Network synchronization, diffusion, and the paradox of heterogeneity

Aging in a weighted ensemble of excitable and self-oscillatory neurons: The role of pairwise and higher-order interactions

We investigate the aging transition in networks of excitable and self-oscillatory units as the fraction of inherently excitable units increases. Two network topologies are considered: a scale-free network with weighted pairwise interactions and a two-dimensional simplicial complex with weighted scale-free pairwise and triadic interactions. Without triadic interactions, the aging transition from collective oscillations to oscillation death (inhomogeneous stationary states) can occur either suddenly or through an intermediate state of partial oscillation. However, when triadic interactions are present, the network becomes less resilient, and the transition occurs without partial oscillation at any coupling strength. Furthermore, we observe the presence of inhomogeneous steady states within the complete oscillation death regime, regardless of the network interaction models.

Synchronization cluster bursting in adaptive oscillator networks

Adaptive dynamical networks are ubiquitous in real-world systems. This paper aims to explore the synchronization dynamics in networks of adaptive oscillators based on a paradigmatic system of adaptively coupled phase oscillators. Our numerical observations reveal the emergence of synchronization cluster bursting, characterized by periodic transitions between cluster synchronization and global synchronization. By investigating a reduced model, the mechanisms underlying synchronization cluster bursting are clarified. We show that a minimal model exhibiting this phenomenon can be reduced to a phase oscillator with complex-valued adaptation. Furthermore, the adaptivity of the system leads to the appearance of additional symmetries, and thus, to the coexistence of stable bursting solutions with very different Kuramoto order parameters.

Dynamics of simplicial SEIRS epidemic model: global asymptotic stability and neural Lyapunov functions

The transmission of infectious diseases on a particular network is ubiquitous in the physical world. Here, we investigate the transmission mechanism of infectious diseases with an incubation period using a networked compartment model that contains simplicial interactions, a typical high-order structure. We establish a simplicial SEIRS model and find that the proportion of infected individuals in equilibrium increases due to the many-body connections, regardless of the type of connections used. We analyze the dynamics of the established model, including existence and local asymptotic stability, and highlight differences from existing models. Significantly, we demonstrate global asymptotic stability using the neural Lyapunov function, a machine learning technique, with both numerical simulations and rigorous analytical arguments. We believe that our model owns the potential to provide valuable insights into transmission mechanisms of infectious diseases on high-order network structures, and that our approach and theory of using neural Lyapunov functions to validate model asymptotic stability can significantly advance investigations on complex dynamics of infectious disease.

Synchronization dynamics of phase oscillators on power grid models

We investigate topological and spectral properties of models of European and US-American power grids and of paradigmatic network models as well as their implications for the synchronization dynamics of phase oscillators with heterogeneous natural frequencies. We employ the complex-valued order parameter-a widely used indicator for phase ordering-to assess the synchronization dynamics and observe the order parameter to exhibit either constant or periodic or non-periodic, possibly chaotic temporal evolutions for a given coupling strength but depending on initial conditions and the systems' disorder. Interestingly, both topological and spectral characteristics of the power grids point to a diminished capability of these networks to support a temporarily stable synchronization dynamics. We find non-trivial commonalities between the synchronization dynamics of oscillators on seemingly opposing topologies.

Does the brain behave like a (complex) network? I. Dynamics

Graph theory is now becoming a standard tool in system-level neuroscience. However, endowing observed brain anatomy and dynamics with a complex network structure does not entail that the brain actually works as a network. Asking whether the brain behaves as a network means asking whether network properties count. From the viewpoint of neurophysiology and, possibly, of brain physics, the most substantial issues a network structure may be instrumental in addressing relate to the influence of network properties on brain dynamics and to whether these properties ultimately explain some aspects of brain function. Here, we address the dynamical implications of complex network, examining which aspects and scales of brain activity may be understood to genuinely behave as a network. To do so, we first define the meaning of networkness, and analyse some of its implications. We then examine ways in which brain anatomy and dynamics can be endowed with a network structure and discuss possible ways in which network structure may be shown to represent a genuine organisational principle of brain activity, rather than just a convenient description of its anatomy and dynamics.

Synchronization enhancement subjected to adaptive blinking coupling

Synchronization holds a significant role, notably within chaotic systems, in various contexts where the coordinated behavior of systems plays a pivotal and indispensable role. Hence, many studies have been dedicated to investigating the underlying mechanism of synchronization of chaotic systems. Networks with time-varying coupling, particularly those with blinking coupling, have been proven essential. The reason is that such coupling schemes introduce dynamic variations that enhance adaptability and robustness, making them applicable in various real-world scenarios. This paper introduces a novel adaptive blinking coupling, wherein the coupling adapts dynamically based on the most influential variable exhibiting the most significant average disparity. To ensure an equitable selection of the most effective coupling at each time instance, the average difference of each variable is normalized to the synchronous solution's range. Due to this adaptive coupling selection, synchronization enhancement is expected to be observed. This hypothesis is assessed within networks of identical systems, encompassing Lorenz, Rössler, Chen, Hindmarsh-Rose, forced Duffing, and forced van der Pol systems. The results demonstrated a substantial improvement in synchronization when employing adaptive blinking coupling, particularly when applying the normalization process.

