Dynamical robustness in complex networks: the crucial role of low-degree nodes

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.

Fine construction of gene coexpression network analysis using GTOM and RECODE detected a critical module of neuroblastoma stages 4 and 4S

BACKGROUND

Stage 4 neuroblastoma (NBL), a solid tumor of childhood, has a poor prognosis. Despite intensive molecular genetic studies, no targetable gene abnormalities have been identified. Stage 4S NBL has a characteristic of spontaneous regression, and elucidation of the mechanistic differences between stages 4 and 4S may improve treatment. Conventional NBL studies have mainly focused on the detection of abnormalities in individual genes and have rarely examined abnormalities in gene networks. While the gene coexpression network is expected to contribute to the detection of network abnormalities, the fragility of the network due to data noise and the extraction of arbitrary topological structures for the high-dimensional network are issues.

RESULTS

The present paper concerns the classification method of stages 4 and 4S NBL patients using highly accurate gene coexpression network analysis based on RNA-sequencing data of transcription factors (TFs). In particular, after applying a noise reduction method RECODE, generalized topological overlapping measure (GTOM), which weighs the connections of nodes in the network structure, succeeded in extracting a cluster of TFs that showed high classification performance for stages 4 and 4S. In addition, we investigated how these clusters correspond to clinical information and to TFs which control the normal adrenal tissue and NBL characters.

CONCLUSIONS

A clustering method is presented for finding intermediate-scale clusters of TFs that give considerable separation performance for distinguishing between stages 4 and 4S. It is suggested that this method is useful as a way to extract factors that contribute to the separation of groups from multiple pieces of information such as gene expression levels.

Transformation starts at the periphery of networks where pushback is less

Complex systems ranging from societies to ecological communities and power grids may be viewed as networks of connected elements. Such systems can go through critical transitions driven by an avalanche of contagious change. Here we ask, where in a complex network such a systemic shift is most likely to start. Intuitively, a central node seems the most likely source of such change. Indeed, topological studies suggest that central nodes can be the Achilles heel for attacks. We argue that the opposite is true for the class of networks in which all nodes tend to follow the state of their neighbors, a category we call two-way pull networks. In this case, a well-connected central node is an unlikely starting point of a systemic shift due to the buffering effect of connected neighbors. As a result, change is most likely to cascade through the network if it spreads first among relatively poorly connected nodes in the periphery. The probability of such initial spread is highest when the perturbation starts from intermediately connected nodes at the periphery, or more specifically, nodes with intermediate degree and relatively low closeness centrality. Our finding is consistent with empirical observations on social innovation, and may be relevant to topics as different as the sources of originality of art, collapse of financial and ecological networks and the onset of psychiatric disorders.

Soil microbial diversity under different types of interference in birch secondary forest in the Greater Khingan Mountains in China

Introduction

Soil microorganisms are an important component of soil ecosystems with an indispensable role in forest ecosystems. We analyzed the soil microbial diversity in birch secondary forest formed by natural restoration or artificial reconstruction after interference by burning, clear cutting, and gradient cutting, and the Betula platyphylla Suk undisturbed forest in the Greater Khingan Mountains in China.

Methods

Illumina high-throughput sequencing technology was used to analyze the characteristics of the soil microbial community during the restoration process of birch secondary forest caused by the different types of interference. The relationships between bacteria and fungi were analyzed. The gene functions of the soil bacterial community and the ecological functions of soil fungi were predicted using PICRUSt and FunGuild, respectively.

Results

At the phylum level, the species and quantity of bacteria were more abundant than that of fungi. At the genus level, no obvious differences in the abundance of bacteria were observed; there were obvious differences in the abundance of fungi. Among the eight sample plots, the artificial larch forest belt had the highest bacterial and fungal alpha diversity, which was slightly higher than undisturbed forest, while the other sample plots were significantly lower. Gradual cutting pure birch forest bacteria and fungi had the highest beta diversity, and artificial larch forest belt bacteria and heavy burn sample plot fungi had the lowest beta diversity. Samples from the cutting and burning sample plots were significantly different from the undisturbed forest at the phylum level of Acidobacteriae, Acidimicrobiia, Mortierellomycetes and Sordariomycetes. We found statistical differences in biomarkers between bacterial and fungal communities in undisturbed forest and artificial larch forest belt and burn sample plots. PICRUSt prediction and FunGuild prediction showed that soil bacterial and fungal communities were rich in gene and ecological functions, respectively. In the microbial network, the stability or anti-interference performance of the fungal community was higher than that of bacteria.

Conclusion

Our data reveal the characteristics of the soil microbial community during the restoration process of Betula platyphylla Suk secondary forest under different types of disturbance, which is of great significance for understanding the role of soil microorganisms in the forest ecological cycle.

Determinants of collective failure in excitable networks

We study collective failures in biologically realistic networks that consist of coupled excitable units. The networks have broad-scale degree distribution, high modularity, and small-world properties, while the excitable dynamics is determined by the paradigmatic FitzHugh-Nagumo model. We consider different coupling strengths, bifurcation distances, and various aging scenarios as potential culprits of collective failure. We find that for intermediate coupling strengths, the network remains globally active the longest if the high-degree nodes are first targets for inactivation. This agrees well with previously published results, which showed that oscillatory networks can be highly fragile to the targeted inactivation of low-degree nodes, especially under weak coupling. However, we also show that the most efficient strategy to enact collective failure does not only non-monotonically depend on the coupling strength, but it also depends on the distance from the bifurcation point to the oscillatory behavior of individual excitable units. Altogether, we provide a comprehensive account of determinants of collective failure in excitable networks, and we hope this will prove useful for better understanding breakdowns in systems that are subject to such dynamics.

