Towards real-world complexity: an introduction to multiplex networks

Many real-world complex systems are best modeled by multiplex networks of interacting network layers. The multiplex network study is one of the newest and hottest themes in the statistical physics of complex networks. Pioneering studies have proven that the multiplexity has broad impact on the system's structure and function. In this Colloquium paper, we present an organized review of the growing body of current literature on multiplex networks by categorizing existing studies broadly according to the type of layer coupling in the problem. Major recent advances in the field are surveyed and some outstanding open challenges and future perspectives will be proposed.Comment: 20 pages, 10 figure

Disease Localization in Multilayer Networks

We present a continuous formulation of epidemic spreading on multilayer networks using a tensorial representation, extending the models of monoplex networks to this context. We derive analytical expressions for the epidemic threshold of the SIS and SIR dynamics, as well as upper and lower bounds for the disease prevalence in the steady state for the SIS scenario. Using the quasi-stationary state method we numerically show the existence of disease localization and the emergence of two or more susceptibility peaks, which are characterized analytically and numerically through the inverse participation ratio. Furthermore, when mapping the critical dynamics to an eigenvalue problem, we observe a characteristic transition in the eigenvalue spectra of the supra-contact tensor as a function of the ratio of two spreading rates: if the rate at which the disease spreads within a layer is comparable to the spreading rate across layers, the individual spectra of each layer merge with the coupling between layers. Finally, we verified the barrier effect, i.e., for three-layer configuration, when the layer with the largest eigenvalue is located at the center of the line, it can effectively act as a barrier to the disease. The formalism introduced here provides a unifying mathematical approach to disease contagion in multiplex systems opening new possibilities for the study of spreading processes.Comment: Revised version. 25 pages and 18 figure

Exploring connections between open quantum systems, relativity and complex quantum networks

