Diversity of information pathways drives sparsity in realworld networks
A. Ghavasieh, M. De Domenico, Nature Physics (2024) Abstract » Read » BibTeX Diversity of information pathways drives sparsity in realworld networksComplex systems must respond to external perturbations and, at the same time, internally distribute information to coordinate their components. Although networked backbones help with the latter, they limit the components’ individual degrees of freedom and reduce their collective dynamical range. Here we show that realworld networks balance the loss of response diversity with gain in information flow. Encoding network states as density matrices, we demonstrate that such a tradeoff mathematically resembles the thermodynamic efficiency characterized by heat and work in physical systems, providing a variational principle to macroscopically explain the sparsity and empirical scaling law observed in hundreds of realworld networks across multiple domains, both analytically and numerically. We show that the emergence of topological features such as modularity, smallworldness and heterogeneity agrees with maximizing the tradeoff between information exchange and response diversity from middle to large temporal scales. Our results suggest that the emergence of some of the most prevalent topological features of realworld networks might have a thermodynamic origin. 

Emergent information dynamics in manybody interconnected systems
W. Merbis, M. De Domenico, Phys. Rev. E 108, 014312 (2023) Abstract » Read » BibTeX Emergent information dynamics in manybody interconnected systemsThe information implicitly represented in the state of physical systems allows for their analysis using analytical techniques from statistical mechanics and information theory. This approach has been successfully applied to complex networks, including biophysical systems such as virushost proteinprotein interactions and wholebrain models in health and disease, drawing inspiration from quantum statistical physics. Here we propose a general mathematical framework for modeling information dynamics on complex networks, where the internal node states are vector valued, allowing each node to carry multiple types of information. This setup is relevant for various biophysical and sociotechnological models of complex systems, ranging from viral dynamics on networks to models of opinion dynamics and social contagion. Instead of focusing on nodenode interactions, we shift our attention to the flow of information between network configurations. We uncover fundamental differences between widely used spin models on networks, such as voter and kinetic dynamics, which cannot be detected through classical nodebased analysis. We illustrate the mathematical framework further through an exemplary application to epidemic spreading on a lowdimensional network. Our model provides an opportunity to adapt powerful analytical methods from quantum manybody systems to study the interplay between structure and dynamics in interconnected systems. 

Generalized network density matrices for analysis of multiscale functional diversity
A. Ghavasieh, M. De Domenico, Phys. Rev. E 107, 044304 (2023) Abstract » Read » BibTeX Generalized network density matrices for analysis of multiscale functional diversityThe network density matrix formalism allows for describing the dynamics of information on top of complex structures and it has been successfully used to analyze, e.g., a system's robustness, perturbations, coarsegraining multilayer networks, characterization of emergent network states, and performing multiscale analysis. However, this framework is usually limited to diffusion dynamics on undirected networks. Here, to overcome some limitations, we propose an approach to derive density matrices based on dynamical systems and information theory, which allows for encapsulating a much wider range of linear and nonlinear dynamics and richer classes of structure, such as directed and signed ones. We use our framework to study the response to local stochastic perturbations of synthetic and empirical networks, including neural systems consisting of excitatory and inhibitory links and generegulatory interactions. Our findings demonstrate that topological complexity does not necessarily lead to functional diversity, i.e., the complex and heterogeneous response to stimuli or perturbations. Instead, functional diversity is a genuine emergent property which cannot be deduced from the knowledge of topological features such as heterogeneity, modularity, the presence of asymmetries, and dynamical properties of a system. 

Dismantling the information flow in complex interconnected systems
A. Ghavasieh, G. Bertagnolli, M. De Domenico, Phys. Rev. Research 5, 013084 (2023) Abstract » Read » BibTeX Dismantling the information flow in complex interconnected systemsMicroscopic structural damage, such as lesions in neural systems or disruptions in urban transportation networks, can impair the dynamics crucial for systems' functionality, such as electrochemical signals or human flows, or any other type of information exchange, respectively, at larger topological scales. Damage is usually modeled by progressive removal of components or connections and, consequently, systems' robustness is assessed in terms of how fast their structure fragments into disconnected subsystems. Yet, this approach fails to capture how damage hinders the propagation of information across scales, since system function can be degraded even in absence of fragmentation—e.g., pathological yet structurally integrated human brain. Here, we probe the response to damage of dynamical processes on the top of complex networks, to study how such an information flow is affected. We find that removal of nodes central for network connectivity might have insignificant effects, challenging the traditional assumption that structural metrics alone are sufficient to gain insights about how complex systems operate. Using a damaging protocol explicitly accounting for flow dynamics, we analyze synthetic and empirical systems, from biological to infrastructural ones, and show that it is possible to drive the system towards functional fragmentation before full structural disintegration. 

