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RACE-OC (Rotation and ACtivity Evolution in Open Clusters) is a project aimed at characterising the rotational and magnetic activity properties of the late-type members of open clusters, stellar associations, and moving groups of different ages. As part of this project, in the present paper we present the results of an investigation of a likely member of the AB Doradus association, the M-type star 2MASS J15594729+4403595.} {In the present study, we aim to reveal the real nature of our target, which turned out to be a hierarchical triple system, to derive the stellar rotation period and surface differential rotation, and to characterise its photospheric magnetic activity.} {We have collected radial velocity and photometric time series, complemented with archive data, to determine the orbital parameters and the rotation period and we have used the spot modelling technique to explore what causes its photometric variability. \rm } {We found 2MASS J15594729 +4403595 to be a hierarchical triple system consisting of a dwarf, SB1 M2, and a companion, M8. The M2 star has a rotation period of P = 0.37\,d, making it the fastest among M-type members of AB Dor. The most relevant result is the detection of a periodic variation in the spotted area on opposite stellar hemispheres, which resembles a sort of Rossby wave or Rieger-like cycles on an extremely short timescale. Another interesting result is the occurrence of a highly significant photometric periodicity, P = 0.443\,d, which may be related to the stellar rotation in terms of either a Rossby wave or surface differential rotation.} {2MASS J15594729+4403595 may be the prototype of a new class of extremely fast rotating stars exibiting short Rieger-like cycles. We shall further explore what may drive these short-duration cycles and we shall also search for similar stars to allow for a statistical analysis., Comment: 14 pages, 12 figures

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 well-known 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 topology-feature 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 between 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.

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 self-similarity. Our results underline the importance of geometric effects on network topology even when the coupling to the underlying space is weak., Comment: 7 pages, 4 figures (Supplementary: 16 pages)

Almost strictly sign regular matrices are sign regular matrices with a special zero pattern and whose nontrivial minors are nonzero. In this paper we provide several properties of almost strictly sign regular rectangular matrices and analyze their QR factorization., Comment: 10 pages

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 $FiD$-Mercator, a model-based ultra-low dimensional reduction technique that integrates node features with network structure to create $D$-dimensional 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 network embedding techniques capable of exploiting such correlations to optimize both network structure and feature association jointly in the future.

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$\times$Similarity $\mathbb{S}^1$/$\mathbb{H}^2$ 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 quasi-geometric regime suggests that the transition between non-geometric and geometric networks is not a sharp one., Comment: 11 pages, 5 figures, 1 table

The architecture of the human connectome supports efficient communication protocols relying either on distances between brain regions or on the intensities of connections. However, none of these protocols combines information about the two or reaches full efficiency. Here, we introduce a continuous spectrum of decentralized routing strategies that combine link weights and the spatial embedding of connectomes to transmit signals. We applied the protocols to individual connectomes in two cohorts, and to cohort archetypes designed to capture weighted connectivity properties. We found that there is an intermediate region, a sweet spot, in which navigation achieves maximum communication efficiency at low transmission cost. Interestingly, this phenomenon is robust and independent of the particular configuration of weights.Our results indicate that the intensity and topology of neural connections and brain geometry interplay to boost communicability, fundamental to support effective responses to external and internal stimuli and the diversity of brain functions., Comment: 49 pages (10 main text, 39 Supplementary Material)

Graph-structured 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 paper, 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 hyperbolic geometric 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., Comment: 11 pages, 10 figures

The geometric renormalization technique for complex networks has successfully revealed the multiscale self-similarity of real network topologies and can be applied to generate replicas at different length scales. In this letter, we extend the geometric renormalization framework to weighted networks, where the intensities of the interactions play a crucial role in their structural organization and function. Our findings demonstrate that weights in real networks exhibit multiscale self-similarity under a renormalization protocol that selects the connections with the maximum weight across increasingly longer length scales. We present a theory that elucidates this symmetry, and that sustains the selection of the maximum weight as a meaningful procedure. Based on our results, scaled-down replicas of weighted networks can be straightforwardly derived, facilitating the investigation of various size-dependent phenomena in downstream applications., Comment: 5 pages, 3 figures

One of the pillars of the geometric approach to networks has been the development of model-based 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 $D$-Mercator, a model-based embedding method that produces multidimensional maps of real networks into the $(D+1)$-hyperbolic space, where the similarity subspace is represented as a $D$-sphere. We used $D$-Mercator 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.

