This talk will present an overview of the theoretical framework developed to unveil the hidden geometry of real complex networks. Since is a complete space, the sequence has a limit. Motivated by definition 1 we explore the following powerhop conjecture, which avoids the limitation to only spatially embedded networks. We model the network as a set of nodes in a distance metric space where.
Let be a cauchy sequence in the sequence of real numbers is a cauchy sequence check it. The geometric nature of weights in real complex networks. Statistical mechanics and its applications 386, 686691 2007. Oct 10, 2007 we demonstrate that the selfsimilarity of some scalefree networks with respect to a simple degreethresholding renormalization scheme finds a natural interpretation in the assumption that network nodes exist in hidden metric spaces. In the remaining sections, we will refer to this model as the hidden metric model hmm. A geometric approach to the structure of complex networks core. This hypothesis, combined with a suitable underlying space, has offered a geometric interpretation of the complex topologies observed in real networks, including scalefree degree distributions, the smallworld effect, strong clustering as a reflection of the triangle inequality, and selfsimilarity. The remaining complexity is encoded into the organization of the magnitude of such interactions i. First, the problem of link prediction is described in section ii. Our work is inspired by the definition of the fractal correlation dimension d 2 on a cloud of points. Geometry appears as the most plausible explanation of the complex architecture we observe in real networks. Jan 18, 2017 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. The model relies on the application of a geometric renormalization protocol which decreases the resolution by coarsegraining and averaging over short similarity distances.
M 2008 selfsimilarity of complex networks and hidden metric spaces. Mar 19, 2018 models of complex networks based on hidden metric spaces 14,15,16,17 open the door to a proper geometric definition of selfsimilarity and scale invariance and to an unfolding of the different. Selfsimilarity of complex networks and hidden metric spaces more. In some cases, such as airport networks, the embedding is explicit, while it is hidden in other cases such as the internet or metabolic networks. The fractal geometry of networks enables casting the selfsimilar symmetries underlying the organization of complex systems under the three pillars of scalin. It is known that realworld networks have lots of common features on structures, dynamic processes and functions.
The s1 class of hidden variable models with underlying metric spaces are able to accurately reproduce the observed topology and self. Dec 16, 2019 network geometry is also able to explain in a very natural way other nontrivial properties, like selfsimilarity 1, 4 and community structure 57, their navigability properties 810, and is the basis for the definition of a renormalization group in complex networks. Modeling and navigation of social information networks in. The hidden geometry of complex networks conference on. Correlation properties and selfsimilarity of renormalization email. Distance closures on complex networks network science. A survey on network embedding ieee transactions on knowledge and data engineering 2019 31 5 833 852 2 serrano m. Therefore, hidden metric spaces offer explanations of two open problems in complex networks science. Aug 18, 2020 strikingly, a geometric network model, where distances are not euclidean, predicts the multiscale properties of connectomes, including selfsimilarity.
The geometric nature of weights in real complex networks ncbi nih. Disorder in complex networks is manifested by the smallworld effect 1 and a highly heterogeneous degree distribution 2, both properties commonly present in real complex networks 3,4. Latent metric spaces with hyperbolic geometry provide a nat. We demonstrate that the selfsimilarity of some scalefree networks with respect to a simple degreethresholding renormalization scheme finds a natural interpretation in the assumption that network nodes exist in hidden metric spaces. Fractals, complex network, physical sciences, scale free network, fuzzy metric space, and hidden variables download. A geometric approach to the renormalization of complex networks. Scalefree degree distributions, clustering, smallworld property, communities, reciprocity and many more topological properties can be explained by the existence of a metric space controlling the network interactions.
Angeles serrano, dmitri krioukov, and marian boguna. Exploring the hidden metric space of complex networks. A geometric approach to the structure of complex networks. Greedy forwarding in dynamic scalefree networks embedded in. Distance closures on complex networks volume 3 issue 2. Modeling the selfsimilarity in complex networks based on. N2 complex networks have been studied extensively owing to their relevance to many real systems such as the worldwide web, the internet, energy landscapes and biological and social networks. Deciphering the global organization of clustering in real.
Multiscale unfolding of real networks by geometric. The scalefree property of complex networks can emerge as a consequence of the exponential expansion of hyperbolic space. The model employs the concept of a hidden metric space hms existing behind every complex network, including the internet. Link prediction in complex networks based on a hidden. It is still an intriguing problem for researchers to explain how and why so many obviously different networks emerge and share common intrinsic characteristics such as short diameter, higher cluster and powerlaw degree distribution. The geometric approach has also been successfully extended to weighted networks 12 and multiplexes, 14.
