Epidemic spreading in scale-free networks

Duality between predictability and reconstructability in complex systems

Predicting the evolution of a large system of units using its structure of interaction is a fundamental problem in complex system theory. And so is the problem of reconstructing the structure of interaction from temporal observations. Here, we find an intricate relationship between predictability and reconstructability using an information-theoretical point of view. We use the mutual information between a random graph and a stochastic process evolving on this random graph to quantify their codependence. Then, we show how the uncertainty coefficients, which are intimately related to that mutual information, quantify our ability to reconstruct a graph from an observed time series, and our ability to predict the evolution of a process from the structure of its interactions. We provide analytical calculations of the uncertainty coefficients for many different systems, including continuous deterministic systems, and describe a numerical procedure when exact calculations are intractable. Interestingly, we find that predictability and reconstructability, even though closely connected by the mutual information, can behave differently, even in a dual manner. We prove how such duality universally emerges when changing the number of steps in the process. Finally, we provide evidence that predictability-reconstruction dualities may exist in dynamical processes on real networks close to criticality.

Learning the mechanisms of network growth

We propose a novel model-selection method for dynamic networks. Our approach involves training a classifier on a large body of synthetic network data. The data is generated by simulating nine state-of-the-art random graph models for dynamic networks, with parameter range chosen to ensure exponential growth of the network size in time. We design a conceptually novel type of dynamic features that count new links received by a group of vertices in a particular time interval. The proposed features are easy to compute, analytically tractable, and interpretable. Our approach achieves a near-perfect classification of synthetic networks, exceeding the state-of-the-art by a large margin. Applying our classification method to real-world citation networks gives credibility to the claims in the literature that models with preferential attachment, fitness and aging fit real-world citation networks best, although sometimes, the predicted model does not involve vertex fitness.

The effectiveness of intervention measures on MERS-CoV transmission by using the contact networks reconstructed from link prediction data

Introduction

Mitigating the spread of infectious diseases is of paramount concern for societal safety, necessitating the development of effective intervention measures. Epidemic simulation is widely used to evaluate the efficacy of such measures, but realistic simulation environments are crucial for meaningful insights. Despite the common use of contact-tracing data to construct realistic networks, they have inherent limitations. This study explores reconstructing simulation networks using link prediction methods as an alternative approach.

Methods

The primary objective of this study is to assess the effectiveness of intervention measures on the reconstructed network, focusing on the 2015 MERS-CoV outbreak in South Korea. Contact-tracing data were acquired, and simulation networks were reconstructed using the graph autoencoder (GAE)-based link prediction method. A scale-free (SF) network was employed for comparison purposes. Epidemic simulations were conducted to evaluate three intervention strategies: Mass Quarantine (MQ), Isolation, and Isolation combined with Acquaintance Quarantine (AQ + Isolation).

Results

Simulation results showed that AQ + Isolation was the most effective intervention on the GAE network, resulting in consistent epidemic curves due to high clustering coefficients. Conversely, MQ and AQ + Isolation were highly effective on the SF network, attributed to its low clustering coefficient and intervention sensitivity. Isolation alone exhibited reduced effectiveness. These findings emphasize the significant impact of network structure on intervention outcomes and suggest a potential overestimation of effectiveness in SF networks. Additionally, they highlight the complementary use of link prediction methods.

Discussion

This innovative methodology provides inspiration for enhancing simulation environments in future endeavors. It also offers valuable insights for informing public health decision-making processes, emphasizing the importance of realistic simulation environments and the potential of link prediction methods.

Markovian Approach for Exploring Competitive Diseases with Heterogeneity-Evidence from COVID-19 and Influenza in China

Due to the complex interactions between multiple infectious diseases, the spreading of diseases in human bodies can vary when people are exposed to multiple sources of infection at the same time. Typically, there is heterogeneity in individuals' responses to diseases, and the transmission routes of different diseases also vary. Therefore, this paper proposes an SIS disease spreading model with individual heterogeneity and transmission route heterogeneity under the simultaneous action of two competitive infectious diseases. We derive the theoretical epidemic spreading threshold using quenched mean-field theory and perform numerical analysis under the Markovian method. Numerical results confirm the reliability of the theoretical threshold and show the inhibitory effect of the proportion of fully competitive individuals on epidemic spreading. The results also show that the diversity of disease transmission routes promotes disease spreading, and this effect gradually weakens when the epidemic spreading rate is high enough. Finally, we find a negative correlation between the theoretical spreading threshold and the average degree of the network. We demonstrate the practical application of the model by comparing simulation outputs to temporal trends of two competitive infectious diseases, COVID-19 and seasonal influenza in China.

Nonlinear spreading behavior across multi-platform social media universe

Understanding how harmful content (mis/disinformation, hate, etc.) manages to spread among online communities within and across social media platforms represents an urgent societal challenge. We develop a non-linear dynamical model for such viral spreading, which accounts for the fact that online communities dynamically interconnect across multiple social media platforms. Our mean-field theory (Effective Medium Theory) compares well to detailed numerical simulations and provides a specific analytic condition for the onset of outbreaks (i.e., system-wide spreading). Even if the infection rate is significantly lower than the recovery rate, it predicts system-wide spreading if online communities create links between them at high rates and the loss of such links (e.g., due to moderator pressure) is low. Policymakers should, therefore, account for these multi-community dynamics when shaping policies against system-wide spreading.

Delay-driven phase transitions in an epidemic model on time-varying networks

A complex networked system typically has a time-varying nature in interactions among its components, which is intrinsically complicated and therefore technically challenging for analysis and control. This paper investigates an epidemic process on a time-varying network with a time delay. First, an averaging theorem is established to approximate the delayed time-varying system using autonomous differential equations for the analysis of system evolution. On this basis, the critical time delay is determined, across which the endemic equilibrium becomes unstable and a phase transition to oscillation in time via Hopf bifurcation will appear. Then, numerical examples are examined, including a periodically time-varying network, a blinking network, and a quasi-periodically time-varying network, which are simulated to verify the theoretical results. Further, it is demonstrated that the existence of time delay can extend the network frequency range to generate Turing patterns, showing a facilitating effect on phase transitions.

