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Understanding diffusive processes in networks is a significant challenge in complexity science. Networks possess a diffusive potential that depends on their topological configuration, but diffusion also relies on the process and initial conditions. This article presents Diffusion Capacity, a concept that measures a node’s potential to diffuse information based on a distance distribution that considers both geodesic and weighted shortest paths and dynamical features of the diffusion process. Diffusion Capacity thoroughly describes the role of individual nodes during a diffusion process and can identify structural modifications that may improve diffusion mechanisms. The article defines Diffusion Capacity for interconnected networks and introduces Relative Gain, which compares the performance of a node in a single structure versus an interconnected one. The method applies to a global climate network constructed from surface air temperature data, revealing a significant change in diffusion capacity around the year 2000, suggesting a loss of the planet’s diffusion capacity that could contribute to the emergence of more frequent climatic events. Research partially supported by Brazilian agencies FAPEMIG, CAPES, and CNPq. P.M.P. acknowledges support from the “Paul and Heidi Brown Preeminent Professorship in ISE, University of Florida”, and RSF 14-41- 00039, Humboldt Research Award (Germany) and LATNA, Higher School of Economics, RF. C.M. acknowledges partial support from Spanish MINECO (PID2021-123994NB-C21) and ICREA ACADEMIA. A.D.- G. knowledges support from the Spanish grants PGC2018-094754-BC22 and PID2021-128005NB-C22, funded by MCIN/AEI/ 10.13039/ 501100011033 and “ERDF A way of making Europe”; and from Generalitat de Catalunya (2021SGR00856). M.G.R acknowledges partial support from FUNDEP.

From grain boundaries and heterojunctions to manipulating 2D materials, solid-solid interfaces play a key role in many technological applications. Understanding and predicting properties of these complex systems present an ongoing and increasingly important challenge. Over the last few decades computer simulation of interfaces has become vastly more powerful and sophisticated. However, theoretical interface screening remains based on largely heuristic methods and is strongly biased to systems that are amenable to modelling within constrained periodic cell approaches. Here we present an unconstrained and generally applicable non-periodic screening approach for systematic exploration of material’s interfaces based on extracting and aligning disks from periodic reference slabs. Our disk interface method directly and accurately describes how interface structure and energetic stability depends on arbitrary relative displacements and twist angles of two interacting surfaces. The resultant detailed and comprehensive energetic stability maps provide a global perspective for understanding and designing interfaces. We confirm the power and utility of our method with respect to the catalytically important TiO2 anatase (101)/(001) and TiO2 anatase (101)/rutile (110) interfaces.

Treballs Finals de Grau de Física, Facultat de Física, Universitat de Barcelona, Curs: 2022, Tutor: Jordi Soriano Fradera, Human brains are often modelled based on primary cultures from rodents, mostly rats, which do not replicate well its complexity. Here, the advantages of neuronal cultures derived from human induced pluripotent stem cells (hiPSCs) are studied and compared with rat and human primary cultures. Data is obtained from spontaneous activity recordings using calcium fluorescence imaging, and analysed in the context of complex networks to obtain the functional connectivity traits of the different conditions. hiPSCs cultures showed rich activity patterns and functional features. Ratcultures, by contrast, had a very rigid activity and poor functional traits. Human primary cultures had a similar behaviour as hiPSCs. Our study shows that hiPSC technology is an excellent tool to model human–like networks in vitro, which may help to better model alterations caused by damage or neurodegeneration

Treballs Finals de Grau de Física, Facultat de Física, Universitat de Barcelona, Curs: 2022, Tutor: Jordi Soriano Fradera Human brains are often modelled based on primary cultures from rodents, mostly rats, which do not replicate well its complexity. Here, the advantages of neuronal cultures derived from human induced pluripotent stem cells (hiPSCs) are studied and compared with rat and human primary cultures. Data is obtained from spontaneous activity recordings using calcium fluorescence imaging, and analysed in the context of complex networks to obtain the functional connectivity traits of the different conditions. hiPSCs cultures showed rich activity patterns and functional features. Ratcultures, by contrast, had a very rigid activity and poor functional traits. Human primary cultures had a similar behaviour as hiPSCs. Our study shows that hiPSC technology is an excellent tool to model human–like networks in vitro, which may help to better model alterations caused by damage or neurodegeneration

The idea of Terraformation comes from the science fiction literature, where humans have the capability of changing a non-habitable planet to an Earth-like one. Nowadays, Nature is changing rapidly, the poles are melting, oceans biodiversity is vanishing due to plastic pollution, and the deserts are advancing at an unstoppable rhythm. This thesis is a first step towards the exploration of new strategies that could serve to change this pernicious tendencies jeopardising ecosystems. We suggest it may not only be possible by adding new species (alien species), but also engineering autochthonous microbial species that are already adapted to the environment. Such engineering may improve their functions and capabilities allowing them to recover the (host) ecosystem upon their re-introduction. These new functionalities should make the microbes be able to induce a bottom-up change in the ecosystem: from the micro-scale (microenvironment) to the macro-scale (even changing the composition of species in the entire the ecosystem). To make this possible, the so-called Terraformation strategy needs to fuse many different fields of knowledge. The focus of this thesis relies on studying the outcome of the interactions between species and their environment (Ecology), on making the desired modifications by means of genetic engineering of the wild-type species (Synthetic Biology), and on monitoring the evaluation of the current ecosystems’ states, testing the possible changes, and predicting the future development of possible interventions (Dynamical Systems). In order to do so, in this thesis, we have gathered the tools provided by these different fields of knowledge. The methodology is based on loops between observation, designing, and prediction. For example, if there is a lack of humidity in semiarid ecosystems, we then propose to engineer e.g. Nostoc sp. to enhace its capability to produce extracellular matrix (increasing water retention). With this framework, we perform a model to understand the different possible dynamics, by means of dynamical equations to evaluate e.g. when a synthetic strain will remain in the ecosystem and the effects it will produce. We have also studied spatial models to predict their ability to modify the spatial organization of vegetation. Transient dynamics depend on the kind of transition underlying the occurring tipping point. For this reason, we have studied different systems: vegetation dynamics with facilitation (typical from drylands), a cooperator-parasite system, and a trophic chain model where different human interventions can be tested (i.e. hunting, deforestation, soil degradation, habitat destruction). All of these systems are shown to promote different types of transitions (i.e. smooth and catastrophic transitions). Each transition has its own dynamical fingerprint and thus knowing them can help monitoring and anticipating these transitions even before they happen, taking advantage of the so-called early warning signals. In this travel, we have found that transients can be an important phenomena in the current changing world. The ecosystems that we observe can be trapped into a seemingly stable regime, but be indeed in an unstable situation driving to a future sudden collapse (Fig 1) For this reason, we need to investigate novel intervention methods able to sustain the current ecosystems, for instance: Terraformation.

