MariĆ”n BoguĆ±Ć” Espinal
Marián Boguñá Espinal (Barcelona, 1967) is a full professor at the Departament de Física de la Matèria Condensada of the Universitat de Barcelona. He graduated in Physics in 1994 and obtained his PhD also in Physics in 1998. In 1999, he moved to the USA to do a postdoctoral stay with Professor George H. Weiss at the National Institutes of Health, Washington DC. After this period, he moved back to Barcelona where, in 2003, he was awarded a Ramón y Cajal fellowship. He got the tenure position at the end of 2008. During this period, he has also spent several months in the USA as invited guest scientist at Indiana University. M. Boguñá has written over 70 publications in major peer reviewed international scientific journals, book chapters, and conference proceedings. Among those, Nature, Nature Physics, Nature Communications, Proceedings of the National Academy of Sciences US, Physical Review Letters, and Physical Review X. He was the chair of the international conference BCNetWORKSHOP 2008 Trends and Perspectives in Complex Networks and has served as a program committee member in many international conferences. In January 2008, he obtained the Outstanding Referee award of the American Physical Society. In December 2010 and 2015, he was awarded as ICREA Academia researcher 2010 and 2015, respectively. Since January 2013 he serves as an editorial board member for Scientific Reports.
Hyperbolic geometry of complex networks
Krioukov, D; Papadopoulos, F; Kitsak, M; Vahdat, A; Boguna, M
Physical Review E
82
36106
(2010)
We develop a geometric framework to study the structure and function of complex networks. We assume that hyperbolic geometry underlies these networks, and we show that with this assumption, heterogeneous degree distributions and strong clustering in complex networks emerge naturally as simple reflections of the negative curvature and metric property of the underlying hyperbolic geometry. Conversely, we show that if a network has some metric structure, and if the network degree distribution is heterogeneous, then the network has an effective hyperbolic geometry underneath. We then establish a mapping between our geometric framework and statistical mechanics of complex networks. This mapping interprets edges in a network as noninteracting fermions whose energies are hyperbolic distances between nodes, while the auxiliary fields coupled to edges are linear functions of these energies or distances. The geometric network ensemble subsumes the standard configuration model and classical random graphs as two limiting cases with degenerate geometric structures. Finally, we show that targeted transport processes without global topology knowledge, made possible by our geometric framework, are maximally efficient, according to all efficiency measures, in networks with strongest heterogeneity and clustering, and that this efficiency is remarkably robust with respect to even catastrophic disturbances and damages to the network structure.
Absence of Epidemic Threshold in ScaleFree Networks with Degree Correlations
Boguna, M; PastorSatorras, R; Vespignani, A
PHYSICAL REVIEW LETTERS
90
28701
(2003)
Random scalefree networks have the peculiar property of being prone to the spreading of infections. Here we provide for the susceptibleinfectedsusceptible model an exact result showing that a scalefree degree distribution with diverging second moment is a sufficient condition to have null epidemic threshold in unstructured networks with either assortative or disassortative mixing. Degree correlations result therefore irrelevant for the epidemic spreading picture in these scalefree networks. The present result is related to the divergence of the average nearest neighbor´s degree, enforced by the degree detailed balance condition.
Extracting the multiscale backbone of complex weighted networks
Serrano, MA; Boguna, M; Vespignani, A
PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA
106
64836488
(2009)
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A large number of complex systems find a natural abstraction in the form of weighted networks whose nodes represent the elements of the system and the weighted edges identify the presence of an interaction and its relative strength. In recent years, the study of an increasing number of largescale networks has highlighted the statistical heterogeneity of their interaction pattern, with degree and weight distributions that vary over many orders of magnitude. These features, along with the large number of elements and links, make the extraction of the truly relevant connections forming the network´s backbone a very challenging problem. More specifically, coarsegraining approaches and filtering techniques come into conflict with the multiscale nature of largescale systems. Here, we define a filtering method that offers a practical procedure to extract the relevant connection backbone in complex multiscale networks, preserving the edges that represent statistically significant deviations with respect to a null model for the local assignment of weights to edges. An important aspect of the method is that it does not belittle smallscale interactions and operates at all scales defined by the weight distribution. We apply our method to realworld network instances and compare the obtained results with alternative backbone extraction techniques.
