Abstract-Several systems can be modeled as sets of interconnected networks or networks with multiple types of connections, here generally called multilayer networks. Spreading processes such as information propagation among users of an online social networks, or the diffusion of pathogens among individuals through their contact network, are fundamental phenomena occurring in these networks. However, while information diffusion in single networks has received considerable attention from various disciplines for over a decade, spreading processes in multilayer networks is still a young research area presenting many challenging research issues. In this paper we review the main models, results and applications of multilayer spreading processes and discuss some promising research directions.

It is well known that many complex systems, both in technology and nature, exhibit hierarchical modularity: smaller modules, each of them providing a certain function, are used within larger modules that perform more complex functions. What is not well understood however is how this hierarchical structure (which is fundamentally a network property) emerges, and how it evolves over time. We propose a modeling framework, referred to as Evo-Lexis, that provides insight to some fundamental questions about evolving hierarchical systems. Evo-Lexis models the most elementary modules of the system as symbols ("sources") and the modules at the highest level of the hierarchy as sequences of those symbols ("targets"). Evo-Lexis computes the optimized adjustment of a given hierarchy when the set of targets changes over time by additions and removals (a process referred to as "incremental design"). In this paper we use computation modeling to show that:-Low-cost and deep hierarchies emerge when the population of target sequences evolves through tinkering and mutation. -Strong selection on the cost of new candidate targets results in reuse of more complex (longer) nodes in an optimized hierarchy. -The bias towards reuse of complex nodes results in an "hourglass architecture" (i.e., few intermediate nodes that cover almost all source-target paths). -With such bias, the core nodes are conserved for relatively long time periods although still being vulnerable to major transitions and punctuated equilibria. -Finally, we analyze the differences in terms of cost and structure between incrementally designed hierarchies and the corresponding "clean-slate" hierarchies which result when the system is designed from scratch after a change.arXiv:1805.04924v2 [cs.NE] 4 Aug 20182. Under what conditions do the emergent hierarchies exhibit the so called "hourglass effect"? Why are few intermediate modules reused much more than others? 3. Do intermediate modules persist during the evolution of hierarchies? Or are there "punctuated equilibria" where the highly reused modules change significantly? 4. Which are the differences in terms of cost and structure between the incrementally designed and the corresponding clean-slate designed hierarchies?The structure of the paper is as follows: In Section 2, we present an overview of Lexis, the static optimization framework that serves as the main building block in Evo-Lexis. 1 In Section 3, we present the components of the Evo-Lexis framework, along with the metrics that we use for the analysis of evolving hierarchies. In Section 4, we evaluate the evolution of hierarchies under different target generation models. Sections 5 and 6 present further analysis regarding the evolvability and major transitions in hierarchies produced using the most full-fledged (MRS) target generation model. Finally, Section 7 focuses on the comparison between clean-slate and incremental design in terms of cost and structure. In Section 8, we review related work in the context of Evo-Lexis. Section 9 discusses the results and p...

Several systems can be modeled as sets of interdependent networks where each network contains distinct nodes. Diffusion processes like the spreading of a disease or the propagation of information constitute fundamental phenomena occurring over such coupled networks. In this paper we propose a new concept of multidimensional epidemic threshold characterizing diffusion processes over interdependent networks, allowing different diffusion rates on the different networks and arbitrary degree distributions. We analytically derive and numerically illustrate the conditions for multilayer epidemics, i.e., the appearance of a giant connected component spanning all the networks. Furthermore, we study the evolution of infection density and diffusion dynamics with extensive simulation experiments on synthetic and real networks.

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Many complex systems, both in technology and nature, exhibit hierarchical modularity: smaller modules, each of them providing a certain function, are used within larger modules that perform more complex functions. Previously, we have proposed a modeling framework, referred to as Evo-Lexis [21], that provides insight to some fundamental questions about evolving hierarchical systems. The predictions of the Evo-Lexis model should be tested using real data from evolving systems in which the outputs can be well represented by sequences. In this paper, we investigate the time series of iGEM synthetic DNA dataset sequences, and whether the resulting iGEM hierarchies exhibit the qualitative properties predicted by the Evo-Lexis framework. Contrary to Evo-Lexis, in iGEM the amount of reuse decreases during the timeline of the dataset. Although this results in development of less cost-efficient and less deep Lexis-DAGs, the dataset exhibits a bias in reusing specific nodes more often than others. This results in the Lexis-DAGs to take the shape of an hourglass with relatively high H-score values and stable set of core nodes. Despite the reuse bias and stability of the core set, the dataset presents a high amount of diversity among the targets which is in line with modeling of Evo-Lexis.