Treballs Finals de Màster en Física dels Sistemes Complexos i Biofísica, Facultat de Física, Universitat de Barcelona. Curs: 2021-2022. Tutora: Luce Prignano, Network theory has demonstrated to be an invaluable tool for studying complex systems, and it has been widely used in many disciplines. Archaeology is no exception. Ancient transportation infrastructures (TIs) are often represented as networks of interacting settlements. However, these various TIs are different in nature, and the network they form requires a different approach. Here we show that ancient TIs as a whole can be characterized by a multiplex. After an analysis of the dynamic performance and structural properties of the individual road and river networks of Southern Etruria and Latium Vetus in the Iron Age (950 − 580 BC), we overlap both TIs by regarding them as layers in a multiplex. We then study the multiplex with the aim to assess how good the interplay between layers is. Our work thus serves as a case study of an empirical multiplex network. We prove that the tools we use are good to characterize ancient TIs as one multiplex and could become very powerful if further studied and refined.

[eng] Complex systems, which involve a massive ammount of components interacting in nontrivial ways and exhibiting properties such as emergence, feedback-loops and self-organization, permeate our world to an unthinkable extent. In order to understand such pervasive systems in nature, science, technology and business one has to abandon unsuitable reductionist approches and develop a deep understanding of the networks of interactions that sustain them. In this respect, this thesis exploits a key discovery provided by network science: the architecture of networks arising in completely different domains are in fact remarkably similar to each other, and thus can be modeled using the same set of mathematical tools. This thesis presents a geometric approach to the study of connectivity patterns and dynamical phenomena in real-world complex networks. The main hypothesis of network geometry states that the architecture of real complex networks has a geometric nature. This has been confirmed during the last decade, with a great number of networks from different domains being successfully mapped into a (hidden) metric space with hyperbolic geometry. Maps have served throughout history as a precise and relevant source of information, at the center of political, economic, and geostrategic decisions. In this thesis, geometric maps of networks play a central role and constitute the underlying thread that weaves the various chapters together. In particular, the work presented in this thesis exploits the current explosion in computing power and the ability of geometric maps to abstract complex topologies in order to unravel fundamental principles underlying the organization and function of real networks. This thesis starts by introducing complex systems and their mathematical description in terms of networks. In the Preliminary Methods section, we review the S1 and H2 models, which provide a geometric explanation of networks, and discuss the embedding methods used to produce geometric maps. The next four chapters contain the results of our investigations about the structure of and the dynamics on real complex networks. These are organized so that each chapter covers a different topic, while the focus shifts from structure to dynamics traversing through the combination of both. First, we concentrate on structure and devote to the diagnosis of communities through the development of a specialized geometric configuration model, which provides an alternative to the S1 model for generating random versions of geometric networks. Second, we advance our knowledge on structure by providing a thorough analysis of the hierarchical organization of networks while uncovering mechanisms that favor cooperation in social dilemmas susceptible to stratified order. In chapter three, we further enmesh structure and function by coupling opinion dynamics on a network with its metric structure in order to unravel the multiscale anatomy of opinion formation processes in real systems. In chapter four we fully shift our attention to the dynamics of networks and explore the navigability of time-evolving graphs in hyperbolic space. Finally, we present our conclusions followed by a list of publications related to this thesis. Along the several chapters of this thesis we discover and develop a range of applications of the S1 / H2 formalism which inform different important and still obscure structural and dynamical features of real networks. Taken together, our findings have important implications for the design and evaluation of efficient routing protocols, for the predictability of opinion formation dynamics, and even for the control and detection of hierarchical power structures. This thesis demonstrates that the future of the geometric approach to network science remains full of opportunities, and the many insights yet to uncover are only bounded by the imagination of those who venture to exploit the geometric paradigm., [cat] Els sistemes complexos són aquells que involucren un nombre elevat de components que interaccionen de manera no trivial. Aquesta tesis comença introduïnt els sistemes complexos i la seva descripció matemàtica en termes de xarxes complexes. Concretament, s’introdueix el paradigma geomètric, que esdevé el punt de referència des del qual es tracten diversos punts importants de l’estructura i la dinàmica de les xarxes complexes reals. A la secció de mètodes es discuteixen els models geomètrics S(1) i H(2), que descriuen les xarxes en termes d’espais mètric subjacents. També s’expliquen les tècniques utilitzades per a crear mapes geomètrics en l’espai hiperbòlic que representen les xarxes reals. Els següents quatre capítols inclouen els resultats de les nostres investigacions. Cada capítol tracta un tema concret que lliga amb el següent de manera que el centre d’atenció passa de l’estructura a la dinàmica. Primerament ens centrem en el diagnòstic de l’estructura de comunitats en xarxes heterogènies. Ho fem mitjançant el desenvolupament d’un model especialitzat de configuració que permet generar rèpliques fidedignes de les xarxes complexes descomptant exclusivament les comunitats de manera geomètrica. Posteriorment, avancem en el coneixement de l’estructura de les xarxes analitzant la seva organització jeràrquica a través d’una nova mesura geomètrica de jerarquia. Aquesta mesura permet descobrir l’ordre jeràrquic de xarxes no direccionals. També revela un mecanisme de filtrat que permet descobrir els grafs que promouen la cooperació entre individus quan apliquen dinàmiques de dilemes socials a les xarxes. Al tercer capítol, estructura i dinàmica es barregen completament per donar lloc a un model de formació d’opinió multiescala que ens permet estudiar com les diferents granularitats a nivell social influencien el procés de consens en sistemes connexos. Finalment, el darrer capítol es focalitza en la dinàmica per complet i s’investiga la navegació efficient de xarxes temporals utilitzant l’espai hiperbòlic.