Robustness Evaluation of the Open Source Product Community Network Considering Different Influential Nodes

With the rapid development of Internet technology, the innovative value and importance of the open source product community (OSPC) is becoming increasingly significant. Ensuring high robustness is essential to the stable development of OSPC with open characteristics. In robustness analysis, degree and betweenness are traditionally used to evaluate the importance of nodes. However, these two indexes are disabled to comprehensively evaluate the influential nodes in the community network. Furthermore, influential users have many followers. The effect of irrational following behavior on network robustness is also worth investigating. To solve these problems, we built a typical OSPC network using a complex network modeling method, analyzed its structural characteristics and proposed an improved method to identify influential nodes by integrating the network topology characteristics indexes. We then proposed a model containing a variety of relevant node loss strategies to simulate the changes in robustness of the OSPC network. The results showed that the proposed method can better distinguish the influential nodes in the network. Furthermore, the network's robustness will be greatly damaged under the node loss strategies considering the influential node loss (i.e., structural hole node loss and opinion leader node loss), and the following effect can greatly change the network robustness. The results verified the feasibility and effectiveness of the proposed robustness analysis model and indexes.

Dynamical robustness of complex networks subject to long-range connectivity

In spite of a few attempts in understanding the dynamical robustness of complex networks, this extremely important subject of research is still in its dawn as compared to the other dynamical processes on networks. We hereby consider the concept of long-range interactions among the dynamical units of complex networks and demonstrate for the first time that such a characteristic can have noteworthy impacts on the dynamical robustness of networked systems, regardless of the underlying network topology. We present a comprehensive analysis of this phenomenon on top of diverse network architectures. Such dynamical damages being able to substantially affect the network performance, determining mechanisms that boost the robustness of networks becomes a fundamental question. In this work, we put forward a prescription based upon self-feedback that can efficiently resurrect global rhythmicity of complex networks composed of active and inactive dynamical units, and thus can enhance the network robustness. We have been able to delineate the whole proposition analytically while dealing with all d -path adjacency matrices, having an excellent agreement with the numerical results. For the numerical computations, we examine scale-free networks, Watts–Strogatz small-world model and also Erdös–Rényi random network, along with Landau–Stuart oscillators for casting the local dynamics.

Measuring the importance of individual units in producing the collective behavior of a complex network

A quantitative evaluation of the contribution of individual units in producing the collective behavior of a complex network can allow us to understand the potential damage to the structure integrity due to the failure of local nodes. Given a time series for each unit, a natural way to do this is to find the information flowing from the unit of concern to the rest of the network. In this study, we show that this flow can be rigorously derived in the setting of a continuous-time dynamical system. With a linear assumption, a maximum likelihood estimator can be obtained, allowing us to estimate it in an easy way. As expected, this "cumulative information flow" does not equal the sum of the information flows to other individual units, reflecting the collective phenomenon that a group is not the addition of individual members. For the purpose of demonstration and validation, we have examined a network made of Stuart-Landau oscillators. Depending on the topology, the computed information flow may differ. In some situations, the most crucial nodes for the network are not the hubs, i.e., nodes with high degrees; they may have low degrees and, if depressed or attacked, will cause the failure of the entire network. This study can help diagnose neural network problems, control epidemic diseases, trace city traffic bottlenecks, identify the potential cause of power grid failure (e.g., the 2003 great power outage that darkened much of North America), build robust computer networks, and so forth.

Dynamic Robustness of Semantic-Based Collaborative Knowledge Network of Open Source Project

Robustness of the collaborative knowledge network (CKN) is critical to the success of open source projects. To study this robustness more comprehensively and accurately, we constructed a weighted CKN based on the semantic analysis of collaborative behavior, where (a) open source designers were the network nodes, (b) collaborative behavior among designers was the edges, and (c) collaborative text content intensity and collaborative frequency intensity were the edge weights. To study the robustness from a dynamic viewpoint, we constructed three CKNs from different stages of the project life cycle: the start-up, growth and maturation stages. The connectivity and collaboration efficiency of the weighted network were then used as robustness evaluation indexes. Further, we designed four edge failure modes based on the behavioral characteristics of open source designers. Finally, we carried out dynamic robustness analysis experiments based on the empirical data of a Local Motors open source car design project. Our results showed that the CKN performed differently at different stages of the project life cycle, and our specific findings could help community managers of open source projects to formulate different network protection strategies at different stages of their projects.

