Opinion-Based Centrality in Multiplex Networks: A Convex Optimization Approach

Most people simultaneously belong to several distinct social networks, in which their relations can be different. They have opinions about certain topics, which they share and spread on these networks, and are influenced by the opinions of other persons. In this paper, we build upon this observation to propose a new nodal centrality measure for multiplex networks. Our measure, called Opinion centrality, is based on a stochastic model representing opinion propagation dynamics in such a network. We formulate an optimization problem consisting in maximizing the opinion of the whole network when controlling an external influence able to affect each node individually. We find a mathematical closed form of this problem, and use its solution to derive our centrality measure. According to the opinion centrality, the more a node is worth investing external influence, and the more it is central. We perform an empirical study of the proposed centrality over a toy network, as well as a collection of real-world networks. Our measure is generally negatively correlated with existing multiplex centrality measures, and highlights different types of nodes, accordingly to its definition

Multilayer Networks

In most natural and engineered systems, a set of entities interact with each other in complicated patterns that can encompass multiple types of relationships, change in time, and include other types of complications. Such systems include multiple subsystems and layers of connectivity, and it is important to take such "multilayer" features into account to try to improve our understanding of complex systems. Consequently, it is necessary to generalize "traditional" network theory by developing (and validating) a framework and associated tools to study multilayer systems in a comprehensive fashion. The origins of such efforts date back several decades and arose in multiple disciplines, and now the study of multilayer networks has become one of the most important directions in network science. In this paper, we discuss the history of multilayer networks (and related concepts) and review the exploding body of work on such networks. To unify the disparate terminology in the large body of recent work, we discuss a general framework for multilayer networks, construct a dictionary of terminology to relate the numerous existing concepts to each other, and provide a thorough discussion that compares, contrasts, and translates between related notions such as multilayer networks, multiplex networks, interdependent networks, networks of networks, and many others. We also survey and discuss existing data sets that can be represented as multilayer networks. We review attempts to generalize single-layer-network diagnostics to multilayer networks. We also discuss the rapidly expanding research on multilayer-network models and notions like community structure, connected components, tensor decompositions, and various types of dynamical processes on multilayer networks. We conclude with a summary and an outlook.Comment: Working paper; 59 pages, 8 figure

Financial stability from a network perspective

This thesis consists of six chapters related to applications of network analysis’ methods for financial stability. The first chapter introduces the network perspective as a new mapping technique for studying and understanding financial markets’architecture. The second chapter breaks down the Colombian sovereign securities market into different layers of interaction corresponding to distinct trading and registering platforms. The third chapter addresses an overlooked issue: How to measure the importance of financial market infrastructures within their corresponding network. The fourth chapter studies the connective and hierarchical structure of the Colombian non-collateralized money market, and uses an information retrieval algorithm for identifying those financial institutions that simultaneously excel at borrowing and lending central bank’s liquidity (i.e. superspreaders). The fifth chapter addresses –for the first time- the question regarding the presence of a modular hierarchy in financial networks, and discusses the main implications for financial stability. The sixth chapter explicitly models the role of financial market infrastructures as financial markets’ “plumbing”, and recognizes that traditional analysis of financial institutions networks is of a virtual or logical nature. The third chapter is published in the Journal of Financial Market Infrastructures (Vol.2 (3), 2014), whereas the fourth chapter is published in the Journal of Financial Stability (Vol.15, 2014)

