A first course in network theory by Ernesto Estrada, Philip A. Knight

By Ernesto Estrada, Philip A. Knight

The learn of community idea is a hugely interdisciplinary box, which has emerged as a massive subject of curiosity in a variety of disciplines starting from physics and arithmetic, to biology and sociology. This ebook promotes the varied nature of the learn of complicated networks by way of balancing the desires of scholars from very various backgrounds. It references the main usual innovations in community idea, provides Read more...

summary: The learn of community thought is a hugely interdisciplinary box, which has emerged as an immense subject of curiosity in a number of disciplines starting from physics and arithmetic, to biology and sociology. This e-book promotes the varied nature of the research of complicated networks by way of balancing the desires of scholars from very various backgrounds. It references the main prevalent innovations in community idea, presents examples in their functions in fixing functional difficulties, and transparent symptoms on the right way to examine their effects. within the first a part of the ebook, scholars and researchers will realize the quantitative and analytical instruments essential to paintings with advanced networks, together with the main uncomplicated suggestions in community and graph idea, linear and matrix algebra, in addition to the actual strategies most often used for learning networks. they are going to additionally locate guideline on a few key abilities resembling tips on how to facts analytic effects and the way to control empirical community information. the majority of the textual content is targeted on educating readers at the Most worthy instruments for contemporary practitioners of community concept. those comprise measure distributions, random networks, community fragments, centrality measures, clusters and groups, communicability, and native and worldwide homes of networks. the combo of concept, instance and strategy which are provided during this textual content, may still prepared the coed to behavior their very own research of networks with self belief and make allowance lecturers to choose applicable examples and difficulties to educate this topic within the school room

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A network is k-partite if its nodes can be partitioned into k sets V1 , V2 , . . , Vk such that if u, v ∈ Vi then there is no edge between them. 14 (i) Trees are bipartite. To show this, pick a node on a tree and colour it black. Then colour all its neighbours white. Colour the nodes adjacent to the white nodes black and repeat until the whole tree is coloured. This could only break down if we encounter a previously coloured node. If this were the case, we would have found a cycle in the network.

We will only include one of each of these pairs (it doesn’t matter which) in the incidence matrix. In this case, you may find in some references that the incidence matrix is defined so that all the nonzero entries are set to one and our definition of the incidence matrix is known as the oriented incidence matrix. There are many different conventions for including loops in incidence matrices. Since we are primarily concerned with simple networks it doesn’t really matter which convention we use. We will look more at the connections between the adjacency and incidence matrices when we look at the spectra of networks.

There are lots and lots of trees! There are nn–2 distinct (up to isomorphism) labelled trees with n nodes. For n = 1, 2, 3, 4, 5, 6 this gives 1, 1, 3, 16, 125, 1296 trees before truly explosive growth sets in. Counting unlabelled trees is much harder, and there is no known formula in terms of the number of nodes but their abundance appears to grow exponentially in n. 14 illustrates a number of trees. 15. The left-hand tree can be labelled in four ways, but only in 12 distinct ways since one half are just the reverse of the other.

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