[eng] Most of the real-world complex systems are best described as complex networks and can be mathematically described as oscillatory systems, coupled with the neighbours through the connections of the network. The flashing of fireflies, the neuronal brain signals or the energy flow through the power grid are some examples. Yoshiki Kuramoto came up with a tractable mathematical model that could capture the phenomenology of collective synchronization by suggesting that oscillators were coupled by a sinusoidal function of their phase differences. Later, Yoshiki Kuramoto together with Hidetsugu Sakaguchi presented a generalization of the previous limit-cycle set of oscillators Kuramoto’s model which incorporated a constant phase lag between oscillators. Subsequent studies of the model included the network structure within the model together with the global shift. For a wide range of the phase lag values, the system becomes synchronized to a resulting frequency, i.e., the dynamics reaches a stationary state. In the original work of Kuramoto and Sakaguchi and in most of the consequent later studies, a uniform distribution of phase lag parameters is customarily assumed. However, the intrinsic properties of nodes – that assuredly represent the constituents of real systems – do not need be identical but distributed non-homogeneously among the population. This thesis contributes to the understanding of the Kuramoto-Sakaguchi model with a generalization for nonhomogeneous phase lag parameter distribution. Considering different scenarios concerning the distribution of the frustration parameter among the oscillators represents a major step towards the extension of the original model and provides significant novel insights into the structure and function of the considered network. The first setting that the present thesis considers consists in perturbing the stationary state of the system by introducing a non-zero phase lag shift into the dynamics of a single node. The aim of this work is to sort the nodes by their potential effect on the whole network when a change on their individual dynamics spreads over the entire oscillatory system by disrupting the otherwise synchronized state. In particular, we define functionability, a novel centrality measure that addresses the question of which are the nodes that, when individually perturbed, are best able to move the system away from the fully synchronized state. This issue may be relevant for the identification of critical nodes that are either beneficial – by enabling access to a broader spectrum of states – or harmful – by destroying the overall synchronization. The second scenario that the present thesis addresses considers a more general configuration in which the phase lag parameter is an intrinsic property of each node, not necessarily zero, and hence exploring the potential heterogeneity of the frustration among oscillators. We obtain the analytical solution of frustration parameters so as to achieve any phase configuration, by linearizing the most general model. We also address the fact that the question ’among all the possible solutions, which is the one that makes the system achieve a particular phase configuration with the minimum required cost?’ is of particular relevance when we consider the plausible real nature of the system. Finally, the homogenous distribution of phase lag parameters is revisited in the last scenario. As studied in the literature, a certain degree of symmetry is an attribute of real-world networks. Nevertheless, beyond structural or topological symmetry, one should consider the fact that real- world networks are exposed to fluctuations or errors, as well as mistaken insertions or removals. In the present thesis, we provide an alternative notion to approximate symmetries, which we call ‘Quasi-Symmetries’ and are defined such that they remain free to impose any invariance of a particular network property and are obtained from the stationary state of the Kuramoto-Sakaguchi model with a homogeneous phase lag distribution. A first contribution is exploring the distributions of structural similarity among all pairs of nodes. Secondly, we define the ‘dual network’, a weighted network –and its corresponding binarized counterpart– that effectively encloses all the information of quasi-symmetries in the original one., [cat] La major part dels sistemes complexos presents en la natura i la societat es poden descriure com a xarxes complexes. Molts d’aquests sistemes es poden modelitzar matemàticament com un sistema oscil·latori, on les unitats queden acoblades amb els components veïns a través de les connexions de la xarxa. Yoshiki Kuramoto i Hidetsugu Sakaguchi van presentar la generalització del ben conegut model d’oscil·ladors de Kuramoto, on s’incorporava un terme de desfasament entre parelles d’oscil·ladors. Aquesta tesi contribueix en la comprensió d’aquest model, tot considerant una distribució no homogènia d’aquest paràmetre de desfasament o frustració. S’han considerat tres escenaris diferents, tots ells donant lloc a resultats que permeten una millor descripció de l’estructura i funció de la xarxa que s’està considerant. Una primera configuració consisteix en pertorbar l’estat estacionari tot introduint un desfasament en la dinàmica d’un node de manera aïllada. Seguidament, definim la funcionabilitat, una mesura de centralitat única que respon a la pregunta de, quins nodes, quan són pertorbats individualment, són més capaços d’allunyar el sistema de l’estat sincronitzat. Aquest fet podria suposar un comportament beneficiós o perjudicial per sistemes reals. El segon escenari considera la configuració més flexible, explorant la potencial heterogeneïtat dels paràmetres de frustració dels diferents nodes. Obtenim la solució analítica d’aquesta distribució per tal d’assolir qualsevol configuració de les fases dels oscil·ladors, a través de la linearització del model. També contestem a la pregunta: “de totes les possibles solucions, quina és la que implica un menor cost per tal d’assolir una configuració en particular?”. Finalment, en l’últim escenari, proporcionem una definició alternativa al concepte de simetria aproximada d’una xarxa, i que anomenem “Quasi simetries”. Aquestes són definides sense imposar invariàncies en les propietats del sistema, sinó que s’obtenen de l’estat estacionari del model de Kuramoto-Sakaguchi model, tot considerant una distribució homogènia dels paràmetres de frustració.

The 2021 Nobel Prize in Physics recognized the fundamental role of complex systems in the natural sciences. In order to celebrate this milestone, this editorial presents the point of view of the editorial board of JPhys Complexity on the achievements, challenges, and future prospects of the field. To distinguish the voice and the opinion of each editor, this editorial consists of a series of editor perspectives and reflections on few selected themes. A comprehensive and multi-faceted view of the field of complexity science emerges. We hope and trust that this open discussion will be of inspiration for future research on complex systems. The authors acknowledge their funding sources, including: Royal Society (IEC\NSFC\191147 (G Bianconi); NSF Grants CCF-1839232, PHY-1806372, DGE-2125899, PHY-2210566 (L.C.); NRF, Grant No. NRF-2014R1A3A2069005 and the KENTECH Research Grant (KRG2021-01-007) (B Kahng); EU H2020 ICT48 project ‘Humane AI Net’ under Contract #952026 and EU Horizon 2020—ERC Synergy Grant 810115 ‘Dynasnet’ (J Kertesz); ICREA Academia Award, Catalan Institution for Research and Advanced Studies (C Masoller); National Science Foundation Award No. DMS-1647351 (A E Motter); Slovenian Research Agency (Grant Nos. P1-0403 and J1-2457) (M Perc); PID2019-106811GB-C31 from MCIN/AEI/10.13039/501100011033 (M Sales Pardo); PACSS (RTI2018-093732-B-C21) and MDM-2017-0711 from MCIN/AEI/10.13039/501100011033, Spain (M San Miguel). FR acknowledges support by the Army Research Office (W911NF-21-1-0194) and by the Air Force Office of Scientific Research (FA9550-21-1-0446). Peer Reviewed Article signat per 19 autors/es : Ginestra Bianconi, Alex Arenas, Jacob Biamonte, Lincoln D Carr, Byungnam Kahng, Janos Kertesz, Jürgen Kurths, Linyuan Lu, Cristina Masoller, Adilson E Motter Matjaz Perc, Filippo Radicchi, Ramakrishna Ramaswamy, Francisco A Rodrigues, Marta Sales-Pardo, Maxi San Miguel, Stefan Thurner and Taha Yasseri.