Popularity versus similarity in growing networks
Fragkiskos Papadopoulos, Maksim Kitsak, M. Ćngeles Serrano, MariĆ”n BoguĆ±Ć”, and Dmitri Krioukov
Nature
489
537540
(2012)
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The principle that ‘popularity is attractive’ underlies preferential attachment, which is a common explanation for the emergence of scaling in growing networks. If new connections are made preferentially to more popular nodes, then the resulting distribution of the number of connections possessed by nodes follows power laws, as observed in many real networks. Preferential attachment has been directly validated for some real networks (including the Internet), and can be a consequence of different underlying processes based on node fitness, ranking, optimization, random walks or duplication. Here we show that popularity is just one dimension of attractiveness; another dimension is similarity. We develop a framework in which new connections optimize certain tradeoffs between popularity and similarity, instead of simply preferring popular nodes.The framework has a geometric interpretation in which popularity preference emerges from local optimization. As opposed to preferential attachment, our optimization framework accurately describes the largescale evolution of technological (the Internet), social (trust relationships between people) and biological (Escherichia coli metabolic) networks, predicting the probability of new links with high precision.The framework that we have developed can thus be used for predicting new links in evolving networks, and provides a different perspective on preferential attachment as an emergent phenomenon.
Class of correlated random networks with hidden variables
Boguna, M; PastorSatorras, R
PHYSICAL REVIEW E
68
36112
(2003)
We study a class of models of correlated random networks in which vertices are characterized by hidden variables controlling the establishment of edges between pairs of vertices. We find analytical expressions for the main topological properties of these models as a function of the distribution of hidden variables and the probability of connecting vertices. The expressions obtained are checked by means of numerical simulations in a particular example. The general model is extended to describe a practical algorithm to generate random networks with an a priori specified correlation structure. We also present an extension of the class, to map nonequilibrium growing networks to networks with hidden variables that represent the time at which each vertex was introduced in the system.
Navigability of complex networks
Boguna, M; Krioukov, D; Claffy, KC
Nature Physics
5
7480
(2009)
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Routing information through networks is a universal phenomenon in both natural and manmade complex systems. When each node has full knowledge of the global network connectivity, finding short communication paths is merely a matter of distributed computation. However, in many real networks, nodes communicate efficiently even without such global intelligence. Here, we show that the peculiar structural characteristics of many complex networks support efficient communication without global knowledge. We also describe a general mechanism that explains this connection between network structure and function. This mechanism relies on the presence of a metric space hidden behind an observable network. Our findings suggest that real networks in nature have underlying metric spaces that remain undiscovered. Their discovery should have practical applications in a wide range of areas where networks are used to model complex systems.
Sustaining the Internet with hyperbolic mapping
BoguĆ±Ć”, M; Papadopoulos, F; Krioukov, D
Nature Communications
1
62
(2010)
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The Internet infrastructure is severely stressed. Rapidly growing overheads associated with the primary function of the Internet—routing information packets between any two computers in the world—cause concerns among Internet experts that the existing Internet routing architecture may not sustain even another decade. In this paper, we present a method to map the Internet to a hyperbolic space. Guided by a constructed map, which we release with this paper, Internet routing exhibits scaling properties that are theoretically close to the best possible, thus resolving serious scaling limitations that the Internet faces today. Besides this immediate practical viability, our network mapping method can provide a different perspective on the community structure in complex networks.
Epidemic spreading on interconnected networks
Anna SaumellMendiola, M. Ćngeles Serrano, and MariĆ”n BoguĆ±Ć”
Physical Review E
86
026106
(2012)
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Many real networks are not isolated from each other but form networks of networks, often interrelated in non trivial ways. Here, we analyze an epidemic spreading process taking place on top of two interconnected complex networks. We develop a heterogeneous mean field approach that allows us to calculate the conditions for the emergence of an endemic state. Interestingly, a global endemic state may arise in the coupled system even though the epidemics is not able to propagate on each network separately, and even when the number of coupling connections is small. Our analytic results are successfully confronted against largescale numerical simulations.