Maximum entropy network states for coalescence processes
A. Ghavasieh, M. De Domenico, (2022) Abstract » Read » Maximum entropy network states for coalescence processesComplex network states are characterized by the interplay between system's structure and dynamics. One way to represent such states is by means of network density matrices, whose von Neumann entropy characterizes the number of distinct microstates compatible with given topology and dynamical evolution. In this Letter, we propose a maximum entropy principle to characterize network states for systems with heterogeneous, generally correlated, connectivity patterns and nontrivial dynamics. We focus on three distinct coalescence processes, widely encountered in the analysis of empirical interconnected systems, and characterize their entropy and transitions between distinct dynamical regimes across distinct temporal scales. Our framework allows one to study the statistical physics of systems that aggregate, such as in transportation infrastructures serving the same geographic area, or correlate, such as interbrain synchrony arising in organisms that socially interact, and active matter that swarm or synchronize. 

Ranking Edges by their Impact on the Spectral Complexity of Information Diffusion over Networks
J. Kazimer, M. De Domenico, P.J. Mucha, D. Taylor, (2022) Abstract » Read » Ranking Edges by their Impact on the Spectral Complexity of Information Diffusion over NetworksDespite the numerous ways now available to quantify which parts or subsystems of a network are most important, there remains a lack of centrality measures that are related to the complexity of information flows and are derived directly from entropy measures. Here, we introduce a ranking of edges based on how each edge's removal would change a system's von Neumann entropy (VNE), which is a spectralentropy measure that has been adapted from quantum information theory to quantify the complexity of information dynamics over networks. We show that a direct calculation of such rankings is computationally inefficient (or unfeasible) for large networks: e.g.\ the scaling is (N3) per edge for networks with N nodes. To overcome this limitation, we employ spectral perturbation theory to estimate VNE perturbations and derive an approximate edgeranking algorithm that is accurate and fast to compute, scaling as (N) per edge. Focusing on a form of VNE that is associated with a transport operator e−βL, where L is a graph Laplacian matrix and β>0 is a diffusion timescale parameter, we apply this approach to diverse applications including a network encoding polarized voting patterns of the 117th U.S. Senate, a multimodal transportation system including roads and metro lines in London, and a multiplex brain network encoding correlated human brain activity. Our experiments highlight situations where the edges that are considered to be most important for information diffusion complexity can dramatically change as one considers short, intermediate and long timescales β for diffusion. 

Statistical physics of network structure and information dynamics
A. Ghavasieh, M. De Domenico, J. of Physics: Complexity 3, 011001 (2022) Abstract » Read » BibTeX Statistical physics of network structure and information dynamicsIn the last two decades, network science has proven to be an invaluable tool for the analysis of empirical systems across a wide spectrum of disciplines, with applications to data structures admitting a representation in terms of complex networks. On the one hand, especially in the last decade, an increasing number of applications based on geometric deep learning have been developed to exploit, at the same time, the rich information content of a complex network and the learning power of deep architectures, highlighting the potential of techniques at the edge between applied math and computer science. On the other hand, studies at the edge of network science and quantum physics are gaining increasing attention, e.g., because of the potential applications to quantum networks for communications, such as the quantum Internet. In this work, we briefly review a novel framework grounded on statistical physics and techniques inspired by quantum statistical mechanics which have been successfully used for the analysis of a variety of complex systems. The advantage of this framework is that it allows one to define a set of informationtheoretic tools which find widely used counterparts in machine learning and quantum information science, while providing a grounded physical interpretation in terms of a statistical field theory of information dynamics. We discuss the most salient theoretical features of this framework and selected applications to protein–protein interaction networks, neuronal systems, social and transportation networks, as well as potential novel applications for quantum network science and machine learning. 

