Real networks often form interacting parts of larger and more complex systems. Examples can be found in different domains, ranging from the Internet to structural and functional brain networks. Here, we show that these multiplex systems are not random combinations of single network layers. Instead, they are organized in specific ways dictated by hidden geometric correlations between the individual layers. We find that these correlations are strong in different real multiplexes, and form a key framework for answering many important questions. Specifically, we show that these geometric correlations facilitate: (i) the definition and detection of multidimensional communities, which are sets of nodes that are simultaneously similar in multiple layers; (ii) accurate trans-layer link prediction, where connections in one layer can be predicted by observing the hidden geometric space of another layer; and (iii) efficient targeted navigation in the multilayer system using only local knowledge, which outperforms navigation in the single layers only if the geometric correlations are sufficiently strong. Our findings uncover fundamental organizing principles behind real multiplexes and can have important applications in diverse domains.
Elements composing complex systems usually interact in several different ways and as such the interaction architecture is well modelled by a multiplex network. However often this architecture is hidden, as one usually only has experimental access to an aggregated projection. A fundamental challenge is thus to determine whether the hidden underlying architecture of complex systems is better modelled as a single interaction layer or results from the aggregation and interplay of multiple layers. Here we show that using local information provided by a random walker navigating the aggregated networ…
Recent progress towards unraveling the hidden geometric organization of real multiplexes revealed significant correlations across the hyperbolic node coordinates in different network layers, which facilitated applications like trans-layer link prediction and mutual navigation. But are geometric correlations alone sufficient to explain the topological relation between the layers of real systems? Here we provide the negative answer to this question. We show that connections in real systems tend to persist from one layer to another irrespectively of their hyperbolic distances. This suggests that …
We show that real multiplex networks are unexpectedly robust against targeted attacks on high degree nodes, and that hidden interlayer geometric correlations predict this robustness. Without geometric correlations, multiplexes exhibit an abrupt breakdown of mutual connectivity, even with interlayer degree correlations. With geometric correlations, we instead observe a multistep cascading process leading into a continuous transition, which apparently becomes fully continuous in the thermodynamic limit. Our results are important for the design of efficient protection strategies and of robust int…
Weak multiplex percolation generalizes percolation to multi-layer networks, represented as networks with a common set of nodes linked by multiple types (colors) of edges. We report a novel discontinuous phase transition in this problem. This anomalous transition occurs in networks of three or more layers without unconnected nodes, $P(0)=0$. Above a critical value of a control parameter, the removal of a tiny fraction $Δ$ of nodes or edges triggers a failure cascade which ends either with the total collapse of the network, or a return to stability with the system essentially intact. The discont…
Jilin University（吉林大学）；Hong Kong Baptist University（香港浸会大学）
Complex network theory aims to model and analyze complex systems that consist of multiple and interdependent components. Among all studies on complex networks, topological structure analysis is of the most fundamental importance, as it represents a natural route to understand the dynamics, as well as to synthesize or optimize the functions, of networks. A broad spectrum of network structural patterns have been respectively reported in the past decade, such as communities, multipartites, hubs, authorities, outliers, bow ties, and others. Here, we show that most individual real-world networks de…
A fundamental problem in network science is to predict how certain individuals are able to initiate new networks to spring up "new ideas". Frequently, these changes in trends are triggered by a few innovators who rapidly impose their ideas through "viral" influence spreading producing cascades of followers fragmenting an old network to create a new one. Typical examples include the raise of scientific ideas or abrupt changes in social media, like the raise of Facebook.com to the detriment of Myspace.com. How this process arises in practice has not been conclusively demonstrated. Here, we show …
In big data times, massive datasets often carry different relationships among the same group of nodes, analyzing on these heterogeneous relationships may give us a window to peek the essential relationships among nodes. In this paper, first of all we propose a new metric "similarity rate" in order to capture the changing rate of similarities between node-pairs though all networks; secondly, we try to use this new metric to uncover essential relationships between node-pairs which essential relationships are often hidden and hard to get. From experiments study of Indonesian Terrorists dataset, t…
In this study we present an extension of the replicator equation with diffusion to multiplex graphs. We derive an exact formula for the diffusion term, which shows that, while diffusion is linear for numbers of agents, it is necessary to account for non-linear terms when working with fractions of individuals. We also derive the transition probabilities that give rise to such macroscopic behavior, completing the bottom-up description. Finally, it shown that the usual assumption of constant population sizes induces a hidden selective pressure in the system.
