A polynomial eigenvalue approach for multiplex networks

We explore the block nature of the matrix representation of multiplex networks, introducing a new formalism to deal with its spectral properties as a function of the inter-layer coupling parameter. This approach allows us to derive interesting results based on an interpretation of the traditional eigenvalue problem. More specifically, we reduce the dimensionality of our matrices but increase the power of the characteristic polynomial, i.e, a polynomial eigenvalue problem. Such an approach may sound counterintuitive at first glance, but it allows us to relate the quadratic problem for a 2-Layer multiplex system with the spectra of the aggregated network and to derive bounds for the spectra, among many other interesting analytical insights. Furthermore, it also permits to directly obtain analytical and numerical insights on the eigenvalue behavior as a function of the coupling between layers. Our study includes the supra-adjacency, supra-Laplacian, and the probability transition matrices, which enable us to put our results under the perspective of structural phases in multiplex networks. We believe that this formalism and the results reported will make it possible to derive new results for multiplex networks in the future.Comment: 15 pages including figures. Submitted for publicatio

On degree-degree correlations in multilayer networks

We propose a generalization of the concept of assortativity based on the tensorial representation of multilayer networks, covering the definitions given in terms of Pearson and Spearman coefficients. Our approach can also be applied to weighted networks and provides information about correlations considering pairs of layers. By analyzing the multilayer representation of the airport transportation network, we show that contrasting results are obtained when the layers are analyzed independently or as an interconnected system. Finally, we study the impact of the level of assortativity and heterogeneity between layers on the spreading of diseases. Our results highlight the need of studying degree-degree correlations on multilayer systems, instead of on aggregated networks.Comment: 8 pages, 3 figure

Layer degradation triggers an abrupt structural transition in multiplex networks

Network robustness is a central point in network science, both from a theoretical and a practical point of view. In this paper, we show that layer degradation, understood as the continuous or discrete loss of links' weight, triggers a structural transition revealed by an abrupt change in the algebraic connectivity of the graph. Unlike traditional single layer networks, multiplex networks exist in two phases, one in which the system is protected from link failures in some of its layers and one in which all the system senses the failure happening in one single layer. We also give the exact critical value of the weight of the intra-layer links at which the transition occurs for continuous layer degradation and its relation to the value of the coupling between layers. This relation allows us to reveal the connection between the transition observed under layer degradation and the one observed under the variation of the coupling between layers.Comment: 8 pages, and 8 figures in Revtex style. Submitted for publicatio

On the onset of synchronization of Kuramoto oscillators in scale-free networks

Despite the great attention devoted to the study of phase oscillators on complex networks in the last two decades, it remains unclear whether scale-free networks exhibit a nonzero critical coupling strength for the onset of synchronization in the thermodynamic limit. Here, we systematically compare predictions from the heterogeneous degree mean-field (HMF) and the quenched mean-field (QMF) approaches to extensive numerical simulations on large networks. We provide compelling evidence that the critical coupling vanishes as the number of oscillators increases for scale-free networks characterized by a power-law degree distribution with an exponent $2 < \gamma \leq 3$, in line with what has been observed for other dynamical processes in such networks. For $\gamma > 3$, we show that the critical coupling remains finite, in agreement with HMF calculations and highlight phenomenological differences between critical properties of phase oscillators and epidemic models on scale-free networks. Finally, we also discuss at length a key choice when studying synchronization phenomena in complex networks, namely, how to normalize the coupling between oscillators

An approach for interactive educational software in a classroom setting

This paper presents a systematic set of activities related to the ergonomic analysis process of interactive educational software in a classroom setting. It uses the User Interface Design (1991-1999 Interface Systems International) with the aim of developing an ergonomic model for Human Computer Interaction (HCI) in the context of interactive educational software. The model should be adaptable to any kind of user and evolutive, i.e., the model should be able to switch to more advanced concepts once the user has learned the previous ones

Disease Localization in Multilayer Networks

We present a continuous formulation of epidemic spreading on multilayer networks using a tensorial representation, extending the models of monoplex networks to this context. We derive analytical expressions for the epidemic threshold of the SIS and SIR dynamics, as well as upper and lower bounds for the disease prevalence in the steady state for the SIS scenario. Using the quasi-stationary state method we numerically show the existence of disease localization and the emergence of two or more susceptibility peaks, which are characterized analytically and numerically through the inverse participation ratio. Furthermore, when mapping the critical dynamics to an eigenvalue problem, we observe a characteristic transition in the eigenvalue spectra of the supra-contact tensor as a function of the ratio of two spreading rates: if the rate at which the disease spreads within a layer is comparable to the spreading rate across layers, the individual spectra of each layer merge with the coupling between layers. Finally, we verified the barrier effect, i.e., for three-layer configuration, when the layer with the largest eigenvalue is located at the center of the line, it can effectively act as a barrier to the disease. The formalism introduced here provides a unifying mathematical approach to disease contagion in multiplex systems opening new possibilities for the study of spreading processes.Comment: Revised version. 25 pages and 18 figure