Neural heterogeneity controls computations in spiking neural networks

The brain is composed of complex networks of interacting neurons that express considerable heterogeneity in their physiology and spiking characteristics. How does this neural heterogeneity influence macroscopic neural dynamics, and how might it contribute to neural computation? In this work, we use a mean-field model to investigate computation in heterogeneous neural networks, by studying how the heterogeneity of cell spiking thresholds affects three key computational functions of a neural population: the gating, encoding, and decoding of neural signals. Our results suggest that heterogeneity serves different computational functions in different cell types. In inhibitory interneurons, varying the degree of spike threshold heterogeneity allows them to gate the propagation of neural signals in a reciprocally coupled excitatory population. Whereas homogeneous interneurons impose synchronized dynamics that narrow the dynamic repertoire of the excitatory neurons, heterogeneous interneurons act as an inhibitory offset while preserving excitatory neuron function. Spike threshold heterogeneity also controls the entrainment properties of neural networks to periodic input, thus affecting the temporal gating of synaptic inputs. Among excitatory neurons, heterogeneity increases the dimensionality of neural dynamics, improving the network's capacity to perform decoding tasks. Conversely, homogeneous networks suffer in their capacity for function generation, but excel at encoding signals via multistable dynamic regimes. Drawing from these findings, we propose intra-cell-type heterogeneity as a mechanism for sculpting the computational properties of local circuits of excitatory and inhibitory spiking neurons, permitting the same canonical microcircuit to be tuned for diverse computational tasks.

Impact of Network Topology on Neural Synchrony in a Model of the Subthalamic Nucleus-Globus Pallidus Circuit

Synchronous neural oscillations within the beta frequency range are observed across the parkinsonian basal ganglia network, including within the subthalamic nucleus (STN) - globus pallidus (GPe) subcircuit. The emergence of pathological synchrony in Parkinson's disease is often attributed to changes in neural properties or connection strength, and less often to the network topology, i.e. the structural arrangement of connections between neurons. This study investigates the relationship between network structure and neural synchrony in a model of the STN-GPe circuit comprised of conductance-based spiking neurons. Changes in net synaptic input were controlled for through a synaptic scaling rule, which facilitated separation of the effects of network structure from net synaptic input. Five topologies were examined as structures for the STN-GPe circuit: Watts-Strogatz, preferential attachment, spatial, stochastic block, k-regular random. Beta band synchrony generally increased as the number of connections increased, however the exact relationship was topology specific. Varying the wiring pattern while maintaining a constant number of connections caused network synchrony to be enhanced or suppressed, demonstrating the ability of purely structural changes to alter synchrony. This relationship was well-captured by the algebraic connectivity of the network, the second smallest eigenvalue of the network's Laplacian matrix. The structure-synchrony relationship was further investigated in a network model designed to emulate the action selection role of the STN-GPe circuit. It was found that increasing the number of connections and/or the overlap of action selection channels could lead to a rapid transition to synchrony, which was also predicted by the algebraic connectivity.

Within-subject reproducibility varies in multi-modal, longitudinal brain networks

Network neuroscience provides important insights into brain function by analyzing complex networks constructed from diffusion Magnetic Resonance Imaging (dMRI), functional MRI (fMRI) and Electro/Magnetoencephalography (E/MEG) data. However, in order to ensure that results are reproducible, we need a better understanding of within- and between-subject variability over long periods of time. Here, we analyze a longitudinal, 8 session, multi-modal (dMRI, and simultaneous EEG-fMRI), and multiple task imaging data set. We first confirm that across all modalities, within-subject reproducibility is higher than between-subject reproducibility. We see high variability in the reproducibility of individual connections, but observe that in EEG-derived networks, during both rest and task, alpha-band connectivity is consistently more reproducible than connectivity in other frequency bands. Structural networks show a higher reliability than functional networks across network statistics, but synchronizability and eigenvector centrality are consistently less reliable than other network measures across all modalities. Finally, we find that structural dMRI networks outperform functional networks in their ability to identify individuals using a fingerprinting analysis. Our results highlight that functional networks likely reflect state-dependent variability not present in structural networks, and that the type of analysis should depend on whether or not one wants to take into account state-dependent fluctuations in connectivity.

Diffusion capacity of single and interconnected networks

Understanding diffusive processes in networks is a significant challenge in complexity science. Networks possess a diffusive potential that depends on their topological configuration, but diffusion also relies on the process and initial conditions. This article presents Diffusion Capacity, a concept that measures a node's potential to diffuse information based on a distance distribution that considers both geodesic and weighted shortest paths and dynamical features of the diffusion process. Diffusion Capacity thoroughly describes the role of individual nodes during a diffusion process and can identify structural modifications that may improve diffusion mechanisms. The article defines Diffusion Capacity for interconnected networks and introduces Relative Gain, which compares the performance of a node in a single structure versus an interconnected one. The method applies to a global climate network constructed from surface air temperature data, revealing a significant change in diffusion capacity around the year 2000, suggesting a loss of the planet's diffusion capacity that could contribute to the emergence of more frequent climatic events.

Optimal time-varying coupling function can enhance synchronization in complex networks

In this paper, we propose a time-varying coupling function that results in enhanced synchronization in complex networks of oscillators. The stability of synchronization can be analyzed by applying the master stability approach, which considers the largest Lyapunov exponent of the linearized variational equations as a function of the network eigenvalues as the master stability function. Here, it is assumed that the oscillators have diffusive single-variable coupling. All possible single-variable couplings are studied for each time interval, and the one with the smallest local Lyapunov exponent is selected. The obtained coupling function leads to a decrease in the critical coupling parameter, resulting in enhanced synchronization. Moreover, synchronization is achieved faster, and its robustness is increased. For illustration, the optimum coupling function is found for three networks of chaotic Rössler, Chen, and Chua systems, revealing enhanced synchronization.