Selfsimilarity of complex networks and hidden metric spaces
Serrano, MA; Krioukov, D; Boguna, M
PHYSICAL REVIEW LETTERS
100
78701
(2008)
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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. Clustering, i.e., cycles of length three, plays a crucial role in this framework as a topological reflection of the triangle inequality in the hidden geometry. We prove that a class of hidden variable models with underlying metric spaces are able to accurately reproduce the selfsimilarity properties that we measured in the real networks. Our findings indicate that hidden geometries underlying these real networks are a plausible explanation for their observed topologies and, in particular, for their selfsimilarity with respect to the degreebased renormalization.
Nature of the Epidemic Threshold for the SusceptibleInfectedSusceptible Dynamics in Networks
MariĆ”n BoguĆ±Ć”, Claudio Castellano, and Romualdo PastorSatorras
Physical Review Letters
111
068701
(2013)
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We develop an analytical approach to the susceptibleinfectedsusceptible epidemic model that allows us to unravel the true origin of the absence of an epidemic threshold in heterogeneous networks. We find that a delicate balance between the number of high degree nodes in the network and the topological distance between them dictates the existence or absence of such a threshold. In particular, smallworld random networks with a degree distribution decaying slower than an exponential have a vanishing epidemic threshold in the thermodynamic limit.
Percolation and Epidemic Thresholds in Clustered Networks
Serrano, MA; Boguna, M
PHYSICAL REVIEW LETTERS
97
88701
(2006)
We develop a theoretical approach to percolation in random clustered networks. We find that, although clustering in scalefree networks can strongly affect some percolation properties, such as the size and the resilience of the giant connected component, it cannot restore a finite percolation threshold. In turn, this implies the absence of an epidemic threshold in this class of networks, thus extending this result to a wide variety of real scalefree networks which shows a high level of transitivity. Our findings are in good agreement with numerical simulations.
Network Cosmology
Dmitri Krioukov, Maksim Kitsak, Robert S. Sinkovits, David Rideout, David Meyer, MariĆ”n BoguĆ±Ć”
Scientific Reports
2
793
(2012)
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Prediction and control of the dynamics of complex networks is a central problem in network science. Structural and dynamical similarities of different real networks suggest that some universal laws might accurately describe the dynamics of these networks, albeit the nature and common origin of such laws remain elusive. Here we show that the causal network representing the largescale structure of spacetime in our accelerating universe is a powerlaw graph with strong clustering, similar to many complex networks such as the Internet, social, or biological networks. We prove that this structural similarity is a consequence of the asymptotic equivalence between the largescale growth dynamics of complex networks and causal networks. This equivalence suggests that unexpectedly similar laws govern the dynamics of complex networks and spacetime in the universe, with implications to network science and cosmology.
Navigating Ultrasmall Worlds in Ultrashort Time
Boguna, M; Krioukov, D
PHYSICAL REVIEW LETTERS
102
58701
(2009)
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Random scalefree networks are ultrasmall worlds. The average length of the shortest paths in networks of size N scales as lnlnN. Here we show that these ultrasmall worlds can be navigated in ultrashort time. Greedy routing on scalefree networks embedded in metric spaces finds paths with the average length scaling also as lnlnN. Greedy routing uses only local information to navigate a network. Nevertheless, it finds asymptotically the shortest paths, a direct computation of which requires global topology knowledge. Our findings imply that the peculiar structure of complex networks ensures that the lack of global topological awareness has asymptotically no impact on the length of communication paths. These results have important consequences for communication systems such as the Internet, where maintaining knowledge of current topology is a major scalability bottleneck.
Hidden geometric correlations in real multiplex networks
KajKolja Kleineberg, MariĆ”n BoguĆ±Ć”, M. Ćngeles Serrano, and Fragkiskos Papadopoulos
Nature Physics
12
1076 1081
(2016)
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Real networks often form interacting parts of larger and more complex systems. Examples can be found in di erent 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 layers. We find that these correlations are significant in di erent real multiplexes, and form a key framework for answering many important questions. Specifically, we show that these geometric correlations facilitate the definition and detection of multidimensional communities, which are sets of nodes that are simultaneously similar in multiple layers. They also enable accurate translayer link prediction, meaning that connections in one layer can be predicted by observing the hidden geometric space of another layer. And they allow e cient targeted navigation in the multilayer system using only local knowledge, outperforming navigation in the single layers only if the geometric correlations are su ciently strong.
