Table of contents

Complexity and network science are nowadays used, or at least invoked, in a variety of scientific researchareas ranging from the analysis of financial systems to social structure and even to medicine. Here I explore some of the possible reasons for this success, the relationship between them and how they might be used in the future.

Existing information-theoretic frameworks based on maximum entropy network ensembles are not able to explain the emergence of heterogeneity in complex networks. Here, we fill this gap of knowledge by developing a classical framework for networks based on finding an optimal trade-off between the information content of a compressed representation of the ensemble and the information content of the actual network ensemble. We introduce a novel classical network ensemble satisfying a set of soft constraints and we find the optimal distribution of the constraints for this ensemble. We show that for the classical network ensemble in which the only constraints are the expected degrees a power-law degree distribution is optimal. Also, we study spatially embedded networks finding that the interactions between nodes naturally lead to non-uniform spread of nodes in the embedding space, leading in some cases to a fractal distribution of nodes. This result is consistent with the so called `blessing of non-uniformity' of data, i.e. the fact that real world data typically do not obey uniform distributions. The pertinent features of real-world air transportation networks are well described by the proposed framework.

The coupling between population growth and transport accessibility has been an elusive problem for more than 60 years now. Due to the lack of theoretical foundations, most of the studies that considered how the evolution of transportation networks impacts the population growth are based on regression analysis in order to identify relevant variables. The recent availability of large amounts of data allows us to envision the construction of new approaches for understanding this coupling between transport and population growth. Here, we use a detailed dataset for about 36 000 municipalities in France from 1968 until now. In the case of large urban areas such as Paris, we show that growth rate statistical variations decay in time and display a trend towards homogenization where local aspects are less relevant. We also show that growth rate differences due to accessibility are very small and can mostly be observed for cities that experienced very large accessibility variations. This suggests that the relevant variable for explaining growth rate variations is not the accessibility but its temporal variation. We propose a model that integrates the stochastic internal variation of the municipalities population and an inter-urban migration term that we show to be proportional to the accessibility variation and has a limited time duration. This model provides a simple theoretical framework that allows to go beyond econometric studies and sheds a new light on the impact of transportation modes on city growth.

Optimizing group performance is one of the principal objectives that underlie human collaboration and prompts humans to share resources with each other. Connectivity between individuals determines how resources can be accessed and shared by the group members, yet, empirical knowledge on the relationship between the topology of the interconnecting network and group performance is scarce. To improve our understanding of this relationship, we created a game in virtual reality where small teams collaborated toward a shared goal. We conducted a series of experiments on 30 groups of three players, who played three rounds of the game, with different network topologies in each round. We hypothesized that higher network connectivity would enhance group performance due to two main factors: individuals' ability to share resources and their arousal. We found that group performance was positively associated with the overall network connectivity, although registering a plateau effect that might be associated with topological features at the node level. Deeper analysis of the group dynamics revealed that group performance was modulated by the connectivity of high and low performers in the group. Our findings provide insight into the intricacies of group structures, toward the design of effective human teams.

Compared with the well-studied topic of human mobility in real geographic space, only a few studies focus on human mobility in virtual space, such as interests, knowledge, ideas, and so on. However, it relates to the issues like public opinion management, knowledge diffusion, and innovation. In this paper, we assume that the interests of a group of online users can span an Euclidean space which is called interest space, and the transfers of user interests can be modelled as Lévy Flight in the interest space. Considering the interaction between users, we assume that the random walkers are not independent but interacting with each other indirectly via the digital resources in the interest space. The proposed model in this paper successfully reproduced a set of scaling laws for describing the growth of attention flow networks of online communities, and obtaining similar ranges of users' scaling exponents with empirical data. Further, we inferred parameters for describing the individual behaviours of the users according to the scaling laws of empirical attention flow network. Our model can not only provide theoretical understanding of human online behaviours but also has broad potential applications such as dissemination and public opinion management, online recommendation, etc.

A key question of collective social behavior is related to the influence of mass media on public opinion. Different approaches have been developed to address quantitatively this issue, ranging from field experiments to mathematical models. In this work we propose a combination of tools involving natural language processing and time series analysis. We compare selected features of mass media news articles with measurable manifestation of public opinion. We apply our analysis to news articles belonging to the 2016 US presidential campaign. We compare variations in polls (as a proxy of public opinion) with changes in the connotation of the news (sentiment) or in the agenda (topics) of a selected group of media outlets. Our results suggest that the sentiment content by itself is not enough to understand the differences in polls, but the combination of topics coverage and sentiment content provides an useful insight of the context in which public opinion varies. The methodology employed in this work is far general and can be easily extended to other topics of interest.

We introduce an oscillatory toy-model with variable frequency governed by a 3rd order equation to shed light on the formation of chimera states in systems of coupled oscillators. The toy-oscillators are constructed as bistable units and depending on the initial conditions their frequency may result in one of the two attracting fixed points, and (two-level synchronization). Numerical simulations demonstrate that when these oscillators are nonlocally coupled in networks, they organize in domains with alternating frequencies. In each domain the oscillators synchronize, while sequential domains follow different modes of synchronization. The border elements between two consecutive domains form the asynchronous domains as they are influenced by both frequencies. This way chimera states are formed via a two-level synchronization scenario. We investigate the influence of the frequency coupling constant and of the coupling range on the chimera morphology and we show that the chimera multiplicity decreases as the coupling range increases.

The frequency spectrum is calculated in the coherent and incoherent domains of this model. In the coherent domains single frequencies ( or ) are observed, while in the incoherent domains both and as well as their superpositions appear. This mechanism of creating domains of alternating frequencies offers a reasonable generic scenario for chimera state formation.

We consider time-dependent relaxation of observables in quantum systems of chaotic and regular type. Using statistical arguments and exact numerical solutions we show that the spread of the initial wave function in the Hilbert space and the main characteristics of evolution of observables have certain generic features. The study compares examples of regular dynamics, a completely chaotic case of the Gaussian orthogonal ensemble, a bosonic system with random interactions, and a fully realistic case of the time evolution of various initial non-stationary states in the nuclear shell model. In the case of the Gaussian orthogonal ensemble we show that the survival probability obtained analytically also fully defines the relaxation timescale of observables. This is not the case in general. Using the realistic nuclear shell model and the quadrupole moment as an observable we demonstrate that the relaxation time is significantly longer than defined by the survival probability of the initial state. The full analysis does not show the presence of an analog of the Lyapunov exponent characteristic for examples of classical chaos.

Complex systems may morph between structures with different dimensionality and degrees of freedom. As a tool for their modelling, nonlinear embeddings are introduced that encompass objects with different dimensionality as a continuous parameter $\kappa \in \mathbb{R}$ is being varied, thus allowing the unification of vectors, matrices and tensors in single mathematical structures. This technique is applied to construct warped models in the passage from supergravity in 10 or 11-dimensional spacetimes to four-dimensional ones. We also show how nonlinear embeddings can be used to connect cellular automata (CAs) to coupled map lattices (CMLs) and to nonlinear partial differential equations, deriving a class of nonlinear diffusion equations. Finally, by means of nonlinear embeddings we introduce CA connections, a class of CMLs that connect any two arbitrary CAs in the limits κ → 0 and κ → ∞ of the embedding.