Persistence in multilayer ecological network consisting of harvested patches

Complex network theory yields a powerful approach to solve the difficulties arising in a major section of ecological systems, prey-predator interaction being one among them. A large variety of ecological systems have been successfully investigated employing the theory of complex networks, and one of the most significant advancements in this theory is the emerging field of multilayer networks. The field of multilayer networks provides a natural framework to accommodate multiple layers of complexities emerging in ecosystems. In this article, we consider prey-predator patches communicating among themselves while being connected by distinct small-world dispersal topologies in two layers of the network. We scrutinize the robustness of the multilayer ecological network sustaining gradually over harvested patches. We thoroughly report the consequences of introducing asymmetries in both interlayer and intralayer dispersal strengths as well as the network topologies on the global persistence of species in the network. Besides numerical simulation, we analytically derive the critical point up to which the network can sustain species in the network. Apart from the results on a purely multiplex framework, we validate our claims for multilayer formalism in which the patches of the layers are different. Interestingly, we observe that due to the interaction between the two layers, species are recovered in the layer that we assume to be extinct initially. Moreover, we find similar results while considering two completely different prey-predator systems, which eventually attests that the outcomes are not model specific.

Best influential spreaders identification using network global structural properties

Influential spreaders are the crucial nodes in a complex network that can act as a controller or a maximizer of a spreading process. For example, we can control the virus propagation in an epidemiological network by controlling the behavior of such influential nodes, and amplify the information propagation in a social network by using them as a maximizer. Many indexing methods have been proposed in the literature to identify the influential spreaders in a network. Nevertheless, we have notice that each individual network holds different connectivity structures that we classify as complete, incomplete, or in-between based on their components and density. These affect the accuracy of existing indexing methods in the identification of the best influential spreaders. Thus, no single indexing strategy is sufficient from all varieties of network connectivity structures. This article proposes a new indexing method Network Global Structure-based Centrality (ngsc) which intelligently combines existing kshell and sum of neighbors' degree methods with knowledge of the network's global structural properties, such as the giant component, average degree, and percolation threshold. The experimental results show that our proposed method yields a better spreading performance of the seed spreaders over a large variety of network connectivity structures, and correlates well with ranking based on an SIR model used as ground truth. It also out-performs contemporary techniques and is competitive with more sophisticated approaches that are computationally cost.

Enhancement of dynamical robustness in a mean-field coupled network through self-feedback delay

The network of self-sustained oscillators plays an important role in exploring complex phenomena in many areas of science and technology. The aging of an oscillator is referred to as turning non-oscillatory due to some local perturbations that might have adverse effects in macroscopic dynamical activities of a network. In this article, we propose an efficient technique to enhance the dynamical activities for a network of coupled oscillators experiencing aging transition. In particular, we present a control mechanism based on delayed negative self-feedback, which can effectively enhance dynamical robustness in a mean-field coupled network of active and inactive oscillators. Even for a small value of delay, robustness gets enhanced to a significant level. In our proposed scheme, the enhancing effect is more pronounced for strong coupling. To our surprise even if all the oscillators perturbed to equilibrium mode were delayed negative self-feedback is able to restore oscillatory activities in the network for strong coupling strength. We demonstrate that our proposed mechanism is independent of coupling topology. For a globally coupled network, we provide numerical and analytical treatment to verify our claim. To show that our scheme is independent of network topology, we also provide numerical results for the local mean-field coupled complex network. Also, for global coupling to establish the generality of our scheme, we validate our results for both Stuart-Landau limit cycle oscillators and chaotic Rössler oscillators.

Aging in global networks with competing attractive-Repulsive interaction

We study the dynamical inactivity of the global network of identical oscillators in the presence of mixed attractive and repulsive coupling. We consider that the oscillators are a priori in all to all attractive coupling and then upon increasing the number of oscillators interacting via repulsive interaction, the whole network attains a steady state at a critical fraction of repulsive nodes, p_c. The macroscopic inactivity of the network is found to follow a typical aging transition due to competition between attractive-repulsive interactions. The analytical expression connecting the coupling strength and p_c is deduced and corroborated with numerical outcomes. We also study the influence of asymmetry in the attractive-repulsive interaction, which leads to symmetry breaking. We detect chimera-like and mixed states for a certain ratio of coupling strengths. We have verified sequential and random modes to choose the repulsive nodes and found that the results are in agreement. The paradigmatic networks with diverse dynamics, viz., limit cycle (Stuart-Landau), chaos (Rössler), and bursting (Hindmarsh-Rose neuron), are analyzed.

Abnormal route to aging transition in a network of coupled oscillators

In this article, we investigate the dynamical robustness in a network of Van der Pol oscillators. In particular, we consider a network of diffusively coupled Van der Pol oscillators to explore the aging transition phenomena. Our investigation reveals that the route to aging transition in a network of Van der Pol oscillator is different from that of typical sinusoidal oscillators such as Stuart-Landau oscillators. Unlike sinusoidal oscillators, the order parameter does not follow smooth second-order phase transition. Rather, we observe an abnormal phase transition of the order parameter due to the sudden appearance of unbounded trajectories at a critical point. We provide detailed bifurcation analysis of such an abnormal phase transition. We show that the boundary crisis of a limit-cycle oscillator is at the helm of such an unusual discontinuous path of aging transition.