We investigate opinion diffusion on complex networks and the interplay between the existence of neutral opinion states and non-trivial network structures. For this purpose, we apply a three-state opinion model based on magnetic-like interactions to modular complex networks, both synthetic and real networks extracted from Twitter. The model allows for tuning the contribution of neutral agents using a neutrality parameter. We also consider social agitation, encoded as a temperature, that accounts for random opinion changes that are beyond the agent neighborhood opinion state. Using this model, we study which topological features influence the formation of consensus, bipartidism, or fragmentation of opinions in three parties, and how the neutrality parameter and the temperature interplay with the network structure.

The design of dissymmetric organic ligands featuring combinations of 1,3-diketone and 2,6-diacetylpyridine coordination pockets has been exploited to produce dinuclear and trinuclear lanthanide-based coordination compounds. These molecules exhibit two or more non-equivalent Ln ions, most remarkably enabling the access to well-defined heterolanthanide compositions. The site-selective disposition of each metal ion within the molecular entities allows the study of each centre individually as a spin-based quantum bit, affording unparalleled versatility for quantum gate design. The inherent weak interaction between the Ln ions permits the performance of multi-qubit quantum logical operations realized through their derived magnetic states, or implementing quantum-error correction protocols. The different studies performed to date on these systems are revised, showing their vast potential within spin-based quantum information processing., The authors thank the European Union's Horizon 2020 research and innovation programmes, the QUANTERA project SUMO, the FET-OPEN FATMOLS (grant 862893; GA and OR), the Spanish MICINN (grants PGC2018-098630-B-I00 (GA and DA) and MAT2017-86826-R (OR)) and the Generalitat de Catalunya for the ICREA Academia 2018 Prize (GA).

La regla 110 dels Autòmats Cel·lulars Elementals (Elementary Cellular Automata, ECA) és especialment interessant ja que és capaç de simular una Màquina de Turing Universal (UTM), donada una codificació d'aquesta en la seva condició inicial. En aquest sentit l'ECA(110), aquest sistema dinàmic tan senzill, és capaç de computació universal. En aquest treball volem investigar si aquests autòmats ECA(110) són capaços de servir com a reservoir en sistemes de reservoir computing dedicats a problemes de classificació, posant èmfasi en el benchmark estàndard MNIST. En aquest treball es portarà a terme una investigació sobre l’ús de la regla 110 dels autòmats cel·lulars elementals com a reservori en problemes de classificació. Actualment hi ha molts sistemes que intenten millorar o provar la classificació d’imatges i en el nostre treball volem centrar-nos en una nova línia d’investigació per analitzar els resultats.

Systems composed of reactive particles diffusing in a network display emergent dynamics. While Fick’s diffusion can lead to Turing patterns, other diffusion schemes might display more complex phenomena. Here we study the death and restoration of collective oscillations in networks of oscillators coupled by random-walk diffusion, which modifies both the original unstable fixed point and the stable limit-cycle, making them topology-dependent. By means of numerical simulations we show that, in some cases, the diffusion-induced heterogeneity stabilizes the initially unstable fixed point via a Hopf bifurcation. Further increasing the coupling strength can moreover restore the oscillations. A numerical stability analysis indicates that this phenomenology corresponds to a case of amplitude death, where the inhomogeneous stabilized solution arises from the interplay of random walk diffusion and heterogeneous topology. Our results are relevant in the fields of epidemic spreading or ecological dispersion, where random walk diffusion is more prevalent. Since Turing’s seminal work, combining local reaction with diffusion is key to understand how patterns can appear in interacting systems. Here, the authors investigate the effect of coupling local dynamics with a non-traditional type of diffusion mechanism, random-walk diffusion, showing that this simple modification can give rise to previously unobserved amplitude death and restoration of collective oscillations

Treballs finals del Màster de Fonaments de Ciència de Dades, Facultat de matemàtiques, Universitat de Barcelona. Curs: 2020-2021. Tutor: Albert Díaz Guilera i Oriol Pujol Vila, [en] The aim of this thesis is to primarily learn to predict trajectories of physical systems by using Machine Learning. We have used the Interaction Network as a base model and introduced tweaks to its structure in the framework of Graph Networks in order to deal with systems without interaction, with force-based interactions and systems governed by the Vicsek model. We have been able to replicate very well systems without interaction and systems governed by a Vicsek model of infinite reach. However the results with force-based systems are mediocre because they need more trainable parameters and training data. The results for the Vicsek model with a finite radius of reach are the worst but we have learned the necessity and the methodology of introducing attention to the base model to deal with this class of problem.

