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Combinatorics
Published in Erchin Serpedin, Thomas Chen, Dinesh Rajan, Mathematical Foundations for SIGNAL PROCESSING, COMMUNICATIONS, AND NETWORKING, 2012
We say two vertices are connected if there is a walk from one to the other. (In view of Theorem 6.7.1, we could replace “walk” by “path.”) This gives us a formal definition of connectivity: a graph is connected if and only if all its pairs of vertices are connected. We can now define a component as the subgraph formed by taking a maximal set of vertices with every pair connected, plus all the edges touching them.
Big Graph Analytics: Techniques, Tools, Challenges, and Applications
Published in Mohiuddin Ahmed, Al-Sakib Khan Pathan, Data Analytics, 2018
Dhananjay Kumar Singh, Pijush Kanti Dutta Pramanik, Prasenjit Choudhury
In a graph, how nodes are connected with each other is defined by connectivity. To determine weakness in utility networks, such as power grid, connectivity analysis can be used. Another example where connectivity analysis is used is in the comparison of connectivity across the networks.
Star structure connectivities of pancake graphs and burnt pancake graphs
Published in International Journal of Parallel, Emergent and Distributed Systems, 2021
Subinur Dilixiati, Eminjan Sabir, Jixiang Meng
An interconnection network can be represented by an undirected graph , where each vertex in V corresponds to a processor, and each edge in E corresponds to a communication link between two processors. The connectivity of a connected graph G is the cardinality of a minimum set of vertices in G, whose removal leaves the remaining graph disconnected or trivial. The connectivity [29] is an important parameter to measure the reliability and fault tolerance of an interconnection network. However, the classic connectivity underrates the robustness of systems since it tacitly assumes that all processors adjacent to a healthy processor can fail synchronously, which is impossible in a real multiprocessor system [1]. To compensate for this shortcoming, conditional connectivity [2], restricted connectivity [3], g-extra connectivity [4], k-restricted connectivity [1], -restricted connectivity [5], k-embedding-connectivity [6] and component connectivity [7] were explored and studied.
Connectivity of two-dimensional assemblies: trusses and roads
Published in Civil Engineering and Environmental Systems, 2021
Our focus, however, is on the notion of connectivity per se, which has also been extensively explored, stemming from an interest to analyse community structure (e.g. Newman 2004). These explorations have been more general and fundamental in nature, with fairly close links to formal graph theory. For example, the concept of graph connectivity is concerned with the number of elements (edges or vertices) that need to be removed in order to disconnect the graph. Nodal degree (the number of elements incident at a joint) and strength (how strongly a node is directly connected to other nodes) are key metrics (Hevey 2018). Connectivity in damaged or disconnected graphs has also been an area of interest (Cartledge and Nelson 2011). Most graph connectivity algorithms follow the ‘max-flow-min-cut’ theorem by Menger (1927), e.g. work by Esfahanian (2013). These algorithms reveal clusters within the graph that have dense intraconnections and sparse interconnections. Such algorithms can also be used to find deeper communities embedded within the resulting clusters. Several methods that can be used to achieve similar results have been collated by Schaeffer (2007). Elms (1983) has proposed a method for identifying the hierarchical structure of connectivity, starting from an association matrix.
Combining multi-criteria and space syntax analysis to assess a pedestrian network: the case of Oporto
Published in Journal of Urban Design, 2018
Mona Jabbari, Fernando Fonseca, Rui Ramos
This paper argues that assessing a pedestrian network requires a street connectivity analysis. In fact, connectivity measures the degree to which dense and diverse urban activities are accessible. The principle is that in order for streets to be walkable, they must be connected. Even if a street provides good conditions for walking, it must be linked with other streets and spaces, otherwise people hardly go there on foot. In the study, connectivity was accessed with space syntax by using the DepthmapX software. Space syntax was adopted because it has various advantages compared to more simple street connectivity measures such as passive graphic notions, namely for calculating movements in network-configured human settlements and functional connectivity in networks (Tianxiang, Dong, and Shoubing 2015). The DepthmapX software performs a set of spatial network analyses designed to understand social processes within the physical environment (Jeong and Ban 2016). Based on the graph theory, the connectivity of a node can be defined as the number of other nodes directly connected to it. The analysis performed with space syntax shows a street connectivity ranging from 1 to 14 (Table 3). Higher space syntax values correspond to streets with many connections (nodes) and vice versa. These values were then normalized between 0.0 and 1.0 by fuzzy logic and inserted in the GIS database. A WLC was calculated again to obtain the final scores by using a weight of 0.5 for the criteria evaluation and 0.5 for street connectivity, as suggested by the experts. The result is an assessment of the pedestrian network, showing the streets most suitable for walking, reflecting not only the conditions provided to pedestrians, but also their connectivity.