Identification of Network Topology Variations Based on Spectral Entropy

Based on the fact that the traditional probability distribution entropy describing a local feature of the system cannot effectively capture the global topology variations of the network, some indicators constructed by the network adjacency matrix and Laplacian matrix come into being. Specifically, these measures are based on the eigenvalues of the scaled Laplace matrix, the eigenvalues of the network communicability matrix, and the spectral entropy based on information diffusion that has been proposed recently, respectively. In this article, we systematically study the dependence of these measures on the topological structure of the network. We prove from various aspects that spectral entropy has a better ability to identify the global topology than the traditional distribution entropy. Furthermore, the indicator based on the eigenvalues of the network communicability matrix achieves good results in some aspects while, overall, the spectral entropy is able to identify network topology variations from a global perspective.

Maximizing synchronizability of networks with community structure based on node similarity

In reality, numerous networks have a community structure characterized by dense intra-community connections and sparse inter-community connections. In this article, strategies are proposed to enhance synchronizability of such networks by rewiring a certain number of inter-community links, where the research scope is complete synchronization on undirected and diffusively coupled dynamic networks. First, we explore the effect of adding links between unconnected nodes with different similarity levels on network synchronizability and find that preferentially adding links between nodes with lower similarity can improve network synchronizability more than that with higher similarity, where node similarity is measured by our improved Asymmetric Katz (AKatz) and Asymmetric Leicht-Holme-Newman (ALHNII) methods from the perspective of link prediction. Additional simulations demonstrate that the node similarity-based link-addition strategy is more effective in enhancing network synchronizability than the node centrality-based methods. Furthermore, we apply the node similarity-based link-addition or deletion strategy as the valid criteria to the rewiring process of inter-community links and then propose a Node Similarity-Based Rewiring Optimization (NSBRO) algorithm, where the optimization process is realized by a modified simulated annealing technique. Simulations show that our proposed method performs better in optimizing synchronization of such networks compared with other centrality-based heuristic methods. Finally, simulations on the Rössler system indicate that the network structure optimized by the NSBRO algorithm also leads to better synchronizability of coupled oscillators.

Network structure from a characterization of interactions in complex systems

Many natural and man-made complex dynamical systems can be represented by networks with vertices representing system units and edges the coupling between vertices. If edges of such a structural network are inaccessible, a widely used approach is to identify them with interactions between vertices, thereby setting up a functional network. However, it is an unsolved issue if and to what extent important properties of a functional network on the global and the local scale match those of the corresponding structural network. We address this issue by deriving functional networks from characterizing interactions in paradigmatic oscillator networks with widely-used time-series-analysis techniques for various factors that alter the collective network dynamics. Surprisingly, we find that particularly key constituents of functional networks—as identified with betweenness and eigenvector centrality—coincide with ground truth to a high degree, while global topological and spectral properties—clustering coefficient, average shortest path length, assortativity, and synchronizability—clearly deviate. We obtain similar concurrences for an empirical network. Our findings are of relevance for various scientific fields and call for conceptual and methodological refinements to further our understanding of the relationship between structure and function of complex dynamical systems.

Brain networks in migraine with and without aura: An exploratory arterial spin labeling MRI study

OBJECTIVES

The aim of this exploratory study was to investigate the underlying pathomechanisms of migraine with aura (MA) and migraine without aura (MO) in the interictal phase using a connectivity analysis.

METHODS

We prospectively enrolled patients who were newly diagnosed with migraine. All patients underwent brain MRI, including diffusion tensor imaging and arterial spin labeling perfusion MRI. We analyzed the differences between patients with MA and those with MO in structural connectivity based on diffusion tensor imaging and functional connectivity based on arterial spin labeling perfusion MRI using a graph theoretical analysis.

RESULTS

We enrolled 58 patients with migraine (11 patients with MA and 47 patients with MO). There were no differences between patients with MA and those with MO in the network measures of global structural connectivity. However, differences in global functional connectivity were found between the two groups. The assortative coefficient was lower in patients with MA than in those with MO (-0.050 vs. -0.012, p = .017). There were no differences in local structural and functional connectivity between patients with MA and those with MO.

CONCLUSION

We found differences in global functional connectivity between patients with MO and those with MA. The study of MA and MO using a connectivity analysis may shed light on migraine pathophysiology. We suggest it is worthwhile to investigate if changes in functional connectivity may serve as novel biomarkers in MA. In this regard, ASL MRI appears to be valuable in the context of network analysis, but further studies are needed to confirm our findings.

Individual risk perception and empirical social structures shape the dynamics of infectious disease outbreaks

The dynamics of a spreading disease and individual behavioral changes are entangled processes that have to be addressed together in order to effectively manage an outbreak. Here, we relate individual risk perception to the adoption of a specific set of control measures, as obtained from an extensive large-scale survey performed via Facebook-involving more than 500,000 respondents from 64 countries-showing that there is a "one-to-one" relationship between perceived epidemic risk and compliance with a set of mitigation rules. We then develop a mathematical model for the spreading of a disease-sharing epidemiological features with COVID-19-that explicitly takes into account non-compliant individual behaviors and evaluates the impact of a population fraction of infectious risk-deniers on the epidemic dynamics. Our modeling study grounds on a wide set of structures, including both synthetic and more than 180 real-world contact patterns, to evaluate, in realistic scenarios, how network features typical of human interaction patterns impact the spread of a disease. In both synthetic and real contact patterns we find that epidemic spreading is hindered for decreasing population fractions of risk-denier individuals. From empirical contact patterns we demonstrate that connectivity heterogeneity and group structure significantly affect the peak of hospitalized population: higher modularity and heterogeneity of social contacts are linked to lower peaks at a fixed fraction of risk-denier individuals while, at the same time, such features increase the relative impact on hospitalizations with respect to the case where everyone correctly perceive the risks.

