Ranking Edges by their Impact on the Spectral Complexity of Information Diffusion over Networks
J. Kazimer, M. De Domenico, P.J. Mucha, D. Taylor, To appear in SIAM Multiscale Modeling and Simulation (2024) 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 O(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 O(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. 

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. 

Robustness and resilience of complex networks
O. Artime, M. Grassia, M. De Domenico, J.P. Gleeson, H.A. Makse, G. Mangioni, M. Perc, F. Radicchi , Nature Reviews Physics (2024) Abstract » Read » BibTeX Robustness and resilience of complex networksComplex networks are ubiquitous: a cell, the human brain, a group of people and the Internet are all examples of interconnected manybody systems characterized by macroscopic properties that cannot be trivially deduced from those of their microscopic constituents. Such systems are exposed to both internal, localized, failures and external disturbances or perturbations. Owing to their interconnected structure, complex systems might be severely degraded, to the point of disintegration or systemic dysfunction. Examples include cascading failures, triggered by an initially localized overload in power systems, and the critical slowing downs of ecosystems which can be driven towards extinction. In recent years, this general phenomenon has been investigated by framing localized and systemic failures in terms of perturbations that can alter the function of a system. We capitalize on this mathematical framework to review theoretical and computational approaches to characterize robustness and resilience of complex networks. We discuss recent approaches to mitigate the impact of perturbations in terms of designing robustness, identifying earlywarning signals and adapting responses. In terms of applications, we compare the performance of the stateoftheart dismantling techniques, highlighting their optimal range of applicability for practical problems, and provide a repository with readytouse scripts, a muchneeded tool set. 

More is different in realworld multilayer networks
M. De Domenico, Nature Physics (2023) Abstract » Read » BibTeX More is different in realworld multilayer networksThe constituents of many complex systems are characterized by nontrivial connectivity patterns and dynamical processes that are well captured by network models. However, most systems are coupled with each other through interdependencies, characterized by relationships among heterogeneous units, or multiplexity, characterized by the coexistence of different kinds of relationships among homogeneous units. Multilayer networks provide the framework to capture the complexity typical of systems of systems, enabling the analysis of biophysical, social and humanmade networks from an integrated perspective. Here I review the most important theoretical developments in the past decade, showing how the layered structure of multilayer networks is responsible for phenomena that cannot be observed from the analysis of subsystems in isolation or from their aggregation, including enhanced diffusion, emergent mesoscale organization and phase transitions. I discuss applications spanning multiple spatial scales, from the cell to the human brain and to ecological and social systems, and offer perspectives and challenges on future research directions. 

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. 

Complex topological features of reservoirs shape learning performances in bioinspired recurrent neural networks
V. d'Andrea, M. Puppin, M. De Domenico, (2022) Abstract » Read » Complex topological features of reservoirs shape learning performances in bioinspired recurrent neural networksRecurrent networks are a special class of artificial neural systems that use their internal states to perform computing tasks for machine learning. One of its stateoftheart developments, i.e. reservoir computing (RC), uses the internal structure  usually a static network with random structure  to map an input signal into a nonlinear dynamical system defined in a higher dimensional space. Reservoirs are characterized by nonlinear interactions among their units and their ability to store information through recurrent loops, allowing to train artificial systems to learn taskspecific dynamics. However, it is fundamentally unknown how the random topology of the reservoir affects the learning performance. Here, we fill this gap by considering a battery of synthetic networks  characterized by different topological features  and 45 empirical connectomes  sampled from brain regions of organisms belonging to 8 different species  to build the reservoir and testing the learning performance against a prediction task with a variety of complex input signals. We find nontrivial correlations between RC performances and both the number of nodes and rank of the covariance matrix of activation states, with performance depending on the nature  stochastic or deterministic  of input signals. Remarkably, the modularity and the link density of the reservoir are found to affect RC performances: these results cannot be predicted by models only accounting for simple topological features of the reservoir. Overall, our findings highlight that the complex topological features characterizing biophysical computing systems such as connectomes can be used to design efficient bioinspired artificial neural networks. 