This thesis is a collection of works focusing on interrelations between the fields of special relativity, open quantum systems and complex networks. As each of the aforementioned fields encompasses a huge variety of topics, the thesis contains a selection of particular connections. The first half of the thesis considers an open quantum systems approach to the description of relativistic phenomena. As the aforementioned phenomena often involve dynamically evolving counterparts, some non-Markovian aspects of open quantum systems are initially explored. Conditions for complete positivity are derived for a type of phase-covariant time-local master equation and then applied to a non-Markovian master equation with heuristically-derived decay rates. This is relevant in cases when the physicality of the master equation cannot be postulated. The first relativistic system considered is a superposition of coherent states in a gravitational gradient. It is shown to exhibit decoherence induced by gravitational time dilation through interaction between inner degrees of freedom and the centre of mass of the superposition. The decoherence is quantified using various, widely-used non-classicality measures. The decoherence rates are then compared to decoherence induced by classical noise to roughly evaluate the experimental precision necessary to detect gravitationally induced decoherence. In the second relativistic system, the Unruh effect is modeled as an open system where an Unruh-deWitt detector is interacting with bosonic fields. A master equation with time-dependent decay rates for the open system is derived by assuming a non-conventional spacetime path profile for the detector. A particular parameter governing the physical evolution is identified. The system is shown to exhibit non-Markovianity within the completely positive domain of the master equation, the latter ensured by the conditions derived before. The second half of the thesis starts with a definition of complex networks, which are yet not a commonly used tool in quantum physics. The concept of pairwise tomography networks is then introduced as a way to represent many-body quantum states. As the number of pairwise connections grows quadratically with the number of nodes, an efficiently scaling measurement scheme to recover the pairwise tomography networks is presented. The scheme and the concept of pairwise tomography networks are demonstrated to be useful in various applications and the results of a proof-of-principle experiment are shown. One application explored is a paradigmatic spin chain model known as the XX model. A pairwise entanglement network representation of the XX model is shown to suggest new phenomena such as gradual establishment of quasi-long range order accompanied by a symmetry regarding single-spin concurrence distributions. The existence of structural classes and a cyclic self-similarity in the state are revealed.Tämä väitöskirja on erikoisen suhteellisuusteorian, avoimien kvanttisysteemien ja kompleksisten verkkojen alojen läpileikkauksiin keskittyvä kokoelma. Sillä jokainen edellämainituista aloista sisältää valtavan moninaisuuden erilaisia käsitteitä, väitöskirjaan sisältyy valikoima tietynlaisia yhteyksiä. Väitöskirjan ensimmäinen puolisko käsittelee avoimien kvanttisysteemien lähestymistapaa relativististen ilmiöiden kuvaamiseen. Edellä mainitut ilmiöt yleensä sisältävät dynaamisesti kehittyviä osia, joten tutkitaan avoimien kvanttisysteemien ei-markovisia näkökulmia. Täyspositiivisuuden ehdot johdetaan tietyntyyppiselle faasi-kovariantille ajassa lokaalille master-yhtälölle. Nämä ehdot sovitellaan ei-markoviselle master-yhtälölle jossa on heuristisesti johdetut hajoamisnopeudet. Tämä on merkityksellistä silloin kun master-yhtälön fysikaalisuus ei ole postuloitavissa. Ensimmäinen tarkasteltu relativistinen systeemi on koherenttien tilojen superpositio gravitationaalisessa gradientissa. Sen osoitetaan ilmentävän gravitaationaalisen aikadilataation aiheuttamaa dekoherenssiä sisäisten vapausasteiden ja massakeskipisteen välisten vuorovaikutusten kautta. Tämä dekoherenssi kvantifioidaan käyttämällä useilla laajassa käytössä olevilla ei-klassisuusmitoilla. Dekoherenssinopeuksia verrataan klassisen melun aiheuttamaan dekoherenssiin alustavan gravitaation indusoiman dekoherenssin kokeelliseen havaintoon tarvittavan mittaustarkkuusarvion tekemiseksi. Toisessa tarkasteltavassa relativistisessa systeemissä, Unruhin ilmiö mallinnetaan avoimena kvanttisysteeminä jossa Unruh-deWitt detektori vuorovaikuttaa bosonikenttien kanssa. Olettamalla detektorille epätavanomainen aika-avaruuspolkuprofiili johdetaan avoimelle kvanttisysteemille master-yhtälö ajasta riippuvin hajoamisnopeuksin. Tunnistetaan erityinen fyysista evoluutiota hallitseva parametri. Systeemin näytetään omaavan ei-markovisuutta master-yhtälön täyspositiivisilla alueilla, täyspositiivisuus taataan aiemmin johdetulla tavalla Väitöskirjan toisen puoliskon alussa määritellään kompleksiset kvantti-verkot, jotka eivät vielä ole laajasti käytettyjä työkaluja kvanttifysiikassa. Esitellään uusi tapa esittää monikappalekvanttitiloja parittaisina tomografiaverkkoina. Parittaisten yhteyksien lukumäärän ollessa neliöllisesti verrannollinen solmupisteiden lukumäärään, esitetään tehokkaasti skaalautuva parittaisia tomografiaverkkoja muodostava mittausskeema. Skeema ja parittaisten tomografiaverkkojen konsepti osoitetaan hyödyllisiksi erilaisissa sovelluksissa ja esitetään periaatetodistuskokeen tulokset. Eräänä sovelluksena esitetään XX mallina tunnettu paradigmaattinen spinketjumalli. XX mallin parittaisen kietoutumisen verkkoesityksen näytetään tuovan ilmi uusia ilmiöitä kuten vaiheittaisen kvasipitkän etäisyyden järjestyksen yksittäisten spinien konkurrenssidistribuutioiden symmetrialla. Paljastetaan tilan rakenteellisten luokkien ja syklisten itse-samankaltaisuuksien olemassaolo

Fundamentals of spreading processes in single and multilayer complex networks

Spreading processes have been largely studied in the literature, both analytically and by means of large-scale numerical simulations. These processes mainly include the propagation of diseases, rumors and information on top of a given population. In the last two decades, with the advent of modern network science, we have witnessed significant advances in this field of research. Here we review the main theoretical and numerical methods developed for the study of spreading processes on complex networked systems. Specifically, we formally define epidemic processes on single and multilayer networks and discuss in detail the main methods used to perform numerical simulations. Throughout the review, we classify spreading processes (disease and rumor models) into two classes according to the nature of time: (i) continuous-time and (ii) cellular automata approach, where the second one can be further divided into synchronous and asynchronous updating schemes. Our revision includes the heterogeneous mean-field, the quenched-mean field, and the pair quenched mean field approaches, as well as their respective simulation techniques, emphasizing similarities and differences among the different techniques. The content presented here offers a whole suite of methods to study epidemic-like processes in complex networks, both for researchers without previous experience in the subject and for experts.Comment: Review article. 73 pages, including 24 figure