A multiplex is a collection of network layers, each representing a specific type of edges. This appears to be a genuine representation for many real-world systems. However, due to a variety of potential factors, such as limited budget and equipment, or physical impossibility, multiplex data can be difficult to observe directly. Often, only partial information on the layer structure of the system is available, whereas the remaining information is in the form of a single-layer network. In this work, we face the problem of reconstructing the hidden multiplex structure of an aggregated network fro…
Complex networks are characterized by latent geometries induced by their topology or by the dynamics on the top of them. In the latter case, different network-driven processes induce distinct geometric features that can be captured by adequate metrics. Random walks, a proxy for a broad spectrum of processes, from simple contagion to metastable synchronization and consensus, have been recently used in [Phys. Rev. Lett. 118, 168301 (2017)] to define the class of diffusion geometry and pinpoint the functional mesoscale organization of complex networks from a genuine geometric perspective. Here, w…
We study urban public transport systems by means of multiplex networks in which stops are represented as nodes and each line is represented by a layer. We determine and visualize public transport network orientations and compare them with street network orientations of the $36$ largest German as well as $18$ selected major European cities. We find that German urban public transport networks are mainly oriented in a direction close to the cardinal east-west axis, which usually coincides with one of two orthogonal preferential directions of the corresponding street network. While this behavior i…
The fundamental idea of embedding a network in a metric space is rooted in the principle of proximity preservation. Nodes are mapped into points of the space with pairwise distance that reflects their proximity in the network. Popular methods employed in network embedding either rely on implicit approximations of the principle of proximity preservation or implement it by enforcing the geometry of the embedding space, thus hindering geometric properties that networks may spontaneously exhibit. Here, we take advantage of a model-free embedding method explicitly devised for preserving pairwise pr…
The topology of many real complex networks has been conjectured to be embedded in hidden metric spaces, where distances between nodes encode their likelihood of being connected. Besides of providing a natural geometrical interpretation of their complex topologies, this hypothesis yields the recipe for sustainable Internet's routing protocols, sheds light on the hierarchical organization of biochemical pathways in cells, and allows for a rich characterization of the evolution of international trade. We present empirical evidence that this geometric interpretation also applies to the weighted or…
Multilevel self-assembly involving small structured groups of nano-particles provides new routes to development of functional materials with a sophisticated architecture. Apart from the inter-particle forces, the geometrical shapes and compatibility of the building blocks are decisive factors in each phase of growth. Therefore, a comprehensive understanding of these processes is essential for the design of large assemblies of desired properties. Here, we introduce a computational model for cooperative self-assembly with simultaneous attachment of structured groups of particles, which can be de…
Long-range social pressure and the evolution of cooperation in multiplex networks
Universidad Politécnica de Madrid（马德里理工大学）；Grupo de Investigación Ingeniería de Organización y Logística (IOL), Departamento de Ingeniería de Organización, Administración de Empresas y Estadística, Escuela Técnica Superior de Ingenieros Industriales, Universidad Politécnica de Madrid（马德里理工大学工业工程师高等学院组织工程、企业管理与统计学系组织工程与物流研究组（IOL））；Grupo Interdisciplinar de Sistemas Complejos (GISC)（复杂系统跨学科研究组（GISC））
We show that the community structure of a network can be used as a coarse version of its embedding in a hidden space with hyperbolic geometry. The finding emerges from a systematic analysis of several real-world and synthetic networks. We take advantage of the analogy for reinterpreting results originally obtained through network hyperbolic embedding in terms of community structure only. First, we show that the robustness of a multiplex network can be controlled by tuning the correlation between the community structures across different layers. Second, we deploy an efficient greedy protocol fo…
Network embedding is a fervid topic in current networks science and observes that most real complex systems can be embedded in hidden metrics space and emerge as the geometrical property, where the geometric distance between nodes determines the likelihood of links connected. Among those, hyperbolic space associated with the structural organization of many real complex systems, it has thus received extensive attention. However, the majority of methods and measurements, recently developed, less take these features into consideration for the asymmetry of links. Here, we discuss how to multiplex …