Validity of annealed approximation in a high-dimensional system

This study investigates the suitability of the annealed approximation in high-dimensional systems characterized by dense networks with quenched link disorder, employing models of coupled oscillators. We demonstrate that dynamic equations governing dense-network systems converge to those of the complete-graph version in the thermodynamic limit, where link disorder fluctuations vanish entirely. Consequently, the annealed-network systems, where fluctuations are attenuated, also exhibit the same dynamic behavior in the thermodynamic limit. However, a significant discrepancy arises in the incoherent (disordered) phase wherein the finite-size behavior becomes critical in determining the steady-state pattern. To explicitly elucidate this discrepancy, we focus on identical oscillators subject to competitive attractive and repulsive couplings. In the incoherent phase of dense networks, we observe the manifestation of random irregular states. In contrast, the annealed approximation yields a symmetric (regular) incoherent state where two oppositely coherent clusters of oscillators coexist, accompanied by the vanishing order parameter. Our findings imply that the annealed approximation should be employed with caution even in dense-network systems, particularly in the disordered phase.

General protocol for predicting outbreaks of infectious diseases in social networks

Epidemic spreading on social networks with quenched connections is strongly influenced by dynamic correlations between connected nodes, posing theoretical challenges in predicting outbreaks of infectious diseases. The quenched connections introduce dynamic correlations, indicating that the infection of one node increases the likelihood of infection among its neighboring nodes. These dynamic correlations pose significant difficulties in developing comprehensive theories for threshold determination. Determining the precise epidemic threshold is pivotal for diseases control. In this study, we propose a general protocol for accurately determining epidemic thresholds by introducing a new set of fundamental conditions, where the number of connections between individuals of each type remains constant in the stationary state, and by devising a rescaling method for infection rates. Our general protocol is applicable to diverse epidemic models, regardless of the number of stages and transmission modes. To validate our protocol's effectiveness, we apply it to two widely recognized standard models, the susceptible-infected-recovered-susceptible model and the contact process model, both of which have eluded precise threshold determination using existing sophisticated theories. Our results offer essential tools to enhance disease control strategies and preparedness in an ever-evolving landscape of infectious diseases.

An application of small-world network on predicting the behavior of infectious disease on campus

Networks haven been widely used to understand the spread of infectious disease. This study examines the properties of small-world networks in modeling infectious disease on campus. Two different small-world models are developed and the behaviors of infectious disease in the models are observed through numerical simulations. The results show that the behavior pattern of infectious disease in a small-world network is different from those in a regular network or a random network. The spread of the infectious disease increases as the proportion of long-distance connections p increasing, which indicates that reducing the contact among people is an effective measure to control the spread of infectious disease. The probability of node position exchange in a network (p2) had no significant effect on the spreading speed, which suggests that reducing human mobility in closed environments does not help control infectious disease. However, the spreading speed is proportional to the number of shared nodes (s), which means reducing connections between different groups and dividing students into separate sections will help to control infectious disease. In the end, the simulating speed of the small-world network is tested and the quadratic relationship between simulation time and the number of nodes may limit the application of the SW network in areas with large populations.

Efficient parameter inference in networked dynamical systems via steady states: A surrogate objective function approach integrating mean-field and nonlinear least squares

In networked dynamical systems, inferring governing parameters is crucial for predicting nodal dynamics, such as gene expression levels, species abundance, or population density. While many parameter estimation techniques rely on time-series data, particularly systems that converge over extreme time ranges, only noisy steady-state data is available, requiring a new approach to infer dynamical parameters from noisy observations of steady states. However, the traditional optimization process is computationally demanding, requiring repeated simulation of coupled ordinary differential equations. To overcome these limitations, we introduce a surrogate objective function that leverages decoupled equations to compute steady states, significantly reducing computational complexity. Furthermore, by optimizing the surrogate objective function, we obtain steady states that more accurately approximate the ground truth than noisy observations and predict future equilibria when topology changes. We empirically demonstrate the effectiveness of the proposed method across ecological, gene regulatory, and epidemic networks. Our approach provides an efficient and effective way to estimate parameters from steady-state data and has the potential to improve predictions in networked dynamical systems.

Transition from time-variant to static networks: Timescale separation in N-intertwined mean-field approximation of susceptible-infectious-susceptible epidemics

We extend the N-intertwined mean-field approximation (NIMFA) for the susceptible-infectious-susceptible (SIS) epidemiological process to time-varying networks. Processes on time-varying networks are often analyzed under the assumption that the process and network evolution happen on different timescales. This approximation is called timescale separation. We investigate timescale separation between disease spreading and topology updates of the network. We introduce the transition times [under T]̲(r) and T[over ¯](r) as the boundaries between the intermediate regime and the annealed (fast changing network) and quenched (static network) regimes, respectively, for a fixed accuracy tolerance r. By analyzing the convergence of static NIMFA processes, we analytically derive upper and lower bounds for T[over ¯](r). Our results provide insights and bounds on the time of convergence to the steady state of the static NIMFA SIS process. We show that, under our assumptions, the upper-transition time T[over ¯](r) is almost entirely determined by the basic reproduction number R_{0} of the network. The value of the upper-transition time T[over ¯](r) around the epidemic threshold is large, which agrees with the current understanding that some real-world epidemics cannot be approximated with the aforementioned timescale separation.

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.

Bridge synergy and simplicial interaction in complex contagions

Modeling complex contagion in networked systems is an important topic in network science, for which various models have been proposed, including the synergistic contagion model that incorporates coherent interference and the simplicial contagion model that involves high-order interactions. Although both models have demonstrated success in investigating complex contagions, their relationship in modeling complex contagions remains unclear. In this study, we compare the synergy and the simplest form of high-order interaction in the simplicial contagion model, known as the triangular one. We analytically show that the triangular interaction and the synergy can be bridged within complex contagions through the joint degree distribution of the network. Monte Carlo simulations are then conducted to compare simplicial and corresponding synergistic contagions on synthetic and real-world networks, the results of which highlight the consistency of these two different contagion processes and thus validate our analysis. Our study sheds light on the deep relationship between the synergy and high-order interactions and enhances our physical understanding of complex contagions in networked systems.