Relay Synchronization in a Weighted Triplex Network

Relay synchronization in multi-layer networks implies inter-layer synchronization between two indirectly connected layers through a relay layer. In this work, we study the relay synchronization in a three-layer multiplex network by introducing degree-based weighting mechanisms. The mechanism of within-layer connectivity may be hubs-repelling or hubs-attracting whenever low-degree or high-degree nodes receive strong influence. We adjust the remote layers to hubs-attracting coupling, whereas the relay layer may be unweighted, hubs-repelling, or hubs-attracting network. We establish that relay synchronization is improved when the relay layer is hubs-repelling compared to the other cases. We determine analytically necessary stability conditions of relay synchronization state using the master stability function approach. Finally, we explore the relation between synchronization and the topological property of the relay layer. We find that a higher clustering coefficient hinders synchronizability, and vice versa. We also look into the intra-layer synchronization in the proposed weighted triplex network and establish that intra-layer synchronization occurs in a wider range when relay layer is hubs-attracting.

Node differentiation dynamics along the route to synchronization in complex networks

Synchronization has been the subject of intense research during decades mainly focused on determining the structural and dynamical conditions driving a set of interacting units to a coherent state globally stable. However, little attention has been paid to the description of the dynamical development of each individual networked unit in the process towards the synchronization of the whole ensemble. In this paper we show how in a network of identical dynamical systems, nodes belonging to the same degree class, differentiate in the same manner, visiting a sequence of states of diverse complexity along the route to synchronization independently on the global network structure. In particular, we observe, just after interaction starts pulling orbits from the initially uncoupled attractor, a general reduction of the complexity of the dynamics of all units being more pronounced in those with higher connectivity. In the weak-coupling regime, when synchronization starts to build up, there is an increase in the dynamical complexity, whose maximum is achieved, in general, first in the hubs due to their earlier synchronization with the mean field. For very strong coupling, just before complete synchronization, we found a hierarchical dynamical differentiation with lower degree nodes being the ones exhibiting the largest complexity departure. We unveil how this differentiation route holds for several models of nonlinear dynamics, including toroidal chaos and how it depends on the coupling function. This study provides insights to understand better strategies for network identification or to devise effective methods for network inference.

Dynamics of hierarchical weighted networks of van der Pol oscillators

We investigate the dynamics of regular fractal-like networks of hierarchically coupled van der Pol oscillators. The hierarchy is imposed in terms of the coupling strengths or link weights. We study the low frequency modes, as well as frequency and phase synchronization, in the network by a process of repeated coarse-graining of oscillator units. At any given stage of this process, we sum over the signals from the oscillator units of a clique to obtain a new oscillating unit. The frequencies and the phases for the coarse-grained oscillators are found to progressively synchronize with the number of coarse-graining steps. Furthermore, the characteristic frequency is found to decrease and finally stabilize to a value that can be tuned via the parameters of the system. We compare our numerical results with those of an approximate analytic solution and find good qualitative agreement. Our study on this idealized model shows how oscillations with a precise frequency can be obtained in systems with heterogeneous couplings. It also demonstrates the effect of imposing a hierarchy in terms of link weights instead of one that is solely topological, where the connectivity between oscillators would be the determining factor, as is usually the case.

Explosive synchronization in temporal networks: A comparative study

We present a comparative study on Explosive Synchronization (ES) in temporal networks consisting of phase oscillators. The temporal nature of the networks is modeled with two configurations: (1) oscillators are allowed to move in a closed two-dimensional box such that they couple with their neighbors and (2) oscillators are static and they randomly switch their coupling partners. Configuration (1) is further studied under two possible scenarios: in the first case, oscillators couple to fixed numbers of neighbors, while, in the other case, they couple to all oscillators lying in their circle of vision. Under these circumstances, we monitor the degrees of temporal networks, velocities, and radius of circle of vision of the oscillators and the probability of forming connections in order to study and compare the critical values of the coupling required to induce ES in the population of phase oscillators.

Predicting the antiepileptic drug response by brain connectivity in newly diagnosed focal epilepsy

OBJECTIVE

Growing evidence has suggested that epilepsy is a disease with alterations in brain connectivity. The aim of this study was to investigate whether the changes in brain connectivity can predict the response to an antiepileptic drug (AED) in patients with a newly diagnosed focal epilepsy of unknown etiology.

METHODS

This observational study was independently performed at two tertiary hospitals (Group A and B). Thirty-eight patients with newly diagnosed focal epilepsy of unknown etiology were enrolled in Group A and 46 patients in Group B. We divided these patients into two groups according to their seizure control after AED treatment: AED good and poor responders. We defined the AED good responders as those in whom had seizure free for at least the last 6 months while AED poor responders who were not. All of the subjects underwent diffusion tensor imaging, and graph theoretical analysis was applied to reveal the brain connectivity. We investigated the difference in the clinical characteristics and network measurements between the two groups.

RESULTS

Of the network measures, the assortativity coefficient in the AED good responders was significantly higher than that in the AED poor responders in both Groups A and B (- 0.0239 vs. - 0.0473, p = 0.0110 in Group A; 0.0173 vs. - 0.0180, p = 0.0024 in Group B). The Kaplan-Meier survival analysis revealed that the time to failure to retain the first AED was significantly longer in the patients with assortative networks (assortativity coefficient > 0) than in those with disassortative networks (assortativity coefficient < 0) in Group B.

CONCLUSION

We demonstrated that the assortativity coefficient differed between patients with newly diagnosed focal epilepsy of unknown etiology according to their AED responses, which suggests that the changes in brain connectivity could be a biomarker for predicting the responses to AED.