However, it is widely believed that complex networks are not. Order is, on the other hand, manifested by the presence of triangles or. Jan 15, 2019 hidden metric spaces of networks have been studied since about 2009. To explain these phenomena, in reference marian boguna et al. The project thoroughly investigates the hypothesis that the observable scalefree structure of complex networks is a consequence of natural evolution that maximizes the efficiency of greedy routing on these hmss. We prove that a class of hidden variable models with underlying metric spaces are able to accurately reproduce the self similarity properties. A metric space is called complete if every cauchy sequence converges to a limit. We show that similarity space defines a notion of scale in realworld networks. Here the authors extend this concept to weighted networks. Jan 12, 20 we are living in a world of various kinds of social information networks with smallworld and scalefree characteristics. If a subset of a metric space is not closed, this subset can not be sequentially compact. Most of the complexity of the socalled complex systems is encoded into the intricate topology of the networks defined by the interactions among their components. Pdf selfsimilarity of complex networks and hidden metric. Fractals, complex network, physical sciences, scale free network, fuzzy metric space, and hidden variables.
Selfsimilarity of complex networks and hidden metric spaces caida. While for networks embedded in a metric space the definitions can be applied almost unchanged 15, 16 this is not the case for the majority of the complex networks we. The geometric approach has also been successfully extended to weighted networks. Selfsimilarity of complex networks and hidden metric spaces. Figure 1 illustrates how an underlying hms influences the topological and functional properties of the graph built on top of it. Studying node centrality based on the hidden hyperbolic. Models of complex networks based on hidden metric spaces 1417 open the door to a proper geometric definition of self similarity and scale invariance and to. Greedy forwarding in dynamic scalefree networks embedded in hyperbolic metric spaces. The hidden metric spaces approach can be exploited beyond network metrics. Hidden geometric correlations in real multiplex networks. Network embedding is a frontier topic in current network science.
We prove that a class of hidden variable models with underlying metric spaces are able to accurately reproduce the selfsimilarity properties. Finally, mapping real complex networks into a hidden metric space has yielded a. Selfsimilarity of complex networks university of miami. Mar 14, 2016 eq 1 can be applied unchanged for networks embedded in a metric space where any of the distance metrics of the space e. Jun 01, 2016 in the next section, some complex networks are investigated and the results show that the minimum number of the boxes used to cover the networks and the box size follows a power law rule, which reflects the selfsimilarity of the complex networks.
The architecture of real complex systems lies between order and disorder, although its precise location is quite difficult to determine. Request pdf selfsimilarity of complex networks and hidden metric spaces we demonstrate that the selfsimilarity of some scalefree networks with respect to a simple degreethresholding. Geometric renormalization unravels selfsimilarity of the. We demonstrate that the selfsimilarity of some scalefree networks with respect to a simple degree thresholding renormalization scheme finds a natural interpretation in the assumption that network nodes exist in hidden metric spaces. Among other things, it will be shown how considering a hidden metric space in mathematical models naturally reproduces the strong clustering and the selfsimilarity of real complex networks, explains the navigability of the internet without an explicit knowledge of its global structure, and allows to shed light on the hierarchical organization. In the context of complex networks, the study of the multiscale problem 18, 19and related concepts like scale invariance and selfsimilarity 20is built upon the renormalization technique of statistical physics 21, 22, which successfully explained the universality of critical behavior in phase transitions 23. Uncovering the hidden geometry behind metabolic networks. We demonstrate that the self similarity of some scalefree networks with respect to a simple degreethresholding renormalization scheme finds a natural. Jan 18, 2017 complex networks have been conjectured to be hidden in metric spaces, which offer geometric interpretation of networks topologies. Aberasturi entitled selfsimilarity of random aggregation trees in hyperbolic spaces be accepted in partia. This modeling approach allows to reproduce a wide range of topological properties observed in real networks, such as selfsimilarity 15, high clustering coe cient 16, scalefree degree distribution 17, 18. Pdf navigability of complex networks marian boguna. Complex networks have been studied extensively owing to their relevance to many real systems such as the worldwide web, the internet, energy landscapes and biological and social networks.
Node similarity and hidden metric spacesour work is inspired by the seminal work of sociologist stanley milgram on the small world problem. Many real world complex networks are embedded in metric spaces. The remainder of this paper is organized as follows. The observed network topology is closely coupled to the hidden space geometry. The scalefree property of complex networks can emerge as a consequence of the. A subset is called net if a metric space is called totally bounded if finite net. Selfsimilarity of human protein interaction networks. We present a geometric renormalization group transformation that unveils a previously unknown selfsimilarity of real networks.
At the formal level, hidden metric spaces are variations of hidden variables 79. Pdf modeling and navigation of social information networks in. The geometric nature of weights in real complex networks antoine allard1,2,m. The observed network topology is coupled to the hidden space geometry in the following way. Hidden metric space network models couple the topology of a net.
In the case of complex networks these various dimensions carry information about many interesting underlying properties such as information diffusion and percolation 1114. Self similarity of complex networks and hidden metric space. Hidden metric spaces and observable network topology. We recommend that the thesis prepared under our supervision by dillon t. Self similarity of complex networks and hidden metric spaces. In the context of complex networks, the study of the multiscale problem 18, 19and related concepts like scale invariance and selfsimilarity 20is built upon the renormalization technique of statistical physics 21, 22, which successfully explained. An asymmetric popularitysimilarity optimization method for. M selfsimilarity of complex networks and hidden metric spaces. While for networks embedded in a metric space the definitions can be applied almost unchanged 15, 16 this is not the case for the majority of the complex networks we study. An asymmetric popularitysimilarity optimization method.
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