[eng] Epidemic modeling has proven to be an essential framework for the study of contagion phenomena in biological, social, and technical systems. Albeit epidemic models have evolved into powerful predictive tools, most assume memoryless agents and independent transmission channels. Nevertheless, many real-life examples are manifestly time-sensitive and show strong correlations. Moreover, recent trends in agent-based modeling support a generalized shift from edge-based descriptions toward node-centric approaches. Here I develop an infection mechanism that is endowed with memory of past exposures and simultaneously incorporates the joint effect of multiple infectious sources. A notion of social reinforcement/inhibition arises organically, without being incorporated explicitly into the model. As a result, the concepts of non-Markovian dynamics and complex contagion become intrinsically coupled. I derive mean-field approximations for random degree-regular networks and perform extensive stochastic simulations for nonhomogeneous networks. The analysis of the SIS model reveals a sophisticated interplay between two memory modes, displayed by a collective memory loss and the dislocation of the critical point into two phase transitions. An intermediate region emerges where the system is either excitable or bistable, exhibiting fundamentally distinct behaviors compared to the customary healthy and endemic phases. Additionally, the transition to the endemic phase becomes hybrid, showing both continuous and discontinuous properties. These results provide renewed insights on the interaction between microscopic mechanisms and topological aspects of the underlying contact networks, and their joint effect on the properties of spreading processes. In particular, this type of modeling approach that combines memory effects and complex contagion could be suitable to describe ecological interactions between biological and social pathogens., [cat] El modelatge epidèmic ha demostrat ser un marc essencial per a l’estudi dels fenòmens de contagi en sistemes biològics, socials i tècnics. Tot i que els models epidèmics han evolucionat cap a potents eines de predicció, la majoria assumeixen agents sense memòria i canals de transmissió independents. No obstant això, molts exemples de la vida real mostren fortes correlacions temporals i estructurals. A més, les tendències recents en la modelització basada en agents donen suport a un canvi generalitzat de les descripcions basades en els enllaços cap a enfocaments on els nodes són centrals. Aquí desenvolupo un mecanisme d’infecció dotat de memòria a exposicions passades i que simultàniament incorpora l’efecte conjunt de múltiples fonts infeccioses. Una noció de reforç/inhibició social sorgeix de manera orgànica, sense incorporar-se explícitament al model. Com a resultat, els conceptes de dinàmica no markoviana i contagi complex s’acoblen intrínsecament. Derivo aproximacions de camp mitjà per a xarxes aleatòries de grau fix i realitzo extenses simulacions estocàstiques per a xarxes no homogènies. L'anàlisi del model SIS revela una interacció sofisticada entre dos modes de memòria, que es manifesta mitjançant una pèrdua de memòria col·lectiva i la dislocació del punt crític en dues transicions de fase. Apareix una regió intermitja on el sistema és excitable o bistable, amb comportaments fonamentalment diferents en comparació amb les fases sanes i endèmiques habituals. A més, la transició a la fase endèmica esdevé híbrida, mostrant propietats contínues i també discontínues. Aquests resultats proporcionen una visió renovada sobre la interacció entre mecanismes microscòpics i aspectes topològics de les xarxes de contacte subjacents, i el seu efecte conjunt sobre les propietats dels processos de propagació. En particular, aquest tipus de modelització que combina efectes de memòria i contagi complex podria ser adequat per descriure interaccions ecològiques entre patògens biològics i socials., [spa] El modelado epidémico ha demostrado ser un marco esencial para el estudio de los fenómenos de contagio en sistemas biológicos, sociales y técnicos. Aunque los modelos epidémicos han evolucionado hacia potentes herramientas de predicción, la mayoría asumen agentes sin memoria y canales de transmisión independientes. Sin embargo, muchos ejemplos de la vida real muestran fuertes correlaciones temporales y estructurales. Además, las tendencias recientes en la modelización basada en agentes apoyan un cambio generalizado de las descripciones basadas en los enlaces hacia enfoques donde los nodos son centrales. Aquí desarrollo un mecanismo de infección dotado de memoria a exposiciones pasadas y que simultáneamente incorpora el efecto conjunto de múltiples fuentes infecciosas. Una noción de refuerzo/inhibición social surge de manera orgánica, sin incorporarse explícitamente al modelo. Como resultado, los conceptos de dinámica no Markoviana y contagio complejo se acoplan intrínsecamente. Derivo aproximaciones de campo medio para redes aleatorias de grado fijo y realizo extensas simulaciones estocásticas para redes no homogéneas. El análisis del modelo SIS revela una interacción sofisticada entre dos modos de memoria, que se manifiesta mediante una pérdida de memoria colectiva y la dislocación del punto crítico en dos transiciones de fase. Aparece una región intermedia donde el sistema es excitable o bistable, con comportamientos fundamentalmente diferentes en comparación con las fases sanas y endémicas habituales. Además, la transición a la fase endémica se convierte en híbrida, mostrando propiedades continuas y también discontinuas. Estos resultados proporcionan una visión renovada sobre la interacción entre mecanismos microscópicos y aspectos topológicos de las redes de contacto subyacentes, y su efecto conjunto sobre las propiedades de los procesos de propagación. En particular, este tipo de modelización que combina efectos de memoria y contagio complejo podría ser adecuado para describir interacciones ecológicas entre patógenos biológicos y sociales.

La ciència de l'esport evoluciona però segueix ancorada en supòsits tàcits, creences infundades i teories poc actualitzades o de qüestionable validesa. Amb un plantejament eminentment deductiu, i d'acord amb les ciències de la complexitat, aquesta tesi es proposa transformar aquests supòsits teòrics i pràctics, i oferir noves hipòtesis d'investigació aplicades a l'entrenament esportiu. En primer lloc, es suggereixen una sèrie d'intervencions orientades a la formació d'entrenadores i entrenadors, així com al desenvolupament de les seves competències per actuar en entorns complexos. En segon lloc, s'ofereix una nova comprensió de l’esportista / equip, de les seves propietats relacionades amb el rendiment esportiu, i dels processos d'entrenament i lesió des de la perspectiva dels sistemes dinàmics complexos. Es proposa per primera vegada: a) una sistematització dels principis teòrics i metodològics de l’entrenament des de la complexitat, b) una redefinició dels constrenyiments de la tasca i de la seva classificació, c) la propietat de nidificació temporal i causalitat circular dels constrenyiments per millorar la comprensió i eficàcia del procés d’entrenament i trencar amb el seu reduccionisme, d) explicar els mecanismes de lesió esportiva en base a la teoria de percolació i e) les mesures de connectivitat com a variables d’estudi en la prevenció de lesions. La ciencia del deporte evoluciona pero sigue anclada en supuestos tácitos, creencias infundadas y teorías poco actualizadas o de cuestionable validez. Con un planteamiento eminentemente deductivo, y en base a las ciencias de la complejidad, esta tesis se propone ofrecer nuevos supuestos teóricos, prácticos, e hipótesis de investigación aplicadas al entrenamiento deportivo. En primer lugar, se sugieren una serie de intervenciones orientadas a la formación de entrenadoras y entrenadores, así como al desarrollo de sus competencias para actuar en entornos complejos. En segundo lugar, se ofrece una comprensión del deportista / equipo, de sus propiedades relacionadas con el rendimiento, y de los procesos de entrenamiento y lesión desde la perspectiva de los sistemas dinámicos complejos. Se proponen por primera vez a) una sistematización de los principios teóricos y metodológicos orientados al entrenamiento desde la complejidad, b) una redefinición de los constreñimientos de la tarea y de su clasificación, c) incorporar la propiedad de anidamiento temporal y causalidad circular de los constreñimientos para mejorar la comprensión y eficacia del proceso de entrenamiento, d) explicar los mecanismos de lesión deportiva en base a la teoría de percolación, y e) las medidas de conectividad como variables de estudio en la prevención de lesiones. Sport science evolves but prejudices and tacit assumptions in the field favour the presence of pseudo-science in education and professional practice. With an eminently deductive approach, and based on the complexity science, this thesis aims to transform these assumptions offering new theoretical, practical principles and new research hypotheses applied to sports training. First, we suggest a series of interventions aimed at training coaches and trainers and developing their skills to act in complex environments. Second, an understanding of the athlete / team, its performance - related properties, and training and injury processes is offered from the perspective of complex dynamic systems. For the first time is proposed, a) a systematization of theoretical and methodological principles oriented to training from complexity science, b) a redefinition of task constraints and its classification c) to incorporate the property of temporal nesting and circular causality of the constraints to improve the understanding and effectiveness of the training process, d) to explain the mechanisms of sports injury based on the percolation theory, and e) to use connectivity measures as study variables in injury prevention.