Multiscale Information Propagation in Emergent Functional Networks
A. Ghavasieh, M. De Domenico, Entropy 23, 1369 (2021) Abstract » Read » BibTeX Multiscale Information Propagation in Emergent Functional NetworksComplex biological systems consist of large numbers of interconnected units, characterized by emergent properties such as collective computation. In spite of all the progress in the last decade, we still lack a deep understanding of how these properties arise from the coupling between the structure and dynamics. Here, we introduce the multiscale emergent functional state, which can be represented as a network where links encode the flow exchange between the nodes, calculated using diffusion processes on top of the network. We analyze the emergent functional state to study the distribution of the flow among components of 92 fungal networks, identifying their functional modules at different scales and, more importantly, demonstrating the importance of functional modules for information content of networks, quantified in terms of network spectral entropy. Our results suggest that the topological complexity of fungal networks guarantees the existence of functional modules at different scales keeping the information entropy, and functional diversity, high. 

Persistence of information flow: a multiscale characterization of human brain
B. Benigni, A. Ghavasieh, A. Corso, V. d'Andrea, M. De Domenico, Network Neuroscience 5, 831 (2021) Abstract » Read » BibTeX Persistence of information flow: a multiscale characterization of human brainInformation exchange in the human brain is crucial for vital tasks and to drive diseases. Neuroimaging techniques allow for the indirect measurement of information flows among brain areas and, consequently, for reconstructing connectomes analyzed through the lens of network science. However, standard analyses usually focus on a small set of network indicators and their joint probability distribution. Here, we propose an informationtheoretic approach for the analysis of synthetic brain networks (based on generative models) and empirical brain networks, and to assess connectome's information capacity at different stages of dementia. Remarkably, our framework accounts for the whole network state, overcoming limitations due to limited sets of descriptors, and is used to probe human connectomes at different scales. We find that the spectral entropy of empirical data lies between two generative models, indicating an interpolation between modular and geometrydriven structural features. In fact, we show that the mesoscale is suitable for characterizing the differences between brain networks and their generative models. Finally, from the analysis of connectomes obtained from healthy and unhealthy subjects, we demonstrate that significant differences between healthy individuals and the ones affected by Alzheimer’s disease arise at the microscale (max. posterior probability smaller than 1%) and at the mesoscale (max. posterior probability smaller than 10%). 

How choosing randomwalk model and network representation matters for flowbased community detection in hypergraphs
A. Eriksson, D. Edler, A. Rojas, M. De Domenico, M. Rosvall, Communications Physics 4, 133 (2021) Abstract » Read » BibTeX How choosing randomwalk model and network representation matters for flowbased community detection in hypergraphsHypergraphs offer an explicit formalism to describe multibody interactions in complex systems. To connect dynamics and function in systems with these higherorder interactions, network scientists have generalised randomwalk models to hypergraphs and studied the multibody effects on flowbased centrality measures. Mapping the largescale structure of those flows requires effective community detection methods applied to cogent network representations. For different hypergraph data and research questions, which combination of randomwalk model and network representation is best? We define unipartite, bipartite, and multilayer network representations of hypergraph flows and explore how they and the underlying randomwalk model change the number, size, depth, and overlap of identified multilevel communities. These results help researchers choose the appropriate modelling approach when mapping flows on hypergraphs. 

Unraveling the effects of multiscale network entanglement on empirical systems
A. Ghavasieh, M. Stella, J. Biamonte, M. De Domenico, Communications Physics 4, 129 (2021) Abstract » Read » BibTeX Unraveling the effects of multiscale network entanglement on empirical systemsComplex systems are large collections of entities that organize themselves into nontrivial structures, represented as networks. One of their key emergent properties is robustness against random failures or targeted attacks —i.e., the networks maintain their integrity under removal of nodes or links. Here, we introduce network entanglement to study network robustness through a multiscale lens, encoded by the time required for information diffusion through the system. Our measure’s foundation lies upon a recently developed statistical field theory for information dynamics within interconnected systems. We show that at the smallest temporal scales, the nodenetwork entanglement reduces to degree, whereas at extremely large scales, it measures the direct role played by each node in keeping the network connected. At the mesoscale, entanglement plays a more important role, measuring the importance of nodes for the transport properties of the system. We use entanglement as a centrality measure capturing the role played by nodes in keeping the overall diversity of the information flow. As an application, we study the disintegration of empirical social, biological and transportation systems, showing that the nodes central for information dynamics are also responsible for keeping the network integrated. 