Functional rich clubs emerging from the diffusion geometry of complex networks
G. Bertagnolli, M. De Domenico, Phys. Rev. Research 4, 033185 (2022) Abstract » Read » BibTeX Functional rich clubs emerging from the diffusion geometry of complex networksReal systems are characterized by complex patterns of interactions between their units, by dynamical processes on them, and by the interplay of the two. It is well known that particular structures affect dynamical processes at different scales. Sometimes richly connected units are connected by costly, longrange links. In the brain, hubs form rich clubs for integrating information between different brain regions, and many biological and social networks show this same structural organization. It remains, however, unclear whether this structural organization alone enables a rapid communication between highly connected nodes or whether a functional rich club may emerge as a combination of direct links and longer paths between rich nodes. Here, we identify functional rich clubs through the diffusion geometry, providing a perspective on richclub phenomena in complex networks. We show that weak structural rich clubs may be functionally stronger, thanks to bridge nodes, while diffusion inside strong structural rich clubs may be damped in modular networks. 

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%). 

Quantifying efficient information exchange in real network flows
G. Bertagnolli, R. Gallotti, M. De Domenico, Communications Physics 4, 125 (2021) Abstract » Read » BibTeX Quantifying efficient information exchange in real network flowsNetwork science enables the effective analysis of real interconnected systems, characterized by a complex interplay between topology and network flows. It is wellknown that the topology of a network affects its resilience to failures or attacks, as well as its functions. Many real systems—such as the Internet, transportation networks and the brain—exchange information, so it is crucial to quantify how efficiently system’s units communicate. Measures of parallel communication efficiency for weighted networks rely on the identification of an ideal version of the system, which currently lacks a universal definition. Consequently, an inattentive choice might hinder a rigorous comparison of network flows across scales or might lead to a descriptor not robust to fluctuations in the topology or the flows. We propose a physicallygrounded estimator of flow efficiency valid for any weighted network, regardless of scale, nature of weights and (missing) metadata, allowing for comparison across disparate systems. Our estimator captures the effect of flows heterogeneity along with topological differences of both synthetic and empirical systems. We also show that cutting the heaviest connections may increase the average efficiency of the system and hence, counterintuively, a sparser network is not necessarily less efficient. 

Interlayer connectivity reconstruction for multilayer brain networks using phase oscillator models
P. Tewarie et al, New Journal of Physics 23, 063065 (2021) Abstract » Read » BibTeX Interlayer connectivity reconstruction for multilayer brain networks using phase oscillator modelsLargescale neurophysiological networks are often reconstructed from bandpass filtered time series derived from magnetoencephalography (MEG) data. Common practice is to reconstruct these networks separately for different frequency bands and to treat them independently. Recent evidence suggests that this separation may be inadequate, as there can be significant coupling between frequency bands (interlayer connectivity). A multilayer network approach offers a solution to analyze frequencyspecific networks in one framework. We propose to use a recently developed network reconstruction method in conjunction with phase oscillator models to estimate interlayer connectivity that optimally fits the empirical data. This approach determines interlayer connectivity based on observed frequencyspecific time series of the phase and a connectome derived from diffusion weighted imaging. The performance of this interlayer reconstruction method was evaluated insilico. Our reconstruction of the underlying interlayer connectivity agreed to very high degree with the ground truth. Subsequently, we applied our method to empirical restingstate MEG data obtained from healthy subjects and reconstructed twolayered networks consisting of either alphatobeta or thetatogamma band connectivity. Our analysis revealed that interlayer connectivity is dominated by a multiplex structure, i.e. by onetoone interactions for both alphatobeta band and thetatogamma band networks. For theta–gamma band networks, we also found a plenitude of interlayer connections between distant nodes, though weaker connectivity relative to the onetoone connections. Our work is an stepping stone towards the identification of interdependencies across frequencyspecific networks. Our results lay the ground for the use of the promising multilayer framework in this field with moreinformed and justified interlayer connections. 

Network Geometry
M. Boguna, I. Bonamassa, M. De Domenico, S. Havlin, D. Krioukov, M.A. Serrano, Nature Reviews Physics 3, 114 (2021) Abstract » Read » BibTeX Network GeometryNetworks are finite metric spaces, with distances defined by the shortest paths between nodes. However, this is not the only form of network geometry: two others are the geometry of latent spaces underlying many networks and the effective geometry induced by dynamical processes in networks. These three approaches to network geometry are intimately related, and all three of them have been found to be exceptionally efficient in discovering fractality, scale invariance, selfsimilarity and other forms of fundamental symmetries in networks. Network geometry is also of great use in a variety of practical applications, from understanding how the brain works to routing in the Internet. We review the most important theoretical and practical developments dealing with these approaches to network geometry and offer perspectives on future research directions and challenges in this frontier in the study of complexity. 

