Abstract: Social interactions are composite, involve different communication layers and evolve in time. However, a rigorous analysis of the whole complexity of social networks has been hindered so far by lack of suitable data. Here we consider both the multi-layer and dynamic nature of social relations by analysing a diverse set of empirical tem- poral multiplex networks. We focus on the measurement and charac- terization of inter-layer correlations to investigate how activity in one layer affects social acts in another layer. We define observables able to detect when genuine correlations are present in empirical data, and single out spurious correlation induced by the bursty nature of human dynamics. We show that such temporal correlations do exist in social interactions, where they act to depress the tendency to con- centrate long stretches of activity on the same layer. They also imply some amount of potential predictability in the connection patterns between layers, and may affect the dynamics of spreading processes unfolding on different layers. Our work sets up a general framework to measure temporal correlations in multiplex networks, and we an- ticipate that it will be of interest to researchers in a broad array of fields.

Abstract: Biological and socio-economic populations can sometimes increase their fitness by identifying and adjusting to more details of the fitness landscape in which they evolve. When does it pay off for the population to use a more detailed classification system? This depends on the particular shape of the fitness matrix in both space and time. Sometimes the population does not need to bother looking closer. It is at least as beneficial to simply let natural selection act as a blind watchmaker. Other times, proactive resource redistribution among fine-grained distinctions pays off, such as done by a portfolio manager or genetic phenotype switches. In order to understand the difference we decompose population fitness and represent the strength of evolutionary selection with the information theoretic metric of relative entropy (Kullback-Leibler divergence). It is a multivariate metric that allows us to analyze selection pressure over multiple taxonomic levels in one single equation. It turns out that difference in relative entropy between different strategies is proportional to the potential to increase population fitness through intervention. As such, differences in relative entropy allow to quantify the potential to increase fitness. An intuitive interpretation is that it quantifies the amount of complementary variety between different population types and corresponding environmental states. The more type fitness is skewed to opposing directions in different environmental states, the proportionally larger the potential benefit of strategic intervention over natural selection. In reference to the literature of bet-hedging and portfolio theory, it is shown that the higher the degree of specialization of different population types to different environmental states, the larger the complementary variety among them, and therefore the larger the potential payoff. Complementary variety is the necessary condition to increase overall fitness through resource redistribution. This suggests developing classification systems of population types and environmental states that contain complementary variety.

Abstract: Variability on the external conditions, such as the temperature, humidity, available nutrients, etc., have important consequences for the dynamics and organization of communities of living systems. Furthermore, the interplay between the timescales of the environment and the intrinsic dynamics plays a fundamental role in many situations. However, most of the mathematical models neglect temporal correlations of the environment. We propose a unifying framework of some precedent available analytical and numerical tools to deal with colored noise, and provide a general scheme to answer some relevant questions concerning population dynamics: quantification of the population growth rate and population density, under which internal and internal conditions the population may become extinct, and in such a case, how much time does it take to disappear. These questions are of fundamental relevance, for instance, in the context of epidemiology, as they provide valuable information for the control and eradication of disease spreading. We test our results in a SIS model in which the infection rate fluctuates in time with environmental conditions.

Abstract: Hierarchy is pervasive in both natural and man-made systems. A significant part of complex networks science is concerned with identifying hierarchical features in observed real world networks, explaining their origins via generative models with simple assembly rules and studying the interplay between features on many scales by studying appropriate random network ensembles. Recently it has been shown that a new topological feature of directed networks termed "trophic coherence" is a prominent in many real world networks ranging from ecological food-webs to gene transcription networks. Trophic coherence characterizes how "layered" a directed network is - the tendency of nodes to form well defined, hierarchically organized groups. In networks with high trophic coherence interactions via directed links start from base nodes with no incoming connections and follow up the hierarchy of nodes in a chain-of-command fashion. In networks with less trophic coherence, however, "shortcuts" may be present that distorts the hierarchy and allows interactions between groups of nodes far apart in the network. This has implications on local topological features and information spread. We present results on how trophic coherence affects the local structure of food-webs. Our findings indicate that trophic coherence reveals that the majority of known food-webs fall into one of two groups - those with relatively little omnivory and those with a lot of omnivory as defined by the presence of feed-forward loops. This result complements previous results by Johnson et al. where a network model with trophic coherence was shown to reproduce most food-web features better than the standard Niche and Cascade models. Additionally, work on studying contagion processes on trophically coherent networks has shown that the level of trophic coherence plays a large role in determining the final size of an outbreak. We briefly discuss numerical results and future work.

Abstract: Synchronization processes are ubiquitous in nature, from the entrainment of circadian rhythms and the synchronous firing of neurons to the flocking of birds or the shoaling of fish. The emergence of collective consensus in networked systems is thus a current focus in biology, physics, chemistry, as well as in social and technological networks. Previous studies have typically focused on total synchronization, where all agents on the network converge to the same dynamics. Here we use tools from graph theory to study the phenomenon of cluster synchronization, where groups of agents converge to several distinct behaviors. We show that cluster synchronization can emerge in networks with certain regularities, as captured via a graph partition called the external equitable partition. Indeed, when the underlying coupling network presents certain regularities, the dynamics can be coarse-grained into clusters by means of external equitable partitions and their associated quotient graphs. We derive conditions and properties of networks under which such clustered behavior emerges, and show that the ensuing dynamics is the result of a localization of the eigenvectors of the associated graph Laplacians. The framework is applied to both linear (consensus dynamics) and non-linear models (generic oscillator models, including the classic Kuramoto model), first in the standard case of networks with positive edges, before being generalized to the case of signed networks with both positive and negative interactions. Furthermore, we demonstrate how our graph-theoretical approach allows us to extend the analysis to cluster synchronization of signed networks (with positive and negative links), which are used to describe social interactions and inhibitory-excitatory interactions in biology.

Abstract: Complex systems may display strong changes in their dynamics: bifurcations, tipping points, phase transitions, etc. Some examples of special relevance are phase transitions in condensed matter systems (magnetism, superconductivity), sudden physiological alterations (strokes, epileptic seizures), economic crisis or climatic changes associated to the global warming such as potential modifications of weather and oceanic circulation patterns. The origin of these sudden changes can be traced down to the interactions between the system components. In most cases, the correlation of the components' dynamics enhances before the transition due to the cooperative phenomena that give raise to the emergence of a new global dynamical state. While understanding the ultimate causes of the change is of great importance, from a practical point of view it is absolutely crucial to count with metrics able to act as early warning signals or precursors of the dynamic transition. This may mean the difference between being able to react preemptively to the change or arriving to it unaware. In this work, we exploit the increase of microscopic correlations when the tipping point gets closer and define a set of early warning metrics using concepts imported from percolation theory on the functional network framework. We show that the functional networks encoding the system dynamics undergo a percolation transition way before the tipping point arrives. Furthermore, the number of clusters of size s peaks way before the percolation transition and, therefore, the sequence of peaks clearly mark the path to the transition. Our warning signals are general, as shown by analyzing three very different types of transitions, they precedes other early warning signals proposed in the literature and are straightforwardly applicable to many real-world complex systems, as proven by the analysis run with them on the South Pacific El Nino Oscillation. Our results have made available online at http://arxiv.org/abs/1601.01978