Understanding the relationship between stay-at-home measures and vaccine shortages: a conventional, heterogeneous, and fractional dynamic approach

In light of the global prevalence of a highly contagious respiratory disease, this study presents a novel approach to address the pressing and unanticipated issues by introducing a modified vaccination and lockdown-centered epidemic model. The rapid spread of the disease is attributed to viral transmissibility, the emergence of new strains (variants), lack of immunization, and human unawareness. This study aims to provide policymakers with crucial insights for making informed decisions regarding lockdown strategies, vaccine availability, and other control measures. The research adopts three types of models: deterministic, heterogeneous, and fractional-order dynamics, on both theoretical and numerical approaches. The heterogeneous network considers varying connectivity and interaction patterns among individuals, while the ABC fractional-order derivatives analyze the impact of integer-order control in different semi-groups. An extensive theoretical analysis is conducted to validate the proposed model. A comprehensive numerical investigation encompasses deterministic, stochastic, and ABC fractional-order derivatives, considering the combined effects of an effective vaccination program and non-pharmaceutical interventions, such as lockdowns and shutdowns. The findings of this research are expected to be valuable for policymakers in different countries, helping them implement dynamic strategies to control and eradicate the epidemic effectively.

Influence of human motion patterns on epidemic spreading dynamics

Extensive real-data indicate that human motion exhibits novel patterns and has a significant impact on the epidemic spreading process. The research on the influence of human motion patterns on epidemic spreading dynamics still lacks a systematic study in network science. Based on an agent-based model, this paper simulates the spread of the disease in the gathered population by combining the susceptible-infected-susceptible epidemic process with human motion patterns, described by moving speed and gathering preference. Our simulation results show that the emergence of a hysteresis loop is observed in the system when the moving speed is slow, particularly when humans prefer to gather; that is, the epidemic prevalence of the systems depends on the fraction of initial seeds. Regardless of the gathering preference, the hysteresis loop disappears when the population moves fast. In addition, our study demonstrates that there is an optimal moving speed for the gathered population, at which the epidemic prevalence reaches its maximum value.

Towards Digital Twin-Oriented Complex Networked Systems: Introducing heterogeneous node features and interaction rules

This study proposes an extendable modelling framework for Digital Twin-Oriented Complex Networked Systems (DT-CNSs) with a goal of generating networks that faithfully represent real-world social networked systems. Modelling process focuses on (i) features of nodes and (ii) interaction rules for creating connections that are built based on individual node's preferences. We conduct experiments on simulation-based DT-CNSs that incorporate various features and rules about network growth and different transmissibilities related to an epidemic spread on these networks. We present a case study on disaster resilience of social networks given an epidemic outbreak by investigating the infection occurrence within specific time and social distance. The experimental results show how different levels of the structural and dynamics complexities, concerned with feature diversity and flexibility of interaction rules respectively, influence network growth and epidemic spread. The analysis revealed that, to achieve maximum disaster resilience, mitigation policies should be targeted at nodes with preferred features as they have higher infection risks and should be the focus of the epidemic control.

Epidemic spreading under game-based self-quarantine behaviors: The different effects of local and global information

During the outbreak of an epidemic, individuals may modify their behaviors in response to external (including local and global) infection-related information. However, the difference between local and global information in influencing the spread of diseases remains inadequately explored. Here, we study a simple epidemic model that incorporates the game-based self-quarantine behavior of individuals, taking into account the influence of local infection status, global disease prevalence, and node heterogeneity (non-identical degree distribution). Our findings reveal that local information can effectively contain an epidemic, even with only a small proportion of individuals opting for self-quarantine. On the other hand, global information can cause infection evolution curves shaking during the declining phase of an epidemic, owing to the synchronous release of nodes with the same degree from the quarantined state. In contrast, the releasing pattern under the local information appears to be more random. This shaking phenomenon can be observed in various types of networks associated with different characteristics. Moreover, it is found that under the proposed game-epidemic framework, a disease is more difficult to spread in heterogeneous networks than in homogeneous networks, which differs from conventional epidemic models.

An agent-based nested model integrating within-host and between-host mechanisms to predict an epidemic

The COVID-19 pandemic has remarkably heightened concerns regarding the prediction of communicable disease spread. This study introduces an innovative agent-based modeling approach. In this model, the quantification of human-to-human transmission aligns with the dynamic variations in the viral load within an individual, termed "within-host" and adheres to the susceptible-infected-recovered (SIR) process, referred to as "between-host." Variations in the viral load over time affect the infectivity between individual agents. This model diverges from the traditional SIR model, which employs a constant transmission probability, by incorporating a dynamic, time-dependent transmission probability influenced by the viral load in a host agent. The proposed model retains the time-integrated transmission probability characteristic of the conventional SIR model. As observed in this model, the overall epidemic size remains consistent with the predictions of the standard SIR model. Nonetheless, compared to predictions based on the classical SIR process, notable differences existed in the peak number of the infected individuals and the timing of this peak. These nontrivial differences are induced by the direct correlation between the time-evolving transmission probability and the viral load within a host agent. The developed model can inform targeted intervention strategies and public health policies by providing detailed insights into disease spread dynamics, crucial for effectively managing epidemics.

Graph theory analysis reveals an assortative pain network vulnerable to attacks

The neural substrate of pain experience has been described as a dense network of connected brain regions. However, the connectivity pattern of these brain regions remains elusive, precluding a deeper understanding of how pain emerges from the structural connectivity. Here, we employ graph theory to systematically characterize the architecture of a comprehensive pain network, including both cortical and subcortical brain areas. This structural brain network consists of 49 nodes denoting pain-related brain areas, linked by edges representing their relative incoming and outgoing axonal projection strengths. Within this network, 63% of brain areas share reciprocal connections, reflecting a dense network. The clustering coefficient, a measurement of the probability that adjacent nodes are connected, indicates that brain areas in the pain network tend to cluster together. Community detection, the process of discovering cohesive groups in complex networks, successfully reveals two known subnetworks that specifically mediate the sensory and affective components of pain, respectively. Assortativity analysis, which evaluates the tendency of nodes to connect with other nodes that have similar features, indicates that the pain network is assortative. Finally, robustness, the resistance of a complex network to failures and perturbations, indicates that the pain network displays a high degree of error tolerance (local failure rarely affects the global information carried by the network) but is vulnerable to attacks (selective removal of hub nodes critically changes network connectivity). Taken together, graph theory analysis unveils an assortative structural pain network in the brain that processes nociceptive information. Furthermore, the vulnerability of this network to attack presents the possibility of alleviating pain by targeting the most connected brain areas in the network.

