By Ernesto Estrada, Philip A. Knight
The research of community thought is a hugely interdisciplinary box, which has emerged as a massive subject of curiosity in quite a few disciplines starting from physics and arithmetic, to biology and sociology. This e-book promotes the varied nature of the learn of complicated networks by means of balancing the desires of scholars from very diversified backgrounds. It references the main established suggestions in community conception, provides examples in their purposes in fixing sensible difficulties, and transparent symptoms on easy methods to examine their effects. within the first a part of the ebook, scholars and researchers will observe the quantitative and analytical instruments essential to paintings with complicated networks, together with the main easy ideas in community and graph idea, linear and matrix algebra, in addition to the actual innovations most often used for learning networks. they are going to additionally locate guideline on a few key talents resembling the right way to facts analytic effects and the way to govern empirical community facts. the majority of the textual content is targeted on teaching readers at the most beneficial instruments for contemporary practitioners of community conception. those comprise measure distributions, random networks, community fragments, centrality measures, clusters and groups, communicability, and native and worldwide houses of networks. the combo of idea, instance and technique which are awarded during this textual content, may still prepared the scholar to behavior their very own research of networks with self assurance and make allowance lecturers to choose applicable examples and difficulties to coach this topic within the classroom. Read more...
summary: The examine of community conception is a hugely interdisciplinary box, which has emerged as a huge subject of curiosity in numerous disciplines starting from physics and arithmetic, to biology and sociology. This ebook promotes the varied nature of the examine of complicated networks through balancing the desires of scholars from very varied backgrounds. It references the main widespread suggestions in community concept, offers examples in their purposes in fixing functional difficulties, and transparent symptoms on the best way to examine their effects. within the first a part of the booklet, scholars and researchers will notice the quantitative and analytical instruments essential to paintings with advanced networks, together with the main uncomplicated ideas in community and graph concept, linear and matrix algebra, in addition to the actual recommendations most often used for learning networks. they're going to additionally locate guide on a few key abilities equivalent to find out how to evidence analytic effects and the way to control empirical community information. the majority of the textual content is concentrated on teaching readers at the most valuable instruments for contemporary practitioners of community idea. those contain measure distributions, random networks, community fragments, centrality measures, clusters and groups, communicability, and native and international houses of networks. the combo of concept, instance and approach which are offered during this textual content, should still prepared the coed to behavior their very own research of networks with self assurance and make allowance academics to choose acceptable examples and difficulties to coach this topic within the school room
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Extra info for A first course in network theory
A network is bipartite if and only if it contains no cycle of odd length. You can start by drawing a bipartite network (following some previous advice). Now try to add a triangle to the network. You will immediately realize that to add a triangle you necessarily need to connect two nodes which are in the same disjoint set of the bipartite network. Thus, the graph to which you have added the triangle is no longer bipartite. The same happens if you try a pentagon or a heptagon. Thus, a key ingredient in your proof should be the fact that the existence of odd cycles necessarily implies connections between nodes in the same set of the bipartition, which necessarily means destroying the bipartivity of the graph.
7 Other general advice There are many other techniques and tricks available for proving results in network theory and it is impossible to cover them all in this chapter. Apart from the cases we have analysed, the use of special cases and extreme examples abound in proofs in network theory. The student should learn as many of these techniques as possible to create an extensive arsenal for solving problems in this area. A recommendation in this direction is that the student tries to remember not only the theorems or statements of the results but also the techniques used in their proofs.
So for a completely heterogeneous set of degrees you may assume that the degrees of the n nodes of your imagined network are 0, 1, 2, . . , n – 2, n – 1. The node with degree n – 1 must be connected to all the other nodes in the network. However, there is a node with degree 0, which contradicts the previous statement. Consequently, you have proved by contradiction that the statement is true. It has become a theorem and you can now be absolutely convinced that such a dreamed network cannot exist.
A first course in network theory by Ernesto Estrada, Philip A. Knight