Manlio De Domenico

Mathematical Formulation of Multilayer Networks

A network representation is useful for describing the structure of a large variety of complex systems. However, most real and engineered systems have multiple subsystems and layers of connectivity, and the data produced by such systems are very rich. Achieving a deep understanding of such systems necessitates generalizing ‘‘traditional’’ network theory, and the newfound deluge of data now makes it possible to test increasingly general frameworks for the study of networks. In particular, although adjacency matrices are useful to describe traditional single-layer networks, such a representation is insufficient for the analysis and description of multiplex and time-dependent networks. One must therefore develop a more general mathematical framework to cope with the challenges posed by multilayer complex systems. In this paper, we introduce a tensorial framework to study multilayer networks, and we discuss the generalization of several important network descriptors and dynamical processes—including degree centrality, clustering coefficients, eigenvector centrality, modularity, von Neumann entropy, and diffusion—for this framework. We examine the impact of different choices in constructing these generalizations, and we illustrate how to obtain known results for the special cases of single-layer and multiplex networks. Our tensorial approach will be helpful for tackling pressing problems in multilayer complex systems, such as inferring who is influencing whom (and by which media) in multichannel social networks and developing routing techniques for multimodal transportation systems.

Navigability of interconnected networks under random failures.

Significance Network theory has been exploited in the last decades to deepen our comprehension of complex systems. However, real-world complex systems exhibit multiple levels of relationships and require modeling by interconnected networks, characterizing interactions on several levels simultaneously. Questions such as “what is the efficiency of exploration of a city using the multiple transportation layers, like subway and bus?” and “what is its resilience to failures?” have to be answered using the multiplex framework. Here, we introduce fundamental mechanisms to perform such exploration, using random walks on multilayer networks, and we show how the topological structure, together with the navigation strategy, influences the efficiency in exploring the whole structure. Assessing the navigability of interconnected networks (transporting information, people, or goods) under eventual random failures is of utmost importance to design and protect critical infrastructures. Random walks are a good proxy to determine this navigability, specifically the coverage time of random walks, which is a measure of the dynamical functionality of the network. Here, we introduce the theoretical tools required to describe random walks in interconnected networks accounting for structure and dynamics inherent to real systems. We develop an analytical approach for the covering time of random walks in interconnected networks and compare it with extensive Monte Carlo simulations. Generally speaking, interconnected networks are more resilient to random failures than their individual layers per se, and we are able to quantify this effect. As an application––which we illustrate by considering the public transport of London––we show how the efficiency in exploring the multiplex critically depends on layers’ topology, interconnection strengths, and walk strategy. Our findings are corroborated by data-driven simulations, where the empirical distribution of check-ins and checks-out is considered and passengers travel along fastest paths in a network affected by real disruptions. These findings are fundamental for further development of searching and navigability strategies in real interconnected systems.Multiplex networks are receiving increasing interests because they allow to model relationships between networked agents on several layers simultaneously. In this letter, we extend well-known random walks to multiplexes and we introduce a new type of walk that can exist only in multiplexes. We derive exact expressions for vertex occupation time and the coverage. Finally, we show how the efficiency in exploring the multiplex critically depends on the underlying topology of layers, the weight of their inter-connections and the strategy adopted to walk.

Ranking in interconnected multilayer networks reveals versatile nodes

Real-world complex systems exhibit multiple levels of relationships. In many cases, they require to be modeled by interconnected multilayer networks, characterizing interactions on several levels simultaneously. It is of crucial importance in many fields, from economics to biology, from urban planning to social sciences, to identify the most (or the less) influent nodes in a network. However, defining the centrality of actors in an interconnected structure is not trivial. In this paper, we capitalize on the tensorial formalism, recently proposed to characterize and investigate this kind of complex topologies, to show how several centrality measures – well-known in the case of standard (“monoplex”) networks – can be extended naturally to the realm of interconnected multiplexes. We consider diagnostics widely used in different fields, e.g., computer science, biology, communication and social sciences, to cite only some of them. We show, both theoretically and numerically, that using the weighted monoplex obtained by aggregating the multilayer network leads, in general, to relevant differences in ranking the nodes by their importance.The determination of the most central agents in complex networks is important because they are responsible for a faster propagation of information, epidemics, failures and congestion, among others. A challenging problem is to identify them in networked systems characterized by different types of interactions, forming interconnected multilayer networks. Here we describe a mathematical framework that allows us to calculate centrality in such networks and rank nodes accordingly, finding the ones that play the most central roles in the cohesion of the whole structure, bridging together different types of relations. These nodes are the most versatile in the multilayer network. We investigate empirical interconnected multilayer networks and show that the approaches based on aggregating--or neglecting--the multilayer structure lead to a wrong identification of the most versatile nodes, overestimating the importance of more marginal agents and demonstrating the power of versatility in predicting their role in diffusive and congestion processes.

