Abstract:
The interactions among the elementary components of many complex systems can be qualitatively different. Such systems are therefore naturally described in terms of multiplex or multi-layer networks, i.e. networks where each layer stands for a different type of interaction between the same set of nodes. There is today a growing interest in understanding when and why a description in terms of a multiplex network is necessary and more informative than a single-layer projection. Here, we contribute to this debate by presenting a comprehensive study of correlations in multiplex networks. Correlations in node properties, especially degree-degree correlations, have been thoroughly studied in single-layer networks. Here we extend this idea to investigate and characterize correlations between the different layers of a multiplex network. Such correlations are intrinsically multiplex, and we first study them empirically by constructing and analyzing several multiplex networks from the real-world. In particular, we introduce various measures to characterize correlations in the activity of the nodes and in their degree at the different layers, and between activities and degrees. We show that real-world networks exhibit indeed non-trivial multiplex correlations. For instance, we find cases where two layers of the same multiplex network are positively correlated in terms of node degrees, while other two layers are negatively correlated. We then focus on constructing synthetic multiplex networks, proposing a series of models to reproduce the correlations observed empirically and/or to assess their relevance.

Abstract:
Our understanding of a variety of phenomena in physics, biology and economics crucially depends on the analysis of multivariate time series. While a wide range of tools and techniques for time series analysis already exist, the increasing availability of massive data structures calls for new approaches for multidimensional signal processing. We present here a non-parametric method to analyse multivariate time series, based on the mapping of a multidimensional time series into a multilayer network, which allows to extract information on a high dimensional dynamical system through the analysis of the structure of the associated multiplex network. The method is simple to implement, general, scalable, does not require ad hoc phase space partitioning, and is thus suitable for the analysis of large, heterogeneous and non-stationary time series. We show that simple structural descriptors of the associated multiplex networks allow to extract and quantify nontrivial properties of coupled chaotic maps, including the transition between different dynamical phases and the onset of various types of synchronization. As a concrete example we then study financial time series, showing that a multiplex network analysis can efficiently discriminate crises from periods of financial stability, where standard methods based on time-series symbolization often fail.

Abstract:
We consider degree-biased random walkers whose probability to move from a node to one of its neighbors of degree $k$ is proportional to $k^{\alpha}$, where $\alpha$ is a tuning parameter. We study both numerically and analytically three types of characteristic times, namely: i) the time the walker needs to come back to the starting node, ii) the time it takes to visit a given node for the first time, and iii) the time it takes to visit all the nodes of the network. We consider a large data set of real-world networks and we show that the value of $\alpha$ which minimizes the three characteristic times is different from the value $\alpha_{\rm min}=-1$ analytically found for uncorrelated networks in the mean-field approximation. In addition to this, we found that assortative networks have preferentially a value of $\alpha_{\rm min}$ in the range $[-1,-0.5]$, while disassortative networks have $\alpha_{\rm min}$ in the range $[-0.5, 0]$. We derive an analytical relation between the degree correlation exponent $\nu$ and the optimal bias value $\alpha_{\rm min}$, which works well for real-world assortative networks. When only local information is available, degree-biased random walks can guarantee smaller characteristic times than the classical unbiased random walks, by means of an appropriate tuning of the motion bias.

Abstract:
Many real-world complex systems consist of a set of elementary units connected by relationships of different kinds. All such systems are better described in terms of multiplex networks, where the links at each layer represent a different type of interaction between the same set of nodes, rather than in terms of (single-layer) networks. In this paper we present a general framework to describe and study multiplex networks, whose links are either unweighted or weighted. In particular we propose a series of measures to characterize the multiplexicity of the systems in terms of: i) basic node and link properties such as the node degree, and the edge overlap and reinforcement, ii) local properties such as the clustering coefficient and the transitivity, iii) global properties related to the navigability of the multiplex across the different layers. The measures we introduce are validated on a genuine multiplex data set of Indonesian terrorists, where information among 78 individuals are recorded with respect to mutual trust, common operations, exchanged communications and business relationships.

