents the average length of the shortest paths instead of their sum. It is generally given by the previous formula multiplied by N-1, where N is the number of nodes in the graph resulting in: : C(x)= \frac. The normalization of closeness simplifies the comparison of nodes in graphs of different sizes. For large graphs, the minus one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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