Graphs
In mathematics, a graph is a collection of points, called vertices, connected by lines or curves, called edges. Graphs are used to represent relationships between objects or data points. They are widely used in various fields such as computer science, social networks, and operations research for modeling and analyzing complex systems.
5 Key excerpts on "Graphs"
eBook - ePubCulture, Curiosity and Communication in Scientific Discovery
The Eye in Ideas
- Nigel Sanitt(Author)
- 2018(Publication Date)
- Routledge(Publisher)
...6 Graph theory Up to now I have described the network model for scientific theorizing in qualitative terms. There is, however, a branch of mathematics which deals with networks at a quantitative level called graph theory. There is an enormous range of types of networks and graph theory provides a mathematical and formal framework for describing, analysing and calculating with networks. All the procedures and characteristics of networks, which I have described, have a ready interpretation in graph theory and many results and theorems in graph theory have a corresponding application in the network model I have proposed. There are no true propositions in science but mathematics is an effective tool both as cement within scientific theories, linking assumptions and predictions, and as a logical foundation for the network structures that constitute the backbone of our understanding. 6.1 What is a graph? There are two different meanings of graph, which have arisen rather confusingly in mathematics. The more commonly known meaning of graph refers to a data plot. The usual example of this type of graph is the familiar diagram of Cartesian coordinates, which consists of two perpendicular axes designated x (horizontal) and y (upright). There are many kinds of graph other than the Cartesian coordinate graph, which was described by René Descartes in 1637. The second meaning of graph, and the one relevant to the present work, was first proposed by the Swiss mathematician Leonhard Euler in 1735. In this sense of the word, a graph is a collection of objects linked together. Pictorially, they are represented by nodes, which may or may not be linked together by lines. 1 If the links are ordered then the lines have arrows on them and are referred to as arcs and the nodes are called vertices; such a graph is said to be directed. Figures 5.1 and 5.2 in the previous chapter are examples of directed Graphs or diGraphs. The vertices or nodes of a graph may be labelled or unlabelled...
eBook - ePubBioinformatics Algorithms
Design and Implementation in Python
- Miguel Rocha, Pedro G. Ferreira(Authors)
- 2018(Publication Date)
- Academic Press(Publisher)
...Chapter 13 Graphs: Concepts and Algorithms Abstract In this chapter, we present the mathematical concept of graph and its computational representation. We address some of the main algorithms over Graphs and develop a set of Python classes to implement different types of Graphs and underlying algorithms. Graphs will be central in the development of algorithms for handling biological networks and genome assembly, tasks addressed in the next chapters. Keywords Graphs; Graph representations; Adjacency lists; Node degree; Paths and distances in Graphs; Graph search algorithms In this chapter, we present the mathematical concept of graph and its computational representation. We address some of the main algorithms over Graphs and develop a set of Python classes to implement different types of Graphs and underlying algorithms. Graphs will be central in the development of algorithms for handling biological networks and genome assembly, tasks addressed in the next chapters. 13.1 Graphs: Definitions and Representations A graph can be defined, in Mathematics, as a set of objects in which some of the pairs of the objects in this set are related. While they can be easily defined and have a simple structure, they are powerful and flexible data structures, with a huge set of applications in computer science and many fields of science and engineering. Formally, a graph G can be defined by two sets: (V, E), where V is the set of objects, named as vertices or nodes of the graph, and E is a set of pairs (u. v) of vertices from V, named edges or arcs, indicating the existence of a relationship between u and v. The edges in E may have an orientation, i.e. the pairs are ordered, in which case the graph is classified as directed, or digraph. Otherwise, the pairs are unordered and the graph is termed undirected...