First principle network models are crucial to make sense of the intricate topology of real complex networks. While modeling efforts have been quite successful in undirected networks, generative models for networks with asymmetric interactions are still not well developed and are unable to reproduce several basic topological properties. This is particularly disconcerting considering that real directed networks are the norm rather than the exception in many natural and human-made complex systems. In this paper, we fill this gap and show how the network geometry paradigm can be elegantly extended to the case of directed networks. We define a maximum entropy ensemble of geometric (directed) random graphs with a given sequence of in- and out-degrees. Beyond these local properties, the ensemble requires only two additional parameters to fix the level of reciprocity and the seven possible types of 3-node 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 complex directed networks.

Turing patterns, arising from the interplay between competing species of diffusive particles, has long been an important concept for describing non-equilibrium self-organization in nature, and has 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 scale-free degree distributions and the small world property, although with an apparent absence of geometric organization. While hints of explicitly geometric patterns in simple network models (e.g Watts-Strogatz) have been found, the question of the exact nature and morphology of geometric Turing patterns in heterogeneous complex networks remains unresolved. In this work, we study the Turing instability in the framework of geometric random graph models, where the network topology is explained by 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., Comment: 16 pages, 8 figures, 1 table

Radiation-induced embrittlement of nuclear steels is one of the main limiting factors for safe long-term operation of nuclear power plants. In support of accurate and safe reactor pressure vessel (RPV) lifetime assessments, we developed a physics-based model that predicts RPV steel hardening and subsequent embrittlement as a consequence of the formation of nano-sized clusters of minor alloying elements. This model is shown to provide reliable assessments of embrittlement for a very wide range of materials, with higher accuracy than industrial correlations. The core of our model is a multiscale modelling tool that predicts the kinetics of solute clustering, given the steel chemical composition and its irradiation conditions. It is based on the observation that the formation of solute clusters ensues from atomic transport driven by radiation-induced mechanisms, differently from classical nucleation-and-growth theories. We then show that the predicted information about solute clustering can be translated into a reliable estimate for radiation-induced embrittlement, via standard hardening laws based on the dispersed barrier model. We demonstrate the validity of our approach by applying it to hundreds of nuclear reactors vessels from all over the world., Comment: 22 pages, 9 figures

Reducing dimension redundancy to find simplifying patterns in high-dimensional 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 networks from different domains and found unexpected regularities, including: tissue-specific 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.

We introduce the Multiscale Voter Model (MVM) to investigate clan influence at multiple scale -- family, neighborhood, political party... -- in opinion formation on real complex networks. Clans, consisting of similar nodes, are constructed using a coarse-graining procedure on network embeddings that allows us to control for the length scale of interactions. We ran numerical simulations to monitor the evolution of MVM dynamics in real and synthetic networks, and identified a transition between a final stage of full consensus and one with mixed binary opinions. The transition depends on the scale of the clans and on the strength of their influence. We found that enhancing group diversity promotes consensus while strong kinship yields to metastable clusters of same opinion. The segregated domains, which signal opinion polarization, are discernible as spatial patterns in the hyperbolic embeddings of the networks. Our multiscale framework can be easily applied to other dynamical processes affected by scale and group influence., Comment: 3 Figures. Requires Supplemental

Clustering $\unicode{x2013}$ the tendency for neighbors of nodes to be connected $\unicode{x2013}$ 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 finite size 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., Comment: 18 pages, 4 figures (Supplementary: 31 pages)

Silicon-substituted hydroxyapatite (SiHA) macroporous scaffolds have been prepared by robocasting. In order to optimize their bone regeneration properties, we have manufactured these scaffolds presenting different microstructures: nanocrystalline and crystalline. Moreover, their surfaces have been decorated with vascular endothelial growth factor (VEGF) to evaluate the potential coupling between vascularization and bone regeneration. In vitro cell culture tests evidence that nanocrystalline SiHA hinders pre-osteblast proliferation, whereas the presence of VEGF enhances the biological functions of both endothelial cells and pre-osteoblasts. The bone regeneration capability has been evaluated using an osteoporotic sheep model. In vivo observations strongly correlate with in vitro cell culture tests. Those scaffolds made of nanocrystalline SiHA were colonized by fibrous tissue, promoted inflammatory response and fostered osteoclast recruitment. These observations discard nanocystalline SiHA as a suitable material for bone regeneration purposes. On the contrary, those scaffolds made of crystalline SiHA and decorated with VEGF exhibited bone regeneration properties, with high ossification degree, thicker trabeculae and higher presence of osteoblasts and blood vessels. Considering these results, macroporous scaffolds made of SiHA and decorated with VEGF are suitable bone grafts for regeneration purposes, even in adverse pathological scenarios such as osteoporosis., Comment: 23 pages

