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The LISA Pathfinder mission was flown to test some key technologies for the in-flight detection of gravitational waves. The mechanism developed to initialise the science phase, releasing the test mass into a geodesic trajectory, produced unexpected test mass states, requiring dedicated in-flight and on-ground test campaigns. Due to the limited information available from the telemetry, the analysis of the results required a combination of experimental and analytical approaches. A dynamic model of the release mechanism, integrated by an impact model, is developed to explain the in-flight test results. The proposed interpretation of the mechanism anomalies may provide useful guidelines for the design and development of the flight units for the forthcoming LISA mission.

The researchers studying complex networks will acquire from this advanced modern book a number of new issues and ideas, not yet touched upon in other reference volumes. The book considers a wide range of networks and processes taking place on them, paying particular attention to the recently developed directions, methods, and techniques. It proposes a statistical mechanics view of random networks based on the concept of statistical ensembles, but approaches and methods of modern graph theory, concerning random graphs, overlap strongly with statistical physics. Hence mathematicians have a good chance to discover interesting things in this book, even though it does not contain mathematical proofs and trades off rigour for comprehension, brevity, and relevance. The book combines features of an advanced textbook, a reference book and a detailed review of the current state of the art. This book will be useful for undergraduate, master, and PhD students and young researchers from physics, multidisciplinary studies, computer science, and applied mathematics wishing to gain a serious insight into the principles of complex networks. The book can be used as a text in university courses on complex networks. It proposes to determined students not only a brief trip to the land of complex networks but an option to stay there forever.

In particular, if all connections are only between layers, then we get a multipartite graph, which we already discussed. In this chapter we proceed further and consider more interesting and more complicated networks having connections both within and between the layers. Distinct types of edges result in different kinds of paths between vertices, and hence to a more complex connectivity. This raises new questions, impossible for ordinary graphs. For example, we can ask about a pair of vertices: how are they interconnected by paths running within particular layers? This rich connectivity enables one to introduce new kinds of connected components playing specific roles in the integrity, robustness, and function of a network.

Many of the problems considered in this book are equivalent to models of interacting spins or agents. For instance, the bond percolation problem is equivalent to the 1-state Potts model (Kasteleyn and Fortuin, 1969; Kasteleyn and Fortuin, 1969), the k-core problem is equivalent to the ferromagnetic Ising model in a heterogeneous magnetic field (Section 6.9), and so on. This chapter touches upon a few basic cooperative models demonstrating various behaviours.

By the late 1970s, the theory of classical random graphs was already well developed, and mathematicians started to search for more general network constructions. In 1978, Edward A. Bender and E. Rodney Canfield published a paper entitled ‘The asymptotic number of labelled graphs with given degree sequences’ in which they described random networks with significantly richer architectures than the Erdős–Rényi graph. Béla Bollobás strictly formulated this generalization of the Erdős–Rényi model in his 1980 paper ‘A probabilistic proof of an asymptotic formula for the number of labelled random graphs’ and named it the configuration model. This generalization turned out to be a major step toward real networks in the post-Erdős epoch.

Milgram’s algorithm is actually the standard one in computer science, belonging to the class of decentralized search algorithms. A number of routing algorithms exploit geographic information about vertices of communication networks (Karp and Kung, 2000). The simplest geographic routing implements the greedy routing algorithm assuming that: (i) each vertex in a network has its geographic coordinate, and (ii) a vertex forwards messages (packets) to that its nearest neighbour in the network, which is geographically closest to the destination.

