Numerical assessment of the percolation threshold using complement networks

Models of percolation processes on networks currently assume locally tree-like structures at low densities, and are derived exactly only in the thermodynamic limit. Finite size effects and the presence of short loops in real systems however cause a deviation between the empirical percolation threshold $p_c$ and its model-predicted value $\pi_c$. Here we show the existence of an empirical linear relation between $p_c$ and $\pi_c$ across a large number of real and model networks. Such a putatively universal relation can then be used to correct the estimated value of $\pi_c$. We further show how to obtain a more precise relation using the concept of the complement graph, by investigating on the connection between the percolation threshold of a network, $p_c$, and that of its complement, $\bar{p}_c$

Multiple structural transitions in interacting networks

Many real-world systems can be modeled as interconnected multilayer networks, namely a set of networks interacting with each other. Here we present a perturbative approach to study the properties of a general class of interconnected networks as inter-network interactions are established. We reveal multiple structural transitions for the algebraic connectivity of such systems, between regimes in which each network layer keeps its independent identity or drives diffusive processes over the whole system, thus generalizing previous results reporting a single transition point. Furthermore we show that, at first order in perturbation theory, the growth of the algebraic connectivity of each layer depends only on the degree configuration of the interaction network (projected on the respective Fiedler vector), and not on the actual interaction topology. Our findings can have important implications in the design of robust interconnected networked system, particularly in the presence of network layers whose integrity is more crucial for the functioning of the entire system. We finally show results of perturbation theory applied to the adjacency matrix of the interconnected network, which can be useful to characterize percolation processes on such systems

Fragility and anomalous susceptibility of weakly interacting networks

Percolation is a fundamental concept that brought new understanding on the robustness properties of complex systems. Here we consider percolation on weakly interacting networks, that is, network layers coupled together by much less interlinks than the connections within each layer. For these kinds of structures, both continuous and abrupt phase transition are observed in the size of the giant component. The continuous (second-order) transition corresponds to the formation of a giant cluster inside one layer, and has a well defined percolation threshold. The abrupt transition instead corresponds to the merger of coexisting giant clusters among different layers, and is characterised by a remarkable uncertainty in the percolation threshold, which in turns causes an anomalous trend in the observed susceptibility. We develop a simple mathematical model able to describe this phenomenon and to estimate the critical threshold for which the abrupt transition is more likely to occur. Remarkably, finite-size scaling analysis in the abrupt region supports the hypothesis of a genuine first-order phase transition

Frequency Analysis of a Nonlinear Hamiltonian System, application to Sound Synthesis

This work arises from the interest to create a connection beetween the two different fields of Sound Synthesis. Historically speaking the first goal of sound synthesis was to imitate the sound of musical instruments. For this reason the most popular kinds of synthesis are meant to recreate the frequency spectrum of a given audio signal. These kinds of synthesis can be distinguished in a simple mathematical way: linear synthesis (LS) and nonlinear synthesis (NLS). The main difference between LS and NLS is the following: linear techniques do not generate any frequencies which were not in the input signals whereas nonlinear techniques can generate new frequencies. Furthermore, from a practical point of view, LS is easy to manipulate, but requires a significant amount of input data in order to create a complex sound. This technique is often adequate for sounds whose harmonics do not vary over time, but unfortunately the spectra of most instruments vary considerably in time. On the other hand NLS allows to easily obtain dynamic spectra, but requires a less intuitive programming compared to LS. Unfortunately the use of NLS (and synthesis in general) with the aim of recreating the dynamic spectrum of a musical instrument ended as soon as advanced samplers were developed together with the growth of computer technology. Nowadays sound synthesis is not meant to imitate the sound of real instruments, but it's rather used to create sounds that are interesting beacuse they're actually really different from those which can be produced by any real instrument. The sound synthesis we investigate in this thesis work is interesting because it's able to generate a very large variety of timbres by using a very small number of external parameters (basically just one). The algorithm that generates the signal is strongly nonlinear, but the control parameters are very few, so this feature should overcome the not-user-friendly aspect of NLS. In order to achieve this goal we have studied the dynamical features of a 2D discrete map derived from a 4D nonlinear Hamiltonian system, whose possible trajectories in phase space are determined from both the initial conditions and an additional external parameter. Every trajectory is interpreted as a discrete-time sequence; each value of this sequence can be written to a DAC (Digital-to-Analog Converter) at a given sample frequency.The result is a real-time audio representation of a trajectory of the dynamical system we're dealing with. This kind of synthesis was studied heuristically in the past and named as Functional Iteration Synthesis (FIS) by A. Di Scipio in 1999. The chosen mathematical map for this purpose was the one derived from the natural 2D Poincaré Section of the Gravitational Billiard, a nonlinear Hamiltonian system which concerns the motion of a point particle moving in a symmetric wedge subject to a constant gravitational field pointing downwards along the direction of the axis of symmetry. The reason of this choice lies in the fact that this system is known to exhibit a remarkably complex behavior despite of having only two degrees of freedom. In fact, in order to achieve an interesting synthesizer using FIS we first need a dynamical system whose phase space is extremely rich in different trajectories, and that's exactly the case of the gravitational billiard. The aim of this thesis work is to give a more rigorous approach to FIS: in order to fulfill this task modern techniques for describing Complex Hamiltonian Systems' dynamics have been applied. The advantages of a FIS synthesizer based on the gravitational billiard map are finally discussed

