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Shortest Path Models and Algorithms
Published in Craig A. Tovey, Linear Optimization and Duality, 2020
The transitive closure of graph G=(V,E) is the graph H=(V,F) defined by (i,j)∈F⇔∃ path P in G from i to j. How can you construct H efficiently, given G?
Getting Social: Graph Theory and Social Network Analysis
Published in Jesús Rogel-Salazar, Advanced Data Science and Analytics with Python, 2020
The ties that connect the nodes in the network can tell us things regarding how strongly (or weakly) the actors interact. We can identify clusters in the network exhibiting characteristics such as homophily, that is the tendency of individuals to associate and bond with similar actors, or transitivity, i.e., when there is a tie from node i to node j, and also from node j to node k the relationship is transitive if there is also a tie from i to k (see Figure 3.3). We can also distinguish cliques which are densely, fully connected components in the network. Homophily → “Birds of a feather flock together”.Transitivity → “Friends of my friends are my friends”.
Eigenvalues and Eigenvectors
Published in James R. Kirkwood, Bessie H. Kirkwood, Elementary Linear Algebra, 2017
James R. Kirkwood, Bessie H. Kirkwood
A relation is an equivalence relation if three conditions hold. These areA matrix is related to itself. (This is called the reflexive property.)If A is related to B, then B is related to A (symmetric property).If A is related to B and B is related to C, then A is related to C (transitive property).
Fifty years of similarity relations: a survey of foundations and applications
Published in International Journal of General Systems, 2022
It is instructive to see first what happens in the crisp case: If R is a reflexive and symmetric crisp relation on a finite set X, it can be represented by a graph with vertices the elements of X and with two vertices x and y connected by an edge when xRy. Its transitive closure is the smallest graph that contains and such that all its connected components are complete subgraphs. This makes an equivalence relation. Figure 1 shows a graph corresponding to a reflexive and symmetric relation and the graph of its corresponding transitive closure.
Food Insecurity, Dietary Quality, and the Role of Social Network Support and Barriers Among Immigrant Latino Adults in New York City
Published in Journal of Hunger & Environmental Nutrition, 2023
Sydney Miller, Kayla de la Haye, Brooke Bell, Alyshia Gálvez, Stephanie Rupp, Karen R. Flórez
Social cohesion was operationalized as network transitivity,42 which represents the phenomena of “a friend of a friend is my friend.” Transitivity is computed by examining the presence or absence of ties between triads of network members, within the respondent’s personal network. In computing this measure, the participant is excluded from the network structure. A triad is considered transitive when all three network members are connected to each other (e.g., A knows B, and B knows C, and A knows C). Total network transitivity is the proportion of all triads that are transitive, expressed as a percentage ranging from 0% to 100%, with a higher score representing greater transitivity.
Diagnosability of arrangement graphs with missing edges under the MM* model
Published in International Journal of Parallel, Emergent and Distributed Systems, 2020
A graph is vertex-transitive if and only if for its any pair of vertices u and v, there exists an automorphism of the graph that maps u to v. A graph is edge-transitive if and only if for its any pair of edges and , there exists an automorphism of the graph that maps to .