The osteogenic and angiogenic responses to metal macroporous scaffolds coated with silicon substituted hydroxyapatite (SiHA) and decorated with vascular endothelial growth factor (VEGF) have been evaluated in vitro and in vivo. Ti6Al4V-ELI scaffolds were prepared by electron beam melting and subsequently coated with Ca10(PO4)5.6(SiO4)0.4(OH)1.6 following a dip coating method. In vitro studies demonstrated that SiHA stimulates the proliferation of MC3T3-E1 pre-osteoblastic cells, whereas the adsorption of VEGF stimulates the proliferation of EC2 mature endothelial cells. In vivo studies were carried out in an osteoporotic sheep model, evidencing that only the simultaneous presence of both components led to a significant increase of new tissue formation in osteoporotic bone., Comment: 39 pages, 8 figures

Hierarchies permeate the structure of real networks, whose nodes can be ranked according to different features. However, networks are far from tree-like structures and the detection of hierarchical ordering remains a challenge, hindered by the small-world property and the presence of a large number of cycles, in particular clustering. Here, we use geometric representations of undirected networks to achieve an enriched interpretation of hierarchy that integrates features defining popularity of nodes and similarity between them, such that the more similar a node is to a less popular neighbor the higher the hierarchical load of the relationship. The geometric approach allows us to measure the local contribution of nodes and links to the hierarchy within a unified framework. Additionally, we propose a link filtering method, the similarity filter, able to extract hierarchical backbones containing the links that represent statistically significant deviations with respect to the maximum entropy null model for geometric heterogeneous networks. We applied our geometric approach to the detection of similarity backbones of real networks in different domains and found that the backbones preserve local topological features at all scales. Interestingly, we also found that similarity backbones favor cooperation in evolutionary dynamics modelling social dilemmas., Comment: 13 pages, 5 figures. Supplementary material available as appendix

Real networks are finite metric spaces. Yet the geometry induced by shortest path distances in a network is definitely not its only geometry. Other forms of network geometry 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 all intimately related, and all three of them have been found to be exceptionally efficient in discovering fractality, scale-invariance, self-similarity, and other forms of fundamental symmetries in networks. Network geometry is also of great utility in a variety of practical applications, ranging from the understanding how the brain works, to routing in the Internet. Here, we review the most important theoretical and practical developments dealing with these approaches to network geometry in the last two decades, and offer perspectives on future research directions and challenges in this novel frontier in the study of complexity.

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 self-similar growth of network structure in the evolution of real systems and present the Geometric Branching Growth model, which predicts this evolution and explains the symmetries observed. The model produces multiscale unfolding of a network in a sequence of scaled-up 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 finite-size scaling to assess critical behavior under random link failures., Comment: 30 pages, 29 figures

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 of spatial network models, only a few reproduce even the most basic properties of real-world networks. Here, we focus on three such properties---sparsity, small worldness, and clustering---and identify the general subclass of spatial homogeneous and heterogeneous network models that are sparse small worlds 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 dependence on spatial distances determines network small worldness and clustering.

Structural connectivity in the brain is typically studied by reducing its observation to a single spatial resolution. However, the brain possesses a rich architecture organized over multiple scales linked to one another. We explored the multiscale organization of human connectomes using datasets of healthy subjects reconstructed at five different resolutions. We found that the structure of the human brain remains self-similar when the resolution of observation is progressively decreased by hierarchical coarse-graining of the anatomical regions. Strikingly, a geometric network model, where distances are not Euclidean, predicts the multiscale properties of connectomes, including self-similarity. The model relies on the application of a geometric renormalization protocol which decreases the resolution by coarse-graining and averaging over short similarity distances. Our results suggest that simple organizing principles underlie the multiscale architecture of human structural brain networks, where the same connectivity law dictates short- and long-range connections between different brain regions over many resolutions. The implications are varied and can be substantial for fundamental debates, such as whether the brain is working near a critical point, as well as for applications including advanced tools to simplify the digital reconstruction and simulation of the brain.