The structure and dynamics of multiplex networks

PhDNetwork science has provided useful answers to research questions in many fields, from biology to social science, from ecology to urban science. The first analyses of networked systems focused on binary networks, where only the topology of the connections were considered. Soon network scientists started considering weighted networks, to represent interactions with different strength, cost, or distance in space and time. Also, connections are not fixed but change over time. This is why in more recent years, a lot of attention has been devoted to temporal or time-varying networks. We now entered the era of multi-layer networks, or multiplex networks, relational systems whose units are connected by different relationships, with links of distinct types embedded in different layers. Multiplexity has been observed in many contexts, from social network analysis to economics, medicine and ecology. The new challenge consists in applying the new tools of multiplex theory to unveil the richness associated to this novel level of complexity. How do agents organise their interactions across layers? How does this affect the dynamics of the system? In the first part of the thesis, we provide a mathematical framework to deal with multiplex networks. We suggest metrics to unveil multiplexity from basic node, layer and edge properties to more complicated structure at the micro- and meso-scale, such as motifs, communities and cores. Measures are validated through the analysis of real-world systems such as social and collaboration networks, transportation systems and the human brain. In the second part of the thesis we focus on dynamical processes taking place on top of multiplex networks, namely biased random walks, opinion dynamics, cultural dynamics and evolutionary game theory. All these examples show how multiplexity is crucial to determine the emergence of unexpected and instrinsically multiplex collective behavior, opening novel perspectives for the field of non-linear dynamics on networks.European Union project LASAGN

Contagion à effet de seuil dans les réseaux complexes

Networks arise frequently in the study of complex systems, since interactions among the components of such systems are critical. Networks can act as a substrate for dynamical process, such as the diffusion of information or disease throughout populations. Network structure can determine the temporal evolution of a dynamical process, including the characteristics of the steady state.The simplest representation of a complex system is an undirected, unweighted, single layer graph. In contrast, real systems exhibit heterogeneity of interaction strength and type. Such systems are frequently represented as weighted multiplex networks, and in this work we incorporate these heterogeneities into a master equation formalism in order to study their effects on spreading processes. We also carry out simulations on synthetic and empirical networks, and show that spreading dynamics, in particular the speed at which contagion spreads via threshold mechanisms, depend non-trivially on these heterogeneities. Further, we show that an important family of networks undergo reentrant phase transitions in the size and frequency of global cascades as a result of these interactions.A challenging feature of real systems is their tendency to evolve over time, since the changing structure of the underlying network is critical to the behaviour of overlying dynamical processes. We show that one aspect of temporality, the observed “burstiness” in interaction patterns, leads to non-monotic changes in the spreading time of threshold driven contagion processes.The above results shed light on the effects of various network heterogeneities, with respect to dynamical processes that evolve on these networks.Les interactions entre les composants des systèmes complexes font émerger différents types de réseaux. Ces réseaux peuvent jouer le rôle d’un substrat pour des processus dynamiques tels que la diffusion d’informations ou de maladies dans des populations. Les structures de ces réseaux déterminent l’évolution d’un processus dynamique, en particulier son régime transitoire, mais aussi les caractéristiques du régime permanent.Les systèmes complexes réels manifestent des intéractions hétérogènes en type et en intensité. Ces systèmes sont représetés comme des réseaux pondérés à plusieurs couches. Dans cette thèse, nous développons une équation maîtresse afin d’intégrer ces hétérogénéités et d’étudier leurs effets sur les processus de diffusion. À l’aide de simulations mettant en jeu des réseaux réels et générés, nous montrons que les dynamiques de diffusion sont liées de manière non triviale à l’hétérogénéité de ces réseaux, en particulier la vitesse de propagation d’une contagion basée sur un effet de seuil. De plus, nous montrons que certaines classes de réseaux sont soumises à des transitions de phase réentrantes fonctions de la taille des “global cascades”.La tendance des réseaux réels à évoluer dans le temps rend difficile la modélisation des processus de diffusion. Nous montrons enfin que la durée de diffusion d’un processus de contagion basé sur un effet de seuil change de manière non-monotone du fait de la présence de“rafales” dans les motifs d’intéractions. L’ensemble de ces résultats mettent en lumière les effets de l’hétérogénéité des réseaux vis-à-vis des processus dynamiques y évoluant

