Resumen de Higher-order dynamics on complex networks
Joan Matamalas Llodrà
Dynamical processes running on complex networks have been largely studied due to their suitability to model very different scenarios. Understanding critical aspects of ecological systems, modeling how traffic flows in a city or modeling relations between species in ecosystem are a few examples of their potential applications. Nevertheless, there are some inherent limitations imposed by the way how these traditionally models are built, that could diminish their analytical power. For instance, consider the case of modeling mobility on a city. There, users can move between different places using different transportation systems, bus, subway, taxi, etc. Embedding this multimodality of systems inside our mobility model is critical to understand how it will behave in situations like congestion or failures since, different transportation modes have different capacities, costs or speeds. However, traditional network representations cannot embed this heterogeneity since the relations that they represent have to share a common definition. To overcome this issue, we should increase the structural complexity order of our model, enabling the use of different kinds of relations.
On this document, we focus on the understanding of how increasing the order of dynamical models, i.e., the amount of information represented, concerning structure, relational correlations, and temporal information and their effects on three topics that profoundly affect our lives: the emergence of cooperation, the spreading of epidemics and, finally, human mobility. We show that this increase on the complexity has a significant contribution to the way how we understand the way these processes work.
This document is organized as follows:
Chapter 1 provides a brief introduction to complex networks, explaining its origins and significant breakthroughs. Then, we move to provide a gentle introduction to the three subjects of study: cooperation, epidemic spreading and human mobility. For each of them, we motivate the importance of the problem, a summary of the principal techniques used to analyze them and, finally, an analysis of their limitations.
In Chapter 2 we concentrate on the modeling of social dilemmas as a medium to study the emergence of cooperative behaviors. Using multilayer networks, hence, increasing the structural order of the model, and numerical simulations, we systematically study the evolution of cooperation in all four games in the T-S plane, considering the density of cooperators, the convergence to equilibrium or the fluctuations of the system. More importantly, we show some remarkable and previously unknown features in the microscopic organization of the strategies, that are responsible for the important differences between cooperative dynamics in monoplex and multiplex networks. Specifically, we analyze the fact that, in the stationary state, there are individuals that play the same strategy in all layers (coherent), and others that do not (incoherent), proving their existence analytically in the harmony game, where only cooperation was expected.
We devote our efforts in Chapter 3 to explain a new way to model how epidemic diseases spread. We provide a new approach based on Markov chains which captures microscopic information about the epidemic properties of the links. The model is validated in synthetic and real networks, yielding an accurate determination of epidemic incidence and critical thresholds. Finally, thanks to the epidemic descriptors at the level of links, we propose a new containment strategy based on the removal of the links that mostly contribute to the spreading of the disease. We show how our method allows a faster containment of epidemic than other used methods, preserving the connectivity of the network, and hence, its functionality.
In Chapter 4, we expose the limitations that traditional mobility models have to capture recurrent patterns while, at the same time, retaining temporal information about the time spent on each location before moving somewhere else, an important issue, since these properties are inherent to human mobility. Then, we introduce an adaptive memory-driven approach to overcome such issues. At variance with the other compared models, it can realistically model conditional waiting times, i.e., the probability to stay in a specific area depending on individuals’ historical movements. Our results demonstrate that, in standard mobility models, the individuals tend to diffuse faster than observed in reality, whereas the predictions of the adaptive memory approach significantly agree with observations. We show that, as a consequence, the incidence and the geographical spread of a disease can be inadequately estimated when standard approaches are used, with crucial implications on resources deployment and policy-making during an epidemic outbreak.
Finally, in Chapter 5, we provide a detailed analysis of the principal conclusions derived from the realization of this work. Analyzing our contributions, but also giving a critical view of our models, and exposing their limitations and possible ways to enhance them. At the end of this chapter, a brief detail of future work is depicted.