Additional repulsion reduces the dynamical resilience in the damaged networks

In this paper, we investigate the dynamical robustness of diffusively coupled oscillatory networks under the influence of an additional repulsive link. Such a dynamical resilience property is realized through the aging process of the damaged network of active and inactive oscillators. The aging process is one type of phase transition, mainly appearing at a critical threshold of a fraction of the inactive oscillator node where the mean oscillation amplitude of the entire network suddenly vanishes. These critical fractions of the failure nodes in the network are broadly used as a measure of network resilience. Here, we analytically derived the critical fraction of the aging process in the dynamical network. We find that the addition of the repulsive link enhances the critical threshold of the aging transition of diffusively coupled oscillators, which indicated that the dynamical robustness of the coupled network decreases with the presence of the repulsive interaction. Furthermore, we investigate the dynamical robustness of the network against the number of deteriorating repulsive links. We observed that a certain percentage of the repulsive link is enabled to produce the aging process in the entire network. Finally, the effect of symmetry-breaking coupling and the targeted inactivation process on the dynamical robustness property of damaged networks were investigated. The analytically obtained results are verified numerically in the network of coupled Stuart-Landau oscillators. These findings may help us to better understand the role of the coupling mechanism on the phase transition in the damaged network.

Cortical Circuit Dynamics Are Homeostatically Tuned to Criticality In Vivo

Homeostatic mechanisms stabilize neuronal activity in vivo, but whether this process gives rise to balanced network dynamics is unknown. Here, we continuously monitored the statistics of network spiking in visual cortical circuits in freely behaving rats for 9 days. Under control conditions in light and dark, networks were robustly organized around criticality, a regime that maximizes information capacity and transmission. When input was perturbed by visual deprivation, network criticality was severely disrupted and subsequently restored to criticality over 48 h. Unexpectedly, the recovery of excitatory dynamics preceded homeostatic plasticity of firing rates by >30 h. We utilized model investigations to manipulate firing rate homeostasis in a cell-type-specific manner at the onset of visual deprivation. Our results suggest that criticality in excitatory networks is established by inhibitory plasticity and architecture. These data establish that criticality is consistent with a homeostatic set point for visual cortical dynamics and suggest a key role for homeostatic regulation of inhibition.

Low pass filtering mechanism enhancing dynamical robustness in coupled oscillatory networks

A network that consists of a set of active and inactive nodes is called a damaged network and this type of network shows an aging effect (degradation of dynamical activity). This dynamical deterioration affects the normal functioning of the network and also its performance. Therefore, it is necessary to design a proper mechanism to avoid undesired dynamical activity like degradation. In this work, an efficient mechanism, called the low pass filtering technique, is proposed to enhance the dynamical activity of damaged networks of coupled oscillators. Using this mechanism, the dynamic behavior of the damaged network of coupled active and inactive dynamical units is improved and the network survivability is ensured. Because a minor deviation of the controlling parameter is sufficient to restore the oscillatory behavior when the entire network undergoes an aging transition. Even when the whole network degrades due to the deterioration of each node, the larger values of the interaction strength and the controlling parameter play a key role in favor of the revival of dynamic activity in the entire network. Our proposed mechanism is very simple and effective to recover the dynamic features of a damaged network. The effectiveness of this technique has been testified in globally coupled and Erdős Rényi random networks of Stuart-Landau oscillators.

Resumption of dynamism in damaged networks of coupled oscillators

Deterioration in dynamical activities may come up naturally or due to environmental influences in a massive portion of biological and physical systems. Such dynamical degradation may have outright effect on the substantive network performance. This requires us to provide some proper prescriptions to overcome undesired circumstances. In this paper, we present a scheme based on external feedback that can efficiently revive dynamism in damaged networks of active and inactive oscillators and thus enhance the network survivability. Both numerical and analytical investigations are performed in order to verify our claim. We also provide a comparative study on the effectiveness of this mechanism for feedbacks to the inactive group or to the active group only. Most importantly, resurrection of dynamical activity is realized even in time-delayed damaged networks, which are considered to be less persistent against deterioration in the form of inactivity in the oscillators. Furthermore, prominence in our approach is substantiated by providing evidence of enhanced network persistence in complex network topologies taking small-world and scale-free architectures, which makes the proposed remedy quite general. Besides the study in the network of Stuart-Landau oscillators, affirmative influence of external feedback has been justified in the network of chaotic Rössler systems as well.

Survivability of a metapopulation under local extinctions

A metapopulation structure in landscape ecology comprises a group of interacting spatially separated subpopulations or patches of the same species that may experience several local extinctions. This makes the investigation of survivability (in the form of global oscillation) of a metapopulation on top of diverse dispersal topologies extremely crucial. However, among various dispersal topologies in ecological networks, which one can provide higher metapopulation survivability under local extinction is still not well explored. In this article, we scrutinize the robustness of an ecological network consisting of prey-predator patches having Holling type I functional response, against progressively extinct population patches. We present a comprehensive study on this while considering global, small-world, and scale-free dispersal of the subpopulations. Furthermore, we extend our work in enhancing survivability in the form of sustained global oscillation by introducing asymmetries in the dispersal rates of the considered species. Our findings affirm that the asynchrony among the patches plays an important role in the survivability of a metapopulation. In order to demonstrate the model independence of the observed phenomenon, we perform a similar analysis for patches exhibiting Holling type II functional response. On the grounds of the obtained results, our work is expected to provide a better perception of the influence of dispersal arrangements on the global survivability of ecological networks.