Multiscale statistical physics of the panviral interactome unravels the systemic nature of SARSCoV2 infections
A. Ghavasieh, S. Bontorin, O. Artime, N. Verstraete, M. De Domenico, Communications Physics 4, 83 (2021) Abstract » Read » BibTeX Multiscale statistical physics of the panviral interactome unravels the systemic nature of SARSCoV2 infectionsProtein–protein interaction networks have been used to investigate the influence of SARSCoV2 viral proteins on the function of human cells, laying out a deeper understanding of COVID–19 and providing ground for applications, such as drug repurposing. Characterizing molecular (dis)similarities between SARSCoV2 and other viral agents allows one to exploit existing information about the alteration of key biological processes due to known viruses for predicting the potential effects of this new virus. Here, we compare the novel coronavirus network against 92 known viruses, from the perspective of statistical physics and computational biology. We show that regulatory spreading patterns, physical features and enriched biological pathways in targeted proteins lead, overall, to meaningful clusters of viruses which, across scales, provide complementary perspectives to better characterize SARSCoV2 and its effects on humans. Our results indicate that the virus responsible for COVID–19 exhibits expected similarities, such as to Influenza A and Human Respiratory Syncytial viruses, and unexpected ones with different infection types and from distant viral families, like HIV1 and Human Herpes virus. Taken together, our findings indicate that COVID–19 is a systemic disease with potential effects on the function of multiple organs and human body subsystems. 

Statistical physics of complex information dynamics
A. Ghavasieh, C. Nicolini, M. De Domenico, Phys. Rev. E 102, 052304 (2020) Abstract » Read » BibTeX Statistical physics of complex information dynamicsThe constituents of a complex system exchange information to function properly. Their signaling dynamics often leads to the appearance of emergent phenomena, such as phase transitions and collective behaviors. While information exchange has been widely modeled by means of distinct spreading processes—such as continuoustime diffusion, random walks, synchronization and consensus—on top of complex networks, a unified and physically grounded framework to study information dynamics and gain insights about the macroscopic effects of microscopic interactions is still eluding us. In this paper, we present this framework in terms of a statistical field theory of information dynamics, unifying a range of dynamical processes governing the evolution of information on top of static or timevarying structures. We show that information operators form a meaningful statistical ensemble and their superposition defines a density matrix that can be used for the analysis of complex dynamics. As a direct application, we show that the von Neumann entropy of the ensemble can be a measure of the functional diversity of complex systems, defined in terms of the functional differentiation of higherorder interactions among their components. Our results suggest that modularity and hierarchy, two key features of empirical complex systems—from the human brain to social and urban networks—play a key role to guarantee functional diversity and, consequently, are favored. 

Enhancing transport properties in interconnected systems without altering their structure
A. Ghavasieh, M. De Domenico, Phys. Rev. Research 2, 013155 (2020) Abstract » Read » BibTeX Enhancing transport properties in interconnected systems without altering their structureUnits of complex systems  such as neurons in the brain or individuals in societies  must communicate efficiently to function properly: e.g., allowing electrochemical signals to travel quickly among functionally connected neuronal areas in the human brain, or allowing for fast navigation of humans and goods in complex transportation landscapes. The coexistence of different types of relationships among the units, entailing a multilayer represention in which types are considered as networks encoded by layers, plays an important role in the quality of information exchange among them. While altering the structure of such systems  e.g., by physically adding (or removing) units, connections or layers  might be costly, coupling the dynamics of subset(s) of layers in a way that reduces the number of redundant diffusion pathways across the multilayer system, can potentially accelerate the overall information flow. To this aim, we introduce a framework for functional reducibility which allow us to enhance transport phenomena in multilayer systems by coupling layers together with respect to dynamics rather than structure. Mathematically, the optimal configuration is obtained by maximizing the deviation of system's entropy from the limit of free and noninteracting layers. Our results provide a transparent procedure to reduce diffusion time and optimize noncompact search processes in empirical multilayer systems, without the cost of altering the underlying structure. 

