Ginestra Bianconi

The structure and dynamics of multilayer networks

Abstract In the past years, network theory has successfully characterized the interaction among the constituents of a variety of complex systems, ranging from biological to technological, and social systems. However, up until recently, attention was almost exclusively given to networks in which all components were treated on equivalent footing, while neglecting all the extra information about the temporal- or context-related properties of the interactions under study. Only in the last years, taking advantage of the enhanced resolution in real data sets, network scientists have directed their interest to the multiplex character of real-world systems, and explicitly considered the time-varying and multilayer nature of networks. We offer here a comprehensive review on both structural and dynamical organization of graphs made of diverse relationships (layers) between its constituents, and cover several relevant issues, from a full redefinition of the basic structural measures, to understanding how the multilayer nature of the network affects processes and dynamics.

Theory of rumour spreading in complex social networks

We introduce a general stochastic model for the spread of rumours, and derive mean-field equations that describe the dynamics of the model on complex social networks (in particular, those mediated by the Internet). We use analytical and numerical solutions of these equations to examine the threshold behaviour and dynamics of the model on several models of such networks: random graphs, uncorrelated scale-free networks and scale-free networks with assortative degree correlations. We show that in both homogeneous networks and random graphs the model exhibits a critical threshold in the rumour spreading rate below which a rumour cannot propagate in the system. In the case of scale-free networks, on the other hand, this threshold becomes vanishingly small in the limit of infinite system size. We find that the initial rate at which a rumour spreads is much higher in scale-free networks than in random graphs, and that the rate at which the spreading proceeds on scale-free networks is further increased when assortative degree correlations are introduced. The impact of degree correlations on the final fraction of nodes that ever hears a rumour, however, depends on the interplay between network topology and the rumour spreading rate. Our results show that scale-free social networks are prone to the spreading of rumours, just as they are to the spreading of infections. They are relevant to the spreading dynamics of chain emails, viral advertising and large-scale information dissemination algorithms on the Internet.

Statistical mechanics of multiplex networks: Entropy and overlap

There is growing interest in multiplex networks where individual nodes take part in several layers of networks simultaneously. This is the case, for example, in social networks where each individual node has different kinds of social ties or transportation systems where each location is connected to another location by different types of transport. Many of these multiplexes are characterized by a significant overlap of the links in different layers. In this paper we introduce a statistical mechanics framework to describe multiplex ensembles. A multiplex is a system formed by N nodes and M layers of interactions where each node belongs to the M layers at the same time. Each layer α is formed by a network G^{α}. Here we introduce the concept of correlated multiplex ensembles in which the existence of a link in one layer is correlated with the existence of a link in another layer. This implies that a typical multiplex of the ensemble can have a significant overlap of the links in the different layers. Moreover, we characterize microcanonical and canonical multiplex ensembles satisfying respectively hard and soft constraints and we discuss how to construct multiplexes in these ensembles. Finally, we provide the expression for the entropy of these ensembles that can be useful to address different inference problems involving multiplexes.

Growing multiplex networks.

We propose a modeling framework for growing multiplexes where a node can belong to different networks. We define new measures for multiplexes and we identify a number of relevant ingredients for modeling their evolution such as the coupling between the different layers and the distribution of node arrival times. The topology of the multiplex changes significantly in the different cases under consideration, with effects of the arrival time of nodes on the degree distribution, average shortest path length, and interdependence.

Entropy measures for networks: Toward an information theory of complex topologies

The quantification of the complexity of networks is, today, a fundamental problem in the physics of complex systems. A possible roadmap to solve the problem is via extending key concepts of information theory to networks. In this Rapid Communication we propose how to define the Shannon entropy of a network ensemble and how it relates to the Gibbs and von Neumann entropies of network ensembles. The quantities we introduce here will play a crucial role for the formulation of null models of networks through maximum-entropy arguments and will contribute to inference problems emerging in the field of complex networks.

Assessing the relevance of node features for network structure

Networks describe a variety of interacting complex systems in social science, biology, and information technology. Usually the nodes of real networks are identified not only by their connections but also by some other characteristics. Examples of characteristics of nodes can be age, gender, or nationality of a person in a social network, the abundance of proteins in the cell taking part in protein-interaction networks, or the geographical position of airports that are connected by directed flights. Integrating the information on the connections of each node with the information about its characteristics is crucial to discriminating between the essential and negligible characteristics of nodes for the structure of the network. In this paper we propose a general indicator Θ, based on entropy measures, to quantify the dependence of a networks structure on a given set of features. We apply this method to social networks of friendships in U.S. schools, to the protein-interaction network of Saccharomyces cerevisiae and to the U.S. airport network, showing that the proposed measure provides information that complements other known measures.