Double percolation phase transition in clustered complex networks
Pol ColomerdeSimĆ³n and MariĆ”n BoguĆ±Ć”
Physical Review X
4
041020
(2014)
The internal organization of complex networks often has striking consequences on either their response to external perturbations or on their dynamical properties. In addition to smallworld and scalefree properties, clustering is the most common topological characteristic observed in many real networked systems. In this paper, we report an extensive numerical study on the effects of clustering on the structural properties of complex networks. Strong clustering in heterogeneous networks induces the emergence of a coreperiphery organization that has a critical effect on the percolation properties of the networks. We observe a novel double phase transition with an intermediate phase in which only the core of the network is percolated and a final phase in which the periphery percolates regardless of the core. This result implies breaking of the same symmetry at two different values of the control parameter, in stark contrast to the modern theory of continuous phase transitions. Inspired by this coreperiphery organization, we introduce a simple model that allows us to analytically prove that such an anomalous phase transition is in fact possible.
The geometric nature of weights in real complex networks
Antoine Allard, M. Ćngeles Serrano, Guillermo GarcĆaPĆ©rez, and MariĆ”n BoguĆ±Ć”
Nature Communications
8
14103
(2017)
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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 character ization of the evolution of international trade. Here we present empirical evidence that this geometric interpretation also applies to the weighted organization of real complex networks. We introduce a very general and versatile model and use it to quantify the level of coupling between their topology, their weights and an underlying metric space. Our model accurately reproduces both their topology and their weights, and our results suggest that the formation of connections and the assignment of their magnitude are ruled by different processes.
Percolation in selfsimilar networks
M. Ćngeles Serrano, Dmitri Krioukov, and MariĆ”n Boguna
Physical Review Letters
106
048701
(2011)
We provide a simple proof that graphs in a general class of selfsimilar networks have zero percolation threshold. The considered selfsimilar networks include random scalefree graphs with given expected node degrees and zero clustering, scalefree graphs with finite clustering and metric structure, growing scalefree networks, and many real networks. The proof and the derivation of the giant component size do not require the assumption that networks are treelike. Our results rely only on the observation that selfsimilar networks possess a hierarchy of nested subgraphs whose average degree grows with their depth in the hierarchy. We conjecture that this property is pivotal for percolation in networks.
Geometric correlations mitigate the extreme vulnerability of multiplex networks against targeted attacks
KajKolja Kleineberg, Lubos Buzna, Fragkiskos Papadopoulos, MariĆ”n BoguĆ±Ć”, and M. Ćngeles Serrano
Physical Review Letters
118
218301
(2017)
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We show that real multiplex networks are unexpectedly robust against targeted attacks on highdegree 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 interacting networks in many domains.
Mercator: uncovering faithful hyperbolic embeddings of complex networks
Guillermo GarcĆaPĆ©rez, Antoine Allard, M. Ćngeles Serrano and MariĆ”n BoguĆ±Ć”
New Journal of Physics
21
123033
(2019)
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We introduce Mercator, a reliable embedding method to map real complex networks into their hyperbolic latent geometry. The method assumes that the structure of networks is well described by the Popularity×Similarity S1/H2 static geometric network model, which can accommodate ar bitrary degree distributions and reproduces many pivotal properties of real networks, including selfsimilarity patterns. The algorithm mixes machine learning and maximum likelihood approaches to infer the coordinates of the nodes in the underlying hyperbolic disk with the best matching be tween the observed network topology and the geometric model. In its fast mode, Mercator uses a modeladjusted machine learning technique performing dimensional reduction to produce a fast and accurate map, whose quality already outperform other embedding algorithms in the literature. In the refined Mercator mode, the fastmode embedding result is taken as an initial condition in a Max imum Likelihood estimation, which significantly improves the quality of the final embedding. Apart from its accuracy as an embedding tool, Mercator has the clear advantage of systematically inferring not only node orderings, or angular positions, but also the hidden degrees and global model parame ters, and has the ability to embed networks with arbitrary degree distributions. Overall, our results suggest that mixing machine learning and maximum likelihood techniques in a modeldependent framework can boost the meaningful mapping of complex networks.