Let us return to the configuration model of uncorrelated networks with a given degree distribution P(q) and exploit its local tree-likeness to explore basic structural features of these networks. The elements of the techniques that we use here were first implemented in graph theory (Pittel 1990; Molloy and Reed, 1995, 1998), and physicists developed it into a convenient and powerful mathematical apparatus (Newman, Strogatz, and Watts, 2001) applicable to various locally tree-like networks, including directed, multipartite, and correlated ones, and many others. For the sake of simplicity, we first consider uncorrelated undirected networks. The generating functions techniques is ideally suited for random trees and tree-like structures (Appendix C). The generating function for the degree distribution is defined as

In this chapter we mainly focus on the results of activation processes in networks and on various combinations of activation and deactivation processes. The bootstrap percolation problem is about the basic activation process on networks, in which vertices can be in active and inactive states. A vertex becomes active when the number of its active neighbours exceeds some threshold; and once active, a vertex never becomes inactive (Adler and Aharony, 1988; Adler, 1991). This is one of the spreading processes with discontinuous phase transitions (Bizhani, Paczuski, and Grassberger, 2012). Let us define bootstrap percolation on undirected graphs in more strict terms. In the initial state, a fraction f of vertices is active (seed vertices). These vertices are chosen uniformly at random. Each inactive vertex becomes active if it has at least kb active nearest neighbours. Here we introduce the subscript ‘b’ to distinguish this threshold from a threshold in the k-core percolation problem.

Spectral properties of networks essentially determine the key features of processes and cooperative phenomena taking place in them. For example, the largest eigenvalue of the adjacency matrix of a network determines the epidemic threshold of the SIS model in the quenched mean-field approximation, while the corresponding eigenvector indicates the probability that a vertex is infective near this threshold (Section 7.4). To gain some intuition about how the spectra of graphs are organized, the reader should review the chain of examples for small sample graphs in Section 2.9. In the present chapter, we focus on the spectra of undirected random networks. For an adjacency matrix, the key quantity of interest is the spectral density

Look at the two graphs in Figure 1.1. Could you say which graph is random, left or right? The typical student’s response is: ‘Of course, the right one!’ This immediate answer is dramatically incorrect. It is actually impossible to say whether a finite graph is random or non-random (deterministic) since one can generate any finite graph by some deterministic algorithm. The example in Figure 1.2 explains what is really a random graph. This graph has a number of different realizations (individual graphs), and each of them occurs with some associated probability. Thus a random graph (random network) is a statistical ensemble of individual graphs, in which each member has its probability of realization. (Note that we do not distinguish the terms ‘graph’, or its generalization—‘hypergraph’, and ‘network’.) In particular, all these probabilities may be equal, which provides a uniform ensemble. In short, a random graph is a statistical ensemble of graphs. In this picture, the result of the measurement of some characteristic of a random graph, an observable, is the average of this characteristic over the statistical ensemble accounting for the probabilities of realization of its members.

Inference is generally about drawing conclusions about the whole on the basis of a sample. Statistical inference is ‘the process of deducing properties of an underlying distribution by analysis of data’ (Zdeborová and Krzakala, 2016). More specifically, in statistical mechanics inference usually implies concluding characteristics of a statistical ensemble (or its model, which is practically the same) on the basis of a sample (Clauset, Moore, and Newman, 2006). Here we touch upon a more restricted problem. Consider a branching process taking place on a given graph, which started from some unknown initial vertex, a root. At some instant, an observer makes a snapshot of this process and records its result—a tree subgraph of the substrate graph. The questions are: is it possible to guess the root from this observation; and, when it is possible, what is the best root-finding algorithm? The answers to these questions depend on the branching process and on the substrate graph. Remarkably, root finding is possible for a wide range of branching processes and substrate graphs.

Here we give an insight into two basic models of equilibrium random networks. Often they are both called the Erdős–Rényi random graph, although, strictly speaking, this name is only for the second model. This qualitative picture is generic for random networks. The general properties of a network are primarily determined by whether or not a giant connected component is present. So the first question about any network should be about the presence and relative size of this component.

Let us put aside random graphs, statistical ensembles, and probabilities for a while in favour of individual graphs while we introduce a few necessary basic notions.1 First we discuss undirected graphs. Two (hyper)graphs are isomorphic iff their vertices are connected in the same way, that is, there exists a permutation of the vertices of the first graph such that the adjacent vertices of this graph, after the permutation, become the adjacent vertices of the second.