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$\times$Similarity $\mathbb{S}^1/\mathbb{H}^2$ static geometric network model, which can accommodate arbitrary degree distributions and reproduces many pivotal properties of real networks, including self-similarity 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 between the observed network topology and the geometric model. In its fast mode, Mercator uses a model-adjusted 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 fast-mode embedding result is taken as an initial condition in a Maximum 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 parameters, 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 model-dependent framework can boost the meaningful mapping of complex networks.

Predicting missing links in real networks is an important problem in network science to which considerable efforts have been devoted, giving as a result a vast plethora of link prediction methods in the literature. In this work, we take a different point of view on the problem and study the theoretical limitations to the predictability of missing links. In particular, we hypothesise that there is an irreducible uncertainty in link prediction on real networks as a consequence of the random nature of their formation process. By considering ensembles defined by well-known network models, we prove analytically that even the best possible link prediction method for an ensemble, given by the ranking of the ensemble connection probabilities, yields a limited precision. This result suggests a theoretical limitation to the predictability of links in real complex networks. Finally, we show that connection probabilities inferred by fitting network models to real networks allow to estimate an upper-bound to the predictability of missing links, and we further propose a method to approximate such bound from incomplete instances of real-world networks.

The increasing integration of world economies, which organize in complex multilayer networks of interactions, is one of the critical factors for the global propagation of economic crises. We adopt the network science approach to quantify shock propagation on the global trade-investment multiplex network. To this aim, we propose a model that couples a Susceptible-Infected-Recovered epidemic spreading dynamics, describing how economic distress propagates between connected countries, with an internal contagion mechanism, describing the spreading of such economic distress within a given country. At the local level, we find that the interplay between trade and financial interactions influences the vulnerabilities of countries to shocks. At the large scale, we find a simple linear relation between the relative magnitude of a shock in a country and its global impact on the whole economic system, albeit the strength of internal contagion is country-dependent and the intercountry propagation dynamics is non-linear. Interestingly, this systemic impact can be predicted on the basis of intra-layer and inter-layer scale factors that we name network multipliers, that are independent of the magnitude of the initial shock. Our model sets-up a quantitative framework to stress-test the robustness of individual countries and of the world economy to propagating crashes.

We introduce a model for the randomization of complex networks with geometric structure. The geometric randomization (GR) model assumes a homogeneous distribution of the nodes in an underlying similarity space and uses rewirings of the links to find configurations that maximize a connection probability akin to that of the $\mathbb{S}^1$ or $\mathbb{H}^2$ geometric network models. However, GR preserves the original degree sequence, as in the configuration model, thus eliminating the fluctuations of the degree cutoff. Moreover, the model does not require the explicit estimation of hidden degree variables, which restricts the number of free parameters to one, controlling the level of clustering in the rewired network. We illustrate the potential of GR as a null model by investigating the effects on modularity that derive from the flattening of geometric communities in both real and synthetic networks. As a result, we find that for real networks the geometric and topological communities are consistent, while for the randomized counterparts, the topological communities detected are attributable to structural constraints induced by the underlying geometric architecture., Comment: 8 pages, 6 figures

Connectomes are spatially embedded networks whose architecture has been shaped by physical constraints and communication needs throughout evolution. Using a decentralized navigation protocol, we investigate the relationship between the structure of the connectomes of different species and their spatial layout. As a navigation strategy, we use greedy routing where nearest neighbors, in terms of geometric distance, are visited. We measure the fraction of successful greedy paths and their length as compared to shortest paths in the topology of connectomes. In Euclidean space, we find a striking difference between the navigability properties of mammalian and non-mammalian species, which implies the inability of Euclidean distances to fully explain the structural organization of their connectomes. In contrast, we find that hyperbolic space, the effective geometry of complex networks, provides almost perfectly navigable maps of connectomes for all species, meaning that hyperbolic distances are exceptionally congruent with the structure of connectomes. Hyperbolic maps therefore offer a quantitative meaningful representation of connectomes that suggests a new cartography of the brain based on the combination of its connectivity with its effective geometry rather than on its anatomy only. Hyperbolic maps also provide a universal framework to study decentralized communication processes in connectomes of different species and at different scales on an equal footing., Comment: 20 pages, 5 figures, 2 supp. tables, 3 supp. appendices, 10 supp. figures