Interdependence and predictability of human mobility and social interactions

Previous studies have shown that human movement is predictable to a certain extent at different geographic scales. The existing prediction techniques exploit only the past history of the person taken into consideration as input of the predictors. In this paper, we show that by means of multivariate nonlinear time series prediction techniques it is possible to increase the forecasting accuracy by considering movements of friends, people, or more in general entities, with correlated mobility patterns (i.e., characterised by high mutual information) as inputs. Finally, we evaluate the proposed techniques on the Nokia Mobile Data Challenge and Cabspotting datasets.

Personalized routing for multitudes in smart cities

Human mobility in a city represents a fascinating complex system that combines social interactions, daily constraints and random explorations. New collections of data that capture human mobility not only help us to understand their underlying patterns but also to design intelligent systems. Bringing us the opportunity to reduce traffic and to develop other applications that make cities more adaptable to human needs. In this paper, we propose an adaptive routing strategy which accounts for individual constraints to recommend personalized routes and, at the same time, for constraints imposed by the collectivity as a whole. Using big data sets recently released during the Telecom Italia Big Data Challenge, we show that our algorithm allows us to reduce the overall traffic in a smart city thanks to synergetic effects, with the participation of individuals in the system, playing a crucial role.

Identifying Modular Flows on Multilayer Networks Reveals Highly Overlapping Organization in Interconnected Systems

To comprehend interconnected systems across the social and natural sciences, researchers have developed many powerful methods to identify functional modules. For example, with interaction data aggr ...

MuxViz: a tool for multilayer analysis and visualization of networks

Multilayer relationships among entities and information about entities must be accompanied by the means to analyse, visualize and obtain insights from such data. We present open-source software (muxViz) that contains a collection of algorithms for the analysis of multilayer networks, which are an important way to represent a large variety of complex systems throughout science and engineering. We demonstrate the ability of muxViz to analyse and interactively visualize multilayer data using empirical genetic, neuronal and transportation networks. Our software is available at https://github.com/manlius/muxViz.

The physics of spreading processes in multilayer networks

Despite the success of traditional network analysis, standard networks provide a limited representation of complex systems, which often include different types of relationships (or ‘multiplexity’) between their components. Such structural complexity has a significant effect on both dynamics and function. Throwing away or aggregating available structural information can generate misleading results and be a major obstacle towards attempts to understand complex systems. The recent multilayer approach for modelling networked systems explicitly allows the incorporation of multiplexity and other features of realistic systems. It allows one to couple different structural relationships by encoding them in a convenient mathematical object. It also allows one to couple different dynamical processes on top of such interconnected structures. The resulting framework plays a crucial role in helping to achieve a thorough, accurate understanding of complex systems. The study of multilayer networks has also revealed new physical phenomena that remain hidden when using ordinary graphs, the traditional network representation. Here we survey progress towards attaining a deeper understanding of spreading processes on multilayer networks, and we highlight some of the physical phenomena related to spreading processes that emerge from multilayer structure. Reshaping network theory to describe the multilayered structures of the real world has formed a focus in complex networks research in recent years. Progress in our understanding of dynamical processes is but one of the fruits of this labour.

Mapping Multiplex Hubs in Human Functional Brain Networks

Typical brain networks consist of many peripheral regions and a few highly central ones, i.e., hubs, playing key functional roles in cerebral inter-regional interactions. Studies have shown that networks, obtained from the analysis of specific frequency components of brain activity, present peculiar architectures with unique profiles of region centrality. However, the identification of hubs in networks built from different frequency bands simultaneously is still a challenging problem, remaining largely unexplored. Here we identify each frequency component with one layer of a multiplex network and face this challenge by exploiting the recent advances in the analysis of multiplex topologies. First, we show that each frequency band carries unique topological information, fundamental to accurately model brain functional networks. We then demonstrate that hubs in the multiplex network, in general different from those ones obtained after discarding or aggregating the measured signals as usual, provide a more accurate map of brains most important functional regions, allowing to distinguish between healthy and schizophrenic populations better than conventional network approaches.

Random walk centrality in interconnected multilayer networks

Abstract Real-world complex systems exhibit multiple levels of relationships. In many cases they require to be modeled as interconnected multilayer networks, characterizing interactions of several types simultaneously. It is of crucial importance in many fields, from economics to biology and from urban planning to social sciences, to identify the most (or the less) influent nodes in a network using centrality measures. However, defining the centrality of actors in interconnected complex networks is not trivial. In this paper, we rely on the tensorial formalism recently proposed to characterize and investigate this kind of complex topologies, and extend two well known random walk centrality measures, the random walk betweenness and closeness centrality, to interconnected multilayer networks. For each of the measures we provide analytical expressions that completely agree with numerically results.