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Conditions for Unique Localizability in Cooperative Localization of Wireless ad hoc and Sensor Networks
Published in Chao Gao, Guorong Zhao, Hassen Fourati, Cooperative Localization and Navigation, 2019
Redundant rigidity is a stronger type of rigidity than (plain) rigidity. A rigid graph G = (V,E) is redundantly rigid if G' = (V,E'), where E' = E\{e} remains rigid for every edge e in E. For example, the graph in Figure 2.4 is not redundantly rigid, because if we remove the edge v2v3, it becomes flexible. On the other hand, the graph in Figure 2.5 is redundantly rigid; that is, if we remove any one of the edges in this graph, it still remains rigid. We note that a redundantly rigid graph does not need to have all the possible edges between the set of given vertices. For example, the edge v3v5 is missing in the graph in Figure 2.5. A graph that has edges between every pair of vertices is called a complete graph. An example of a complete graph is shown in Figure 2.6. We note that a complete graph G = (V,E) with |V| > 3 is always a redundantly rigid graph.
G
Published in Philip A. Laplante, Comprehensive Dictionary of Electrical Engineering, 2018
set of variables or non-terminals, P a set of production rules and S V is called the start symbol. Terminals are the symbols with which the strings of the language are made. For example, T may be the English alphabet, the ASCII character set, or the set {0, 1}. Non-terminals are symbols that are replaced by a string of zero or more terminals and non-terminals. The production rules specify which strings can be used to replace non-terminals. The symbol S is the first symbol with which every production starts. All the strings that can be generated from S using rules in P are said to be in L(G), the language of G. For example, let G be the grammar (T, V, P, S) such that T = {a, b} V = {S} where S is the start symbol and P = {S aSb, S } where is the empty string. The language L(G) is the set of all strings of the form a n bn . The symbol signifies that S can be replaced with whatever follows the arrow. Gram-Schmidt orthogonalization a recursive procedure for whitening (decorrelating) a sequence of random vectors. See whitening filter. grant signal a control signal on a bus that gives permission to control the bus to another module. graph a couple G = (E, V ) where V is a set of nodes and E V × V is a set of edges. Graphs are widely used in modeling networks, circuits, and software. graph search an optimization technique used to find the minimum cost path from a starting point to a goal point, through a graph of interconnected nodes. Each link between nodes has an associated path cost, which must be selected based on the problem of interest. See optimization.
Graph Edit Distance—Theory, Algorithms, and Applications
Published in Olivier Lézoray, Leo Grady, Image Processing and Analysis with Graphs, 2012
Without loss of generality, Definition 13.2.1 can be seen as the definition of a labeled graph. Notice that there is not any restriction concerning the nature of the labels of nodes and edges. That is, the label alphabet is not constrained at all. L may be defined as a vector space (i.e. L=ℝn) or simply as a finite or infinite set of discrete labels (i.e., L = {α, β, γ, ⋯ }). The set of labels L can also include the null label (often represented by ε). If all the nodes and edges are labeled with the same null label, the graph is considered as unlabeled. A weighted graph is a special type of labeled graph in which each node is labeled with the null label and each edge (υi, υj) is labeled with a real number or weight wij, usually belonging, but not restricted, to the interval [0, 1]. An unweighted graph can be seen as a particular instance of a weighted graph where ∀(vi,vj) ∈ E, wij = 1. The label of an element ● (where ● can be a node or an edge) is denoted by L(●).
A new criterion for defining the failure of a fractured rock mass slope based on the strength reduction method
Published in Geomatics, Natural Hazards and Risk, 2020
Yuan Wei, Tan Hanhua, Niu Jiandong, Peng Shu, Xue Yanyu, Wang Wei, Sun Xiaoyun
A graph is a mathematical structure consisting of a set of vertices and a set of edges connecting the vertices, which could be represented by a function to model the interactions among nodes and depict a fixed topology of nodes. In this function, denotes a nonempty vector quantity containing nodes, denotes a binary relation pairing two random nodes and denotes a weighted adjacency matrix. It is assumed that the adjacency elements in the matrix are utilized to represent the edges of graph and that the values of all the elements are positive (i.e. >0) except when the subscripts are equal (i.e. )—these elements are excluded from the graph—or when two nodes have no paired relationship (Ge et al. 2017). Figure 1 is provided as an example to illustrate an undirected graph in the plane. As shown in Figure 1, graph contains four vertices and five edges, which are expressed as follows:
Dual topology for partitioning of water distribution networks considering actual valve locations
Published in Urban Water Journal, 2019
G. F. Santonastaso, A. Di Nardo, E. Creaco
In all the applications of this work, network graphs were considered as unweighted graphs. This means that each connection between nodes and segments, looking at the original and dual topologies respectively, was assigned a unitary weight. However, the formulation can be generalized to consider a generic weight ω at each connection, the value of which can be defined in many ways. In this case, matrix A (or A’) should be constructed by considering at each row –ω0.5 and ω0.5, instead of −1 and 1, in correspondence to the upstream and downstream nodes (or segments) of the associated connection, respectively. The other meaningful matrixes would then change as a result. For instance, the generic element (i, j) of matrix B (or B’) would be equal to the sum of the weights ω between the i-th node (or segment) and the j-th node (or segment), rather than equal to the total number of connections between the two nodes (or segments).
Spatiotemporal traffic forecasting: review and proposed directions
Published in Transport Reviews, 2018
Alireza Ermagun, David Levinson
Two links are parallel if they connect the same pair of nodes.Two links are adjacent if they share a common node.A link is loop if its two nodes are the same.A graph is simple if it has no parallel links or loops.A graph is directed if its links show direction.A graph is connected if at least one link exists between every pair of nodes.A ring network is a closed path where every node has exactly two links incident with it.