Information routing is one of the main tasks in many complex networks with a communication function. Maps produced by embedding the networks in hyperbolic space can assist this task enabling the implementation of efficient navigation strategies. However, only static maps have been considered so far, while navigation in more realistic situations, where the network structure may vary in time, remain largely unexplored. Here, we analyze the navigability of real networks by using greedy routing in hyperbolic space, where the nodes are subject to a stochastic activation-inactivation dynamics. We find that such dynamics enhances navigability with respect to the static case. Interestingly, there exists an optimal intermediate activation value, which ensures the best trade-off between the increase in the number of successful paths and a limited growth of their length. Contrary to expectations, the enhanced navigability is robust even when the most connected nodes inactivate with very high probability. Finally, our results indicate that some real networks are ultranavigable and remain highly navigable even if the network structure is extremely unsteady. These findings have important implications for the design and evaluation of efficient routing protocols that account for the temporal nature of real complex networks., Comment: 10 pages, 4 figures. Includes Supplemental Information

The $\mathbb{S}^1$ model has been a central geometric model in the development of the field of network geometry. It has been mainly studied in its homogeneous regime, in which angular coordinates are independently and uniformly scattered on the circle. We now investigate if the model can generate networks with targeted topological features and soft communities, that is, heterogeneous angular distributions. Under these circumstances, hidden degrees must depend on angular coordinates and we propose a method to estimate them. We conclude that the model can be topologically invariant with respect to the soft-community structure. Our results might have important implications, both in expanding the scope of the model beyond the independent hidden variables limit and in the embedding of real-world networks.

The most frequent form of pairwise synthetic lethality (SL) in metabolic networks is known as plasticity synthetic lethality (PSL). It occurs when the simultaneous inhibition of paired functional and silent metabolic reactions or genes is lethal, while the default of the functional reaction or gene in the pair is backed up by the activation of the silent one. Based on a complex systems approach and by using computational techniques on bacterial genome-scale metabolic reconstructions, we found that the failure of the functional PSL partner triggers a critical reorganization of fluxes to ensure viability in the mutant which not only affects the SL pair but a significant fraction of other interconnected reactions forming what we call a SL cluster. Interestingly, SL clusters show a strong entanglement both in terms of silent coessential reactions, which band together to form backup systems, and of other functional and silent reactions in the metabolic network. This strong overlap, also detected at the level of genes, mitigates the acquired vulnerabilities and increased structural and functional costs that pay for the robustness provided by essential plasticity. Finally, the participation of coessential reactions and genes in different SL clusters is very heterogeneous and those at the intersection of many SL clusters could serve as supertargets for more efficient drug action in the treatment of complex diseases and to elucidate improved strategies directed to reduce undesired resistance to chemicals in pathogens., Comment: 10 pages, 4 figures, 15 supplementary figures

Multiple scales coexist in complex networks. However, the small world property makes them strongly entangled. This turns the elucidation of length scales and symmetries a defiant challenge. Here, we define a geometric renormalization group for complex networks and use the technique to investigate networks as viewed at different scales. We find that real networks embedded in a hidden metric space show geometric scaling, in agreement with the renormalizability of the underlying geometric model. This allows us to unfold real scale-free networks in a self-similar multilayer shell which unveils the coexisting scales and their interplay. The multiscale unfolding offers a basis for a new approach to explore critical phenomena and universality in complex networks, and affords us immediate practical applications, like high-fidelity smaller-scale replicas of large networks and a multiscale navigation protocol in hyperbolic space which boosts the success of single-layer versions.