Critical Phenomena in Multilayer Networks

Network science has provided a set of powerful theoretical results for the description of critical phenomena in complex systems. These represent fundamental tools for the design and control of real networked systems with concrete and practical applications. Many of these results have been, however, formulated in the framework of independent networks, i.e., closed systems that do not interact with nor depend on other networks. This is a weak hypothesis because in the realworld networks cannot be considered as independent entities. For instance, critical infrastructure are very often and intentionally coupled together: the functioning of the networks of water and food supply, communications, fuel, financial transactions and power generation and transmission depend one on the other. For this reason, many of the classical results of network science have been questioned, and recent research has provided evidence that many of the results valid for isolated networks are indeed not verified for interacting (or so called multilayer) networked systems. With this thesis we present new results in the field of both linear and nonlinear dynamical processes in the context of multilayer networks, namely diffusion and percolation. While diffusion is naturally related to the concept of walks and navigability of a network, percolation addresses the concept of stability and resilience of a network under random and targeted attacks

Percolation in networks with local homeostatic plasticity

Percolation is a process that impairs network connectedness by deactivating links or nodes. This process features a phase transition that resembles paradigmatic critical transitions in epidemic spreading, biological networks, traffic and transportation systems. Some biological systems, such as networks of neural cells, actively respond to percolation-like damage, which enables these structures to maintain their function after degradation and aging. Here we study percolation in networks that actively respond to link damage by adopting a mechanism resembling synaptic scaling in neurons. We explain critical transitions in such active networks and show that these structures are more resilient to damage as they are able to maintain a stronger connectedness and ability to spread information. Moreover, we uncover the role of local rescaling strategies in biological networks and indicate a possibility of designing smart infrastructures with improved robustness to perturbations

Percolation in networks with local homeostatic plasticity

Percolation is a process that impairs network connectedness by deactivating links or nodes. This process features a phase transition that resembles paradigmatic critical transitions in epidemic spreading, biological networks, traffic and transportation systems. Some biological systems, such as networks of neural cells, actively respond to percolation-like damage, which enables these structures to maintain their function after degradation and aging. Here we study percolation in networks that actively respond to link damage by adopting a mechanism resembling synaptic scaling in neurons. We explain critical transitions in such active networks and show that these structures are more resilient to damage as they are able to maintain a stronger connectedness and ability to spread information. Moreover, we uncover the role of local rescaling strategies in biological networks and indicate a possibility of designing smart infrastructures with improved robustness to perturbations

Homophily in the adoption of digital proximity tracing apps shapes the evolution of epidemics

We study how homophily of human physical interactions affects the impact of digital proximity tracing on the epidemic evolution. Analytical and numerical results show the existence of different dynamical regimes with respect to the mixing rate between adopters and nonadopters, revealing a rich phenomenology in terms of the reproduction number as well as the attack rate. We corroborate our findings with Monte Carlo simulations on different real contact networks. Our results indicate that depending on infectivity and adoption, mixing between adopters can be beneficial as well as detrimental for disease control.G.B. acknowledges financial support from the European Union's Horizon 2020 research and innovation program under the Marie Skłodowska-Curie Grant Agreement No. 945413 and from the Universitat Rovira i Virgili (URV). B.S. acknowledges financial support from the European Unions Horizon 2020 research and innovation program under the Marie Skłodowska-Curie Grant Agreement No. 713679 and from the Universitat Rovira i Virgili (URV). A.A. acknowledges support by Ministerio de Economía y Competitividad (Grants No. PGC2018-094754-B-C21 and No. FIS2015-71582-C2-1), Generalitat de Catalunya (Grants No. 2017SGR-896 and No. 2020PANDE00098), Universitat Rovira i Virgili (Grant No. 2019PFR-URV-B2-41), ICREA Academia, and the James S. McDonnell Foundation (Grant No. 220020325). We thank L. Arola-Fernández and A. Cardillo for helpful comments and suggestions.Peer ReviewedPostprint (author's final draft