Centrality anomalies in complex networks as a result of model over-simplification

Tremendous advances have been made in our understanding of the properties and evolution of complex networks. These advances were initially driven by information-poor empirical networks and theoretical analysis of unweighted and undirected graphs. Recently, information-rich empirical data complex networks supported the development of more sophisticated models that include edge directionality and weight properties, and multiple layers. Many studies still focus on unweighted undirected description of networks, prompting an essential question: how to identify when a model is simpler than it must be? Here, we argue that the presence of centrality anomalies in complex networks is a result of model over-simplification. Specifically, we investigate the well-known anomaly in betweenness centrality for transportation networks, according to which highly connected nodes are not necessarily the most central. Using a broad class of network models with weights and spatial constraints and four large data sets of transportation networks, we show that the unweighted projection of the structure of these networks can exhibit a significant fraction of anomalous nodes compared to a random null model. However, the weighted projection of these networks, compared with an appropriated null model, significantly reduces the fraction of anomalies observed, suggesting that centrality anomalies are a symptom of model over-simplification. Because lack of information-rich data is a common challenge when dealing with complex networks and can cause anomalies that misestimate the role of nodes in the system, we argue that sufficiently sophisticated models be used when anomalies are detected.Comment: 14 pages, including 9 figures. APS style. Accepted for publication in New Journal of Physic

Product renovation and shared ownership: sustainable routes to satisfying the world's growing demand for goods

It has been estimated that by 2030 the number of people who are wealthy enough to be considered as middle class consumers will have tripled. This will have a dramatic impact on the demands for primary materials and energy. Much work has been carried out on sustainable ways of meeting the World’s energy demands and some work has been carried out on the sustainable production and consumption of goods. It has been estimated that with improvements in design and manufacturing it is possible to reduce the primary material requirements by 30% to produce the current demand for goods. Whilst this is a crucial step on the production side, there will still be a doubling of primary material requirements by the end of the century because of an absolute rise in demand for goods and services. It is therefore clear that the consumption of products must also be explored. This is a key areas of research for the UK INDEMAND centre, which is investigating ways of reducing the UK’s industrial energy demand and demand for energy intensive materials. Our ongoing work shows that two strategies would result in considerable reductions in the demand for primary materials: product longevity and using goods more intensively (which may requires increased durability). Product longevity and durability are not new ideas, but ones that can be applied across a raft of goods as methods of reducing the consumption of materials. With long life products there is a potential risk of outdated design and obsolescence, consequently there is a need to ensure upgradability and adaptability are incorporated at the design stage. If products last longer, then the production of new products can be diverted to emerging markets rather than the market for replacement goods. There are many goods which are only used occasionally; these goods do not normally wear out. The total demand for such could be drastically reduced if they were shared with other people. Sharing of goods has traditionally been conducted between friends or by hiring equipment. The use of modern communication systems and social media could enable the development of sharing co-ops and swap spaces that will increase the utilisation of goods and hence reduce the demand for new goods. This could also increase access to a range of goods for those on low incomes. From a series of workshops it has been found that the principal challenges are sociological rather than technological. This paper contains a discussion of these challenges and explores possible futures where these two strategies have been adopted. In addition, the barriers and opportunities that these strategies offer for consumers and businesses are identified, and areas where government policy could be instigated to bring about change are highlighted

Product longevity and shared ownership: Sustainable routes to satisfying the world’s growing demand for goods&nbsp;

It has been estimated that by 2030 the number of people who are wealthy enough to be significant consumers will have tripled. This will have a dramatic impact on the demands for primary materials and energy. It has been estimated that with improvements in design and manufacturing it is possible to maintain the current level of production using 70% of the current primary material consumed. Even with these improvements on the production side, there will still be a doubling of primary material requirements by the end of the century, with accompanying rises in industrial energy demand, if the rise in demand for goods and services is to be met. It is therefore clear that the consumption of products must also be explored. Product longevity and using goods more intensively are two strategies which could reduce the demand for new goods. If products last longer, then manufacturing output can concentrate on emerging markets rather than the market for replacement goods. There are many goods which are infrequently used, these seldom wear out. The total demand for such could be drastically reduced if they we re shared with other people. Sharing of goods has traditionally been conducted between friends or by hiring equipment, but modern communication systems and social media could increase the opportunities to share goods. Sharing goods also increases access to a range of goods for those on low incomes. From a series of workshops it has been found that the principal challenges are sociological rather than technological. This paper contains a discussion of these challenges and explores possible futures where these two strategies have been adopted. In addition, the barriers and opportunities that these strategies offer for 548 AIMS Energy Volume 3, Issue 4, 547-561. consumers and businesses are identified, and areas where government policy could be instigated to bring about change are highlighted