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 interacting networks in many domains., Comment: Supplementary Materials and Videos: https://koljakleineberg.wordpress.com/2017/02/08/new-paper-geometric-correlations-mitigate-the-extreme-vulnerability-of-multiplex-networks-against-targeted-attacks/

We intended to compile the most complete catalog of bona fide members and candidate members of the beta Pictoris association, and to measure their rotation periods and basic properties from our own observations, public archives, and exploring the literature. We carried out a multi-observatories campaign to get our own photometric time series and collected all archived public photometric data time series for the stars in our catalog. Each time series was analyzed with the Lomb-Scargle and CLEAN periodograms to search for the stellar rotation periods. We complemented the measured rotational properties with detailed information on multiplicity, membership, and projected rotational velocity available in the literature and discussed star by star. We measured the rotation periods of 112 out of 117 among bona fide members and candidate members of the beta Pictoris association and, whenever possible, we also measured the luminosity, radius, and inclination of the stellar rotation axis. This represents to date the largest catalog of rotation periods of any young loose stellar association. We provided an extensive catalog of rotation periods together with other relevant basic properties useful to explore a number of open issues, such as the causes of spread of rotation periods among coeval stars, evolution of angular momentum, and lithium-rotation connection., Comment: Forthcoming article, Received: 20 June 2016 / Accepted: 09 September 2016; 40 pages, 2 figures. The online figures A1-A73 are available at CDS

Alloreactivity following stem cell transplantation (SCT) is difficult to predict in patients undergoing transplantation from HLA matched donors. In this study we performed whole exome sequencing of SCT donor-recipient pairs (DRP). This allowed determination of entire library of alloreactive peptide sequences which would bind HLA class I molecules in each DRP. Utilizing the HLA binding affinity (IC50) and tissue expression levels of the parent proteins, an aggregate donor T cell response to the recipient alloreactive peptides was calculated using a vector-operator dynamical system model. Marked variability in the simulated CD8+ T cell responses was observed in all the donor recipient pairs.

The combination of bistability and noise is ubiquitous in complex systems, from biological to social interactions, and has important implications for their functioning and resilience. We analyze a simple three-state model for bistability in networks under varying unbiased noise. In a fully connected network increasing noise yields a collapse of bistability to an unpolarised state. In contrast, in complex networks noise can abruptly switch the polarization state in an irreversible way. When two networks are combined through increasing multiplex coupling, one is dominant and progressively imposes its state on the other, offsetting or promoting the ability of noise to switch polarization. Our results show that dynamical correlations and asymmetry in dynamical processes in networks are sufficient for allowing unbiased noise to produce abrupt irreversible transitions between extremes, which can be neutralized or enhanced by multiplex coupling., Comment: main text: 5 pages, 4 figures supplement: 15 pages, 15 figures

The NJL model for the octet baryons, using proper time regularization to simulate some of the features of confinement, is solved self-consistently in nuclear matter. This provides an alternative framework to the MIT bag model which has been used in the quark-meson coupling model. After fitting the parameters of the model to the saturation properties of symmetric nuclear matter the model is used to explore the equation of state of pure neutron matter as well as nuclear matter at densities relevant to heavy ion collisions. With a view to future studies of high mass neutron stars, the binding of hyperons is also explored., Comment: 7 pages and 4 figures

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., Comment: Supplementary Materials available at http://www.nature.com/nphys/journal/v12/n11/extref/nphys3812-s1.pdf

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 organisation 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., Comment: Major revisions since the previous version. 9 pages, 4 figures (Supplementary: 33 pages, 41 figures)

Here, we present the World Trade Atlas 1870-2013, a collection of annual world trade maps in which distance combines economic size and the different dimensions that affect international trade beyond mere geography. Trade distances, which are based on a gravity model predicting the existence of significant trade channels, are such that the closer countries are in trade space, the greater their chance of becoming connected. The atlas provides us with information regarding the long-term evolution of the international trade system and demonstrates that, in terms of trade, the world is not flat but hyperbolic, as a reflection of its complex architecture. The departure from flatness has been increasing since World War I, meaning that differences in trade distances are growing and trade networks are becoming more hierarchical. Smaller-scale economies are moving away from other countries except for the largest economies; meanwhile those large economies are increasing their chances of becoming connected worldwide. At the same time, Preferential Trade Agreements do not fit in perfectly with natural communities within the trade space and have not necessarily reduced internal trade barriers. We discuss an interpretation in terms of globalization, hierarchization, and localization; three simultaneous forces that shape the international trade system., Comment: World Trade Atlas 1870-2013 interactive tool at http://morfeo.ffn.ub.edu/wta1870-2013