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

A Convex Framework for Epidemic Control in Networks

With networks becoming pervasive, research attention on dynamics of epidemic models in networked populations has increased. While a number of well understood epidemic spreading models have been developed, little to no attention has been paid to epidemic control strategies; beyond heuristics usually based on network centrality measures. Since epidemic control resources are typically limited, the problem of optimally allocating resources to control an outbreak becomes of interest. Existing literature considered homogeneous networks, limited the discussion to undirected networks, and largely proposed network centrality-based resource allocation strategies. In this thesis, we consider the well-known Susceptible-Infected-Susceptible spreading model and study the problem of minimum cost resource allocation to control an epidemic outbreak in a networked population. First, we briefly present a heuristic that outperforms network centrality-based algorithms on a stylized version of the problem previously studied in the literature. We then solve the epidemic control problem via a convex optimization framework on weighted, directed networks comprising heterogeneous nodes. Based on our spreading model, we express the problem of controlling an epidemic outbreak in terms of spectral conditions involving the Perron-Frobenius eigenvalue. This enables formulation of the epidemic control problem as a Geometric Program (GP), for which we derive a convex characterization guaranteeing existence of an optimal solution. We consider two formulations of the epidemic control problem -- the first seeks an optimal vaccine and antidote allocation strategy given a constraint on the rate at which the epidemic comes under control. The second formulation seeks to find an optimal allocation strategy given a budget on the resources. The solution framework for both formulations also allows for control of an epidemic outbreak on networks that are not necessarily strongly connected. The thesis further proposes a fully distributed solution to the epidemic control problem via a Distributed Alternating Direction Method of Multipliers (ADMM) algorithm. Our distributed solution enables each node to locally compute its optimum allocation of vaccines and antidotes needed to collectively globally contain the spread of an outbreak, via local exchange of information with its neighbors. Contrasting previous literature, our problem is a constrained optimization problem associated with a directed network comprising non-identical agents. For the different problem formulations considered, illustrations that validate our solutions are presented. This thesis, in sum, proposes a paradigm shift from heuristics towards a convex framework for contagion control in networked populations

Detection of Core-Periphery Structure in Networks Using Spectral Methods and Geodesic Paths

We introduce several novel and computationally efficient methods for detecting "core--periphery structure" in networks. Core--periphery structure is a type of mesoscale structure that includes densely-connected core vertices and sparsely-connected peripheral vertices. Core vertices tend to be well-connected both among themselves and to peripheral vertices, which tend not to be well-connected to other vertices. Our first method, which is based on transportation in networks, aggregates information from many geodesic paths in a network and yields a score for each vertex that reflects the likelihood that a vertex is a core vertex. Our second method is based on a low-rank approximation of a network's adjacency matrix, which can often be expressed as a tensor-product matrix. Our third approach uses the bottom eigenvector of the random-walk Laplacian to infer a coreness score and a classification into core and peripheral vertices. We also design an objective function to (1) help classify vertices into core or peripheral vertices and (2) provide a goodness-of-fit criterion for classifications into core versus peripheral vertices. To examine the performance of our methods, we apply our algorithms to both synthetically-generated networks and a variety of networks constructed from real-world data sets.Comment: This article is part of EJAM's December 2016 special issue on "Network Analysis and Modelling" (available at https://www.cambridge.org/core/journals/european-journal-of-applied-mathematics/issue/journal-ejm-volume-27-issue-6/D245C89CABF55DBF573BB412F7651ADB

The rise and fall of countries in the global value chains

Countries become global leaders by controlling international and domestic transactions connecting geographically dispersed production stages. We model global trade as a multi-layer network and study its power structure by investigating the tendency of eigenvector centrality to concentrate on a small fraction of countries, a phenomenon called localization transition. We show that the market underwent a significant drop in power concentration precisely in 2007 just before the global financial crisis. That year marked an inflection point at which new winners and losers emerged and a remarkable reversal of leading role took place between the two major economies, the US and China. We uncover the hierarchical structure of global trade and the contribution of individual industries to variations in countries’ economic dominance. We also examine the crucial role that domestic trade played in leading China to overtake the US as the world’s dominant trading nation. There is an important lesson that countries can draw on how to turn early signals of upcoming downturns into opportunities for growth. Our study shows that, despite the hardships they inflict, shocks to the economy can also be seen as strategic windows countries can seize to become leading nations and leapfrog other economies in a changing geopolitical landscape