The effect an individual node or edge can have on a network’s structure and dynamics depends not only on the scale of its activity, but also on its position within the network, as well as the activity of neighbouring nodes and edges. In the study of complex network, many analyses of these interrelations have focused on network spectra, aiming to gain structural information from the network adjacency matrix. These include numerous studies of epidemic processes, where the removal of an edge linking communities can help stem the spread of a disease. Alternatively, many other studies have considered stability through a perturbative approach, defining node importance as the relative change in the c largest eigenvalues of the adjacency matrix upon the node’s removal.
In a recent paper, LML External Fellow Fabio Caccioli, working with Isobel Seabrook and Paolo Barucca, apply this edge-based perturbation approach in the context of networks of financial institutions. As they note, previous work has considered the propagation of distress or financial default in such networks, but has not explicitly measured how an individual node or edge affects the structure of the network in general. Here the authors focus on the leading eigenvalue because it determines, for instance, the stability of spreading processes on social networks or financial shocks on inter-bank networks. Their study focusses on the temporal behaviour of the network in relation to structural importance, and aims to address two questions: 1) Can this approach quantify the extent to which an edge affects the overall network structure, and 2) does this provide information on the network’s temporal evolution?
As the researchers observe, the leading eigenvalue provides an indication of stability in terms of dynamical processes occurring on the network. In the current study, they consider the derivatives of the network’s leading eigenvalue with respect to individual edges and present evidence that this measure could be a useful indicator in understanding temporal changes in network structure. They illustrate this in applications to five real-world networks, demonstrating that the elementwise derivative of the leading eigenvalue can be predictive of subsequent change. This has potential implications for understanding network stability, as a system experiencing more changes to edges of structural dominance could see a reinforcing effect, leading to instability. The authors suggest these methods may be useful in classifying financial asset systems as a part of improved regulatory practices and policy making.
The paper is available at https://epjdatascience.springeropen.com/articles/10.1140/epjds/s13688-021-00279-6