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\chapter{Conclusions}\label{C:conclusions}

In this thesis, we have studied problems arising from the calculation of node centrality rankings, specifically those based on functions of matrices.  We introduced the total communicability of a node $i$, given by the $i$th row sum of the exponential of the adjacency matrix of the network, as a viable centrality method.  We demonstrated that the rankings provided by total communicability are often in strong agreement with those produced by subgraph centrality, especially for highly ranked nodes.  This was shown through extensive numerical tests on both synthetic and real-world networks.  These tests also demonstrated that total communicability is as good as subgraph centrality in identifying essential nodes in a Yeast PPI network \cite{BKlym13,Proteome,Proteomics}.  We also showed that total communicability scores can be estimated extremely quickly using Krylov subspace methods, allowing total communicability to be used to rank nodes on very large networks where the use of subgraph centrality is infeasible.  Additionally, we introduced the total communicability of a network, given by the average node total communicability, as a global measure of network connectivity.  Future work in this area includes providing more precise guidelines on when subgraph centrality and total communicability rankings will be in strong agreement as well as investigating the use of total network communicability in the design of communication networks.

We also analyzed the relationship between four parameterized centrality measures based on the matrix exponential and resolvent with degree and eigenvector centrality.  The centrality measures examined were subgraph centrality and total communicability (given by $[e^{\beta A}]_{ii}$ and $[e^{\beta A}{\bf 1}]_i$, respectively, with $\beta > 0$) and resolvent and Katz centrality (given by $[(I-\alpha A)^{-1}]_{ii}$ and $[(I-\alpha A)^{-1}{\bf 1}]_i$, $0 < \alpha < \frac{1}{\lambda_1}$).  We proved that as $\alpha$ and $\beta$ approach 0, the rankings produced by all four centrality measures converge to those produced by degree centrality.  As $\alpha$ approaches $\frac{1}{\lambda_1}$ and $\beta$ approaches infinity, the rankings converge to those produced by eigenvector centrality.  The speed of this convergence is based on the size of the spectral gap $\lambda_1 - |\lambda_2|$.  These results help to explain the observed correlation between degree and eigenvector centrality on many real-world complex networks.  We also conducted extensive numerical experiments to determine how the convergence to degree and eigenvector centrality rankings behaved on real-world networks.  Based on these experiments, we were able to provide guidelines on the choice of $\alpha$ and $\beta$ to ensure the rankings produced are as different as possible from those produced by degree and eigenvector centrality.

We propose an extension of the measure of subgraph centrality to directed networks based on a bipartization of the network and explicitly determine the exponential of this bipartized network.  We show how this extension can be used to rank hubs and authorities using the diagonal entries of the exponential of the bipartized network.  We also show that the off-diagonal entries  contain information about the ability of pairs of nodes to communicate in various manners.  We used this method to rank hubs and authorities on a variety of directed web networks and compared the rankings to those produced by HITS, PageRank, and Reverse PageRank.  Our proposed method is more expensive than the other methods tested.  Although there are currently methods to estimate the centralities based on Gaussian quadrature, they are still too expensive to use on very large networks.  Future work includes the investigation of techniques to more quickly estimate these centralities and their use in the study of real-world applications.

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%\section{Motivation}
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%In this thesis we are interested in efficient methods for the numerical solution of saddle point systems arising from finite element discretization of two important classes of fluid dynamic problems, namely the \emph{incompressible Navier-Stokes equations} and the \emph{Brinkman equations}. 

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