Essays on the economics of networks

Networks (collections of nodes or vertices and graphs capturing their linkages) are a common object of study across a range of fields includ- ing economics, statistics and computer science. Network analysis is often based around capturing the overall structure of the network by some reduced set of parameters. Canonically, this has focused on the notion of centrality. There are many measures of centrality, mostly based around statistical analysis of the linkages between nodes on the network. However, another common approach has been through the use of eigenfunction analysis of the centrality matrix. My the- sis focuses on eigencentrality as a property, paying particular focus to equilibrium behaviour when the network structure is fixed. This occurs when nodes are either passive, such as for web-searches or queueing models or when they represent active optimizing agents in network games. The major contribution of my thesis is in the applica- tion of relatively recent innovations in matrix derivatives to centrality measurements and equilibria within games that are function of those measurements. I present a series of new results on the stability of eigencentrality measures and provide some examples of applications to a number of real world examples

Partition functions:Zeros, unstable dynamics and complexity

This thesis considers the complexity of approximating the partition functions of the ferromagnetic Ising model and of the hard-core model (independence polynomial) within the class of bounded degree graphs. It is known that the absence of zeros essentially implies that approximation is easy. In this thesis the inverse is proved: the presence of zeros implies that approximation is #P hard. The most important step of the proof is relating both the "zero parameters" and the "#P hard parameters" to the set of parameters around which a related set of functions, namely the occupation ratios, behaves chaotically. The first two chapters contain the proof of the main theorem for the ferromagnetic Ising model and the independence polynomial respectively. Chapters 3 and 4 concern the set of zeros of the independence polynomial for bounded degree graphs. In Chapter 3 it is shown that zeros of Cayley trees are not extremal within the set of zeros of all bounded degree graphs, something that was previously conjectured. In Chapter 4 a very precise description of the set of zeros is given as the degree bound goes to infinity