Network geometry
MariĆ”n BoguĆ±Ć”, Ivan Bonamassa, Manlio De Domenico, Shlomo Havlin, Dmitri Krioukov, and M. Ćngeles Serrano
Nature Reviews Physics
3
114 135
(2021)
Networks are finite metric spaces, with distances defined by the shortest paths between nodes. However, this is not the only form of network geometry: two others are the geometry of latent spaces underlying many networks and the effective geometry induced by dynamical processes in networks. These three approaches to network geometry are intimately related, and all three of them have been found to be exceptionally efficient in discovering fractality, scale invariance, selfsimilarity and other forms of fundamental symmetries in networks. Network geometry is also of great use in a variety of practical applications, from understanding how the brain works to routing in the Internet. We review the most important theoretical and practical developments dealing with these approaches to network geometry and offer perspectives on future research directions and challenges in this frontier in the study of complexity.
Small worlds and clustering in spatial networks
MariĆ”n BoguĆ±Ć”, Dmitri Krioukov, Pedro Almagro, and M. Ćngeles Serrano
Physical Review Research
2
023040
(2020)
Networks with underlying metric spaces attract increasing research attention in network science,statistical physics, applied mathematics, computer science, sociology, and other fields. This attention is further amplified by the current surge of activity in graph embedding. In the vast realm ofspatial network models, only a few reproduce even the most basic properties of realworld networks.Here, we focus on three such properties—sparsity, small worldness, and clustering—and identify thegeneral subclass of spatial homogeneous and heterogeneous network models that are sparse smallworlds and that have nonzero clustering in the thermodynamic limit. We rely on the maximum entropy approach where network links correspond to noninteracting fermions whose energy dependenceon spatial distances determines network small worldness and clustering.
Memoryinduced complex contagion in epidemic spreading
Xavier R. Hoffmann and MariĆ”n BoguĆ±Ć”
New Journal of Physics
21
033034
(2019)
Albeit epidemic models have evolved into powerful predictive tools for the spread of diseases andopinions, most assume memoryless agents and independent transmission channels. We develop an infection mechanism that is endowed with memory of past exposures and simultaneously incorporatesthe joint effect of multiple infectious sources. Analytic equations and simulations of the susceptibleinfectedsusceptible model in unstructured substrates reveal the emergence of an additional phasethat separates the usual healthy and endemic ones. This intermediate phase shows fundamentallydistinct characteristics, and the system exhibits either excitability or an exotic variant of bistability. Moreover, the transition to endemicity presents hybrid aspects. These features are the product of anintricate balance between two memory modes and indicate that nonMarkovian effects significantlyalter the properties of spreading processes
Quantifying Human Engagement into Playful Activities
David Reguera, Pol ColomerdeSimĆ³n, IvĆ”n Encinas, Manel Sort, Jan Wedekind, and MariĆ”n BoguĆ±Ć”
Sci. Rep
10
4145
(2020)
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Engaging in playful activities, such as playing a musical instrument, learning a language, or per forming sports, is a fundamental aspect of human life. We present a quantitative empirical analysis of the engagement dynamics into playful activities. We do so by analyzing the behavior of millions of players of casual video games and discover a scaling law governing the engagement dynamics. This powerlaw behavior is indicative of a multiplicative (i.e., happy gethappier) mechanism of engagement characterized by a set of critical exponents. We also find, depending on the critical exponents, that there is a phase transition between the standard case where all individuals even tually quit the activity and another phase where a finite fraction of individuals never abandon the activity. The behavior that we have uncovered in this work might not be restricted only to human interaction with videogames. Instead, we believe it reflects a more general and profound behavior of how humans become engaged in challenging activities with intrinsic rewards.
Scaling up real networks by geometric branching growth
Muhua Zheng, Guillermo GarcĆaPĆ©rez, MariĆ”n BoguĆ±Ć”, and M. Ćngeles Serrano
Proc. Nat. Acad. Sci.