Abnormal synchronization of functional and structural networks in schizophrenia

Synchronization is believed to play an important role in information processing of the brain. Mounting evidence supports the hypothesis that schizophrenia is related to impaired neural synchrony. However, most previous studies characterize brain synchronization from the perspective of temporal coordination of distributed neural activity, rather than network properties. Our aim was to investigate the network synchronization alterations in schizophrenia using publically available data. Resting-state functional magnetic resonance imaging (fMRI) and diffusion tensor imaging (DTI) were performed in 96 schizophrenia patients and 120 healthy controls. The whole-brain functional and structural networks were constructed and analyzed using graph theoretical approaches. Inter-group differences in network synchronization were investigated. Both the binary and weighted functional networks of schizophrenia patients exhibited decreased synchronizability (increased eigenratio) than those of healthy controls. With respect to the structural binary networks, schizophrenia patients showed a trend towards excessive synchronizability (decreased eigenratio). In addition, the excessive synchronizability of the structural binary networks was associated with more severe negative symptoms in schizophrenia patients. Our findings provide novel biological evidence that schizophrenia involves a disruption of neural synchrony from the perspective of network properties.

Optimal global synchronization of partially forced Kuramoto oscillators

We consider the problem of global synchronization in a large random network of Kuramoto oscillators where some of them are subject to an external periodically driven force. We explore a recently proposed dimensional reduction approach and introduce an effective two-dimensional description for the problem. From the dimensionally reduced model, we obtain analytical predictions for some critical parameters necessary for the onset of a globally synchronized state in the system. Moreover, the low dimensional model also allows us to introduce an optimization scheme for the problem. Our main conclusion, which has been corroborated by exhaustive numerical simulations, is that for a given large random network of Kuramoto oscillators, with random natural frequencies ω_i, such that a fraction of them is subject to an external periodic force with frequency Ω, the best global synchronization properties correspond to the case where the fraction of the forced oscillators is chosen to be those ones such that |ω_i-Ω| is maximal. Our results might shed some light on the structure and evolution of natural systems for which the presence or the absence of global synchronization is a desired property. Some properties of the optimal forced networks and their relation to recent results in the literature are also discussed.

Next Generation Networks: Featuring the Potential Role of Emerging Applications in Translational Oncology

Nowadays, networks are pervasively used as examples of models suitable to mathematically represent and visualize the complexity of systems associated with many diseases, including cancer. In the cancer context, the concept of network entropy has guided many studies focused on comparing equilibrium to disequilibrium (i.e., perturbed) conditions. Since these conditions reflect both structural and dynamic properties of network interaction maps, the derived topological characterizations offer precious support to conduct cancer inference. Recent innovative directions have emerged in network medicine addressing especially experimental omics approaches integrated with a variety of other data, from molecular to clinical and also electronic records, bioimaging etc. This work considers a few theoretically relevant concepts likely to impact the future of applications in personalized/precision/translational oncology. The focus goes to specific properties of networks that are still not commonly utilized or studied in the oncological domain, and they are: controllability, synchronization and symmetry. The examples here provided take inspiration from the consideration of metastatic processes, especially their progression through stages and their hallmark characteristics. Casting these processes into computational frameworks and identifying network states with specific modular configurations may be extremely useful to interpret or even understand dysregulation patterns underlying cancer, and associated events (onset, progression) and disease phenotypes.

Perturbation analysis and comparison of network synchronization methods

In many networked systems, synchronization is important and useful, and how to enhance synchronizability is an interesting problem. Based on the matrix perturbation theory, we analyze five methods of network synchronization enhancement, including the link removal, node removal, dividing hub node, pull control, and pinning control methods, and obtain explicit expressions for eigenvalue changes. By these comparisons, we find that, among all these methods, the pull control method is remarkable, as it can extend the synchronization (coupling strength) region from both the left and right sides, for any controlled node. Extensive simulation results are given to support the accuracy of the perturbation-based analysis.

Identifying Key Genes of Liver Cancer by Networking of Multiple Data Sets

Liver cancer is one of the deadliest cancers in the world. To find effective therapies for this cancer, it is indispensable to identify key genes, which may play critical roles in the incidence of the liver cancer. To identify key genes of the liver cancer with high accuracy, we integrated multiple microarray gene expression data sets to compute common differentially expressed genes, which will result more accurate than those from individual data set. To find the main functions or pathways that these genes are involved in, some enrichment analyses were performed including functional enrichment analysis, pathway enrichment analysis, and disease association study. Based on these genes, a protein-protein interaction network was constructed and analyzed to identify key genes of the liver cancer by combining the local and global influence of nodes in the network. The identified key genes, such as TOP2A, ESR1, and KMO, have been demonstrated to be key biomarkers of the liver cancer in many publications. All the results suggest that our method can effectively identify key genes of the liver cancer. Moreover, our method can be applied to other types of data sets to select key genes of other complex diseases.

Spectral properties of complex networks

This review presents an account of the major works done on spectra of adjacency matrices drawn on networks and the basic understanding attained so far. We have divided the review under three sections: (a) extremal eigenvalues, (b) bulk part of the spectrum, and (c) degenerate eigenvalues, based on the intrinsic properties of eigenvalues and the phenomena they capture. We have reviewed the works done for spectra of various popular model networks, such as the Erdős-Rényi random networks, scale-free networks, 1-d lattice, small-world networks, and various different real-world networks. Additionally, potential applications of spectral properties for natural processes have been reviewed.

