Optimal navigability of weighted human brain connectomes in physical space.

Communication protocols in the brain connectome describe how to transfer information from one region to another. Typically, these protocols hinge on either the spatial distances between brain regions or the intensity of their connections. Yet, none of them combine both factors to achieve optimal efficiency. Here, we introduce a continuous spectrum of decentralized routing strategies that integrates link weights and the spatial embedding of connectomes to route signal transmission. We implemented the protocols on connectomes from individuals in two cohorts and on group-representative connectomes designed to capture weighted connectivity properties. We identified an intermediate domain of routing strategies, a sweet spot, where navigation achieves maximum communication efficiency at low transmission cost. This phenomenon is robust and independent of the particular configuration of weights. Our findings suggest an interplay between the intensity of neural connections and their topology and geometry that amplifies communicability, where weights play the role of noise in a stochastic resonance phenomenon. Such enhancement may support more effective responses to external and internal stimuli, underscoring the intricate diversity of brain functions.

The D-Mercator method for the multidimensional hyperbolic embedding of real networks.

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.

Detecting the ultra low dimensionality of real networks.

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.

Erratum: Precision as a measure of predictability of missing links in real networks [Phys. Rev. E 101, 052318 (2020)].

This corrects the article DOI: 10.1103/PhysRevE.101.052318.

Scaling up real networks by geometric branching growth.

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-the journal-citation network and the world trade web-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.

Geometric renormalization unravels self-similarity of the multiscale human connectome.

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.

Precision as a measure of predictability of missing links in real networks.

Predicting missing links in real networks is an important open 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 focus on predictability instead of prediction. By considering ensembles defined by well-known network models, we prove analytically that even the best possible link prediction method, given by the ensemble connection probabilities, yields a limited precision that depends quantitatively on the topological properties-such as degree heterogeneity, clustering, and community structure-of the ensemble. This suggests an absolute limitation to the predictability of missing links in real networks, due to the irreducible uncertainty arising from the random nature of link formation processes. We show that a predictability limit can be estimated in real networks, and we propose a method to approximate such a bound from real-world networks with missing links. The predictability limit gives a benchmark to gauge the quality of link prediction methods in real networks.

Navigable maps of structural brain networks across species.

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.

The interconnected wealth of nations: Shock propagation on global trade-investment multiplex 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 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 inter-country propagation dynamics is non-linear. Interestingly, this systemic impact can be associated to 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.

Metabolic plasticity in synthetic lethal mutants: Viability at higher cost.

The most frequent form of pairwise synthetic lethality (SL) in metabolic networks is known as plasticity synthetic lethality. It occurs when the simultaneous inhibition of paired functional and silent metabolic reactions or genes is lethal, while the default of the functional partner is backed up by the activation of the silent one. Using computational techniques on bacterial genome-scale metabolic reconstructions, we found that the failure of the functional 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 reactions and genes. This strong overlap 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.

Navigability of temporal networks in hyperbolic space.

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, remains 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.

Geometric Correlations Mitigate the Extreme Vulnerability of Multiplex Networks against Targeted Attacks.

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.

Noise-induced polarization switching in complex networks.

The combination of bistability and noise is ubiquitous in complex systems, from biology to social interactions, and has important implications for their functioning and resilience. Here we use a simple three-state dynamical process, in which nodes go from one pole to another through an intermediate state, to show that noise can induce polarization switching in bistable systems if dynamical correlations are significant. In large, fully connected networks, where dynamical correlations can be neglected, increasing noise yields a collapse of bistability to an unpolarized configuration where the three possible states of the nodes are equally likely. In contrast, increased noise induces abrupt and irreversible polarization switching in sparsely connected networks. In multiplexes, where each layer can have a different polarization tendency, one layer is dominant and progressively imposes its polarization state on the other, offsetting or promoting the ability of noise to switch its polarization. Overall, we show that the interplay of noise and dynamical correlations can yield discontinuous transitions between extremes, which cannot be explained by a simple mean-field description.

Detecting the Significant Flux Backbone of Escherichia coli metabolism.

The heterogeneity of computationally predicted reaction fluxes in metabolic networks within a single flux state can be exploited to detect their significant flux backbone. Here, we disclose the backbone of Escherichia coli, and compare it with the backbones of other bacteria. We find that, in general, the core of the backbones is mainly composed of reactions in energy metabolism corresponding to ancient pathways. In E. coli, the synthesis of nucleotides and the metabolism of lipids form smaller cores which rely critically on energy metabolism. Moreover, the consideration of different media 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.

The geometric nature of weights in real complex networks.

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. 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.

The hidden hyperbolic geometry of international trade: World Trade Atlas 1870-2013.

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, 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.

Rescue of endemic states in interconnected networks with adaptive coupling.

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 when the two layers are interconnected but not in each layer separately, 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 -and therefore on the interconnected system- 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 rewiring amplifies finite-size effects, preventing the disease transmission between finite networks, as there is a non zero probability that the epidemics stays confined in only one network during its lifetime.

Mapping high-growth phenotypes in the flux space of microbial metabolism.

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, such as 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 (FFPs), 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 behaviours, 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 behaviour and capabilities, and so it opens new avenues for biotechnological and biomedical applications.

Regulation of burstiness by network-driven activation.

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 find that hubs tend to be less susceptible to the networked regulatory effects than low degree nodes. Our results have important implications for the analysis and engineering of bursty activity in a range of systems, from communication networks to transcription and translation of genes into proteins in cells.

Simulating non-Markovian stochastic processes.

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 and efficient algorithm like the Gillespie algorithm for Markovian processes, with the difference being 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 nonexponential interevent 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 because it can be implemented very easily in cases (such as for delays coupled to the evolution of the system) in which other algorithms do not work or need adapted versions that are less efficient in computational terms.