118(21)
e2018994118
(2021)
Real networks often grow through the sequential addition of new nodes that connect to older ones in the graph. However, many real systems evolve through the branching of fundamental units, whether those be scientific fields, countries, or species. Here, we provide empirical evidence for selfsimilar growth of network structure in the evolution of real systems—the journalcitation network and the world trade web—and present the geomet ric branching growth model, which predicts this evolution and explains the symmetries observed. The model produces multiscale unfolding of a network in a sequence of scaledup replicas preserving network features, including clustering and community structure, at all scales. Practical applications in real instances include the tuning of network size for best response to external influence and finitesize scaling to assess critical behavior under random link failures.
An anomalous topological phase transition in spatial random graphs
Jasper van der Kolk, M. Ćngeles Serrano, and MariĆ”n BoguĆ±Ć”
Communications Physics
5
245
(2022)
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Clustering–the tendency for neighbors of nodes to be connected–quantifies the coupling of a complex network to its latent metric space. In random geometric graphs, clustering undergoes a continuous phase transition, separating a phase with finite clustering from a regime where clustering vanishes in the thermodynamic limit. We prove this geometric to nongeometric phase transition to be topological in nature, with anomalous features such as diverging entropy as well as atypical finitesize scaling behavior of clustering. Moreover, a slow decay of clustering in the nongeometric phase implies that some real networks with relatively high levels of clustering may be better described in this regime.
Detecting the ultra low dimensionality of real networks
Pedro Almagro, MariĆ”n BoguĆ±Ć”, and M. Ćngeles Serrano
Nature Communications
13
6096
(2022)
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Reducing dimension redundancy to find simplifying patterns in highdimensional datasets and complex networks has become a major endeavor in many scientific fields. However, detecting the dimensionality of their latent space is challenging but necessary to generate efficient embeddings to be used in a multitude of downstream tasks. Here, we propose a method to infer the dimensionality of networks without the need for any a priori spatial embedding. Due to the ability of hyperbolic geometry to capture the complex connectivity of real networks, we detect ultra low dimensionality far below values reported using other approaches. We applied our method to real networksfrom different domains and found unexpected regularities, including: tissuespecific biomolecular networks being extremely low dimensional; brain connectomes being close to the three dimensions of their anatomical embedding; and social networks and the Internet requiring slightly higher dimensionality. Beyond paving the way towards an ultra efficient dimensional reduction, our findings help address fundamental issues that hinge on dimensionality, such as universality in critical behavior.
The Shortest Path to Network Geometry
M. Ćngeles Serrano and MariĆ”n BoguĆ±Ć”
Elements in Structure and Dynamics of Complex Networks. Cambridge University Press
(2021)
Real networks comprise from hundreds to millions of interacting elements and permeate all contexts, from technology to biology to society. All of them display nontrivial connectivity patterns, including the smallworld phenomenon, making nodes to be separated by a small number of intermediate links. As a consequence, networks present an apparent lack of metric structure and are difficult to map. Yet, many networks have a hidden geometry that enables meaningful maps in the twodimensional hyperbolic plane. The discovery of such hidden geometry and the understanding of its role have become fundamental questions in network science giving rise to the field of network geometry. This Element reviews fundamental models and methods for the geometric description of real networks with a focus on applications of real network maps, including decentralized routing protocols, geometric community detection, and the selfsimilar multiscale unfolding of networks by geometric renormalization.
Emergence of geometric Turing patterns in complex networks
Jasper van der Kolk, Guillermo GarcĆaPĆ©rez, Nikos E. Kouvaris, M. Ćngeles Serrano, and MariĆ”n BoguĆ±Ć”
Physical Review X
13
021038
(2023)
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Turing patterns, arising from the interplay between competing species of diffusive particles, have long been an important concept for describing nonequilibrium selforganization in nature and have been extensively investigated in many chemical and biological systems. Historically, these patterns have been studied in extended systems and lattices. Recently, the Turing instability was found to produce topological patterns in networks with scalefree degree distributions and the smallworld property, although with an apparent absence of geometric organization. While hints of explicitly geometric patterns in simple network models (e.g., WattsStrogatz) have been found, the question of the exact nature and morphology of geometric Turing patterns in heterogeneous complex networks remained unresolved. In this work, we study the Turing instability in the framework of geometric random graph models, where the network topology is explained using an underlying geometric space. We demonstrate that not only can geometric patterns be observed, their wavelength can also be estimated by studying the eigenvectors of the annealed graph Laplacian. Finally, we show that Turing patterns can be found in geometric embeddings of real networks. These results indicate that there is a profound connection between the function of a network and its hidden geometry, even when the associated dynamical processes are exclusively determined by the network topology.