Treballs Finals de Grau de Física, Facultat de Física, Universitat de Barcelona, Curs: 2021, Tutor: Jordi Soriano Fradera, Attacks and random failures are powerful damaging tools to know the robustness of a network. Here, a damage analysis was implemented in two different systems. In a first system, we characterized the different node centrality measures for a dolphins’ social network. The results revealed that those nodes with a higher degree have a higher contribution to the exchanging infor-mation capacity of the network than those with higher betweenness centrality. In a second system, attacks and random failures were used in the functional network of two neuronal cultures, one with a homogeneous distribution of neurons and another with an anisotropic, patterned structure. Results showed that both networks were resilient to random failures, but that the patterned network was sensitive to attacks. This suggests that the pattern network is fragiler and more dependent of nodes with high centrality

El diseño de sistemas urbanos sostenibles es un objetivo de la mayoría de las grandes ciudades. El crecimiento demográfico y los nuevos modelos de consumo han afectado al modo en que nos relacionamos con los ecosistemas urbanos. En este contexto, se han propuesto diferentes medidas que reduzcan los efectos derivados del tráfico de vehículos tanto a nivel económico como sanitario provocadas por la polución. Ante esta necesidad y la complejidad existente en este tipo de sistemas, surge la ciencia de las ciudades como una disciplina cuyo objetivo es entender la estructura y comportamiento de los sistemas urbanos. En este Trabajo de Fin de Máster se busca desarrollar un algoritmo que logre establecer qué puntos dentro de una ciudad presentan un mayor riesgo de congestión a partir de asunciones que permitan modelar de manera más realista el comportamiento de los usuarios de las redes de transporte. Para ello se ha modificado la definición y cálculo de la centralidad de intermediación con el objetivo de que este considere rutas subóptimas próximas a la mínima como rutas aceptables. Finalmente, se ha estudiado cómo esta nueva medida de centralidad se comporta aplicada sobre redes de transporte reales. The design of sustainable urban systems is a goal of most large cities. Population growth and new consumption patterns have affected the way we relate to urban ecosystems. In this context, different measures have been proposed to reduce the economic and health effects of vehicle traffic caused by pollution. Given this need and the existing complexity in this type of system, the science of cities arises as a discipline whose objective is to understand the structure and behavior of urban systems. In this Master's Thesis, the aim is to develop an algorithm to establish which points within a city present a higher risk of congestion based on assumptions that allow more realistic modeling of the behavior of users of transport networks. To this end, the definition and calculation of the betweenness centrality have been modified so that it considers suboptimal routes close to the minimum as acceptable routes. Finally, it has been studied how this new centrality measure behaves when applied to real transport networks. El disseny de sistemes urbans sostenibles és un objectiu de la majoria de les grans ciutats. El creixement demogràfic i els nous models de consum han afectat la manera en què ens relacionem amb els ecosistemes urbans. En aquest context, s'han proposat diferents mesures que redueixin els efectes derivats del trànsit de vehicles tant a nivell econòmic com sanitari provocades per la pol·lució. Davant aquesta necessitat i la complexitat existent en aquesta mena de sistemes, sorgeix la ciència de les ciutats com una disciplina l'objectiu de la qual és entendre l'estructura i comportament dels sistemes urbans. En aquest Treball de Fi de Màster es busca desenvolupar un algorisme que aconsegueixi establir quins punts dins d'una ciutat presenten un major risc de congestió a partir d'assumpcions que permetin modelar de manera més realista el comportament dels usuaris de les xarxes de transport. Per a això s'ha modificat la definició i càlcul de la centralitat d'intermediació amb l'objectiu que aquest consideri rutes subòptimes pròximes a la mínima com a rutes acceptables. Finalment, s'ha estudiat com aquesta nova mesura de centralitat es comporta aplicada sobre xarxes de transport reals.

The ligand HL (6-[3-oxo-3-(2-hydroxyphenyl)propionyl]pyridine-2-carboxylic acid), which exhibits two different coordination pockets, has been exploited to engender and study energy transfer (ET) in two dinuclear [LnLn′] analogues of interest, [EuYb] and [NdYb]. Their structural and physical properties have been compared with newly synthesised analogues featuring no possible ET ([EuLu], [NdLu], and [GdYb]) and with the corresponding homometallic [EuEu] and [NdNd] analogues, which have been previously reported. Photophysical data suggest that ET between Eu and Yb does not occur to a significant extent, whereas emission from Yb originates from sensitisation of the ligand. In contrast, energy migration seems to be occurring between the two Nd centres in [NdNd], as well as in [NdYb], in which Yb luminescence is thus, in part, sensitised by ET from Nd. This study shows the versatility of this molecular platform to further the investigation of lanthanide-to-lanthanide ET phenomena in defined molecular systems., This research was supported by the Spanish MINECO through grants CTQ2015-68370-P and PGC2018-098630-B-I00 (G.A., D.A. and V.V.) and MAT2017-86826-R (O.R.), as well as through the Juan de la Cierva program IJCI-2016-29901 (D.A.) and the Aragón government (DGA, consolidated group PLATON E31_17R). G.A. thanks the Generalitat de Catalunya for the ICREA Academia 2018 prize and QUANTERA for project SUMO (through Spanish grant PCI2018-093106). This research used resources of the Advanced Light Source, which is a DOE Office of Science User Facility under contract no. DE-AC02-05CH11231.

Treballs Finals de Grau de Física, Facultat de Física, Universitat de Barcelona, Curs: 2021, Tutor: Albert Díaz-Guilera, The aim of this work is to analyze the network influence in a model of aging that has been proposed recently. The model is based in the interdependency of nodes affected by damage and repair processes. While the original work was limited to analyze random graphs and Barab´asi-Albert networks, we propose a general framework based on an exponent dependent preferential attachment rule for the construction of the network, and the two cases analyzed previously correspond to particular values of the exponent. We provide a discussion on the biological plausibility of the different models, which now include a sublinear and superlinear dependence on the degrees

Treballs Finals de Grau de Física, Facultat de Física, Universitat de Barcelona, Curs: 2021, Tutor: Josep Perelló, We did a behavioural study of persons playing collective games. We’ve seen how players interact with each other when they have to accomplish a shared goal and completed a study of how the in-game unequalities are distributed using the Gini Coefficient. Finally, we did a study (through Mutual Information) of how the betting profile of the individuals matched with others and how much of this match was due to sociodemographical traits

