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Evaluation of Bridge Road Network Security Based on PageRank Algorithm
Published in Airong Chen, Xin Ruan, Dan M. Frangopol, Life-Cycle Civil Engineering: Innovation, Theory and Practice, 2021
Y.C. Yue, K. Jia, Z.Y. Wang, G.H. Peng, D.Y. Xiong, X. Lu
Degrees is one of the simplest and most direct indicators to characterize the attributes of a single node. The degree of a node in a network refers to the number of nodes directly connected to it in the network, which is recorded as K [4]. It represents the local direct influence of a node in the network. The more nodes a node is directly connected to, the higher the degree value, the greater its influence, and the more important it is in the network [5]. In a directed graph, the degree of a node is divided into an out degree and an in degree. The out degree is the total number of nodes pointed to by a node, and the in degree is the total number of nodes pointed to that node. Summing the jth row of the adjacency matrix is the out degree of the j node; Summing column j is the in degree of node j. In an undirected graph, the degree of out and degree of in are not distinguished.
Finding Clusters
Published in Wendy L. Martinez, Angel R. Martinez, Jeffrey L. Solka, Exploratory Data Analysis with MATLAB®, 2017
Wendy L. Martinez, Angel R. Martinez, Jeffrey L. Solka
This approach to clustering proceeds by first forming the graph using our n data points as vertices. The edge weights are obtained using the interpoint distances. As we will see in the next example, one way to specify a graph in MATLAB is to use an adjacency matrix. An adjacency matrix is a square symmetric matrix where the rows and columns correspond to nodes in the graph. A nonzero entry of one represents an edge between node i and j in an unweighted graph. In MATLAB, an adjacency matrix can have a nonzero entry greater than one, indicating the edge weight. Once we have a graph, we can find a minimum spanning tree. Next, we remove the k largest edges from this tree. This will divide the graph into components or subgraphs. Each of these subgraphs will constitute a cluster.
Graph Algorithms I
Published in R. Balakrishnan, Sriraman Sridharan, Discrete Mathematics, 2019
R. Balakrishnan, Sriraman Sridharan
Which representation to choose: adjacency matrix or adjacency linked list?: The answer to this question depends on the operations performed by the algorithm at hand. If a graph is “sparse,” that is, the graph does not have too many arcs, then the adjacency list representation may be suitable. Suppose an algorithm on graph frequently needs to test the existence of an arc from a vertex to another vertex, then the adjacency matrix will be appropriate since accessing an arbitrary entry of a matrix can be done in constant time. With the adjacency matrix, since the initialization itself takes O(n2) time, the importance of algorithms may be diminished with complexity O(n log n) or O(n3/2).
Virtual Power Plant Partition Strategy Based on Network Representation
Published in Electric Power Components and Systems, 2022
Network representation learning can preserve network topology and node information, and embed network nodes into low dimensional vector space. This method makes the original network easier to deal with in the new vector space, which is convenient for further analysis of the network. Network representation learning has the following advantages [21]: It is convenient for calculation and storage. Traditionally, an adjacency matrix is used to store graphs, but for large graphs, adjacency matrices have high dimensionality and high spatial complexity. Therefore, the network representation learning method is used to represent the nodes in the network in a low dimensional dense vector space and can maintain the relevant structure and characteristics of the original network.Without manual extraction of network features, heterogeneous information can be projected into the same dimension space to facilitate downstream calculation.
Command-filter-based distributed containment control of nonlinear multi-agent systems with actuator failures
Published in International Journal of Control, 2018
Guozeng Cui, Shengyuan Xu, Qian Ma, Ze Li, Yuming Chu
Let be a weighted directed graph with the set of nodes , and the set of directed edges . The ordered pair of node (j, i) denotes a directed edge, and it means that node i can receive information from node j. is the neighbour set of node i. The elements of the adjacency matrix related to are defined as aij = 1 if and only if there is a directed edge (j, i) in ; otherwise, aij = 0. Throughout this paper, it is assumed that aii = 0. A directed path is a sequence of nodes 1, 2,… , r such that , i ∈ {1, 2,… , r − 1}. A directed graph has a directed spanning tree if there exists at least one node called root node which has a directed path to all the other nodes. The Laplacian matrix of graph is defined as , where ; .
Prescribed performance distributed consensus control for nonlinear multi-agent systems with unknown dead-zone input
Published in International Journal of Control, 2018
Guozeng Cui, Shengyuan Xu, Qian Ma, Yongmin Li, Zhengqiang Zhang
In this paper, a directed graph is used to describe the information communication among followers, where denotes a nonempty set of nodes, and is a set of directed edges. The ordered pair of node (j, i) denotes a directed edge, and it implies that node i can access information from node j. is the neighbour set of node i. The elements of the adjacency matrix associated with are defined as aij = 1 if and only if there is a directed edge (j, i) in ; otherwise, aij = 0. It is assumed that aii = 0. is the Laplacian matrix, where ; is the diagonal element of the degree matrix . If there is at least one node named root node which has a directed path to all the other nodes, we call this directed graph includes a directed spanning tree.