The science of complex networks has been explosively expanding over the last twenty-plus years, resulting in a number of new research directions. Some have already reached maturity, some became less exciting for the research community with time, and none of the directions has disappeared. Figure 14.1 gives an idea of how this field developed in the respect of objects, phenomena, and processes, putting aside applications. One can see the recent trends and the topics attracting particular attention and efforts of numerous academicians, whose studies on complex networks were essentially induced and multiplied by three major issues: the Big data problem, the Artificial Intelligence algorithms, and the COVID-19 pandemic. We suggest that these key problems, in addition to the growing needs of biotechnology and pharmaceutical industries, will largely determine the progress of the science of complex networks in the forthcoming years.

LISA Pathfinder is a technology demonstrator space mission, aimed at testing key technologies for detecting gravitational waves in space. The mission is the precursor of LISA, the first space gravitational waves observatory, whose launch is scheduled for 2034. The LISA Pathfinder scientific payload includes two gravitational reference sensors (GRSs), each one containing a test mass (TM), which is the sensing body of the experiment. A mission critical task is to set each TM into a pure geodesic motion, i.e. guaranteeing an extremely low acceleration noise in the sub-Hertz frequency bandwidth. The grabbing positioning and release mechanism (GPRM), responsible for the injection of the TM into a geodesic trajectory, was widely tested on ground, with the limitations imposed by the 1-g environment. The experiments showed that the mechanism, working in its nominal conditions, is capable of releasing the TM into free-fall fulfilling the very strict constraint imposed on the TM residual velocity, in order to allow its capture on behalf of the electrostatic actuation. However, the first in-flight releases produced unexpected residual velocity components, for both the TMs. Moreover, all the residual velocity components were greater than maximum value set by the requirements. The main suspect is that unexpected contacts took place between the TM and the surroundings bodies. As a consequence, ad hoc manual release procedures had to be adopted for the few following injections performed during the nominal mission. These procedures still resulted in non compliant TM states which were captured only after impacts. However, such procedures seem not practicable for LISA, both for the limited repeatability of the system and for the unmanageable time lag of the telemetry/telecommand signals (about 4400 s). For this reason, at the end of the mission, the GPRM was deeply tested in-flight, performing a large number of releases, according to different strategies. The tests were carried out in order to understand the unexpected dynamics and limit its effects on the final injection. Some risk mitigation maneuvers have been tested aimed at minimizing the vibration of the system at the release and improving the alignment between the mechanism and the TM. However, no overall optimal release strategy to be implemented in LISA could be found, because the two GPRMs behaved differently., This work has been made possible by the LISA Pathfinder mission, which is part of the space-science programme of the European Space Agency.The French contribution has been supported by the CNES (Accord Specific de projet CNES 1316634/CNRS 103747), the CNRS, the Observatoire de Paris and the University Paris-Diderot.E. Plagnol and H. Inchauspé would also like to acknowledge the financial support of the UnivEarthS Labex program at Sorbonne Paris Cité (ANR-10-LABX-0023 and ANR-11-IDEX-0005-02).The Albert-Einstein-Institut acknowledges the support of the German Space Agency, DLR. The work is supported by the Federal Ministry for Economic Affairs and Energy based on a resolution of the German Bundestag (FKZ 50OQ0501 and FKZ 50OQ1601).The Italian contribution has been supported by Agenzia Spaziale Italiana and Istituto Nazionale di Fisica Nucleare.The Spanish contribution has been supported by contracts AYA2010-15709 (MICINN), ESP2013-47637-P, and ESP2015-67234-P (MINECO).M. Nofrarias acknowledges support from Fundacion General CSIC (Programa ComFuturo).F. Rivas acknowledges an FPI contract (MINECO). The Swiss contribution acknowledges the support of the Swiss Space Office (SSO) via the PRODEX Programme of ESA. L. Ferraioli is supported by the Swiss National Science Foundation.The UK groups wish to acknowledge support from the United Kingdom Space Agency (UKSA), the University of Glasgow, the University of Birmingham,Imperial College, and the Scottish Universities Physics Alliance (SUPA). J. I. Thorpe and J. Slutsky acknowledge the support of the US National Aeronautics and Space Administration (NASA).