Multilayer connector hub mapping reveals key brain regions supporting expressive language
B. Williamson, M. De Domenico, D. Kadis, Brain Connectivity 11, 45 (2020) Abstract » Read » BibTeX Multilayer connector hub mapping reveals key brain regions supporting expressive languageHow components of the distributed brain networks that support cognition participate in typical functioning remains a largely unanswered question. An important subgroup of regions in the larger network are connector hubs, which are areas that are highly connected to several other functionallyspecialized sets of regions, and are likely important for sensorimotor integration. The present study attempts to characterize connector hubs involved in typical expressive language functioning using a datadriven, multimodal, full multilayer MEG connectivitybased pipeline. Twelve adolescents, 1618 years of age (5 male) participated in this study. Participants underwent MEG scanning during a verb generation task. MEG and structural connectivity were calculated at the wholebrain level. Amplitudeamplitude coupling (AAC) was used to compute functional connections both within and between discrete frequency bins. AAC values were then multiplied by a binary structural connectivity matrix, then entered into full multilayer network analysis. Initially, hubs were defined based on multilayer versatility and subsequently reranked by a novel measure called delta centrality on interconnectedness (DCI). DCI is defined as the percent change in interfrequency interconnectedness after removal of a hub. We resolved regions that are important for betweenfrequency communication among other areas during expressive language, with several potential theoretical and clinical applications that can be generalized to other cognitive domains. Our multilayer, datadriven framework captures nonlinear connections that span across scales that are often missed in conventional analyses. The present study suggests that crucial hubs may be conduits for interfrequency communication between action and perception systems that are crucial for typical functioning. 

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. 

Multilayer network modeling of integrated biological systems
M. De Domenico, Physics of Life Reviews 24, 149152 (2018) Abstract » Read » BibTeX Multilayer network modeling of integrated biological systemsBiological systems, from a cell to the human brain, are inherently complex. A powerful representation of such systems, described by an intricate web of relationships across multiple scales, is provided by complex networks. Recently, several studies are highlighting how simple networks – obtained by aggregating or neglecting temporal or categorical description of biological data – are not able to account for the richness of information characterizing biological systems. More complex models, namely multilayer networks, are needed to account for interdependencies, often varying across time, of biological interacting units within a cell, a tissue or parts of an organism. Gosak et al. [1] review the most recent advances in the application of multilayer networks for modeling complex biological systems, from molecular interactions within a cell to neuronal connectivity of the human brain. 

Diffusion Geometry Unravels the Emergence of Functional Clusters in Collective Phenomena
M. De Domenico, Phys. Rev. Lett. 118, 168301 (2017) Abstract » Read » BibTeX Diffusion Geometry Unravels the Emergence of Functional Clusters in Collective PhenomenaCollective phenomena emerge from the interaction of natural or artificial units with a complex organization. The interplay between structural patterns and dynamics might induce functional clusters that, in general, are different from topological ones. In biological systems, like the human brain, the overall functionality is often favored by the interplay between connectivity and synchronization dynamics, with functional clusters that do not coincide with anatomical modules in most cases. In social, sociotechnical, and engineering systems, the quest for consensus favors the emergence of clusters. Despite the unquestionable evidence for mesoscale organization of many complex systems and the heterogeneity of their interconnectivity, a way to predict and identify the emergence of functional modules in collective phenomena continues to elude us. Here, we propose an approach based on random walk dynamics to define the diffusion distance between any pair of units in a networked system. Such a metric allows us to exploit the underlying diffusion geometry to provide a unifying framework for the intimate relationship between metastable synchronization, consensus, and random search dynamics in complex networks, pinpointing the functional mesoscale organization of synthetic and biological systems. 

Multilayer modeling and analysis of human brain networks
M. De Domenico, GigaScience 6, 18 (2017) Abstract » Read » BibTeX Multilayer modeling and analysis of human brain networksUnderstanding how the human brain is structured, and how its architecture is related to the function, is of paramount importance for a variety of applications, including, but not limited to, new ways to prevent, deal with and cure brain diseases, such as Alzheimer’s or Parkinson’s, and psychiatric disorders, such as Schizophrenia. The recent advances in structural and functional neuroimaging, together with the increasing attitude to interdisciplinary approaches involving computer science, mathematics and physics, are fostering interesting results from computational neuroscience, that are quite often based on the analysis of complex network representation of human brain. In the last years, this representation experienced a theoretical and computational revolution that are breaching neuroscience, allowing to cope with the increasing complexity of human brain across multiple scales and in multiple dimensions, and to model structural and functional connectivity from new perspectives, often combined with each other. In this work, we will review the main achievements obtained from interdisciplinary research based on magnetic resonance imaging and establishing, de facto, the birth of multilayer network analysis and modeling of human brain. 