Given the complexity, size, cost, and importance of network infrastructures like the internet, telecommunication, transportation, and social networks, it is not surprising the attention that research on network analysis and protection is receiving. Today¿s world of technological advances creates high expectations in all those areas that humans and computers can manage. Thus, in the context of network science, we expect networks to keep their performance and be adaptative and resilient to diverse circumstances. As all networks are subject to attacks and collapse, safety protocols should be designed to provide tools to mitigate possible network failures. Many networks suffer attacks periodically: Twitter, Amazon, eBay have experienced denial-of-service for hours and even days, power grids have undergone massive blackouts, and similar incidents have happened for many other real networks. The failure usually propagates from one component to another in a cascade manner, known as cascade failure. Cascading failure studies investigate potential solutions to make networks more robust. There are some vital nodes within each network that have a profound influence on cascading phenomena. In this TFM, giving focus to cascade failure analysis, we explore several network structural properties that might impact the network performance when it is under attack, in particular k-core measure. Moreover, a recently proposed deterministic hierarchical graph family, which appears to be a good model for real networks, has been examined under attacks along with some other graph models. By comparing their relative performance, we can assess if this network can be a model to improve future network designs. Our findings show that networks with a high minimum k-core and negative degree-degree correlation (disassortative) perform better under attacks than assortative networks with low minimum k-core. To enhance the robustness of existing networks, we can modify them in a way to have less assortativity and such that their minimum k-core value is increased by about 1-core to 2-core. Regarding the deterministic hierarchical graph family, by considering a simple control strategy along with an 18% extra load capacity for each component, the network would be a proper choice for future network design since it can handle all attacks, according to the standard established model considered in this TFM.

Study, design and implementation of a system in a two-wheeled vehicle capable of maintaining balance without the need for a third point of support. Estudio, diseño e implementación de un sistema en un vehículo de dos ruedas capaz de mantener el equilibrio sin necesidad de un tercer punto de soporte. Estudi, disseny i implementació d'un sistema en un vehicle de dues rodes capaç de mantenir l'equilibri sense necessitat d'un tercer punt de suport.

Recent progress in the field of human induced pluripotent stem cells (iPSCs) has led to the efficient production of human neuronal cell models for in vitro study. This has the potential to enable the understanding of live human cellular and network function which is otherwise not possible. However, a major challenge is the generation of reproducible neural networks together with the ability to interrogate and record at the single cell level. A promising aid is the use of biomaterial scaffolds that would enable the development and guidance of neuronal networks in physiologically relevant architectures and dimensionality. The optimal scaffold material would need to be precisely fabricated with submicron resolution, be optically transparent, and biocompatible. Two-photon polymerisation (2PP) enables precise microfabrication of three-dimensional structures. In this study, we report the identification of two biomaterials that support the growth and differentiation of human iPSC-derived neural progenitors into functional neuronal networks. Furthermore, these materials can be patterned to induce alignment of neuronal processes and enable the optical interrogation of individual cells. 2PP scaffolds with tailored topographies therefore provide an effective method of producing defined in vitro human neural networks for application in influencing neurite guidance and complex network activity. publishedVersion

Complex networks are essentially heterogeneous not only in the basic properties of the constituent nodes, such as their degree, but also in the effects that these have on the global dynamical properties of the network. Networks of coupled identical phase oscillators are good examples for analyzing these effects, since an overall synchronized state can be considered a reference state. A small variation of intrinsic node parameters may cause the system to move away from synchronization, and a new phase-locked stationary state can be achieved. We propose a measure of phase dispersion that quantifies the functional response of the system to a given local perturbation. As a particular implementation, we propose a variation of the standard Kuramoto model in which the nodes of a complex network interact with their neighboring nodes, by including a node-dependent frustration parameter. The final stationary phase-locked state now depends on the particular frustration parameter at each node and also on the network topology. We exploit this scenario by introducing individual frustration parameters and measuring what their effect on the whole network is, measured in terms of the phase dispersion, which depends only on the topology of the network and on the choice of the particular node that is perturbed. This enables us to define a characteristic of the node, its functionability, that can be computed analytically in terms of the network topology. Finally, we provide a thorough comparison with other centrality measures.

Networks with underlying metric spaces attract increasing research attention in network science, statistical physics, applied mathematics, computer science, sociology, and other fields. This attention is further amplified by the current surge of activity in graph embedding. In the vast realm of spatial network models, only a few reproduce even the most basic properties of real-world networks. Here, we focus on three such properties---sparsity, small worldness, and clustering---and identify the general subclass of spatial homogeneous and heterogeneous network models that are sparse small worlds and that have nonzero clustering in the thermodynamic limit. We rely on the maximum entropy approach where network links correspond to noninteracting fermions whose energy dependence on spatial distances determines network small worldness and clustering.

Network Science provides a universal formalism for modelling and studying complex systems based on pairwise interactions between agents. However, many real networks in the social, biological or computer sciences involve interactions among more than two agents, having thus an inherent structure of a simplicial complex. We propose new notions of higher-order degrees of adjacency for simplices in a simplicial complex, allowing any dimensional comparison among them and their faces, which as far as we know were lacked in the literature. We introduce multi-parameter boundary and coboundary operators in an oriented simplicial complex and also a novel multi-combinatorial Laplacian is defined, which generalises the graph and combinatorial Laplacian. To illustrate the potential applications of these theoretical results, we perform a structural analysis of higher-order connectivity in simplicial-complex networks by studying the associated distributions with these simplicial degrees in 17 real-world datasets coming from different domains such as coauthor networks, cosponsoring Congress bills, contacts in schools, drug abuse warning networks, e-mail networks or publications and users in online forums. We find rich and diverse higher-order connectivity structures and observe that datasets of the same type reflect similar higher-order collaboration patterns. Furthermore, we show that if we use what we have called the maximal simplicial degree (which counts the distinct maximal communities in which our simplex and all its strict sub-communities are contained), then its degree distribution is, in general, surprisingly different from the classical node degree distribution., 52 pages, 12 figures, 4 tables

Treballs Finals de Grau de Física, Facultat de Física, Universitat de Barcelona, Curs: 2019, Tutor: Giancarlo Franzese, The study of water confined at stacked phospholipid bilayers is important to under- stand basic functions of biological membranes. Within this framework and by means of all-atom molecular dynamics simulations, we analyze the water-membrane and water-water interactions as function of the local distance of water molecules from the membrane. We find that both can be represented as coarse-grained potentials, useful in large-scale simulations