We study the Susceptible-Infected-Susceptible model of epidemic spreading on two layers of networks interconnected by adaptive links, which are rewired at random to avoid contacts between infected and susceptible nodes at the interlayer. We find that the rewiring reduces the effective connectivity for the transmission of the disease between layers, and may even totally decouple the networks. Weak endemic states, in which the epidemics spreads only if the two layers are interconnected, show a transition from the endemic to the healthy phase when the rewiring overcomes a threshold value that depends on the infection rate, the strength of the coupling and the mean connectivity of the networks. In the strong endemic scenario, in which the epidemics is able to spread on each separate network, the prevalence in each layer decreases when increasing the rewiring, arriving to single network values only in the limit of infinitely fast rewiring. We also find that finite-size effects are amplified by the rewiring, as there is a finite probability that the epidemics stays confined in only one network during its lifetime., Comment: 15 pages, 11 figures

Stem cell transplants may be considered as dynamical systems to allow sequence differences across the exomes of the transplant donors and recipients to be used to simulate an alloreactive T cell response. Whole exome sequencing was performed on HLA matched stem cell transplant donor-recipient pairs, and the nucleotide sequence differences translated to peptides. The binding affinity of the peptides to the relevant HLA in each pair was determined. The resulting array of peptide-HLA binding affinity values in each patient was used to simulate an alloreactive donor derived T cell repertoire. This simulated T cell repertoire reproduces a number of features of clinically observed T cell repertoire. The simulated, alloreactive T cell repertoire was markedly different in HLA matched stem cell transplant donors and recipients and demonstrates a possible correlation with survival., Comment: 50 pages (including references and supplementary materials), 12 figures, 3 supplementary figures

We deduce and discuss the implications of self-similarity for the stability in terms of robustness to failure of multiplexes, depending on interlayer degree correlations. First, we define self-similarity of multiplexes and we illustrate the concept in practice using the configuration model ensemble. Circumscribing robustness to survival of the mutually percolated state, we find a new explanation based on self-similarity both for the observed fragility of interconnected systems of networks and for their robustness to failure when interlayer degree correlations are present. Extending the self-similarity arguments, we show that interlayer degree correlations can change completely the stability properties of self-similar multiplexes, so that they can even recover a zero percolation threshold and a continuous transition in the thermodynamic limit, qualitatively exhibiting thus the ordinary stability attributes of noninteracting networks. We confirm these results with numerical simulations., Comment: arXiv admin note: text overlap with arXiv:1010.5793

The heterogeneity of reaction fluxes present in a metabolic network within a single flux state can be exploited to construct the so-called backbone as a reduced version of metabolism. The backbone maintains all significant fluxes producing or consuming metabolites while displaying a substantially decreased number of interconnections and, hence, it becomes a useful tool to extract primary metabolic routes. Here, we disclose the metabolic backbone of Escherichia coli using the computationally predicted fluxes which maximize the growth rate in glucose minimal medium, and we compare it with the backbone of Mycoplasma pneumoniae, a much simpler organism. We find that the central core in both backbones is mainly composed of reactions in ancient pathways, still playing at present a key role in energy metabolism. In E. coli, the analysis of the backbone reveals that the synthesis of nucleotides and the metabolism of lipids form smaller cores which rely critically on energy metabolism; but not conversely. At the same time, an analysis of the dependence of this backbone on media composition leads to the identification of pathways sensitive to environmental changes. The metabolic backbone of an organism is thus useful to trace simultaneously both its evolution and adaptation fingerprints.

We prove that complex networks of interactions have the capacity to regulate and buffer unpredictable fluctuations in production events. We show that non-bursty network-driven activation dynamics can effectively regulate the level of burstiness in the production of nodes, which can be enhanced or reduced. Burstiness can be induced even when the endogenous inter-event time distribution of nodes' production is non-bursty. We found that hubs tend to be less controllable than low degree nodes, which are more susceptible to the networked regulatory effects. Our results have important implications for the analysis and engineering of bursty activity in a range of systems, from telecommunication networks to transcription and translation of genes into proteins in cells.

Experimental and empirical observations on cell metabolism cannot be understood as a whole without their integration into a consistent systematic framework. However, the characterization of metabolic flux phenotypes is typically reduced to the study of a single optimal state, like maximum biomass yield that is by far the most common assumption. Here we confront optimal growth solutions to the whole set of feasible flux phenotypes (FFP), which provides a benchmark to assess the likelihood of optimal and high-growth states and their agreement with experimental results. In addition, FFP maps are able to uncover metabolic behaviors, such as aerobic fermentation accompanying exponential growth on sugars at nutrient excess conditions, that are unreachable using standard models based on optimality principles. The information content of the full FFP space provides us with a map to explore and evaluate metabolic behavior and capabilities, and so it opens new avenues for biotechnological and biomedical applications.