Heat-Bath and Metropolis Dynamics in Ising-like Models on Directed Regular Random Graphs

Using a single-site mean-field approximation (MFA) and Monte Carlo simulations, we examine Ising-like models on directed regular random graphs. The models are directed-network implementations of the Ising model, Ising model with absorbing states, and majority voter models. When these nonequilibrium models are driven by the heat-bath dynamics, their stationary characteristics, such as magnetization, are correctly reproduced by MFA as confirmed by Monte Carlo simulations. It turns out that MFA reproduces the same result as the generating functional analysis that is expected to provide the exact description of such models. We argue that on directed regular random graphs, the neighbors of a given vertex are typically uncorrelated, and that is why MFA for models with heat-bath dynamics provides their exact description. For models with Metropolis dynamics, certain additional correlations become relevant, and MFA, which neglects these correlations, is less accurate. Models with heat-bath dynamics undergo continuous phase transition, and at the critical point, the power-law time decay of the order parameter exhibits the behavior of the Ising mean-field universality class. Analogous phase transitions for models with Metropolis dynamics are discontinuous.

Edge-based compartmental modeling for the spread of cholera on random networks: A case study in Somalia

Cholera remains a major public health problem that threatens human health worldwide and its severity is continuing. In this paper, an edge-based model for cholera transmission on random networks is proposed and investigated. The model assumes that two communities share a common water source and includes three transmission routes, namely intra- and inter-community human-to-human transmission as well as water-to-human transmission. Intra-community human-to-human contacts are modeled through a random contact network, while both inter-community and water-to-human transmission are modeled through external nodes that reach each individual in the network to the same extent. The basic reproduction number and the equations of the final epidemic size are obtained. In addition, our study considers the cholera situation in Banadir, which is one of the most severely infected regions in Somalia, during the period (2019-2021). According to the geographical location, two adjacent districts are selected and our model fits well with the real data on the monthly cumulative cholera cases of these two districts during the above-mentioned period. From the perspective of network topology, cutting off high-risk contacts by supervising, isolating, quarantining and closing places with high-degree cholera-infected individuals to reduce degree heterogeneity is an effective measure to control cholera transmission. Our findings might offer some useful insights on cholera control.

Global efficiency and network structure of urban traffic flows: A percolation-based empirical analysis

Making the connection between the function and structure of networked systems is one of the fundamental issues in complex systems and network science. Urban traffic flows are related to various problems in cities and can be represented as a network of local traffic flows. To identify an empirical relation between the function and network structure of urban traffic flows, we construct a time-varying traffic flow network of a megacity, Seoul, and analyze its global efficiency with a percolation-based approach. Comparing the real-world traffic flow network with its corresponding null-model network having a randomized structure, we show that the real-world network is less efficient than its null-model network during rush hour, yet more efficient during non-rush hour. We observe that in the real-world network, links with the highest betweenness tend to have lower quality during rush hour compared to links with lower betweenness, but higher quality during non-rush hour. Since the top betweenness links tend to be the bridges that connect the network together, their congestion has a stronger impact on the network's global efficiency. Our results suggest that the spatial structure of traffic flow networks is important to understand their function.

On the adoption of nonpharmaceutical interventions during the pandemic: An evolutionary game model

The adoption of behavioral nonpharmaceutical interventions (NPIs) among the public is essential for tackling the COVID-19 pandemic, yet presents challenges due to the complexity of human behaviors. A large body of literature has utilized classic game theory to investigate the population's decisions regarding the adoption of interventions, where the static solution concept such as the Nash equilibrium is studied. However, individual adoption behavior is not static, instead it is a dynamic process that involves the strategic interactions with other counterparts over time. The study of quantitatively analyzing the dynamics on precautionary behavior during an outbreak is rather scarce. This article fills the research gap by developing an evolutionary game-theoretic framework to model the dynamics of population behavior on the adoption of NPI. We construct the two-group asymmetric game, where behavioral change for each group is characterized by replicator equations. Sensitivity analyses are performed to examine the long-term stability of equilibrium points with respect to perturbation of model parameters. We found that the limiting behavior of intervention adoption in the population consists of only pure strategies in a game setting, indicating that the evolutionary outcome is that everyone either takes up the preventive measure or not. We also applied the framework to examine the mask-wearing behavior, and validated with actual data. Overall, this article provides insights into population dynamics on the adoption of intervention strategy during the outbreak, which can be beneficial for policy makers to better understand the evolutionary trajectory of population behavior.

Scalable parallel and distributed simulation of an epidemic on a graph

We propose an algorithm to simulate Markovian SIS epidemics with homogeneous rates and pairwise interactions on a fixed undirected graph, assuming a distributed memory model of parallel programming and limited bandwidth. This setup can represent a broad class of simulation tasks with compartmental models. Existing solutions for such tasks are sequential by nature. We provide an innovative solution that makes trade-offs between statistical faithfulness and parallelism possible. We offer an implementation of the algorithm in the form of pseudocode in the Appendix. Also, we analyze its algorithmic complexity and its induced dynamical system. Finally, we design experiments to show its scalability and faithfulness. In our experiments, we discover that graph structures that admit good partitioning schemes, such as the ones with clear community structures, together with the correct application of a graph partitioning method, can lead to better scalability and faithfulness. We believe this algorithm offers a way of scaling out, allowing researchers to run simulation tasks at a scale that was not accessible before. Furthermore, we believe this algorithm lays a solid foundation for extensions to more advanced epidemic simulations and graph dynamics in other fields.

A hypernetwork-based urn model for explaining collective dynamics

The topological characterization of complex systems has significantly contributed to our understanding of the principles of collective dynamics. However, the representation of general complex networks is not enough for explaining certain problems, such as collective actions. Considering the effectiveness of hypernetworks on modeling real-world complex networks, in this paper, we proposed a hypernetwork-based Pólya urn model that considers the effect of group identity. The mathematical deduction and simulation experiments show that social influence provides a strong imitation environment for individuals, which can prevent the dynamics from being self-correcting. Additionally, the unpredictability of the social system increases with growing social influence, and the effect of group identity can moderate market inequality caused by individual preference and social influence. The present work provides a modeling basis for a better understanding of the logic of collective dynamics.