Kuramoto dilemma alleviated by optimizing connectivity and rationality

Recently, Antonioni and Cardillo proposed a coevolutionary model based on the intertwining of oscillator synchronization and evolutionary game theory [Phys. Rev. Lett. 118, 238301 (2017)PRLTAO0031-900710.1103/PhysRevLett.118.238301], in which each Kuramoto oscillator can decide whether to interact or not with its neighbors, and all oscillators can receive some benefits from the local synchronization, but those who choose to interact must pay a cost. Oscillators are allowed to update their strategies according to payoff difference, wherein the strategy of an oscillator who has obtained higher payoff is more likely to be followed. Utilizing this coevolutionary model, we find that the global synchronization level reaches the highest level when the average degree of the underlying interaction network is moderate. We also study how synchronization is affected by the individual rationality in choosing strategy.

Control energy of complex networks towards distinct mixture states

Controlling complex networked systems is a real-world puzzle that remains largely unsolved. Despite recent progress in understanding the structural characteristics of network control energy, target state and system dynamics have not been explored. We examine how varying the final state mixture affects the control energy of canonical and conformity-incorporated dynamical systems. We find that the control energy required to drive a network to an identical final state is lower than that required to arrive a non-identical final state. We also demonstrate that it is easier to achieve full control in a conformity-based dynamical network. Finally we determine the optimal control strategy in terms of the network hierarchical structure. Our work offers a realistic understanding of the control energy within the final state mixture and sheds light on controlling complex systems.

Synchronization invariance under network structural transformations

Synchronization processes are ubiquitous despite the many connectivity patterns that complex systems can show. Usually, the emergence of synchrony is a macroscopic observable; however, the microscopic details of the system, as, e.g., the underlying network of interactions, is many times partially or totally unknown. We already know that different interaction structures can give rise to a common functionality, understood as a common macroscopic observable. Building upon this fact, here we propose network transformations that keep the collective behavior of a large system of Kuramoto oscillators invariant. We derive a method based on information theory principles, that allows us to adjust the weights of the structural interactions to map random homogeneous in-degree networks into random heterogeneous networks and vice versa, keeping synchronization values invariant. The results of the proposed transformations reveal an interesting principle; heterogeneous networks can be mapped to homogeneous ones with local information, but the reverse process needs to exploit higher-order information. The formalism provides analytical insight to tackle real complex scenarios when dealing with uncertainty in the measurements of the underlying connectivity structure.

Association schemes perspective of microbubble cluster in ultrasonic fields

Dynamics of a cluster of chaotic oscillators on a network are studied using coupled maps. By introducing the association schemes, we obtain coupling strength in the adjacency matrices form, which satisfies Markov matrices property. We remark that in general, the stability region of the cluster of oscillators at the synchronization state is characterized by Lyapunov exponent which can be defined based on the N-coupled map. As a detailed physical example, dynamics of microbubble cluster in an ultrasonic field are studied using coupled maps. Microbubble cluster dynamics have an indicative highly active nonlinear phenomenon, were not easy to be explained. In this paper, a cluster of microbubbles with a thin elastic shell based on the modified Keller-Herring equation in an ultrasonic field is demonstrated in the framework of the globally coupled map. On the other hand, a relation between the microbubble elements is replaced by a relation between the vertices. Based on this method, the stability region of microbubbles pulsations at complete synchronization state has been obtained analytically. In this way, distances between microbubbles as coupling strength play the crucial role. In the stability region, we thus observe that the problem of study of dynamics of N-microbubble oscillators reduce to that of a single microbubble. Therefore, the important parameters of the isolated microbubble such as applied pressure, driving frequency and the initial radius have effective behavior on the synchronization state.

Competitive influence maximization and enhancement of synchronization in populations of non-identical Kuramoto oscillators

Many networked systems have evolved to optimize performance of function. Much literature has considered optimization of networks by central planning, but investigations of network formation amongst agents connecting to achieve non-aligned goals are comparatively rare. Here we consider the dynamics of synchronization in populations of coupled non-identical oscillators and analyze adaptations in which individual nodes attempt to rewire network topology to optimize node-specific aims. We demonstrate that, even though individual nodes’ goals differ very widely, rewiring rules in which each node attempts to connect to the rest of the network in such a way as to maximize its influence on the system can enhance synchronization of the collective. The observed speed-up of consensus finding in this competitive dynamics might explain enhanced synchronization in real world systems and shed light on mechanisms for improved consensus finding in society.

Asymmetry-induced synchronization in oscillator networks

A scenario has recently been reported in which in order to stabilize complete synchronization of an oscillator network-a symmetric state-the symmetry of the system itself has to be broken by making the oscillators nonidentical. But how often does such behavior-which we term asymmetry-induced synchronization (AISync)-occur in oscillator networks? Here we present the first general scheme for constructing AISync systems and demonstrate that this behavior is the norm rather than the exception in a wide class of physical systems that can be seen as multilayer networks. Since a symmetric network in complete synchrony is the basic building block of cluster synchronization in more general networks, AISync should be common also in facilitating cluster synchronization by breaking the symmetry of the cluster subnetworks.

New Insights on Temporal Lobe Epilepsy Based on Plasticity-Related Network Changes and High-Order Statistics

Epilepsy is a disorder of the brain characterized by the predisposition to generate recurrent unprovoked seizures, which involves reshaping of neuronal circuitries based on intense neuronal activity. In this review, we first detailed the regulation of plasticity-associated genes, such as ARC, GAP-43, PSD-95, synapsin, and synaptophysin. Indeed, reshaping of neuronal connectivity after the primary, acute epileptogenesis event increases the excitability of the temporal lobe. Herein, we also discussed the heterogeneity of neuronal populations regarding the number of synaptic connections, which in the theoretical field is commonly referred as degree. Employing integrate-and-fire neuronal model, we determined that in addition to increased synaptic strength, degree correlations might play essential and unsuspected roles in the control of network activity. Indeed, assortativity, which can be described as a condition where high-degree correlations are observed, increases the excitability of neural networks. In this review, we summarized recent topics in the field, and data were discussed according to newly developed or unusual tools, as provided by mathematical graph analysis and high-order statistics. With this, we were able to present new foundations for the pathological activity observed in temporal lobe epilepsy.