Natural numbers can be divided in two non-overlapping infinite sets, primes and composites, with composites factorizing into primes. Despite their apparent simplicity, the elucidation of the architecture of natural numbers with primes as building blocks remains elusive. Here, we propose a new approach to decoding the architecture of natural numbers based on complex networks and stochastic processes theory. We introduce a parameter-free non-Markovian dynamical model that naturally generates random primes and their relation with composite numbers with remarkable accuracy. Our model satisfies the prime number theorem as an emerging property and a refined version of Cram\'er's conjecture about the statistics of gaps between consecutive primes that seems closer to reality than the original Cram\'er's version. Regarding composites, the model helps us to derive the prime factors counting function, giving the probability of distinct prime factors for any integer. Probabilistic models like ours can help to get deeper insights about primes and the complex architecture of natural numbers.

We present a simple and general framework to simulate statistically correct realizations of a system of non-Markovian discrete stochastic processes. We give the exact analytical solution and a practical an efficient algorithm alike the Gillespie algorithm for Markovian processes, with the difference that now the occurrence rates of the events depend on the time elapsed since the event last took place. We use our non-Markovian generalized Gillespie stochastic simulation methodology to investigate the effects of non-exponential inter-event time distributions in the susceptible-infected-susceptible model of epidemic spreading. Strikingly, our results unveil the drastic effects that very subtle differences in the modeling of non-Markovian processes have on the global behavior of complex systems, with important implications for their understanding and prediction. We also assess our generalized Gillespie algorithm on a system of biochemical reactions with time delays. As compared to other existing methods, we find that the generalized Gillespie algorithm is the most general as it can be implemented very easily in cases, like for delays coupled to the evolution of the system, where other algorithms do not work or need adapted versions, less efficient in computational terms., Comment: Improvement of the algorithm, new results, and a major reorganization of the paper thanks to our coauthors L. Lafuerza and R. Toral

We unravel how functional plasticity and redundancy are essential mechanisms underlying the ability to survive of metabolic networks. We perform an exhaustive computational screening of synthetic lethal reaction pairs in Escherichia coli in a minimal medium and we find that synthetic lethal pairs divide in two different groups depending on whether the synthetic lethal interaction works as a backup or as a parallel use mechanism, the first corresponding to essential plasticity and the second to essential redundancy. In E. coli, the analysis of pathways entanglement through essential redundancy supports the view that synthetic lethality affects preferentially a single function or pathway. In contrast, essential plasticity, the dominant class, tends to be inter-pathway but strongly localized and unveils Cell Envelope Biosynthesis as an essential backup for Membrane Lipid Metabolism. When comparing E. coli and Mycoplasma pneumoniae, we find that the metabolic networks of the two organisms exhibit a large difference in the relative importance of plasticity and redundancy which is consistent with the conjecture that plasticity is a sophisticated mechanism that requires a complex organization. Finally, coessential reaction pairs are explored in different environmental conditions to uncover the interplay between the two mechanisms. We find that synthetic lethal interactions and their classification in plasticity and redundancy are basically insensitive to medium composition, and are highly conserved even when the environment is enriched with nonessential compounds or overconstrained to decrease maximum biomass formation., Comment: 22 pages, 4 figures

We uncover the global organization of clustering in real complex networks. As it happens with other fundamental properties of networks such as the degree distribution, we find that real networks are neither completely random nor ordered with respect to clustering, although they tend to be closer to maximally random architectures. We reach this conclusion by comparing the global structure of clustering in real networks with that in maximally random and in maximally ordered clustered graphs. The former are produced with an exponential random graph model that maintains correlations among adjacent edges at the minimum needed to conform with the expected clustering spectrum; the later with a random model that arranges triangles in cliques inducing highly ordered structures. To compare the global organization of clustering in real and model networks, we compute $m$-core landscapes, where the $m$-core is defined, akin to the $k$-core, as the maximal subgraph with edges participating at least in $m$ triangles. This property defines a set of nested subgraphs that, contrarily to $k$-cores, is able to distinguish between hierarchical and modular architectures. To visualize the $m$-core decomposition we developed the LaNet-vi 3.0 tool.