Analysis of the Effectiveness of Public Health Measures on COVID-19 Transmission

In this study, we investigate the COVID-19 epidemics in Brazilian cities, using early-time approximations of the SIR model in networks and combining the VAR (vector autoregressive) model with machine learning techniques. Different from other works, the underlying network was constructed by inputting real-world data on local COVID-19 cases reported by Brazilian cities into a regularized VAR model. This model estimates directional COVID-19 transmission channels (connections or links between nodes) of each pair of cities (vertices or nodes) using spectral network analysis. Despite the simple epidemiological model, our predictions align well with the real COVID-19 dynamics across Brazilian municipalities, using data only up until May 2020. Given the rising number of infectious people in Brazil-a possible indicator of a second wave-these early-time approximations could be valuable in gauging the magnitude of the next contagion peak. We further examine the effect of public health policies, including social isolation and mask usage, by creating counterfactual scenarios to quantify the human impact of these public health measures in reducing peak COVID-19 cases. We discover that the effectiveness of social isolation and mask usage varies significantly across cities. We hope our study will support the development of future public health measures.

Scale-free dynamics of COVID-19 in a Brazilian city

A common basis to address the dynamics of directly transmitted infectious diseases, such as COVID-19, are compartmental (or SIR) models. SIR models typically assume homogenous population mixing, a simplification that is convenient but unrealistic. Here we validate an existing model of a scale-free fractal infection process using high-resolution data on COVID-19 spread in São Caetano, Brazil. We find that transmission can be described by a network in which each infectious individual has a small number of susceptible contacts, of the order of 2-5. This model parameter correlated tightly with physical distancing measured by mobile phone data, such that in periods of greater distancing the model recovered a lower average number of contacts, and vice versa. We show that the SIR model is a special case of our scale-free fractal process model in which the parameter that reflects population structure is set at unity, indicating homogeneous mixing. Our more general framework better explained the dynamics of COVID-19 in São Caetano, used fewer parameters than a standard SIR model and accounted for geographically localized clusters of disease. Our model requires further validation in other locations and with other directly transmitted infectious agents.

Spreading Dynamics of Capital Flow Transfer in Complex Financial Networks

The financial system, a complex network, operates primarily through the exchange of capital, where the role of information is critical. This study utilizes the transfer entropy method to examine the strength and direction of information flow among different capital flow time series and investigate the community structure within the transfer networks. Moreover, the spreading dynamics of the capital flow transfer networks are observed, and the importance and traveling time of each node are explored. The results imply a dominant role for the food and drink industry within the Chinese market, with increased attention towards the computer industry starting in 2014. The community structure of the capital flow transfer networks significantly differs from those constructed from stock prices, with the main sector predominantly encompassing industry leaders favored by primary funds with robust capital flow connections. The average traveling time from sectors such as food and drink, coal, and utilities to other sectors is the shortest, and the dynamic flow between these sectors displays a significant role. These findings highlight that comprehension of information flow and community structure within the financial system can offer valuable insights into market dynamics and help to identify key sectors and companies.

Complexity in the dengue spreading: A network analysis approach

In an increasingly interconnected society, preventing epidemics has become a major challenge. Numerous infectious diseases spread between individuals by a vector, creating bipartite networks of infection with the characteristics of complex networks. In the case of dengue, a mosquito-borne disease, these infection networks include a vector-the Aedes aegypti mosquito-which has expanded its endemic area due to climate change. In this scenario, innovative approaches are essential to help public agents in the fight against the disease. Using an agent-based model, we investigated the network morphology of a dengue endemic region considering four different serotypes and a small population. The degree, betweenness, and closeness distributions are evaluated for the bipartite networks, considering the interactions up to the second order for each serotype. We observed scale-free features and heavy tails in the degree distribution and betweenness and quantified the decay of the degree distribution with a q-Gaussian fit function. The simulation results indicate that the spread of dengue is primarily driven by human-to-human and human-to-mosquito interaction, reinforcing the importance of controlling the vector to prevent episodes of epidemic outbreaks.

Social network structure and the spread of complex contagions from a population genetics perspective

Ideas, behaviors, and opinions spread through social networks. If the probability of spreading to a new individual is a nonlinear function of the fraction of the individuals' affected neighbors, such a spreading process becomes a "complex contagion." This nonlinearity does not typically appear with physically spreading infections, but instead can emerge when the concept that is spreading is subject to game theoretical considerations (e.g., for choices of strategy or behavior) or psychological effects such as social reinforcement and other forms of peer influence (e.g., for ideas, preferences, or opinions). Here we study how the stochastic dynamics of such complex contagions are affected by the underlying network structure. Motivated by simulations of complex contagions on real social networks, we present a framework for analyzing the statistics of contagions with arbitrary nonlinear adoption probabilities based on the mathematical tools of population genetics. The central idea is to use an effective lower-dimensional diffusion process to approximate the statistics of the contagion. This leads to a tradeoff between the effects of "selection" (microscopic tendencies for an idea to spread or die out), random drift, and network structure. Our framework illustrates intuitively several key properties of complex contagions: stronger community structure and network sparsity can significantly enhance the spread, while broad degree distributions dampen the effect of selection compared to random drift. Finally, we show that some structural features can exhibit critical values that demarcate regimes where global contagions become possible for networks of arbitrary size. Our results draw parallels between the competition of genes in a population and memes in a world of minds and ideas. Our tools provide insight into the spread of information, behaviors, and ideas via social influence, and highlight the role of macroscopic network structure in determining their fate.