In many systems consisting of interacting subsystems, the complex interactions between elements can be represented using multilayer networks. However percolation, key to understanding connectivity and robustness, is not trivially generalised to multiple layers. This Element describes a generalisation of percolation to multilayer networks: weak multiplex percolation. A node belongs to a connected component if at least one of its neighbours in each layer is in this component. The authors fully describe the critical phenomena of this process. In two layers with finite second moments of the degree distributions the authors observe an unusual continuous transition with quadratic growth above the threshold. When the second moments diverge, the singularity is determined by the asymptotics of the degree distributions, creating a rich set of critical behaviours. In three or more layers the authors find a discontinuous hybrid transition which persists even in highly heterogeneous degree distributions, becoming continuous only when the powerlaw exponent reaches $1+1/(M-1)$ for $M$ layers.

We introduce a compartmental model SEIAHRV (Susceptible, Exposed, Infected, Asymptomatic, Hospitalized, Recovered, Vaccinated) with age structure for the spread of the SARAS-CoV virus. In order to model current different vaccines we use compartments for individuals vaccinated with one and two doses without vaccine failure and a compartment for vaccinated individual with vaccine failure. The model allows to consider any number of different vaccines with different efficacies and delays between doses. Contacts among age groups are modeled by a contact matrix and the contagion matrix is obtained from a probability of contagion pc per contact. The model uses known epidemiological parameters and the time dependent probability pc is obtained by fitting the model output to the series of deaths in each locality, and reflects non-pharmaceutical interventions. As a benchmark the output of the model is compared to two good quality serological surveys, and applied to study the evolution of the COVID-19 pandemic in the main Brazilian cities with a total population of more than one million. We also discuss with some detail the case of the city of Manaus which raised special attention due to a previous report of We also estimate the attack rate, the total proportion of cases (symptomatic and asymptomatic) with respect to the total population, for all Brazilian states since the beginning of the COVID-19 pandemic. We argue that the model present here is relevant to assessing present policies not only in Brazil but also in any place where good serological surveys are not available. published

We report strong evidence of the importance of contact hubs (or superspreaders) in mitigating the current COVID-19 pandemic. Contact hubs have a much larger number of contacts than the average in the population, and play a key role on the effectiveness of vaccination strategies. By using an age-structures compartmental SEIAHRV (Susceptible, Exposed, Infected symptomatic, Asymptomatic, Hospitalized, Recovered, Vaccinated) model, calibrated from available demographic and COVID-19 incidence, and considering separately those individuals with a much greater number of contacts than the average in the population, we show that carefully choosing who will compose the first group to be vaccinated can impact positively the total death toll and the demand for health services. This is even more relevant in countries with a lack of basic resources for proper vaccination and a significant reduction in social isolation. In order to demonstrate our approach we show the effect of hypothetical vaccination scenarios in two countries of very different scales and mitigation policies, Brazil and Portugal.

We introduce a compartmental model with age structure to study the dynamics of the SARS-COV−2 pandemic. The contagion matrix in the model is given by the product of a probability per contact with a contact matrix explicitly taking into account the contact structure among different age groups. The probability of contagion per contact is considered as time dependent to represent non-pharmaceutical interventions, and is fitted from the time series of deaths. The approach is used to study the evolution of the COVID−19 pandemic in the main Brazilian cities and compared to two good quality serological surveys. We also discuss with some detail the case of the city of Manaus which raised special attention due to a previous report of three-quarters attack rate by the end of 2020. We discuss estimates for Manaus and all Brazilian cities with a total population of more than one million. We also estimate the attack rate with respect to the total population, in each Brazilian state by January, 1 st 2021 and May, 23 2021.

