Miguel Romance

Evolutionary games defined at the network mesoscale: The Public Goods game

The evolutionary dynamics of the Public Goods game addresses the emergence of cooperation within groups of individuals. However, the Public Goods game on large populations of interconnected individuals has been usually modeled without any knowledge about their group structure. In this paper, by focusing on collaboration networks, we show that it is possible to include the mesoscopic information about the structure of the real groups by means of a bipartite graph. We compare the results with the projected (coauthor) and the original bipartite graphs and show that cooperation is enhanced by the mesoscopic structure contained. We conclude by analyzing the influence of the size of the groups in the evolutionary success of cooperation.

Effective measurement of network vulnerability under random and intentional attacks

The study of the security and stability of complex networks plays a central role in reducing the risk and consequences of attacks or disfunctions of any type. The concept of vulnerability helps to measure the response of complex networks subjected to attacks on vertices and edges and it allows to spot the critical component of a network in order to improve its security. We introduce an accurate and computable definition of network vulnerability which is directly connected with its topology and we analyze its basic properties. We discuss the relationship of the vulnerability with other parameters of the network and we illustrate this with some examples.

Controlling centrality in complex networks

Spectral centrality measures allow to identify influential individuals in social groups, to rank Web pages by popularity, and even to determine the impact of scientific researches. The centrality score of a node within a network crucially depends on the entire pattern of connections, so that the usual approach is to compute node centralities once the network structure is assigned. We face here with the inverse problem, that is, we study how to modify the centrality scores of the nodes by acting on the structure of a given network. We show that there exist particular subsets of nodes, called controlling sets, which can assign any prescribed set of centrality values to all the nodes of a graph, by cooperatively tuning the weights of their out-going links. We found that many large networks from the real world have surprisingly small controlling sets, containing even less than 5 – 10% of the nodes.

Multiscale vulnerability of complex networks

We present a novel approach to quantify the vulnerability of a complex network, i.e., the capacity of a graph to maintain its functional performance under random damages or malicious attacks. The proposed measure represents a multiscale evaluation of vulnerability, and makes use of combined powers of the links betweenness. We show that the proposed approach is able to properly describe some cases for which earlier measures of vulnerability fail. The relevant applications of our method for technological network design are outlined.

John's Decomposition of the Identity in the Non-Convex Case

We prove an extension of the classical Johns Theorem, that characterises the ellipsoid of maximal volume position inside a convex body by the existence of some kind of decomposition of the identity, obtaining some results for maximal volume position of a compact and connected set inside a convex set with nonempty interior. By using those results we give some estimates for the outer volume ratio of bodies not necessarily convex.

Analytical relationships between metric and centrality measures of a network and its dual

The centrality and efficiency measures of a network G are strongly related to the respective measures on the dual G^@? and the bipartite B(G) associated networks. We show some relationships between the Bonacich centralities c(G), c(G^@?) and c(B(G)) and between the efficiencies E(G) and E(G^@?) and we compute the behavior of these parameters in some examples.

Structural Vulnerability and Robustness in Complex Networks: Different Approaches and Relationships Between them

The concept of vulnerability in the context of complex networks quantifies the capacity of a network to maintain its functional performance under random damages, malicious attacks, or malfunctions of any kind. Different types of networks and different applications suggest different approaches to the concept of networks structural vulnerability depending on the aspect we focus upon. In this introductory chapter, we discuss some different approaches and relationships amongst them.

Improvements in performance and security for complex networks

It is important to know how to improve a network when dealing with performance and security. We characterize the optimal design of a network for efficiency or vulnerability giving the extreme networks for those parameters. Moreover, we give an algorithm to improve the vulnerability of a given network when adding a new link. For efficiency, we present two facts that have to be considered in order to improve it when adding a new link, and we calculate bounds for the increment.

A Perron–Frobenius theory for block matrices associated to a multiplex network

Abstract The uniqueness of the Perron vector of a nonnegative block matrix associated to a multiplex network is discussed. The conclusions come from the relationships between the irreducibility of some nonnegative block matrix associated to a multiplex network and the irreducibility of the corresponding matrices to each layer as well as the irreducibility of the adjacency matrix of the projection network. In addition the computation of that Perron vector in terms of the Perron vectors of the blocks is also addressed. Finally we present the precise relations that allow to express the Perron eigenvector of the multiplex network in terms of the Perron eigenvectors of its layers.

Local estimates for eigenvector-like centralities of complex networks

We will analyze several centrality measures by giving a general framework that includes the Bonacich centrality, PageRank centrality or in-degree vector, among others. We will get some local scale estimators for such global measures by giving some geometrical characterizations and some deviation results that help to quantify the error of approximating a spectral centrality by a local estimator.