Abstract:
We present a novel way to characterize the structure of complex networks by studying the statistical properties of the trajectories of random walks over them. We consider time series corresponding to different properties of the nodes visited by the walkers. We show that the analysis of the fluctuations of these time series allows to define a set of characteristic exponents which capture the local and global organization of a network. This approach provides a way of solving two classical problems in network science, namely the systematic classification of networks, and the identification of the salient properties of growing networks. The results contribute to the construction of a unifying framework for the investigation of the structure and dynamics of complex systems.

Abstract:
In the social sciences, the debate over the structural foundations of social capital has long vacillated between two positions on the relative benefits associated with two types of social structures: closed structures, rich in third-party relationships, and open structures, rich in structural holes and brokerage opportunities. In this paper, we engage with this debate by focusing on the measures typically used for formalising the two conceptions of social capital: clustering and effective size. We show that these two measures are simply two sides of the same coin, as they can be expressed one in terms of the other through a simple functional relation. Building on this relation, we then attempt to reconcile closed and open structures by proposing a new measure, Simmelian brokerage, that captures opportunities of brokerage between otherwise disconnected cohesive groups of contacts. Implications of our findings for research on social capital and complex networks are discussed.

Abstract:
Biased random walks on complex networks are a particular type of walks whose motion is biased on properties of the destination node, such as its degree. In recent years they have been exploited to design efficient strategies to explore a network, for instance by constructing maximally mixing trajectories or by sampling homogeneously the nodes. In multiplex networks, the nodes are related through different types of links (layers or communication channels), and the presence of connections at different layers multiplies the number of possible paths in the graph. In this work we introduce biased random walks on multiplex networks and provide analytical solutions for their long-term properties such as the stationary distribution and the entropy rate. We focus on degree-biased walks and distinguish between two subclasses of random walks: extensive biased walks consider the properties of each node separately at each layer, intensive biased walks deal instead with intrinsically multiplex variables. We study the effect of different structural properties, including the number of layers, the presence and sign of inter-layer degree correlations, and the redundancy of edges across layers, on the steady-state behaviour of the walkers, and we investigate how to design an efficient exploration of the system. Finally, we apply our results to the case of a multidimensional social network and to a multimodal transportation system, showing how an appropriate tuning of the bias parameters towards nodes which are truly multiplex allows to obtain a good trade-off between a maximal entropy rate and a homogeneous sampling of the nodes of the network.

Abstract:
In this paper we present a formal description of PROSA, a P2P resource management system heavily inspired by social networks. Social networks have been deeply studied in the last two decades in order to understand how communities of people arise and grow. It is a widely known result that networks of social relationships usually evolves to small–worlds, i.e. networks where nodes are strongly connected to neighbours and separated from all other nodes by a small amount of hops. This work shows that algorithms implemented into PROSA allow to obtain an efficient small–world P2P network. We also show how taking advantage of PROSA structure it is possible to effectively answer queries. In particular, the so–called query recall for PROSA is estimated and compared to that obtained in SETS [1] and GES [2].

Abstract:
Urbanisation is a fundamental phenomenon whose quantitative characterisation is still inadequate. We report here the empirical analysis of a unique data set regarding almost 200 years of evolution of the road network in a large area located north of Milan (Italy). We find that urbanisation is characterised by the homogenisation of cell shapes, and by the stability throughout time of high-centrality roads which constitute the backbone of the urban structure, confirming the importance of historical paths. We show quantitatively that the growth of the network is governed by two elementary processes: (i) `densification', corresponding to an increase in the local density of roads around existing urban centres and (ii) `exploration', whereby new roads trigger the spatial evolution of the urbanisation front. The empirical identification of such simple elementary mechanisms suggests the existence of general, simple properties of urbanisation and opens new directions for its modelling and quantitative description.

Abstract:
Understanding dynamical processes on networks is an important area of research in complex systems, with far reaching implications and applications in many real-world cases. Here we introduce and study a model of intertwined dynamics on interconnected networks, inspired by the human brain, which consists of bidirectionally coupled synchronization and energy transport processes. Remarkably, the proposed model allows the emergence of spontaneous switch-like synchronization transitions driven by the energy transport dynamics, which qualitatively mirror the transitions observed in human brain dynamics between resting-state and cognitive activity. We provide a steady-state analytical explanation for the observed behavior and show that the switch-like transition is robust over a wide range of model parameters and network topologies. Finally, we suggest that the complexity inherent in other interconnected dynamical processes might be responsible for various other emergent behaviors observed in natural systems.