Robustness of coupled oscillator networks with heterogeneous natural frequencies

Robustness of coupled oscillator networks against local degradation of oscillators has been intensively studied in this decade. The oscillation behavior on the whole network is typically reduced with an increase in the fraction of degraded (inactive) oscillators. The critical fraction of inactive oscillators, at which a transition from an oscillatory to a quiescent state occurs, has been used as a measure for the network robustness. The larger (smaller) this measure is, the more robust (fragile) the oscillatory behavior on the network is. Most previous studies have used oscillators with identical natural frequencies, for which the oscillators are necessarily synchronized and thereby the analysis is simple. In contrast, we focus on the effect of heterogeneity in the natural frequencies on the network robustness. First, we analytically derive the robustness measure for the coupled oscillator models with heterogeneous natural frequencies under some conditions. Then, we show that increasing the heterogeneity in natural frequencies makes the network fragile. Moreover, we discuss the optimal parameter condition to maximize the network robustness.

Aging transition in systems of oscillators with global distributed-delay coupling

We consider a globally coupled network of active (oscillatory) and inactive (nonoscillatory) oscillators with distributed-delay coupling. Conditions for aging transition, associated with suppression of oscillations, are derived for uniform and gamma delay distributions in terms of coupling parameters and the proportion of inactive oscillators. The results suggest that for the uniform distribution increasing the width of distribution for the same mean delay allows aging transition to happen for a smaller coupling strength and a smaller proportion of inactive elements. For gamma distribution with sufficiently large mean time delay, it may be possible to achieve aging transition for an arbitrary proportion of inactive oscillators, as long as the coupling strength lies in a certain range.

Threshold-activated transport stabilizes chaotic populations to steady states

We explore Random Scale-Free networks of populations, modelled by chaotic Ricker maps, connected by transport that is triggered when population density in a patch is in excess of a critical threshold level. Our central result is that threshold-activated dispersal leads to stable fixed populations, for a wide range of threshold levels. Further, suppression of chaos is facilitated when the threshold-activated migration is more rapid than the intrinsic population dynamics of a patch. Additionally, networks with large number of nodes open to the environment, readily yield stable steady states. Lastly we demonstrate that in networks with very few open nodes, the degree and betweeness centrality of the node open to the environment has a pronounced influence on control. All qualitative trends are corroborated by quantitative measures, reflecting the efficiency of control, and the width of the steady state window.

Boundary in the dynamic phase of globally coupled oscillatory and excitable units

There is a crucial boundary between dynamic phase 1 and dynamic phase 2 of globally coupled oscillatory and excitable units, where the mean field is constant and oscillates in the former and the latter, respectively. This boundary is theoretically derived here for a large population of dynamical units, each having only a phase variable, where it is assumed that both the coupling strength and the distribution width of bifurcation parameters are equally small. This theory, which is applicable only if all or most of the units are intrinsically oscillatory, is confirmed to agree with simulation results for two different distribution densities.

Long-term variability of importance of brain regions in evolving epileptic brain networks

We investigate the temporal and spatial variability of the importance of brain regions in evolving epileptic brain networks. We construct these networks from multiday, multichannel electroencephalographic data recorded from 17 epilepsy patients and use centrality indices to assess the importance of brain regions. Time-resolved indications of highest importance fluctuate over time to a greater or lesser extent, however, with some periodic temporal structure that can mostly be attributed to phenomena unrelated to the disease. In contrast, relevant aspects of the epileptic process contribute only marginally. Indications of highest importance also exhibit pronounced alternations between various brain regions that are of relevance for studies aiming at an improved understanding of the epileptic process with graph-theoretical approaches. Nonetheless, these findings may guide new developments for individualized diagnosis, treatment, and control.

Network Anatomy Controlling Abrupt-like Percolation Transition

We virtually dissect complex networks in order to understand their internal structure, just as doctors do with the bodies of animals. Our novel method classifies network links into four categories: bone, fat, cartilage, and muscle, based on network connectivity. We derive an efficient percolation strategy from this new viewpoint of network anatomy, which enables abrupt-like percolation transition through removal of a small amount of cartilage links, which play a crucial role in network connectivity. Furthermore, we find nontrivial scaling laws in the relationships between four types of links in each cluster and evaluate power exponents, which characterize network structures as seen in the real large-scale network of trading business firms and in the Erdős-Rényi network. Finally, we observe changes in the transition point for random bond percolation process, demonstrating that the addition of muscle links enhances network robustness, while fat links are irrelevant. These findings aid in controlling the percolation transition for an arbitrary network.

Anomalous Metapopulation Dynamics on Scale-Free Networks

We model transport of individuals across a heterogeneous scale-free network where a few weakly connected nodes exhibit heavy-tailed residence times. Using the empirical law of the axiom of cumulative inertia and fractional analysis, we show that "anomalous cumulative inertia" overpowers highly connected nodes in attracting network individuals. This fundamentally challenges the classical result that individuals tend to accumulate in high-order nodes. The derived residence time distribution has a nontrivial U shape which we encounter empirically across human residence and employment times.

