By Rudenskaya O.G.

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Extra info for 4-Quasiperiodic Functions on Graphs and Hypergraphs

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For various values of c > δ there exist nonempty pairwise-stable networks, but they are not star networks: they must be such that each player has at least two links. This model makes it clear that there are situations in which individual incentives are not aligned with overall societal benefits. While this connections model is highly stylized, it still captures some basic insights about the payoffs from networked relationships, and it shows that we can model the incentives that underlie network formation and see when the resultant networks are efficient.

This is just one of many possible measures of efficiency and societal welfare, which are well-studied subjects in philosophy and economics. How we measure efficiency has important consequences in network analysis and is discussed in more detail in Chapter 6. 9 Gain in total utility from changing a line into a star. all players arranged in a “star” network. That is, there should be some central player who is connected to each other player, so that one player has n − 1 links and each of the other players has 1 link.

Even though both networks have approximately an average degree of 2, they are clearly very different in structure. The degree distribution reflects some aspect of the difference in that the circle is regular, so that every node has exactly two links, while in the binary tree almost half of the nodes have degree 3 and nearly half have degree 1 (the exception is the root node, which has degree 2). However, we need other measures to clearly distinguish these networks. For instance, the diameter of 13.

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4-Quasiperiodic Functions on Graphs and Hypergraphs by Rudenskaya O.G.

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