Evolutionary dynamics on sequential temporal networks

Population structure is a well-known catalyst for the evolution of cooperation and has traditionally been considered to be static in the course of evolution. Conversely, real-world populations, such as microbiome communities and online social networks, frequently show a progression from tiny, active groups to huge, stable communities, which is insufficient to be captured by constant structures. Here, we propose sequential temporal networks to characterize growing networked populations, and we extend the theory of evolutionary games to these temporal networks with arbitrary structures and growth rules. We derive analytical rules under which a sequential temporal network has a higher fixation probability for cooperation than its static counterpart. Under neutral drift, the rule is simply a function of the increment of nodes and edges in each time step. But if the selection is weak, the rule is related to coalescence times on networks. In this case, we propose a mean-field approximation to calculate fixation probabilities and critical benefit-to-cost ratios with lower calculation complexity. Numerical simulations in empirical datasets also prove the cooperation-promoting effect of population growth. Our research stresses the significance of population growth in the real world and provides a high-accuracy approximation approach for analyzing the evolution in real-life systems.

Efficient simulations of epidemic models with tensor networks: Application to the one-dimensional susceptible-infected-susceptible model

The contact process is an emblematic model of a nonequilibrium system, containing a phase transition between inactive and active dynamical regimes. In the epidemiological context, the model is known as the susceptible-infected-susceptible model, and it is widely used to describe contagious spreading. In this work, we demonstrate how accurate and efficient representations of the full probability distribution over all configurations of the contact process on a one-dimensional chain can be obtained by means of matrix product states (MPSs). We modify and adapt MPS methods from many-body quantum systems to study the classical distributions of the driven contact process at late times. We give accurate and efficient results for the distribution of large gaps, and illustrate the advantage of our methods over Monte Carlo simulations. Furthermore, we study the large deviation statistics of the dynamical activity, defined as the total number of configuration changes along a trajectory, and investigate quantum-inspired entropic measures, based on the second Rényi entropy.

Active querying approach to epidemic source detection on contact networks

The problem of identifying the source of an epidemic (also called patient zero) given a network of contacts and a set of infected individuals has attracted interest from a broad range of research communities. The successful and timely identification of the source can prevent a lot of harm as the number of possible infection routes can be narrowed down and potentially infected individuals can be isolated. Previous research on this topic often assumes that it is possible to observe the state of a substantial fraction of individuals in the network before attempting to identify the source. We, on the contrary, assume that observing the state of individuals in the network is costly or difficult and, hence, only the state of one or few individuals is initially observed. Moreover, we presume that not only the source is unknown, but also the duration for which the epidemic has evolved. From this more general problem setting a need to query the state of other (so far unobserved) individuals arises. In analogy with active learning, this leads us to formulate the active querying problem. In the active querying problem, we alternate between a source inference step and a querying step. For the source inference step, we rely on existing work but take a Bayesian perspective by putting a prior on the duration of the epidemic. In the querying step, we aim to query the states of individuals that provide the most information about the source of the epidemic, and to this end, we propose strategies inspired by the active learning literature. Our results are strongly in favor of a querying strategy that selects individuals for whom the disagreement between individual predictions, made by all possible sources separately, and a consensus prediction is maximal. Our approach is flexible and, in particular, can be applied to static as well as temporal networks. To demonstrate our approach's practical importance, we experiment with three empirical (temporal) contact networks: a network of pig movements, a network of sexual contacts, and a network of face-to-face contacts between residents of a village in Malawi. The results show that active querying strategies can lead to substantially improved source inference results as compared to baseline heuristics. In fact, querying only a small fraction of nodes in a network is often enough to achieve a source inference performance comparable to a situation where the infection states of all nodes are known.

Critical and steady-state characteristics of delay propagation in an airport network

In this work, we established a density equation for delayed airports to investigate the horizontal propagation mechanism of delays among airports in an airport network. We explored the critical conditions, steady-state features, and scale of the delay propagation, and designed a simulation system to verify the accuracy of the results. The results indicated that, due to the no-table scale-free feature of an airport network, the critical value of delay propagation is extremely small, and delays are prone to propagate among airports. Furthermore, as delay propagation reaches a steady state in an aviation network, the degree value of the node becomes highly correlated with its delay state. Hub airports with high degree values are the most prone to being affected by delay propagation. In addition, the number of airports that are initially delayed influences the time required for delay propagation to reach a steady state. Specifically, if there are fewer initially delayed airports, a longer time is required to reach a steady state. In the steady state, the delay ratios of airports with different degree values in the network converge to a balance point. The delay degree of the node is highly positively correlated with the delay propagation rate in the network, but negatively related to the degree distribution index of the network.

Emergent information dynamics in many-body interconnected systems

The information implicitly represented in the state of physical systems allows for their analysis using analytical techniques from statistical mechanics and information theory. This approach has been successfully applied to complex networks, including biophysical systems such as virus-host protein-protein interactions and whole-brain models in health and disease, drawing inspiration from quantum statistical physics. Here we propose a general mathematical framework for modeling information dynamics on complex networks, where the internal node states are vector valued, allowing each node to carry multiple types of information. This setup is relevant for various biophysical and sociotechnological models of complex systems, ranging from viral dynamics on networks to models of opinion dynamics and social contagion. Instead of focusing on node-node interactions, we shift our attention to the flow of information between network configurations. We uncover fundamental differences between widely used spin models on networks, such as voter and kinetic dynamics, which cannot be detected through classical node-based analysis. We illustrate the mathematical framework further through an exemplary application to epidemic spreading on a low-dimensional network. Our model provides an opportunity to adapt powerful analytical methods from quantum many-body systems to study the interplay between structure and dynamics in interconnected systems.

Epidemic spreading on multi-layer networks with active nodes

Investigations on spreading dynamics based on complex networks have received widespread attention these years due to the COVID-19 epidemic, which are conducive to corresponding prevention policies. As for the COVID-19 epidemic itself, the latent time and mobile crowds are two important and inescapable factors that contribute to the significant prevalence. Focusing on these two factors, this paper systematically investigates the epidemic spreading in multiple spaces with mobile crowds. Specifically, we propose a SEIS (Susceptible-Exposed-Infected-Susceptible) model that considers the latent time based on a multi-layer network with active nodes which indicate the mobile crowds. The steady-state equations and epidemic threshold of the SEIS model are deduced and discussed. And by comprehensively discussing the key model parameters, we find that (1) due to the latent time, there is a "cumulative effect" on the infected, leading to the "peaks" or "shoulders" of the curves of the infected individuals, and the system can switch among three states with the relative parameter combinations changing; (2) the minimal mobile crowds can also cause the significant prevalence of the epidemic at the steady state, which is suggested by the zero-point phase change in the proportional curves of infected individuals. These results can provide a theoretical basis for formulating epidemic prevention policies.

