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Preliminaries
Published in Yongzhao Hua, Xiwang Dong, Ren Zhang, Formation Tracking Control for Heterogeneous Swarm Systems, 2022
Yongzhao Hua, Xiwang Dong, Ren Zhang
A graph G is said to be undirected if εij∈E implies εji∈E and wij=wji. An undirected graph is connected if there is an undirected path between every pair of distinct nodes. A directed graph is said to have a spanning tree if there is a root node which has at least one directed path to every other node. A directed graph is strongly connected if there is a directed path from every node to every other node. Two graph examples are given in Fig. 2.1, where the first one is a connected undirected graph and the second one is a directed graph with spanning tree.
Graph Algorithms I
Published in R. Balakrishnan, Sriraman Sridharan, Discrete Mathematics, 2019
R. Balakrishnan, Sriraman Sridharan
Consider a directed graph without circuits. Geometrically, a topological sorttopological sort means laying the vertices of a directed graph without circuits in a horizontal manner from left to right such that each arc of the graph goes from left to right. In other words, if the graph has n vertices, we are interested in assigning the integers 1, 2, …, n to the vertices of the graph such that if there is an arc from the vertex i to the vertex j, then we have the inequality i < j.
The Directional Discrete Cosine Transform
Published in Humberto Ochoa-Domínguez, K. R. Rao, Discrete Cosine Transform, 2019
Humberto Ochoa-Domínguez, K. R. Rao
The graphs can be undirectedUndirected graph and directedDirected graph. An undirected graph has no directed edges. Figures 7.3 thru 7.5 are examples of undirected graphs. In directed graphs, edges have a specific direction. Figure 7.6 shows an example of a directed graph.
Distributed constrained consensus for discrete multi-agent systems with additive noises
Published in International Journal of Systems Science, 2022
Huifang Sun, Xiaowen Wang, Jing Zhang, Shuai Liu
Define a directed path as a sequence of edges of the form . A directed graph is strongly connected if there exists a directed path from each node to every other node. An undirected graph is called connected if there is an undirected path between every two nodes. A directed spanning tree of contains a node set and an edge set which is a subset of . Note that the directed graph has a directed spanning tree iff the graph has at least one node with a directed path to every other node. A balanced directed graph is strongly connected iff it contains a spanning tree.
Incorporating individual preference and network influence on choice behavior of electric vehicle sharing using agent-based model
Published in International Journal of Sustainable Transportation, 2020
Wang Ning, Jiahui Guo, Xiang Liu, Huizhong Pan
The social network analysis mainly investigates and studies the relational data, rather than the attribute data in the traditional questionnaires (Scott, 2016). Relational data represent the ties and connections between nodes in a relatively closed network, usually expressed with dichotomous or polytomous variable, corresponding to the presence or strength of the links respectively. The social network data are managed and processed by matrix. The elements of the first row (or the first column) represent each node (each node means each respondent in our study) in the network. In the dichotomous asymmetric matrix, the element with the value 1 indicates that the link is transmitted from the corresponding column node to the corresponding row node, while the element with the value 0 shows that there is no such connection between the two nodes. Scholars also use socio-gram to express the structure of the social network data of small scale. The socio-gram concerns about the collection of elements and the connections among them. The elements are called node, and the connections are called links. The directed graph is used if links are transmitted from one node to another, while the undirected graph is used if the presence of links instead of the direction of the links is concerned about. Figure 1 shows the matrix and the socio-gram of the same network. In the social network, B and C are reciprocated ties; C to A and A to B are unidirectional ties.
Travel Plans in Public Transit Networks Using Artificial Intelligence Planning Models
Published in Applied Artificial Intelligence, 2019
Fernando Elizalde-Ramírez, Romeo Sanchez Nigenda, Iris A. Martínez-Salazar, Yasmín Á. Ríos-Solís
Travel planning in a public transit network greatly differs from the general road planning problem. In a static network with fixed costs, a road planning problem is equivalent to the problem of finding, in a directed graph, the shortest path between two nodes (Dijkstra 1959; Schulz, Wagner, and Weihe 2000). As mentioned earlier, one important difference of public transportation networks with respect to general road networks is that they are time and space dependent. That is, users can travel some parts of the network at specific times, given bus schedules and their locations. Therefore, two of the main challenges in public transit networks are to model the transit timetabling information (Bast et al. 2014), and to define the optimization metrics to traverse such time-space sensitive networks (López and Lozano 2014).