Geometric description of clustering in directed networks
Antoine Allard, M. Ćngeles Serrano, and MariĆ”n BoguĆ±Ć”
Nature Physics
https://doi.org/10.1038/s41567023022466
(2023)
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Firstprinciple network models are crucial to understanding the intricate topology of real complex networks. Although modelling efforts have been quite successful in undirected networks, generative models for networks with asymmetric interactions are still not well developed and unable to reproduce several basic topological properties. Progress in this direction is of particular interest, as real directed networks are the norm rather than the exception in many natural and humanmade complex systems. Here we show how the network geometry paradigm can be extended to the case of directed networks. We define a maximum entropy ensemble of random geometric directed graphs with a given sequence of indegrees and outdegrees. Beyond these local properties, the ensemble requires only two additional parameters to fix the levels of reciprocity and the frequency of the seven possible types of threenode cycles in directed networks. A systematic comparison with several representative empirical datasets shows that fixing the level of reciprocity alongside the coupling with an underlying geometry is able to reproduce the wide diversity of clustering patterns observed in real directed complex networks.
DMercator: multidimensional hyperbolic embedding of real networks
Robert Jankowski, Antoine Allard, MariĆ”n BoguĆ±Ć”, and M. Ćngeles Serrano
Nature Communications
14
7585
(2023)
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One of the pillars of the geometric approach to networks has been the development of modelbased mapping tools that embed real networks in its latent geometry. In particular, the tool Mercator embeds networks into the hyperbolic plane. However, some real networks are better described by the multidimensional formulation of the underlying geometric model. Here, we introduce DMercator, a modelbased embedding method that produces multidimensional maps of real networks into the (D + 1)hyperbolic space, where the similarity subspace is represented as a Dsphere. We used DMercator to produce multidimensional hyperbolic maps of real networks and estimated their intrinsic dimensionality in terms of navigability and community structure. Multidimensional representations of real networks are instrumental in the identification of factors that determine connectivity and in elucidating fundamental issues that hinge on dimensionality, such as the presence of universality in critical behavior.
Featureenriched network geometry explains graphstructured data
Roya Aliakbarisani, M. Ćngeles Serrano, and MariĆ”n BoguĆ±Ć”
arXiv:2307.14198
(2023)
Graphstructured data provide a comprehensive description of complex systems, encompassing not only the interactions among nodes but also the intrinsic features that characterize these nodes. These features play a fundamental role in the formation of links within the network, making them valuable for extracting meaningful topological information. Notably, features are at the core of deep learning techniques such as Graph Convolutional Neural Networks (GCNs) and offer great utility in tasks like node classification, link prediction, and graph clustering. In this letter, we present a comprehensive framework that treats features as tangible entities and establishes a bipartite graph connecting nodes and features. By assuming that nodes sharing similarities should also share features, we introduce a geometric similarity space where both nodes and features coexist, shaping the structure of both the node network and the bipartite network of nodes and features. Through this framework, we can identify correlations between nodes and features in real data and generate synthetic datasets that mimic the topological properties of their connectivity patterns. The approach provides insights into the inner workings of GCNs by revealing the intricate structure of the data.
Random graphs and real networks with weak geometric coupling
Jasper van der Kolk, M. Ćngeles Serrano, and MariĆ”n BoguĆ±Ć”
Physical Review Research
6
013337
(2024)
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Geometry can be used to explain many properties commonly observed in real networks. It is therefore often assumed that real networks, especially those with high average local clustering, live in an underlying hidden geometric space. However, it has been shown that finite size effects can also induce substantial clustering, even when the coupling to this space is weak or non existent. In this paper, we study the weakly geometric regime, where clustering is absent in the thermodynamic limit but present in finite systems. Extending Mercator, a network embedding tool based on the Popularity×Similarity S1/H2 static geometric network model, we show that, even when the coupling to the geometric space is weak, geometric information can be recovered from the connectivity alone. The fact that several real networks are best described in this quasigeometric regime suggests that the transition between nongeometric and geometric networks is not a sharp one.