Objetivos: Avaliar sinais e sintomas descritos na literatura como potenciais desencadeadores ou resultantes de bruxismo e a prevalência dos mesmos em alunos do 5.º ano do curso Mestrado Integrado em Medicina Dentária do Instituto Universitário de Ciências da Saúde (IUCS) – CESPU. Avaliar se a idade, género, toma de antidepressivos, dificuldade em dormir à noite, sensibilidade dentária, ansiedade, depressão e stress são fatores associados ao bruxismo, assim como se o bruxismo está associado a sinais e sintomas isolados das disfunções temporomandibulares, lesões de abfração e desgaste dentário nos alunos. Métodos: 211 alunos foram convidados a preencher um inquérito com questões acerca dos sinais e sintomas referidos. As avaliações dos distúrbios psicológicos e da presença dos sinais e sintomas mais comuns das disfunções temporomandibulares foram feitas através da Escala de Ansiedade, Depressão e Stress – 21 e do Questionário Anamnésico de Fonseca, respetivamente. Clinicamente, foram registadas as lesões de abfração e desgaste dentário de acordo com o Índice de Desgaste Dentário de Smith e Knight. Foram considerados bruxómanos prováveis os alunos que apresentassem desgaste oclusal nos pré-molares. Resultados: Nestes alunos, a taxa de prevalência de bruxismo provável foi 57,1% e os fatores associados foram ter idade superior a 28 anos, lesões de abfração e desgaste incisal/oclusal ≥1 segundo o Índice de Desgaste Dentário de Smith e Knight. Conclusões: Os fatores associados ao bruxismo foram idade superior a 28 anos, lesões de abfração e desgaste incisal/oclusal ≥1. O bruxismo foi considerado como fator de risco das lesões de abfração e desgaste incisal/oclusal ≥1.

We studied the impact of field heterogeneity on entrainment in a system of uniformly interacting phase oscillators. Field heterogeneity is shown to induce dynamical heterogeneity in the system. In effect, the heterogeneous field partitions the system into interacting groups of oscillators that feel the same local field strength and phase. Based on numerical and analytical analysis of the explicit dynamical equations derived from the periodically forced Kuramoto model, we found that the heterogeneous field can disrupt entrainment at different field frequencies when compared to the homogeneous field. This transition occurs when the phase- and frequency-locked synchronization between groups of oscillators is broken at a critical field frequency, causing each group to enter a new dynamical state (disrupted state). Strikingly, it is shown that disrupted dynamics can differ between groups., 10 pages, 7 figures

Message-passing theories have proved to be invaluable tools in studying percolation, non-recurrent epidemics and similar dynamical processes on real-world networks. At the heart of the message-passing method is the nonbacktracking matrix whose largest eigenvalue, the corresponding eigenvector, and the closely related nonbacktracking centrality play a central role in determining how the given dynamical model behaves. Here we propose a degree-class-based method to approximate these quantities using a smaller matrix related to the joint degree-degree distribution of neighbouring nodes. Our findings suggest that in most networks degree-degree correlations beyond nearest neighbour are actually not strong, and our first-order description already results in accurate estimates, particularly when message-passing itself is a good approximation to the original model in question, that is when the number of short cycles in the network is sufficiently low. We show that localization of the nonbacktracking centrality is also captured well by our scheme, particularly in large networks. Our method provides an alternative to working with the full nonbacktracking matrix in very large networks where this may not be possible due to memory limitations., 11 pages, 6 figures; supplemental material

Coronavirus disease 2019 (COVID-19) pandemic, a virus infection caused by the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) virus, has impacted all countries of the world, and the main 2021’s challenge is clearly vaccinating the greater number of persons, in the shortest time span, for a maximal reduction in the number of deaths and in the significant economic impacts. Large-scale vaccination aimed to achieve herd immunity poses many logistic and social difficulties [1], with different vaccine candidates and designs [2,3], and vaccination priorities will determine the evolution of the current COVID-19 pandemic. In this paper we explicitly propose an alternative vaccination protocol that can be more effective than those already being deployed, as the ones in the European Union [4] and in the United States [5]. We report strong evidence based on an epidemiological model for the importance of contact hubs (or superspreaders), having a much larger average number of contacts than in the rest of the population [6-11], on the effectiveness of the vaccination strategy. We show that carefully choosing who will be in the first group to be vaccinated can significantly impact on both health services demand and total death toll, by increasing the overall numbers of lives saved and of hospitalizations. We argue that the approach here considered, which does not coincide with current proposals, and given the current conditions with a lack of basic resources for proper vaccination in several countries, and with a significant reduction in mobility and social isolation restrictions, should be considered by all authorities participating in the design of COVID-19 vaccination with the intent of maximising the number of human lives saved.

