Julio Flores

Eigenvector centrality of nodes in multiplex networks

We extend the concept of eigenvector centrality to multiplex networks, and introduce several alternative parameters that quantify the importance of nodes in a multi-layered networked system, including the definition of vectorial-type centralities. In addition, we rigorously show that, under reasonable conditions, such centrality measures exist and are unique. Computer experiments and simulations demonstrate that the proposed measures provide substantially different results when applied to the same multiplex structure, and highlight the non-trivial relationships between the different measures of centrality introduced.

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.

A mathematical model for networks with structures in the mesoscale

The new concept of multilevel network is introduced in order to embody some topological properties of complex systems with structures in the mesoscale, which are not completely captured by the classical models. This new model, which generalizes the hyper-network and hyper-structure models, fits perfectly with several real-life complex systems, including social and public transportation networks. We present an analysis of the structural properties of the multilevel network, including the clustering and the metric structures. Some analytical relationships amongst the efficiency and clustering coefficient of this new model and the corresponding parameters of the underlying network are obtained. Finally, some random models for multilevel networks are given to illustrate how different multilevel structures can produce similar underlying networks and therefore that the mesoscale structure should be taken into account in many applications.

Domination by positive disjointly strictly singular operators

We prove that each positive operator from a Banach lattice E to a Banach lattice F with a disjointly strictly singular majorant is itself disjointly strictly singular provided the norm on F is order continuous. We prove as well that if S : E --> E is dominated by a disjointly strictly singular operator, then S-2 is disjointly strictly singular.

Domination by Positive Narrow Operators

We prove that each positive operator from a Köthe function-space E(μ) to a Banach lattice F with a narrow majorant is itself narrow provided the norm on F is order continuous. We also prove that every l2-strictly singular regular operator from Lp[0,1], 1≤p < ∞, to a Banach lattice F is narrow, provided F has an order continuous norm.

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.

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.

Structural properties of the line-graphs associated to directed networks

The centrality and efficiency measures of an undirected network

Disjointly Homogeneous Banach Lattices and Applications

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On the α-nonbacktracking centrality for complex networks: Existence and limit cases

were shown by the authors to be strongly related to the respective measures on the associated line graph