Featureaware ultralow dimensional reduction of real networks
Robert Jankowski, Pegah Hozhabrierdi, MariĆ”n BoguĆ±Ć”, and M. Ćngeles Serrano
arXiv:2401.09368
(2024)
In existing models and embedding methods of networked systems, node features describing their qualities are usually overlooked in favor of focusing solely on node connectivity. This study introduces FiDMercator, a modelbased ultralow dimensional reduction technique that integrates node features with network structure to create Ddimensional maps of complex networks in a hyperbolic space. This embedding method efficiently uses features as an initial condition, guiding the search of nodes’ coordinates towards an optimal solution. The research reveals that downstream task performance improves with the correlation between network connectivity and features, emphasizing the importance of such correlation for enhancing the description and predictability of real networks. Simultaneously, hyperbolic embedding’s ability to reproduce local network properties remains unaffected by the inclusion of features. The findings highlight the necessity for developing novel network embedding techniques capable of exploiting such correlations to optimize both network structure and feature association jointly in the future.
Measuring spatial distances in causal sets via causal overlaps
MariĆ”n BoguĆ±Ć” and Dmitri Krioukov
Physical Review D
110
024008
(2024)
Causal set theory is perhaps the most minimalistic approach to quantum gravity, in the sense that it makes next to zero assumptions about the structure of spacetime below the Planck scale. Yet even with this minimalism, the continuum limit is still a major challenge in causal sets. One aspect of this challenge is the measurement of distances in causal sets. While the definition and estimation of timelike distances is relatively straightforward, dealing with spacelike distances is much more problematic. Here we introduce an approach to measure distances between spacelike separated events based on their causal overlap. We show that the distance estimation errors in this approach vanish in the continuum limit even for smallest distances of the order of the Planck length. These results are expected to inform the causal set geometrogenesis in general, and in particular the development of evolving causal set models in which space emerges from causal dynamics.
Renormalization of networks with weak geometric coupling
Jasper van der Kolk, MariĆ”n BoguĆ±Ć”, and M. Ćngeles Serrano
arXiv:2403.12663
(2024)
The Renormalization Group is crucial for understanding systems across scales, including complex networks. Renormalizing networks via network geometry, a framework in which their topology is based on the location of nodes in a hidden metric space, is one of the foundational approaches. However, the current methods assume that the geometric coupling is strong, neglecting weak coupling in many real networks. This paper extends renormalization to weak geometric coupling, showing that geometric information is essential to preserve selfsimilarity. Our results underline the importance of geometric effects on network topology even when the coupling to the underlying space is weak
Hyperbolic Benchmarking Unveils Network TopologyFeature Relationship in GNN Performance
Roya Aliakbarisani, Robert Jankowski, M. Ćngeles Serrano, and MariĆ”n BoguĆ±Ć”
arXiv:2406.02772
(2024)
Graph Neural Networks (GNNs) have excelled in predicting graph properties in various applications ranging from identifying trends in social networks to drug discovery and malware detection. With the abundance of new architectures and increased complexity, GNNs are becoming highly specialized when tested on a few wellknown datasets. However, how the performance of GNNs depends on the topological and features properties of graphs is still an open question. In this work, we introduce a comprehensive benchmarking framework for graph machine learning, focusing on the performance of GNNs across varied network structures. Utilizing the geometric soft configuration model in hyperbolic space, we generate synthetic networks with realistic topological properties and node feature vectors. This approach enables us to assess the impact of network properties, such as topologyfeature correlation, degree distributions, local density of triangles (or clustering), and homophily, on the effectiveness of different GNN architectures. Our results highlight the dependency of model performance on the interplay be tween network structure and node features, providing insights for model selection in various scenarios. This study contributes to the field by offering a versatile tool for evaluating GNNs, thereby assisting in developing and selecting suitable models based on specific data characteristics.

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