Diffusion Geometry Unravels the Emergence of Functional Clusters in Collective Phenomena

Collective 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.

Measure for degree heterogeneity in complex networks and its application to recurrence network analysis

We propose a novel measure of degree heterogeneity, for unweighted and undirected complex networks, which requires only the degree distribution of the network for its computation. We show that the proposed measure can be applied to all types of network topology with ease and increases with the diversity of node degrees in the network. The measure is applied to compute the heterogeneity of synthetic (both random and scale free (SF)) and real-world networks with its value normalized in the interval [Formula: see text]. To define the measure, we introduce a limiting network whose heterogeneity can be expressed analytically with the value tending to 1 as the size of the network N tends to infinity. We numerically study the variation of heterogeneity for random graphs (as a function of p and N) and for SF networks with γ and N as variables. Finally, as a specific application, we show that the proposed measure can be used to compare the heterogeneity of recurrence networks constructed from the time series of several low-dimensional chaotic attractors, thereby providing a single index to compare the structural complexity of chaotic attractors.

Chaotic Griffiths Phase with Anomalous Lyapunov Spectra in Coupled Map Networks

Dynamics of coupled chaotic oscillators on a network are studied using coupled maps. Within a broad range of parameter values representing the coupling strength or the degree of elements, the system repeats formation and split of coherent clusters. The distribution of the cluster size follows a power law with the exponent α, which changes with the parameter values. The number of positive Lyapunov exponents and their spectra are scaled anomalously with the power of the system size with the exponent β, which also changes with the parameters. The scaling relation α∼2(β+1) is uncovered, which is universal independent of parameters and among random networks.

Distress Propagation in Complex Networks: The Case of Non-Linear DebtRank

We consider a dynamical model of distress propagation on complex networks, which we apply to the study of financial contagion in networks of banks connected to each other by direct exposures. The model that we consider is an extension of the DebtRank algorithm, recently introduced in the literature. The mechanics of distress propagation is very simple: When a bank suffers a loss, distress propagates to its creditors, who in turn suffer losses, and so on. The original DebtRank assumes that losses are propagated linearly between connected banks. Here we relax this assumption and introduce a one-parameter family of non-linear propagation functions. As a case study, we apply this algorithm to a data-set of 183 European banks, and we study how the stability of the system depends on the non-linearity parameter under different stress-test scenarios. We find that the system is characterized by a transition between a regime where small shocks can be amplified and a regime where shocks do not propagate, and that the overall stability of the system increases between 2008 and 2013.

Attack Vulnerability of Network Controllability

Controllability of complex networks has attracted much attention, and understanding the robustness of network controllability against potential attacks and failures is of practical significance. In this paper, we systematically investigate the attack vulnerability of network controllability for the canonical model networks as well as the real-world networks subject to attacks on nodes and edges. The attack strategies are selected based on degree and betweenness centralities calculated for either the initial network or the current network during the removal, among which random failure is as a comparison. It is found that the node-based strategies are often more harmful to the network controllability than the edge-based ones, and so are the recalculated strategies than their counterparts. The Barabási-Albert scale-free model, which has a highly biased structure, proves to be the most vulnerable of the tested model networks. In contrast, the Erdős-Rényi random model, which lacks structural bias, exhibits much better robustness to both node-based and edge-based attacks. We also survey the control robustness of 25 real-world networks, and the numerical results show that most real networks are control robust to random node failures, which has not been observed in the model networks. And the recalculated betweenness-based strategy is the most efficient way to harm the controllability of real-world networks. Besides, we find that the edge degree is not a good quantity to measure the importance of an edge in terms of network controllability.

Network-complement transitions, symmetries, and cluster synchronization

Synchronization in networks of coupled oscillators is known to be largely determined by the spectral and symmetry properties of the interaction network. Here, we leverage this relation to study a class of networks for which the threshold coupling strength for global synchronization is the lowest among all networks with the same number of nodes and links. These networks, defined as being uniform, complete, and multi-partite (UCM), appear at each of an infinite sequence of network-complement transitions in a larger class of networks characterized by having near-optimal thresholds for global synchronization. We show that the distinct symmetry structure of the UCM networks, which by design are optimized for global synchronizability, often leads to formation of clusters of synchronous oscillators, and that such states can coexist with the state of global synchronization.

The Role of Node Heterogeneity in the Coupled Spreading of Epidemics and Awareness

Exploring the interplay between information spreading and epidemic spreading is a topic that has been receiving increasing attention. As an efficient means of depicting the spreading of information, which manifests as a cascade phenomenon, awareness cascading is utilized to investigate this coupled transmission. Because in reality, different individuals facing the same epidemic will exhibit distinct behaviors according to their own experiences and attributes, it is important for us to consider the heterogeneity of individuals. Consequently, we propose a heterogeneous spreading model. To describe the heterogeneity, two of the most important but radically different methods for this purpose, the degree and k-core measures, are studied in this paper through three models based on different assumptions. Adopting a Markov chain approach, we succeed in predicting the epidemic threshold trend. Furthermore, we find that when the k-core measure is used to classify individuals, the spreading process is robust to these models, meaning that regardless of the model used, the spreading process is nearly identical at the macroscopic level. In addition, the k-core measure leads to a much larger final epidemic size than the degree measure. These results are cross-checked through numerous simulations, not only of a synthetic network but also of a real multiplex network. The presented findings provide a better understanding of k-core individuals and reveal the importance of considering network structure when investigating various dynamic processes.