Dynamic versus static biomarkers in cancer immune checkpoint blockade: unravelling complexity

Recently, there has been a coordinated effort from academic institutions and the pharmaceutical industry to identify biomarkers that can predict responses to immune checkpoint blockade in cancer. Several biomarkers have been identified; however, none has reliably predicted response in a sufficiently rigorous manner for routine use. Here, we argue that the therapeutic response to immune checkpoint blockade is a critical state transition of a complex system. Such systems are highly sensitive to initial conditions, and critical transitions are notoriously difficult to predict far in advance. Nevertheless, warning signals can be detected closer to the tipping point. Advances in mathematics and network biology are starting to make it possible to identify such warning signals. We propose that these dynamic biomarkers could prove to be useful in distinguishing responding from non-responding patients, as well as facilitate the identification of new therapeutic targets for combination therapy.

Robustness and fragility in coupled oscillator networks under targeted attacks

The dynamical tolerance of coupled oscillator networks against local failures is studied. As the fraction of failed oscillator nodes gradually increases, the mean oscillation amplitude in the entire network decreases and then suddenly vanishes at a critical fraction as a phase transition. This critical fraction, widely used as a measure of the network robustness, was analytically derived for random failures but not for targeted attacks so far. Here we derive the general formula for the critical fraction, which can be applied to both random failures and targeted attacks. We consider the effects of targeting oscillator nodes based on their degrees. First we deal with coupled identical oscillators with homogeneous edge weights. Then our theory is applied to networks with heterogeneous edge weights and to those with nonidentical oscillators. The analytical results are validated by numerical experiments. Our results reveal the key factors governing the robustness and fragility of oscillator networks.

Comparison of microbial communities of activated sludge and membrane biofilm in 10 full-scale membrane bioreactors

Operation of membrane bioreactors (MBRs) for wastewater treatment is hampered by the membrane biofouling resulting from microbial activities. However, the knowledge of the microbial ecology of both biofilm and activated sludge in MBRs has not been sufficient. In this study, we scrutinized microbial communities of biofilm and activated sludge from 10 full-scale MBR plants. Overall, Flavobacterium, Dechloromonas and Nitrospira were abundant in order of abundance in biofilm, whereas Dechloromonas, Flavobacterium and Haliscomenobacter in activated sludge. Community structure was analyzed in either biofilm or activated sludge. Among MBRs, as expected, not only diversity of microbial community but also its composition was different from one another (p < 0.05). Between the biofilm and activated sludge, community composition made significant difference, but its diversity measures (i.e., alpha diversity, e.g., richness, diversity and evenness) did not (p > 0.05). Effects of ten environmental factors on community change were investigated using Spearman correlation. MLSS, HRT, F/M ratio and SADm explained the variation of microbial composition in the biofilm, whereas only MLSS did in the activated sludge. Microbial networks were constructed with the 10 environmental factors. The network results revealed that there were different topological characteristics between the biofilm and activated sludge networks, in which each of the 4 factors had different associations with microbial nodes. These results indicated that the different microbial associations were responsible for the variation of community composition between the biofilm and activated sludge.

Bifurcation and scaling at the aging transition boundary in globally coupled excitable and oscillatory units

Following a previous paper [Phys. Rev. E 88, 052907 (2013)PLEEE81539-375510.1103/PhysRevE.88.052907], we study in detail the mechanism of aging transition in globally and diffusively coupled excitable and oscillatory units. Here two of the three models taken up in the earlier work are used, each composed of a large number of units with their bifurcation parameters forming a uniform distribution. The control parameters are the coupling strength and the average of the bifurcation parameters. The present work is mostly devoted to a region of the phase diagram near the aging transition boundary with the coupling strength greater than its critical value for the onset of bistability and hysteresis. The bifurcation structure of each system at the aging transition boundary is investigated theoretically as well as numerically. Moreover, we show that critical scaling laws of order parameters S and M used in the previous paper are different depending on which region of the coupling strength to be chosen.

A small-world network model of facial emotion recognition

Various models have been proposed to increase understanding of the cognitive basis of facial emotions. Despite those efforts, interactions between facial emotions have received minimal attention. If collective behaviours relating to each facial emotion in the comprehensive cognitive system could be assumed, specific facial emotion relationship patterns might emerge. In this study, we demonstrate that the frameworks of complex networks can effectively capture those patterns. We generate 81 facial emotion images (6 prototypes and 75 morphs) and then ask participants to rate degrees of similarity in 3240 facial emotion pairs in a paired comparison task. A facial emotion network constructed on the basis of similarity clearly forms a small-world network, which features an extremely short average network distance and close connectivity. Further, even if two facial emotions have opposing valences, they are connected within only two steps. In addition, we show that intermediary morphs are crucial for maintaining full network integration, whereas prototypes are not at all important. These results suggest the existence of collective behaviours in the cognitive systems of facial emotions and also describe why people can efficiently recognize facial emotions in terms of information transmission and propagation. For comparison, we construct three simulated networks--one based on the categorical model, one based on the dimensional model, and one random network. The results reveal that small-world connectivity in facial emotion networks is apparently different from those networks, suggesting that a small-world network is the most suitable model for capturing the cognitive basis of facial emotions.