Optimisation and information-theoretic principles in multiplex networks.

PhD ThesesThe multiplex network paradigm has proven very helpful in the study of many real-world complex systems, by allowing to retain full information about all the different possible kinds of relationships among the elements of a system. As a result, new non-trivial structural patterns have been found in diverse multi-dimensional networked systems, from transportation networks to the human brain. However, the analysis of multiplex structural and dynamical properties often requires more sophisticated algorithms and takes longer time to run compared to traditional single network methods. As a consequence, relying on a multiplex formulation should be the outcome of a trade-off between the level of information and the resources required to store it. In the first part of the thesis, we address the problem of quantifying and comparing the amount of information contained in multiplex networks. We propose an algorithmic informationtheoretic approach to evaluate the complexity of multiplex networks, by assessing to which extent a given multiplex representation of a system is more informative than a single-layer graph. Then, we demonstrate that the same measure is able to detect redundancy in a multiplex network and to obtain meaningful lower-dimensional representations of a system. We finally show that such method allows us to retain most of the structural complexity of the original system as well as the salient characteristics determining the behaviour of dynamical processes happening on it. In the second part of the thesis, we shift the focus to the modelling and analysis of some structural features of real-world multiplex systems throughout optimisation principles. We demonstrate that Pareto optimal principles provide remarkable tools not only to model real-world multiplex transportation systems but also to characterise the robustness of multiplex systems against targeted attacks in the context of optimal percolation

Interacting Spreading Processes in Multilayer Networks: A Systematic Review

© 2013 IEEE. The world of network science is fascinating and filled with complex phenomena that we aspire to understand. One of them is the dynamics of spreading processes over complex networked structures. Building the knowledge-base in the field where we can face more than one spreading process propagating over a network that has more than one layer is a challenging task, as the complexity comes both from the environment in which the spread happens and from characteristics and interplay of spreads' propagation. As this cross-disciplinary field bringing together computer science, network science, biology and physics has rapidly grown over the last decade, there is a need to comprehensively review the current state-of-the-art and offer to the research community a roadmap that helps to organise the future research in this area. Thus, this survey is a first attempt to present the current landscape of the multi-processes spread over multilayer networks and to suggest the potential ways forward

Disease localization in multilayer networks

We present a continuous formulation of epidemic spreading on multilayer networks using a tensorial representation, extending the models of monoplex networks to this context. We derive analytical expressions for the epidemic threshold of the susceptible-infected-susceptible (SIS) and susceptibleinfected- recovered dynamics, as well as upper and lower bounds for the disease prevalence in the steady state for the SIS scenario. Using the quasistationary state method, we numerically show the existence of disease localization and the emergence of two or more susceptibility peaks, which are characterized analytically and numerically through the inverse participation ratio. At variance with what is observed in single-layer networks, we show that disease localization takes place on the layers and not on the nodes of a given layer. Furthermore, when mapping the critical dynamics to an eigenvalue problem, we observe a characteristic transition in the eigenvalue spectra of the supra-contact tensor as a function of the ratio of two spreading rates: If the rate at which the disease spreads within a layer is comparable to the spreading rate across layers, the individual spectra of each layer merge with the coupling between layers. Finally, we report on an interesting phenomenon, the barrier effect; i.e., for a three-layer configuration, when the layer with the lowest eigenvalue is located at the center of the line, it can effectively act as a barrier to the disease. The formalism introduced here provides a unifying mathematical approach to disease contagion in multiplex systems, opening new possibilities for the study of spreading processes