The thesis studies from an analytical and computational perspective, and by using reaction-diffusion equations, the spatiotemporal evolution of different populations. First, the dynamics of the T7 bacteriophage infecting the E. coli bacteria is studied. By adding the delayed time in diffusion and reaction terms, as well as new mathematical terms biologically sound, we can achieve results that accurately match the experimental propagation speeds. Secondly, different mathematical models are proposed to correctly understand the expansion of VSV in Glioblastoma. The only model capable of this explanation is the system which understands the delay time for the processes of diffusion and reaction. Finally, the Neolithic transition through Europe is explained by studying ancient genetic DNA samples alongside mathematical simulations. Focusing on haplogroup K, the model is built by analyzing the two Neolithic diffusion mechanisms: demic and cultural. The simulations show that the transition is basically demic, with only 2% of the Neolithic farmers interacting culturally Aquesta tesi estudia des d’un punt de analític i computacional, gràcies a les equacions de reacció-difusió, l’evolució espaciotemporal de diferents poblacions que interactuen entre elles. El primer article estudia la dinàmica del bacteriòfag T7 infectant el bacteri E. coli. Gràcies a la incorporació del temps de retard en els termes de difusió i reacció, així com de nous termes matemàtics amb sentit biològic, aconseguim uns resultats que s’ajusten millor a les velocitats de propagació. El segon article aplica diferents models matemàtics per entendre millor l’expansió del VSV en Glioblastomes. L'únic model capaç d'explicar de manera correcte el sistema té en compte el temps de retard per als processos de difusió i reacció. L’últim article explica la transició del Neolític a través d’Europa utilitzant mostres genètiques antigues i simulacions matemàtiques. Centrant-nos en l’haplogrup K, el model es construeix tenint en compte els dos mecanismes de difusió neolítica: dèmica i cultural. Les simulacions mostren que la transició és bàsicament dèmica, on només el 2% dels neolítics interaccionen culturalment

Treballs Finals de Grau de Física, Facultat de Física, Universitat de Barcelona, Curs: 2019, Tutor: Josep Perelló Palou, We study the GPS data set of a collection of 260 students from up to 10 schools in the Barcelona Metropolitan Area on their trip from house to school. Participants collected the data with the help of a mobile application. After visualizing and cleaning the data, the main dynamical properties have been studied. We firstly study the difuusion of the movement and then we describe the instantaneous velocity with a log-Normal distribution, proposing a stochastic process to model it. We also obtain two typical stopping times for participants' stops duration. In addition, we calculate the orientation of the movement for each school and for the entire data set. Finally, we describe the change of orientation with a family of symmetric distributions on the circle, comparing the results for the different schools. We thus provide a first analysis to very valuable data to model pedestrian mobility with specific origin-destination journeys. Results found can also help to improve urban planing in the schools surroundings.

Diversity, understood as the variety of different elements or configurations that an extensive system has, is a crucial property that allows maintaining the system's functionality in a changing environment, where failures, random events or malicious attacks are often unavoidable. Despite the relevance of preserving diversity in the context of ecology, biology, transport, finances, etc., the elements or configurations that more contribute to the diversity are often unknown, and thus, they can not be protected against failures or environmental crises. This is due to the fact that there is no generic framework that allows identifying which elements or configurations have crucial roles in preserving the diversity of the system. Existing methods treat the level of heterogeneity of a system as a measure of its diversity, being unsuitable when systems are composed of a large number of elements with different attributes and types of interactions. Besides, with limited resources, one needs to find the best preservation policy, i.e., one needs to solve an optimization problem. Here we aim to bridge this gap by developing a metric between labeled graphs to compute the diversity of the system, which allows identifying the most relevant components, based on their contribution to a global diversity value. The proposed framework is suitable for large multiplex structures, which are constituted by a set of elements represented as nodes, which have different types of interactions, represented as layers. The proposed method allows us to find, in a genetic network (HIV-1), the elements with the highest diversity values, while in a European airline network, we systematically identify the companies that maximize (and those that less compromise) the variety of options for routes connecting different airports., Comment: 14 pages, 7 figures

The preparation of heterometallic, lanthanide-only complexes is an extremely difficult synthetic challenge. By a ligand-based strategy, a complete isostructural series of dinuclear heterometallic [LnPr] complexes has been synthesized and structurally characterized. The two different coordination sites featured in this molecular entity allow study of the preferences of the praseodymium ion for a specific position depending on the ionic radii of the accompanying lanthanide partner. The purity of each heterometallic moiety has been evaluated in the solid state and in solution by means of crystallographic and spectrometric methods, respectively, revealing the limits of this strategy for ions with similar sizes. DFT calculations have been carried out to support the experimental results, confirming the nature of the site-selective lanthanide distribution. The predictable selectivity of this system has been exploited to assess the magnetic properties of the [DyPr] and [LuPr] derivatives, showing that the origin of the slow dynamics observed in the former arises from the dysprosium ion., G.A. thanks the Generalitat de Catalunya for the prize ICREA Academia 2008 and 2013 and the ERC for a Starting Grant (258060 FuncMolQIP). The authors acknowledge funding by the Spanish MINECO through CTQ2012-32247 and CTQ2015-68370-P (G.A., L.A.B., D.A., and V.V.), MAT2014-53961-R and MAT2017-86826-R (O.R.), and CTQ2017-88777-R (C.B. and J.G.-F.) as well as through the Juan de la Cierva program IJCI-2016-29901 (D.A.) and the Severo Ochoa Excellence Accreditation SEV-2013-0319 (C.B. and J.G.-F.). We also thank the Generalitat de Catalunya for financial support through the CERCA Program and 2017 SGR 290 grant.

Symmetries in physical theories denote invariance under some transformation, such as self-similarity under a change of scale. The renormalization group provides a powerful framework to study these symmetries, leading to a better understanding of the universal properties of phase transitions. However, the small-world property of complex networks complicates application of the renormalization group by introducing correlations between coexisting scales. Here, we provide a framework for the investigation of complex networks at different resolutions. The approach is based on geometric representations, which have been shown to sustain network navigability and to reveal the mechanisms that govern network structure and evolution. We define a geometric renormalization group for networks by embedding them into an underlying hidden metric space. We find that real scale-free networks show geometric scaling under this renormalization group transformation. We unfold the networks in a self-similar multilayer shell that distinguishes the coexisting scales and their interactions. This in turn offers a basis for exploring critical phenomena and universality in complex networks. It also affords us immediate practical applications, including high-fidelity smaller-scale replicas of large networks and a multiscale navigation protocol in hyperbolic space, which betters those on single layers. Complex networks are not obviously renormalizable, as different length scales coexist. Embedding networks in a geometrical space allows the definition of a renormalization group that can be used to construct smaller-scale replicas of large networks.