Core-like groups result in invalidation of identifying super-spreader by k-shell decomposition

Identifying the most influential spreaders is an important issue in understanding and controlling spreading processes on complex networks. Recent studies showed that nodes located in the core of a network as identified by the k-shell decomposition are the most influential spreaders. However, through a great deal of numerical simulations, we observe that not in all real networks do nodes in high shells are very influential: in some networks the core nodes are the most influential which we call true core, while in others nodes in high shells, even the innermost core, are not good spreaders which we call core-like group. By analyzing the k-core structure of the networks, we find that the true core of a network links diversely to the shells of the network, while the core-like group links very locally within the group. For nodes in the core-like group, the k-shell index cannot reflect their location importance in the network. We further introduce a measure based on the link diversity of shells to effectively distinguish the true core and core-like group, and identify core-like groups throughout the networks. Our findings help to better understand the structural features of real networks and influential nodes.

Robustness of oscillatory behavior in correlated networks

Understanding network robustness against failures of network units is useful for preventing large-scale breakdowns and damages in real-world networked systems. The tolerance of networked systems whose functions are maintained by collective dynamical behavior of the network units has recently been analyzed in the framework called dynamical robustness of complex networks. The effect of network structure on the dynamical robustness has been examined with various types of network topology, but the role of network assortativity, or degree–degree correlations, is still unclear. Here we study the dynamical robustness of correlated (assortative and disassortative) networks consisting of diffusively coupled oscillators. Numerical analyses for the correlated networks with Poisson and power-law degree distributions show that network assortativity enhances the dynamical robustness of the oscillator networks but the impact of network disassortativity depends on the detailed network connectivity. Furthermore, we theoretically analyze the dynamical robustness of correlated bimodal networks with two-peak degree distributions and show the positive impact of the network assortativity.

Random and targeted interventions for epidemic control in metapopulation models

In general, different countries and communities respond to epidemics in accordance with their own control plans and protocols. However, owing to global human migration and mobility, strategic planning for epidemic control measures through the collaboration of relevant public health administrations is gaining importance for mitigating and containing large-scale epidemics. Here, we present a framework to evaluate the effectiveness of random (non-strategic) and targeted (strategic) epidemic interventions for spatially separated patches in metapopulation models. For a random intervention, we analytically derive the critical fraction of patches that receive epidemic interventions, above which epidemics are successfully contained. The analysis shows that the heterogeneity of patch connectivity makes it difficult to contain epidemics under the random intervention. We demonstrate that, particularly in such heterogeneously connected networks, targeted interventions are considerably effective compared to the random intervention. Our framework is useful for identifying the target areas where epidemic control measures should be focused.

Robustness to noise in synchronization of complex networks

In this report, we investigate dynamical robustness of a complex network to noise injected through one of its nodes. We focus on synchronization of coupled nonlinear systems and, as a special instance, we address the classical consensus protocol for linear integrators. We establish an exact closed-form expression of the synchronization error for the consensus protocol and an approximate result for chaotic units. While structural robustness is known to be significantly affected by attacks targeted to network hubs, our results posit that dynamical robustness is controlled by both the topology of the network and the dynamics of the units. We provide examples where hubs perform better or worse than isolated nodes.

Variation of critical point of aging transition in a networked oscillators system

In this work, we study the variation of critical point in aging transition in a networked system consisting of both active and inactive oscillators. By theoretical analysis and numerical simulations, we show that the critical point of aging transition actually is determined by the (normalized) cross links between active and inactive subpopulations of oscillators. This reveals how specific configuration of active and inactive oscillators in the network can lead to the variation of transition point. In particular, we investigate how different strategies of targeted inactivation influence the transition point based on the theory. Our results theoretically explain why the low-degree nodes are crucial regarding dynamical robustness in such systems.

Dynamical robustness of coupled heterogeneous oscillators

We study tolerance of dynamic behavior in networks of coupled heterogeneous oscillators to deterioration of the individual oscillator components. As the deterioration proceeds with reduction in dynamic behavior of the oscillators, an order parameter evaluating the level of global oscillation decreases and then vanishes at a certain critical point. We present a method to analytically derive a general formula for this critical point and an approximate formula for the order parameter in the vicinity of the critical point in networks of coupled Stuart-Landau oscillators. Using the critical point as a measure for dynamical robustness of oscillator networks, we show that the more heterogeneous the oscillator components are, the more robust the oscillatory behavior of the network is to the component deterioration. This property is confirmed also in networks of Morris-Lecar neuron models coupled through electrical synapses. Our approach could provide a useful framework for theoretically understanding the role of population heterogeneity in robustness of biological networks.

Strong impact of temporal resolution on the structure of an ecological network

Most ecological networks are analysed as static structures, where all observed species and links are present simultaneously. However, this is over-simplified, because networks are temporally dynamical. We resolved an arctic, entire-season plant-flower visitor network into a temporal series of 1-day networks and compared the properties with its static equivalent based on data pooled over the entire season. Several properties differed. The nested link pattern in the static network was blurred in the dynamical version, because the characteristic long nestedness tail of flower–visitor specialists got stunted in the dynamical networks. This tail comprised a small food web of pollinators, parasitoids and hyper-parasitoids. The dynamical network had strong time delays in the transmission of direct and indirect effects among species. Twenty percent of all indirect links were impossible in the dynamical network. Consequently, properties and thus also robustness of ecological networks cannot be deduced from the static topology alone.