The H-index of a network node and its relation to degree and coreness

Identifying influential nodes in dynamical processes is crucial in understanding network structure and function. Degree, H-index and coreness are widely used metrics, but previously treated as unrelated. Here we show their relation by constructing an operator , in terms of which degree, H-index and coreness are the initial, intermediate and steady states of the sequences, respectively. We obtain a family of H-indices that can be used to measure a node's importance. We also prove that the convergence to coreness can be guaranteed even under an asynchronous updating process, allowing a decentralized local method of calculating a node's coreness in large-scale evolving networks. Numerical analyses of the susceptible-infected-removed spreading dynamics on disparate real networks suggest that the H-index is a good tradeoff that in many cases can better quantify node influence than either degree or coreness.

Identifying influential nodes in networks is important for the understanding of their structure and function, but there are several so far unrelated measures to assess this. Here, the authors unfold relations among knows criteria and construct a family of indices that interpolate between degree and coreness.

Optimal synchronization of Kuramoto oscillators: A dimensional reduction approach

A recently proposed dimensional reduction approach for studying synchronization in the Kuramoto model is employed to build optimal network topologies to favor or to suppress synchronization. The approach is based in the introduction of a collective coordinate for the time evolution of the phase locked oscillators, in the spirit of the Ott-Antonsen ansatz. We show that the optimal synchronization of a Kuramoto network demands the maximization of the quadratic function ω(T)Lω, where ω stands for the vector of the natural frequencies of the oscillators and L for the network Laplacian matrix. Many recently obtained numerical results can be reobtained analytically and in a simpler way from our maximization condition. A computationally efficient hill climb rewiring algorithm is proposed to generate networks with optimal synchronization properties. Our approach can be easily adapted to the case of the Kuramoto models with both attractive and repulsive interactions, and again many recent numerical results can be rederived in a simpler and clearer analytical manner.

Enhancing the stability of the synchronization of multivariable coupled oscillators

Synchronization processes in populations of identical networked oscillators are the focus of intense studies in physical, biological, technological, and social systems. Here we analyze the stability of the synchronization of a network of oscillators coupled through different variables. Under the assumption of an equal topology of connections for all variables, the master stability function formalism allows assessing and quantifying the stability properties of the synchronization manifold when the coupling is transferred from one variable to another. We report on the existence of an optimal coupling transference that maximizes the stability of the synchronous state in a network of Rössler-like oscillators. Finally, we design an experimental implementation (using nonlinear electronic circuits) which grounds the robustness of the theoretical predictions against parameter mismatches, as well as against intrinsic noise of the system.

Time-dependent degree-degree correlations in epileptic brain networks: from assortative to dissortative mixing

We investigate the long-term evolution of degree-degree correlations (assortativity) in functional brain networks from epilepsy patients. Functional networks are derived from continuous multi-day, multi-channel electroencephalographic data, which capture a wide range of physiological and pathophysiological activities. In contrast to previous studies which all reported functional brain networks to be assortative on average, even in case of various neurological and neurodegenerative disorders, we observe large fluctuations in time-resolved degree-degree correlations ranging from assortative to dissortative mixing. Moreover, in some patients these fluctuations exhibit some periodic temporal structure which can be attributed, to a large extent, to daily rhythms. Relevant aspects of the epileptic process, particularly possible pre-seizure alterations, contribute marginally to the observed long-term fluctuations. Our findings suggest that physiological and pathophysiological activity may modify functional brain networks in a different and process-specific way. We evaluate factors that possibly influence the long-term evolution of degree-degree correlations.

Synchronizability of random rectangular graphs

Random rectangular graphs (RRGs) represent a generalization of the random geometric graphs in which the nodes are embedded into hyperrectangles instead of on hypercubes. The synchronizability of RRG model is studied. Both upper and lower bounds of the eigenratio of the network Laplacian matrix are determined analytically. It is proven that as the rectangular network is more elongated, the network becomes harder to synchronize. The synchronization processing behavior of a RRG network of chaotic Lorenz system nodes is numerically investigated, showing complete consistence with the theoretical results.

Graph Theory-Based Approach for Stability Analysis of Stochastic Coupled Systems With Lévy Noise on Networks

In this paper, a novel class of stochastic coupled systems with Lévy noise on networks (SCSLNNs) is presented. Both white noise and Lévy noise are considered in the networks. By exploiting graph theory and Lyapunov stability theory, criteria ensuring p th moment exponential stability and stability in probability of these SCSLNNs are established, respectively. These principles are closely related to the topology of the network and the perturbation intensity of white noise and Lévy noise. Moreover, to verify the theoretical results, stochastic coupled oscillators with Lévy noise on a network and stochastic Volterra predator-prey system with Lévy noise are performed. Finally, a numerical example about oscillators' network is provided to illustrate the feasibility of our analytical results.

Transitions between dynamical behaviors of oscillator networks induced by diversity of nodes and edges

We study the impact of dynamical and structural heterogeneity on the collective dynamics of large small-world networks of pulse-coupled integrate-and-fire oscillators endowed with refractory periods and time delay. Depending on the choice of homogeneous control parameters (here, refractoriness and coupling strength), these networks exhibit a large spectrum of dynamical behaviors, including asynchronous, partially synchronous, and fully synchronous states. Networks exhibit transitions between these dynamical behaviors upon introducing heterogeneity. We show that the probability for a network to exhibit a certain dynamical behavior (network susceptibility) is affected differently by dynamical and structural heterogeneity and depends on the respective homogeneous dynamics.