Effects of noise on the critical points of Turing instability in complex ecosystems

Noise is ubiquitous in natural and artificial systems. In a noisy environment, the interactions among nodes may fluctuate randomly, leading to more complicated interactions. In this paper we focus on the effects of noise and network topology on the Turing pattern of ecological networks with activator-inhibitor structure, which may be interpreted as prey-predator interactions. Based on the stability theory of stochastic differential equations, a sufficient condition for the uniform state is derived. The analytical results indicate that noise is beneficial for the uniform state. When the ratio between the diffusion coefficients of the predator and prey increases, the ecosystems can exhibit a transition from a uniform stable state to a Turing pattern, while when the ratio decreases, the ecosystems transit from a Turing pattern to a uniform stable state. There are two crucial critical points in Turing patterns, forward and backward. We find that both forward and backward critical points increase as the noise intensity increases. This means that noise favors a stable homogeneous state compared to a state with a heterogeneous pattern, which is consistent with the analytical results. In addition, noise can weaken the hysteresis phenomenon and even eliminate it in some cases. Furthermore, we report that network topology plays an important role in modulating the uniform state of ecosystems, such as the size of prey-predator systems, the network connectivity, and the strength of interaction.

Droplet finite-size scaling of the majority-vote model on scale-free networks

We discuss the majority vote model coupled with scale-free networks and investigate its critical behavior. Previous studies point to a nonuniversal behavior of the majority vote model, where the critical exponents depend on the connectivity. At the same time, the effective dimension D_{eff} is unity for a degree distribution exponent 5/2<γ<7/2. We introduce a finite-size theory of the majority vote model for uncorrelated networks and present generalized scaling relations with good agreement with Monte Carlo simulation results. Our finite-size approach has two sources of size dependence: an external field representing the influence of the mass media on consensus formation and the scale-free network cutoff. The critical exponents are nonuniversal, dependent on the degree distribution exponent, precisely when 5/2<γ<7/2. For γ≥7/2, the model is in the same universality class as the majority vote model on Erdős-Rényi random graphs. However, for γ=7/2, the critical behavior includes additional logarithmic corrections.

Modeling the effectiveness of targeting Rift Valley fever virus vaccination using imperfect network information

Livestock movements contribute to the spread of several infectious diseases. Data on livestock movements can therefore be harnessed to guide policy on targeted interventions for controlling infectious livestock diseases, including Rift Valley fever (RVF)-a vaccine-preventable arboviral fever. Detailed livestock movement data are known to be useful for targeting control efforts including vaccination. These data are available in many countries, however, such data are generally lacking in others, including many in East Africa, where multiple RVF outbreaks have been reported in recent years. Available movement data are imperfect, and the impact of this uncertainty in the utility of movement data on informing targeting of vaccination is not fully understood. Here, we used a network simulation model to describe the spread of RVF within and between 398 wards in northern Tanzania connected by cattle movements, on which we evaluated the impact of targeting vaccination using imperfect movement data. We show that pre-emptive vaccination guided by only market movement permit data could prevent large outbreaks. Targeted control (either by the risk of RVF introduction or onward transmission) at any level of imperfect movement information is preferred over random vaccination, and any improvement in information reliability is advantageous to their effectiveness. Our modeling approach demonstrates how targeted interventions can be effectively used to inform animal and public health policies for disease control planning. This is particularly valuable in settings where detailed data on livestock movements are either unavailable or imperfect due to resource limitations in data collection, as well as challenges associated with poor compliance.

The spread of infectious diseases from a physics perspective

This article deals with the spread of infectious diseases from a physics perspective. It considers a population as a network of nodes representing the population members, linked by network edges representing the (social) contacts of the individual population members. Infections spread along these edges from one node (member) to another. This article presents a novel, modified version of the SIR compartmental model, able to account for typical network effects and percolation phenomena. The model is successfully tested against the results of simulations based on Monte-Carlo methods. Expressions for the (basic) reproduction numbers in terms of the model parameters are presented, and justify some mild criticisms on the widely spread interpretation of reproduction numbers as being the number of secondary infections due to a single active infection. Throughout the article, special emphasis is laid on understanding, and on the interpretation of phenomena in terms of concepts borrowed from condensed-matter and statistical physics, which reveals some interesting analogies. Percolation effects are of particular interest in this respect and they are the subject of a detailed investigation. The concept of herd immunity (its definition and nature) is intensively dealt with as well, also in the context of large-scale vaccination campaigns and waning immunity. This article elucidates how the onset of herd-immunity can be considered as a second-order phase transition in which percolation effects play a crucial role, thus corroborating, in a more pictorial/intuitive way, earlier viewpoints on this matter. An exact criterium for the most relevant form of herd-immunity to occur can be derived in terms of the model parameters. The analyses presented in this article provide insight in how various measures to prevent an epidemic spread of an infection work, how they can be optimized and what potentially deceptive issues have to be considered when such measures are either implemented or scaled down.

Distinguishing Simple and Complex Contagion Processes on Networks

Contagion processes on networks, including disease spreading, information diffusion, or social behaviors propagation, can be modeled as simple contagion, i.e., as a contagion process involving one connection at a time, or as complex contagion, in which multiple interactions are needed for a contagion event. Empirical data on spreading processes, however, even when available, do not easily allow us to uncover which of these underlying contagion mechanisms is at work. We propose a strategy to discriminate between these mechanisms upon the observation of a single instance of a spreading process. The strategy is based on the observation of the order in which network nodes are infected, and on its correlations with their local topology: these correlations differ between processes of simple contagion, processes involving threshold mechanisms, and processes driven by group interactions (i.e., by "higher-order" mechanisms). Our results improve our understanding of contagion processes and provide a method using only limited information to distinguish between several possible contagion mechanisms.