Onset of dynamic activity in globally coupled excitable and oscillatory units

The onset of dynamic activity is studied in heterogeneous populations of globally coupled units such that each unit is either excitable or oscillatory depending on its parameter. By varying the mean of the parameter distribution as well as coupling strength, we show that all or part of the populations studied have common features: similar phase diagrams, hysteresis near or at the onset of dynamic activity, and a fractional power law obeyed by an order parameter. A simplified model is proposed to explain these results.

The diminishing role of hubs in dynamical processes on complex networks

It is notoriously difficult to predict the behaviour of a complex self-organizing system, where the interactions among dynamical units form a heterogeneous topology. Even if the dynamics of each microscopic unit is known, a real understanding of their contributions to the macroscopic system behaviour is still lacking. Here, we develop information-theoretical methods to distinguish the contribution of each individual unit to the collective out-of-equilibrium dynamics. We show that for a system of units connected by a network of interaction potentials with an arbitrary degree distribution, highly connected units have less impact on the system dynamics when compared with intermediately connected units. In an equilibrium setting, the hubs are often found to dictate the long-term behaviour. However, we find both analytically and experimentally that the instantaneous states of these units have a short-lasting effect on the state trajectory of the entire system. We present qualitative evidence of this phenomenon from empirical findings about a social network of product recommendations, a protein–protein interaction network and a neural network, suggesting that it might indeed be a widespread property in nature.

Efficient recovery of dynamic behavior in coupled oscillator networks

We study an effective method to recover dynamic activity in coupled oscillator networks that have been damaged and lost oscillatory dynamics owing to some inactivated or deteriorated oscillator elements. Recovery of the dynamic behavior can be achieved by newly connecting intact oscillators to the network. We analytically and numerically examine the proportion of the oscillators that are needed to be supported by intact oscillators for recovery of oscillation dynamics. Our results show that it can be more effective to preferentially support active oscillators in the damaged network than to preferentially support inactivated ones. The conditions for this counterintuitive result are discussed. Our framework could be a theoretical foundation for understanding regeneration of oscillatory dynamics in physical and biological systems.

Evaluating roles of nodes in optimal allocation of vaccines with economic considerations

Since the allocation of vaccines is often constrained by limited resources, designing an economical vaccination strategy is a fundamental goal of the epidemiological modelling. In this study, with the objective of reducing costs, we determine the optimal allocation of vaccines for a general class of infectious diseases that spread mainly via contact. We use an optimization routine to identify the roles of nodes with distinct degrees as depending on the cost of treatment to that of vaccination (relative cost of treatment). The optimal allocation drives vaccination priority to medium-degree nodes at a low relative cost of treatment or to high-degree nodes at a high relative cost of treatment. According to the presented results, we may adjust the vaccination priority in the face of an endemic situation.

A network perspective on unraveling the role of TRP channels in biology and disease

Transient receptor potential (TRP) channels are a large family of non-selective cation channels that mediate numerous physiological and pathophysiological processes; however, still largely unknown are the underlying molecular mechanisms. With data generated on an unprecedented scale, network-based approaches have been revolutionizing the way in which we understand biology and disease, discover disease genes, and develop therapeutic strategies. These circumstances have created opportunities to encounter TRP channel research to data-intensive science. In this review, we provide an introduction of network-based approaches in biomedical science, describe the current state of TRP channel network biology, and discuss the future direction of TRP channel research. Network perspective will facilitate the discovery of latent roles and underlying mechanisms of TRP channels in biology and disease.

Temporal dynamics and impact of event interactions in cyber-social populations

The advance of information technologies provides powerful measures to digitize social interactions and facilitate quantitative investigations. To explore large-scale indoor interactions of a social population, we analyze 18 715 users' Wi-Fi access logs recorded in a Chinese university campus during 3 months, and define event interaction (EI) to characterize the concurrent interactions of multiple users inferred by their geographic coincidences-co-locating in the same small region at the same time. We propose three rules to construct a transmission graph, which depicts the topological and temporal features of event interactions. The vertex dynamics of transmission graph tells that the active durations of EIs fall into the truncated power-law distributions, which is independent on the number of involved individuals. The edge dynamics of transmission graph reports that the transmission durations present a truncated power-law pattern independent on the daily and weekly periodicities. Besides, in the aggregated transmission graph, low-degree vertices previously neglected in the aggregated static networks may participate in the large-degree EIs, which is verified by three data sets covering different sizes of social populations with various rendezvouses. This work highlights the temporal significance of event interactions in cyber-social populations.

The human functional brain network demonstrates structural and dynamical resilience to targeted attack

In recent years, the field of network science has enabled researchers to represent the highly complex interactions in the brain in an approachable yet quantitative manner. One exciting finding since the advent of brain network research was that the brain network can withstand extensive damage, even to highly connected regions. However, these highly connected nodes may not be the most critical regions of the brain network, and it is unclear how the network dynamics are impacted by removal of these key nodes. This work seeks to further investigate the resilience of the human functional brain network. Network attack experiments were conducted on voxel-wise functional brain networks and region-of-interest (ROI) networks of 5 healthy volunteers. Networks were attacked at key nodes using several criteria for assessing node importance, and the impact on network structure and dynamics was evaluated. The findings presented here echo previous findings that the functional human brain network is highly resilient to targeted attacks, both in terms of network structure and dynamics.