Complex Networks: from Classical to Quantum
J. Biamonte, M. Faccin, M. De Domenico, Communications Physics 2, 53 (2019) Abstract » Read » BibTeX Complex Networks: from Classical to QuantumRecent progress in applying complex network theory to problems faced in quantum information and computation has resulted in a beneficial crossover between two fields. Complex network methods have successfully been used to characterize quantum walk and transport models, entangled communication networks, graphtheoretic models of emergent spacetime and in detecting mesoscale structure in quantum systems. Information physics is setting the stage for a theory of complex and networked systems with quantum informationinspired methods appearing in complex network science, including informationtheoretic distance and correlation measures for network characterization. Novel quantum induced effects have been predicted in random graphswhere edges represent entangled linksand quantum computer algorithms have recently been proposed to offer superpolynomial enhancement for several network and graph theoretic problems. Here we review the results at the cutting edge, pinpointing the similarities and reconciling the differences found in the series of results at the intersection of these two fields. 

Distance entropy cartography characterises centrality in complex networks
M. Stella, M. De Domenico, Entropy 20, 268 (2018) Abstract » Read » BibTeX Distance entropy cartography characterises centrality in complex networksWe introduce distance entropy as a measure of homogeneity in the distribution of path lengths between a given node and its neighbours in a complex network. Distance entropy defines a new centrality measure whose properties are investigated for a variety of synthetic network models. By coupling distance entropy information with closeness centrality, we introduce a network cartography which allows one to reduce the degeneracy of ranking based on closeness alone. We apply this methodology to the empirical multiplex lexical network encoding the linguistic relationships known to English speaking toddlers. We show that the distance entropy cartography better predicts how children learn words compared to closeness centrality. Our results highlight the importance of distance entropy for gaining insights from distance patterns in complex networks. 

Spectral entropies as informationtheoretic tools for complex network comparison
M. De Domenico, J. Biamonte, Phys. Rev. X 6, 041062 (2016) Abstract » Read » BibTeX Spectral entropies as informationtheoretic tools for complex network comparisonAny physical system can be viewed from the perspective that information is implicitly represented in its state. However, the quantification of this information when it comes to complex networks has remained largely elusive. In this work, we use techniques inspired by quantum statistical mechanics to define an entropy measure for complex networks and to develop a set of informationtheoretic tools, based on network spectral properties, such as Renyi qentropy, generalized KullbackLeibler and JensenShannon divergences, the latter allowing us to define a natural distance measure between complex networks. First we show that by minimizing the KullbackLeibler divergence between an observed network and a parametric network model, inference of model parameter(s) by means of maximumlikelihood estimation can be achieved and model selection can be performed appropriate information criteria. Second, we show that the informationtheoretic metric quantifies the distance between pairs of networks and we can use it, for instance, to cluster the layers of a multilayer system. By applying this framework to networks corresponding to sites of the human microbiome, we perform hierarchical cluster analysis and recover with high accuracy existing communitybased associations. Our results imply that spectral based statistical inference in complex networks results in demonstrably superior performance as well as a conceptual backbone, filling a gap towards a network information theory. 

Structural reducibility of multilayer networks
M. De Domenico, V. Nicosia, A. Arenas, V. Latora, Nature Communications 6, 6864 (2015) Abstract » Read » BibTeX Structural reducibility of multilayer networksMany complex systems can be represented as networks composed by distinct layers, interacting and depending on each others. For example, in biology, a good description of the full proteinprotein interactome requires, for some organisms, up to seven distinct network layers, with thousands of proteinprotein interactions each. A fundamental open question is then how much information is really necessary to accurately represent the structure of a multilayer complex system, and if and when some of the layers can indeed be aggregated. Here we introduce a method, based on information theory, to reduce the number of layers in multilayer networks, while minimizing information loss. We validate our approach on a set of synthetic benchmarks, and prove its applicability to an extended data set of proteingenetic interactions, showing cases where a strong reduction is possible and cases where it is not. Using this method we can describe complex systems with an optimal tradeoff between accuracy and complexity.  ScienceDaily 