Researcher incentives: EU cash goes to the sticky and attractive
M. De Domenico, A. Arenas, Nature 531, 580 (2016) Abstract » Read » BibTeX Researcher incentives: EU cash goes to the sticky and attractiveWe find that winning European research money (Nature 530, 33; 2016) is, among other things, contingent on how well national governments manage to retain their own scientists and to attract others from abroad. We analysed statistical indicators of scientists' mobility in the European Union for 200714 (erc.europa.eu and go.nature.com/bpeylu) to determine the 'attractiveness' of different countries to scientists from abroad, as well as their 'stickiness' in preventing native researchers from moving away (known as brain drain). We quantified attractiveness and stickiness as the relative difference between incoming or remaining researchers, respectively, and outgoing ones. For both measures, we found that the higher the value, the better are that country's chances of securing European research funding. The United Kingdom and Sweden are high scorers in both; Italy is among the lowest (further details from M. de D. and A. A.; see go.nature.com/wyvtls). We conclude that there is a 'rich get richer' effect for countries that have high attractiveness and stickiness scores. Those nations also boast a high gross domestic product per capita, and tend to invest more in research and development. This means that they can lure and retain the best researchers by providing competitive salaries and a guaranteed future in research.  Full text 

Mapping multiplex hubs in human functional brain network
M. De Domenico, S. Sasai, A. Arenas, Frontiers in Neuroscience 10, 326 (2016) Abstract » Read » BibTeX Mapping multiplex hubs in human functional brain networkTypical brain networks consist of many peripheral regions and a few highly central ones, i.e. hubs, playing key functional roles in cerebral interregional interactions. Studies have shown that networks, obtained from the analysis of specific frequency components of brain activity, present peculiar architectures with unique profiles of region centrality. However, the identification of hubs in networks built from different frequency bands simultaneously is still a challenging problem, remaining largely unexplored. Here we identify each frequency component with one layer of a multiplex network and face this challenge by exploiting the recent advances in the analysis of multiplex topologies. First, we show that each frequency band carries unique topological information, fundamental to accurately model brain functional networks. We then demonstrate that hubs in the multiplex network, in general different from those ones obtained after discarding or aggregating the measured signals as usual, provide a more accurate map of brain's most important functional regions, allowing to distinguish between healthy and schizophrenic populations better than conventional network approaches. 

MuxViz: a tool for multilayer analysis and visualization of networks
M. De Domenico, Mason A. Porter, A. Arenas, Journal of Complex Networks 3, 159 (2015) Abstract » Read » BibTeX MuxViz: a tool for multilayer analysis and visualization of networksMultilayer relationships among entities and information about entities must be accompanied by the means to analyse, visualize and obtain insights from such data. We present opensource software (muxViz) that contains a collection of algorithms for the analysis of multilayer networks, which are an important way to represent a large variety of complex systems throughout science and engineering. We demonstrate the ability of muxViz to analyse and interactively visualize multilayer data using empirical genetic, neuronal and transportation networks. Our software is available at https://github.com/manlius/muxViz. 

Emergence of assortative mixing between clusters of cultured neurons
S. Teller, C. Granell, M. De Domenico, J. Soriano, S. Gomez, A. Arenas, PLOS Comput. Biol. 10(9), e1003796 (2014) Abstract » Read » BibTeX Emergence of assortative mixing between clusters of cultured neuronsThe analysis of the activity of neuronal cultures is considered to be a good proxy of the functional connectivity of in vivo neuronal tissues. Thus, the functional complex network inferred from activity patterns is a promising way to unravel the interplay between structure and functionality of neuronal systems. Here, we monitor the spontaneous selfsustained dynamics in neuronal cultures formed by interconnected aggregates of neurons (clusters). Dynamics is characterized by the fast activation of groups of clusters in sequences termed bursts. The analysis of the time delays between clusters' activations within the bursts allows the reconstruction of the directed functional connectivity of the network. We propose a method to statistically infer this connectivity and analyze the resulting properties of the associated complex networks. Surprisingly enough, in contrast to what has been reported for many biological networks, the clustered neuronal cultures present assortative mixing connectivity values, meaning that there is a preference for clusters to link to other clusters that share similar functional connectivity, as well as a richclub core, which shapes a ‘connectivity backbone’ in the network. These results point out that the grouping of neurons and the assortative connectivity between clusters are intrinsic survival mechanisms of the culture. 

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