Data, measurement and empirical methods in the science of science

The advent of large-scale datasets that trace the workings of science has encouraged researchers from many different disciplinary backgrounds to turn scientific methods into science itself, cultivating a rapidly expanding 'science of science'. This Review considers this growing, multidisciplinary literature through the lens of data, measurement and empirical methods. We discuss the purposes, strengths and limitations of major empirical approaches, seeking to increase understanding of the field's diverse methodologies and expand researchers' toolkits. Overall, new empirical developments provide enormous capacity to test traditional beliefs and conceptual frameworks about science, discover factors associated with scientific productivity, predict scientific outcomes and design policies that facilitate scientific progress.

Influential nodes identification method based on adaptive adjustment of voting ability

Influential nodes identification technology is one of the important topics which has been widely applied to logistics node location, social information dissemination, transportation network carrying, biological virus dissemination, power network anti-destruction, etc. At present, a large number of influential nodes identification methods have been studied, but the algorithms that are simple to execute, have high accuracy and can be better applied to real networks are still the focus of research. Therefore, due to the advantages of simple to execute in voting mechanism, a novel algorithm based on adaptive adjustment of voting ability (AAVA) to identify the influential nodes is presented by considering the local attributes of node and the voting contribution of its neighbor nodes, to solve the problem of low accuracy and discrimination of the existing algorithms. This proposed algorithm uses the similarity between the voting node and the voted node to dynamically adjust its voting ability without setting any parameters, so that a node can contribute different voting abilities to different neighbor nodes. To verify the performance of AAVA algorithm, the running results of 13 algorithms are analyzed and compared on 10 different networks with the SIR model as a reference. The experimental results show that the influential nodes identified by AAVA have high consistency with SIR model in Top-10 nodes and Kendall correlation, and have better infection effect of the network. Therefore, it is proved that AAV algorithm has high accuracy and effectiveness, and can be applied to real complex networks of different types and sizes.

Dimension reduction in higher-order contagious phenomena

We investigate epidemic spreading in a deterministic susceptible-infected-susceptible model on uncorrelated heterogeneous networks with higher-order interactions. We provide a recipe for the construction of one-dimensional reduced model (resilience function) of the N-dimensional susceptible-infected-susceptible dynamics in the presence of higher-order interactions. Utilizing this reduction process, we are able to capture the microscopic and macroscopic behavior of infectious networks. We find that the microscopic state of nodes (fraction of stable healthy individual of each node) inversely scales with their degree, and it becomes diminished due to the presence of higher-order interactions. In this case, we analytically obtain that the macroscopic state of the system (fraction of infectious or healthy population) undergoes abrupt transition. Additionally, we quantify the network's resilience, i.e., how the topological changes affect the stable infected population. Finally, we provide an alternative framework of dimension reduction based on the spectral analysis of the network, which can identify the critical onset of the disease in the presence or absence of higher-order interactions. Both reduction methods can be extended for a large class of dynamical models.

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.

Coupled disease-vaccination behavior dynamic analysis and its application in COVID-19 pandemic

Predicting the evolutionary dynamics of the COVID-19 pandemic is a complex challenge. The complexity increases when the vaccination process dynamic is also considered. In addition, when applying a voluntary vaccination policy, the simultaneous behavioral evolution of individuals who decide whether and when to vaccinate must be included. In this paper, a coupled disease-vaccination behavior dynamic model is introduced to study the coevolution of individual vaccination strategies and infection spreading. We study disease transmission by a mean-field compartment model and introduce a non-linear infection rate that takes into account the simultaneity of interactions. Besides, the evolutionary game theory is used to investigate the contemporary evolution of vaccination strategies. Our findings suggest that sharing information with the entire population about the negative and positive consequences of infection and vaccination is beneficial as it boosts behaviors that can reduce the final epidemic size. Finally, we validate our transmission mechanism on real data from the COVID-19 pandemic in France.

Accelerating network layouts using graph neural networks

Graph layout algorithms used in network visualization represent the first and the most widely used tool to unveil the inner structure and the behavior of complex networks. Current network visualization software relies on the force-directed layout (FDL) algorithm, whose high computational complexity makes the visualization of large real networks computationally prohibitive and traps large graphs into high energy configurations, resulting in hard-to-interpret "hairball" layouts. Here we use Graph Neural Networks (GNN) to accelerate FDL, showing that deep learning can address both limitations of FDL: it offers a 10 to 100 fold improvement in speed while also yielding layouts which are more informative. We analytically derive the speedup offered by GNN, relating it to the number of outliers in the eigenspectrum of the adjacency matrix, predicting that GNNs are particularly effective for networks with communities and local regularities. Finally, we use GNN to generate a three-dimensional layout of the Internet, and introduce additional measures to assess the layout quality and its interpretability, exploring the algorithm's ability to separate communities and the link-length distribution. The novel use of deep neural networks can help accelerate other network-based optimization problems as well, with applications from reaction-diffusion systems to epidemics.

Graph theory analysis reveals an assortative pain network vulnerable to attacks

The neural substrate of pain experience has been described as a dense network of connected brain regions. However, the connectivity pattern of these brain regions remains elusive, precluding a deeper understanding of how pain emerges from the structural connectivity. Here, we use graph theory to systematically characterize the architecture of a comprehensive pain network, including both cortical and subcortical brain areas. This structural brain network consists of 49 nodes denoting pain-related brain areas, linked by edges representing their relative incoming and outgoing axonal projection strengths. Sixty-three percent of brain areas in this structural pain network share reciprocal connections, reflecting a dense network. The clustering coefficient, a measurement of the probability that adjacent nodes are connected, indicates that brain areas in the pain network tend to cluster together. Community detection, the process of discovering cohesive groups in complex networks, successfully reveals two known subnetworks that specifically mediate the sensory and affective components of pain, respectively. Assortativity analysis, which evaluates the tendency of nodes to connect with other nodes with similar features, indicates that the pain network is assortative. Finally, robustness, the resistance of a complex network to failures and perturbations, indicates that the pain network displays a high degree of error tolerance (local failure rarely affects the global information carried by the network) but is vulnerable to attacks (selective removal of hub nodes critically changes network connectivity). Taken together, graph theory analysis unveils an assortative structural pain network in the brain processing nociceptive information, and the vulnerability of this network to attack opens up the possibility of alleviating pain by targeting the most connected brain areas in the network.