A complete bifurcation analysis of explicit dynamical equations for the periodically forced Kuramoto model was performed in [L. M. Childs and S. H. Strogatz. Chaos 18 , 043128 (2008)], identifying all bifurcations within the model. We show that the phase diagram predicted by this analysis is incomplete. Our numerical analysis of the equations reveals that the model can also undergo an abrupt phase transition from oscillations to wobbly rotations of the order parameter under increasing field frequency or decreasing field strength. This transition was not revealed by bifurcation analysis because it is not caused by a bifurcation, and can neither be classified as first nor second-order since it does not display critical phenomena characteristic of either transition. We discuss the topological origin of this transition and show that it is determined by a singular point in the order-parameter space., 5 pages, 4 figures

Dependency links in single-layer networks offer a convenient way of modeling nonlocal percolation effects in networked systems where certain pairs of nodes are only able to function together. We study the percolation properties of the weak variant of this model: nodes with dependency neighbours may continue to function if at least one of their dependency neighbours is active. We show that this relaxation of the dependency rule allows for more robust structures and a rich variety of critical phenomena, as percolation is not determined strictly by finite dependency clusters. We study Erd\"os-R\'enyi and random scale-free networks with an underlying Erd\"os-R\'enyi network of dependency links. We identify a special "cusp" point above which the system is always stable, irrespective of the density of dependency links. We find continuous and discontinuous hybrid percolation transitions, separated by a tricritical point for Erd\"os-R\'enyi networks. For scale-free networks with a finite degree cutoff we observe the appearance of a critical point and corresponding double transitions in a certain range of the degree distribution exponent. We show that at a special point in the parameter space, where the critical point emerges, the giant viable cluster has the unusual critical singularity $S-S_c \propto (p-p_c)^{1/4}$. We study the robustness of networks where connectivity degrees and dependency degrees are correlated and find that scale-free networks are able to retain their high resilience for strong enough positive correlation, i.e., when hubs are protected by greater redundancy., Comment: 18 pages, 10 figures

Compression, filtering, and cryptography, as well as the sampling of complex systems, can be seen as processing information. A large initial configuration or input space is nontrivially mapped to a smaller set of output or final states. We explored the statistics of filtering of simple patterns on a number of deterministic and random graphs as a tractable example of such information processing in complex systems. In this problem, multiple inputs map to the same output, and the statistics of filtering is represented by the distribution of this degeneracy. For a few simple filter patterns on a ring, we obtained an exact solution of the problem and numerically described more difficult filter setups. For each of the filter patterns and networks, we found three key numbers that essentially describe the statistics of filtering and compared them for different networks. Our results for networks with diverse architectures are essentially determined by two factors: whether the graphs structure is deterministic or random and the vertex degree. We find that filtering in random graphs produces much richer statistics than in deterministic graphs, reflecting the greater complexity of such graphs. Increasing the graph&rsquo, s degree reduces this statistical richness, while being at its maximum at the smallest degree not equal to two. A filter pattern with a strong dependence on the neighbourhood of a node is much more sensitive to these effects.

We describe the critical behavior of weak multiplex percolation, a generalization of percolation to multiplex or interdependent networks. A node can determine its active or inactive status simply by referencing neighboring nodes. This is not the case for the more commonly studied generalization of percolation to multiplex networks, the mutually connected clusters, which requires an interconnecting path within each layer between any two vertices in the giant mutually connected component. We study the emergence of a giant connected component of active nodes under the weak percolation rule, finding several non-typical phenomena. In two layers, the giant component emerges with a continuos phase transition, but with quadratic growth above the critical threshold. In three or more layers, a discontinuous hybrid transition occurs, similar to that found in the giant mutually connected component. In networks with asymptotically powerlaw degree distributions, defined by the decay exponent $\gamma$, the discontinuity vanishes but at $\gamma=1.5$ in three layers, more generally at $\gamma = 1+ 1/(M-1)$ in $M$ layers., Comment: 11 pages, 6 figures