Evolution and control of oxygen order in a cuprate superconductor

The disposition of defects in metal oxides is a key attribute exploited for applications from fuel cells and catalysts to superconducting devices and memristors. The most typical defects are mobile excess oxygens and oxygen vacancies, which can be manipulated by a variety of thermal protocols as well as optical and d.c. electric fields. Here we report the X-ray writing of high-quality superconducting regions, derived from defect ordering, in the superoxygenated layered cuprate, La₂CuO(4+y). Irradiation of a poor superconductor prepared by rapid thermal quenching results first in the growth of ordered regions, with an enhancement of superconductivity becoming visible only after a waiting time, as is characteristic of other systems such as ferroelectrics, where strain must be accommodated for order to become extended. However, in La₂CuO(4+y), we are able to resolve all aspects of the growth of (oxygen) intercalant order, including an extraordinary excursion from low to high and back to low anisotropy of the ordered regions. We can also clearly associate the onset of high-quality superconductivity with defect ordering in two dimensions. Additional experiments with small beams demonstrate a photoresist-free, single-step strategy for writing functional materials.

Inhomogeneity of charge-density-wave order and quenched disorder in a high-Tc superconductor.

It has recently been established that the high-transition-temperature (high-Tc) superconducting state coexists with short-range charge-density-wave order and quenched disorder arising from dopants and strain. This complex, multiscale phase separation invites the development of theories of high-temperature superconductivity that include complexity. The nature of the spatial interplay between charge and dopant order that provides a basis for nanoscale phase separation remains a key open question, because experiments have yet to probe the unknown spatial distribution at both the nanoscale and mesoscale (between atomic and macroscopic scale). Here we report micro X-ray diffraction imaging of the spatial distribution of both short-range charge-density-wave ‘puddles’ (domains with only a few wavelengths) and quenched disorder in HgBa2CuO4 + y, the single-layer cuprate with the highest Tc, 95 kelvin (refs 26, 27, 28). We found that the charge-density-wave puddles, like the steam bubbles in boiling water, have a fat-tailed size distribution that is typical of self-organization near a critical point. However, the quenched disorder, which arises from oxygen interstitials, has a distribution that is contrary to the usually assumed random, uncorrelated distribution. The interstitial-oxygen-rich domains are spatially anticorrelated with the charge-density-wave domains, because higher doping does not favour the stripy charge-density-wave puddles, leading to a complex emergent geometry of the spatial landscape for superconductivity.

The entropy of randomized network ensembles

Randomized network ensembles are the null models of real networks and are extensively used to compare a real system to a null hypothesis. In this paper we study network ensembles with the same degree distribution, the same degree correlations and the same community structure of any given real network. We characterize these randomized network ensembles by their entropy, i.e. the normalized logarithm of the total number of networks which are part of these ensembles. We estimate the entropy of randomized ensembles starting from a large set of real directed and undirected networks. We propose entropy as an indicator to assess the role of each structural feature in a given real network. We observe that the ensembles with fixed scale-free degree distribution have smaller entropy than the ensembles with homogeneous degree distribution indicating a higher level of order in scale-free networks.

Entropy of network ensembles

In this paper we generalize the concept of random networks to describe network ensembles with nontrivial features by a statistical mechanics approach. This framework is able to describe undirected and directed network ensembles as well as weighted network ensembles. These networks might have nontrivial community structure or, in the case of networks embedded in a given space, they might have a link probability with a nontrivial dependence on the distance between the nodes. These ensembles are characterized by their entropy, which evaluates the cardinality of networks in the ensemble. In particular, in this paper we define and evaluate the structural entropy, i.e., the entropy of the ensembles of undirected uncorrelated simple networks with given degree sequence. We stress the apparent paradox that scale-free degree distributions are characterized by having small structural entropy while they are so widely encountered in natural, social, and technological complex systems. We propose a solution to the paradox by proving that scale-free degree distributions are the most likely degree distribution with the corresponding value of the structural entropy. Finally, the general framework we present in this paper is able to describe microcanonical ensembles of networks as well as canonical or hidden-variable network ensembles with significant implications for the formulation of network-constructing algorithms.