Identifying modular flows on multilayer networks reveals highly overlapping organization in interconnected systems
M. De Domenico, A. Lancichinetti, A. Arenas, M. Rosvall, Phys. Rev. X 5, 011027 (2015) Abstract » Read » BibTeX Identifying modular flows on multilayer networks reveals highly overlapping organization in interconnected systemsUnveiling the community structure of networks is a powerful methodology to comprehend interconnected systems across the social and natural sciences. To identify different types of functional modules in interaction data aggregated in a single network layer, researchers have developed many powerful methods. For example, flowbased methods have proven useful for identifying modular dynamics in weighted and directed networks that capture constraints on flow in the systems they represent. However, many networked systems consist of agents or components that exhibit multiple layers of interactions. Inevitably, representing this intricate network of networks as a single aggregated network leads to information loss and may obscure the actual organization. Here we propose a method based on compression of network flows that can identify modular flows in nonaggregated multilayer networks. Our numerical experiments on synthetic networks show that the method can accurately identify modules that cannot be identified in aggregated networks or by analyzing the layers separately. We capitalize on our findings and reveal the community structure of two multilayer collaboration networks: scientists affiliated to the Pierre Auger Observatory and scientists publishing works on networks on the arXiv. Compared to conventional aggregated methods, the multilayer method reveals smaller modules with more overlap that better capture the actual organization.  Science 

Mathematical Formulation of MultiLayer Networks
M. De Domenico, A. SoleRibalta, E. Cozzo, M. Kivela, Y. Moreno, M. A. Porter, S. Gomez, A. Arenas, Phys. Rev. X 3, 041022 (2013) Abstract » Read » BibTeX Mathematical Formulation of MultiLayer NetworksA network representation is useful for describing the structure of a large variety of complex systems. However, most real and engineered systems have multiple subsystems and layers of connectivity, and the data produced by such systems is very rich. Achieving a deep understanding of such systems necessitates generalizing "traditional" network theory, and the newfound deluge of data now makes it possible to test increasingly general frameworks for the study of networks. In particular, although adjacency matrices are useful to describe traditional singlelayer networks, such a representation is insufficient for the analysis and description of multiplex and timedependent networks. One must therefore develop a more general mathematical framework to cope with the challenges posed by multilayer complex systems. In this paper, we introduce a tensorial framework to study multilayer networks, and we discuss the generalization of several important network descriptors and dynamical processes including degree centrality, clustering coefficients, eigenvector centrality, modularity, Von Neumann entropy, and diffusion for this framework. We examine the impact of different choices in constructing these generalizations, and we illustrate how to obtain known results for the special cases of singlelayer and multiplex networks. Our tensorial approach will be helpful for tackling pressing problems in multilayer complex systems, such as inferring who is influencing whom (and by which media) in multichannel social networks and developing routing techniques for multimodal transportation systems. 

Chaos and scaling in daily river flow
M. De Domenico, M. Ali Ghorbani, Submitted (2011) Abstract » Read » Chaos and scaling in daily river flowAdequate knowledge of the nature of river flow process is crucial for proper planning and management of our water resources and environment. This study attempts to detect the salient characteristics of flow dynamics of the Karoon River in Iran. Daily discharge series observed over a period of six years (19992004) is analyzed to examine the chaotic and scaling characteristics of the flow dynamics. The presence of chaos is investigated through the correlation dimension and Lyapunov exponent methods, while the Hurst exponent and Renyi dimension analyses are performed to explore the scaling characteristics. The low correlation dimension (2.60 + 0.07) and the positive largest Lyapunov exponent (0.014 + 0.001) suggest the presence of lowdimensional chaos; they also imply that the flow dynamics are dominantly governed by three variables and can be reliably predicted up to 48 days (i.e. prediction horizon). Results from the Hurst exponent and Renyi dimension analyses reveal the multifractal character of the flow dynamics, with persistent and antipersistent behaviors observed at different time scales. 

Entropic Approach to Multiscale Clustering Analysis
M. De Domenico, A. Insolia, Entropy 14, 865 (2012) Abstract » Read » BibTeX Entropic Approach to Multiscale Clustering AnalysisRecently, a novel method has been introduced to estimate the statistical significance of clustering in the direction distribution of objects. The method involves a multiscale procedure, based on the Kullback–Leibler divergence and the Gumbel statistics of extreme values, providing high discrimination power, even in presence of strong background isotropic contamination. It is shown that the method is: (i) semianalytical, drastically reducing computation time; (ii) very sensitive to small, medium and large scale clustering; (iii) not biased against the null hypothesis. Applications to the physics of ultrahigh energy cosmic rays, as a cosmological probe, are presented and discussed. 

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