eBook - ePub- Stephen P Borgatti, Martin G Everett, Jeffrey C Johnson(Authors)
- 2018(Publication Date)
- SAGE Publications Ltd(Publisher)
...2 Mathematical Foundations Learning Outcomes Represent networks in graph-theoretic language Identify paths, walks, trails and components Formulate networks in matrix terms Compute and interpret multiplication of adjacency matrices Don’t forget to visit the website at https://study.sagepub.com/borgatti2e 2.1 Introduction As should be evident from Chapter 1, social network analysis is a social science. The actors we study are typically individuals (specifically humans, but also other social species such as apes and dolphins) or organizations (such as corporations). But networks are encountered in many other fields as well, including physics, ecology, chemistry, neurology, genetics and computer science. What these instances of network analysis have in common is an underpinning in a branch of discrete mathematics called graph theory. In this chapter we introduce the terminology and basic conceptual building blocks of graph theory. In addition, we present a short introduction to matrices, which can also be used to represent networks, and matrix algebra, which has proved very useful in network analysis. 2.2 Graphs One way of conceptualizing networks mathematically is as Graphs. The term ‘graph’ here does not refer to a diagram but rather a mathematical object (Harary, 1969). A graph G (V, E) consists of a set of vertices V (also called nodes or points), and a set of edges E (or links or lines). The edges connect pairs of vertices. To express that an edge connecting vertices u and v exists in a graph G, we write (u, v) ∈ E (G). If we think of G as a binary relation, then we could also write uGv. For example, if G represents the ‘likes’ relation, the uGv would indicate that u likes v. When two vertices are joined by an edge, we say the vertices are adjacent. So, adjacent just means ‘has a tie’. If an edge connects A with B, and another edge connects A with C, we say that the two edges are incident upon A...
eBook - ePub- Gary Chartrand(Author)
- 2012(Publication Date)
- Dover Publications(Publisher)
...*Chapter 10 Graphs and Other Mathematics Graph theory has close relationships with several mathematical areas. In this chapter we consider three of these areas: matrices, topology, and groups. 10.1 Graphs and Matrices A graph is completely determined by its vertex set and by a knowledge of which pairs of vertices are adjacent. This same information can easily be given by a matrix. In fact, much of graph theory could be investigated as a subject within matrix theory. There are certain advantages to this approach, since matrices can serve as computer input to work a variety of computations. On the other hand, there are several disadvantages to representing Graphs as matrices, for this destroys the visual aspect of Graphs, and the recognition of certain graphical properties in matrices is not necessarily simpler than in the diagram of a graph. Let G be a graph of order p with vertices denoted by v 1, v 2,..., v p. Then the adjacency matrix A = A (G) = [ a ij ] is that p x p matrix in which a ij = 1 if v i and v j are adjacent and a ij = 0 otherwise. Thus, the matrix A is a (0, 1) matrix (i.e., every entry of A is 0 or 1); the main diagonal of A consists entirely of 0’s (i.e., a ii = 0 for i = 1, 2,..., p); and A is symmetric (i.e., a ij = a ji for 1 ≤ i ≤ p and 1 ≤ j ≤ p). A graph G and its adjacency matrix are given in Figure 10.1. Figure 10.1 Note that the adjacency matrix of a graph G ordinarily depends on how we label the vertices. For example, the graph G of Figure 10.1 is shown again in Figure 10.2 with a different labeling, resulting in a different adjacency matrix. Although the matrices of Figures 10.1 and 10.2 are unequal (as matrices), they represent isomorphic Graphs. Figure 10.2 One interesting property of an adjacency matrix of a graph concerns the entries of its various powers...
- Bruce B. Frey(Author)
- 2018(Publication Date)
- SAGE Publications, Inc(Publisher)
...Irene Kaimi Irene Kaimi Kaimi, Irene Christoforos Mamas Christoforos Mamas Mamas, Christoforos Graphical Modeling Graphical modeling 746 750 Graphical Modeling Graphical modeling uses Graphs, which present the different ways the variables in a model depend on each other, to represent and visualize the model. The model’s variables can be simply associated or be connected through causal relationships. The resulting displays rely on probability and graph theory, graph algorithms and machine learning; as such, they connect concepts from statistics and computer science. A wide range of different types of graphical models and methods have been developed in a variety of areas including, but not limited to, medical diagnosis, image understanding, speech recognition, and natural language processing. The use of graphical models can also enable understanding of social and technical features of organizations and structures. In education, such systems may extend from the classroom unit to the school and from the educational system of a country to the educational systems of several countries. Visualization and interpretation of the underlying structures between members of these systems can help in identifying isolated members, which potentially share common characteristics. This in turn can lead to the introduction of improved policies and practices, so that the educational and social needs of all (or groups of the) corresponding members (e.g., students, schools, and educational systems) are better met. This entry presents some of the basic ideas of graphical modeling and then illustrates the concepts in the context of social network analysis. Some Probability Concepts Probabilities are used in everyday life and determine our decision-making processes. For example, the chance of rain informs one’s plans for the weekend...
