Structural importance and evolution: an application to financial transaction networks

A fundamental problem in the study of networks is the identification of important nodes. This is typically achieved using centrality metrics, which rank nodes in terms of their position in the network. This approach works well for static networks, that do not change over time, but does not consider the dynamics of the network. Here we propose instead to measure the importance of a node based on how much a change to its strength will impact the global structure of the network, which we measure in terms of the spectrum of its adjacency matrix. We apply our method to the identification of important nodes in equity transaction networks and show that, while it can still be computed from a static network, our measure is a good predictor of nodes subsequently transacting. This implies that static representations of temporal networks can contain information about their dynamics

Influencers in Dynamic Financial Networks

To monitor risk in temporal financial networks, an understanding of how individual behaviours affect the temporal evolution of networks is needed. This is typically achieved using centrality and importance metrics, which rank nodes in terms of their position in the network. This approach works well for static networks, that do not change over time, but does not consider the dynamics of the network. In addition to this, current methods are often unable to capture the complex, often sparse and disconnected structures of financial transaction networks. This thesis addresses these gaps by considering importance from a dynamical perspective, first by using spectral perturbations to derive measures of importance for nodes and edges, then adapting these methods to incorporate a structural awareness. I complement these methods with a generative model for transaction networks that captures how individual behaviours give rise to the key properties of these networks, offering new methods to add to the regulatory toolkit. My contributions are made across three studies which complement each other in their findings. Study 1: \begin{itemize} \item I define a structural importance metric for the edges of a network, based on perturbing the adjacency matrix and observing the resultant change in its largest eigenvalues. \item I combine this with a model of network evolution where this metric controls the scale and probabilities of subsequent edge changes. This allows me to consider how edge importance relates to subsequent edge behaviour. \item I use this model alongside an exercise to predict subsequent change from edge importance. Using this I demonstrate how the model parameters are related to the capability of predicting whether an edge will change from its importance. \end{itemize} Study 2: \begin{itemize} \item I extend my measure of edge importance to measure the importance of nodes, and to capture complex community structures through the use of additional components of the eigenspectrum. \item While computed from a static network, my measure of node importance outperforms other centrality measures as a predictor of nodes subsequently transacting. This implies that static representations of temporal networks can contain information about their dynamics. \end{itemize} Study 3: \begin{itemize} \item I contrast the snapshot based methods used in the first two studies by modelling the dynamic of transactions between counterparties using both univariate and multivariate Hawkes processes, which capture the non-linear `bursty’ behaviour of transaction sequences. \item I find that the frequency of transactions between counterparties increases the likelihood of them to transact in the future, and that univariate and multivariate Hawkes processes show promise as generative models for transaction sequences. \item Hawkes processes also perform well when used to model buys and sells through a central clearing counterparty when considered as a bivariate process, but not as well when these are modelled as individual univariate processes. This indicates that mutual excitation between buys and sells is present in these markets. \end{itemize} The observations presented in this thesis provide new insights into the behaviour of equities markets, which until now have mainly been studied via price information. The metrics I propose offer a new potential to identify important traders and transactions in complex trading networks. The models I propose provide a null model over which a user could detect outlying transactions and could also be used to generate synthetic data for sharing purposes

The gravity of an edge

Abstract ■■■ We describe a methodology for characterizing the relative structural importance of an arbitrary network edge by exploiting the properties of a k-shortest path algorithm. We introduce the metric Edge Gravity, measuring how often an edge occurs in any possible network path, as well as k-Gravity, a lower bound based on paths enumerated while solving the k-shortest path problem. The methodology is demonstrated using Granovetter’s original strength of weak ties network examples as well as the well-known Florentine families of the Italian Renaissance and the Krebs 2001 terrorist networks. The relationship to edge betweenness is established. It is shown that important edges, i.e. ones with a high Edge Gravity, are not necessarily adjacent to nodes of importance as identified by standard centrality metrics, and that key nodes, i.e. ones with high centrality, often have their importance bolstered by being adjacent to bridges to nowhere–e.g. ones with low Edge Gravity. It is also demonstrated that Edge Gravity distinguishes critically important bridges or local bridges from those of lesser structural importance