Epidemic spreading in a hierarchical social network

Epidemic evolutionarily stable strategies within an age-structured host population

To understand infectious disease dynamics, we need to understand the inextricably intertwined nature of the ecology and evolution of pathogens and hosts. Epidemiological dynamics of many infectious diseases have highlighted the importance of considering the demographics of the societies in which they spread, particularly with respect to age structure. In addition, the waves of the recent COVID-19 pandemic driven by variant replacements at an unprecedented speed show that it is vital to consider the evolutionary aspects. The classic trade-off theory of virulence addresses aspects of pathogen evolution, but here we explore in more detail the possibility of society-specific evolutionarily stable strategies (ESS) during an unfolding pandemic. Theory posits the existence under some conditions of an ESS representing the evolutionary endpoint of change. By using a demographically realistic model incorporating infection rates that vary with age, we outline which evolutionary scenarios are plausible. Focusing on the rate of infection and duration of infectivity, we ask whether an ESS exists, what characterizes it, and as a result which long-term public-health consequences may be expected. We demonstrate that the ESS of an evolving pathogen depends upon the background age-dependent frailty and mortality rates. Our findings shed important light on the plausible long-term trajectories of highly evolvable novel pathogens.

Modeling outbreaks of COVID-19 in China: The impact of vaccination and other control measures on curbing the epidemic

This study aims to examine the development trend of COVID-19 in China and propose a model to assess the impacts of various prevention and control measures in combating the COVID-19 pandemic. Using COVID-19 cases reported by the National Health Commission of China from January 2, 2020, to January 2, 2022, we established a Susceptible-Exposed-Infected-Asymptomatic-Quarantined-Vaccinated-Hospitalized-Removed (SEIAQVHR) model to calculate the COVID-19 transmission rate and Rt effective reproduction number, and assess prevention and control measures. Additionally, we built a stochastic model to explore the development of the COVID-19 epidemic. We modeled the incidence trends in five outbreaks between 2020 and 2022. Some important features of the COVID-19 epidemic are mirrored in the estimates based on our SEIAQVHR model. Our model indicates that an infected index case entering the community has a 50%-60% chance to cause a COVID-19 outbreak. Wearing masks and getting vaccinated were the most effective measures among all the prevention and control measures. Specifically targeting asymptomatic individuals had no significant impact on the spread of COVID-19. By adjusting prevention and control parameters, we suggest that increasing the rates of effective vaccination and mask-wearing can significantly reduce COVID-19 cases in China. Our stochastic model analysis provides a useful tool for understanding the COVID-19 epidemic in China.

Intervention effect of targeted workplace closures may be approximated by single-layered networks in an individual-based model of COVID-19 control

Individual-based models of infectious disease dynamics commonly use network structures to represent human interactions. Network structures can vary in complexity, from single-layered with homogeneous mixing to multi-layered with clustering and layer-specific contact weights. Here we assessed policy-relevant consequences of network choice by simulating different network structures within an established individual-based model of SARS-CoV-2 dynamics. We determined the clustering coefficient of each network structure and compared this to several epidemiological outcomes, such as cumulative and peak infections. High-clustered networks estimate fewer cumulative infections and peak infections than less-clustered networks when transmission probabilities are equal. However, by altering transmission probabilities, we find that high-clustered networks can essentially recover the dynamics of low-clustered networks. We further assessed the effect of workplace closures as a layer-targeted intervention on epidemiological outcomes and found in this scenario a single-layered network provides a sufficient approximation of intervention effect relative to a multi-layered network when layer-specific contact weightings are equal. Overall, network structure choice within models should consider the knowledge of contact weights in different environments and pathogen mode of transmission to avoid over- or under-estimating disease burden and impact of interventions.

Impact of community structure on the spread of epidemics on time-varying multiplex networks

Community structure plays a crucial role in realistic networks and different communities can be created by groups of interest and activity events, and exploring the impact of community properties on collective dynamics is an active topic in the field of network science. Here, we propose a new coupled model with different time scales for online social networks and offline epidemic spreading networks, in which community structure is added into online social networks to investigate its role in the interacting dynamics between information diffusion and epidemic spreading. We obtain the analytical equations of epidemic threshold by MMC (Microscopic Markov Chain) method and conduct a large quantities of numerical simulations using Monte Carlo simulations in order to verify the accuracy of the MMC method, and more valuable insights are also obtained. The results indicate that an increase in the probability of the mobility of an individual can delay the spread of epidemic-related information in the network, as well as delaying the time of the peak of the infection density in the network. However, an increase in the contact ability of mobile individuals produces a facilitating effect on the spread of epidemics. Finally, it is also found that the stronger the acceptance of an individual to information coming from a different community, the lower the infection density in the network, which suggests that it has an inhibitory effect on the disease spreading.

Understanding underlying physical mechanism reveals early warning indicators and key elements for adaptive infections disease networks

The study of infectious diseases holds significant scientific and societal importance, yet current research on the mechanisms of disease emergence and prediction methods still face challenging issues. This research uses the landscape and flux theoretical framework to reveal the non-equilibrium dynamics of adaptive infectious diseases and uncover its underlying physical mechanism. This allows the quantification of dynamics, characterizing the system with two basins of attraction determined by gradient and rotational flux forces. Quantification of entropy production rates provides insights into the system deviating from equilibrium and associated dissipative costs. The study identifies early warning indicators for the critical transition, emphasizing the advantage of observing time irreversibility from time series over theoretical entropy production and flux. The presence of rotational flux leads to an irreversible pathway between disease states. Through global sensitivity analysis, we identified the key factors influencing infectious diseases. In summary, this research offers valuable insights into infectious disease dynamics and presents a practical approach for predicting the onset of critical transition, addressing existing research gaps.

Impact of simplicial complexes on epidemic spreading in partially mapping activity-driven multiplex networks

Over the past decade, the coupled spread of information and epidemic on multiplex networks has become an active and interesting topic. Recently, it has been shown that stationary and pairwise interactions have limitations in describing inter-individual interactions , and thus, the introduction of higher-order representation is significant. To this end, we present a new two-layer activity-driven network epidemic model, which considers the partial mapping relationship among nodes across two layers and simultaneously introduces simplicial complexes into one layer, to investigate the effect of 2-simplex and inter-layer mapping rate on epidemic transmission. In this model, the top network, called the virtual information layer, characterizes information dissemination in online social networks, where information can be diffused through simplicial complexes and/or pairwise interactions. The bottom network, named as the physical contact layer, denotes the spread of infectious diseases in real-world social networks. It is noteworthy that the correspondence among nodes between two networks is not one-to-one but partial mapping. Then, a theoretical analysis using the microscopic Markov chain (MMC) method is performed to obtain the outbreak threshold of epidemics, and extensive Monte Carlo (MC) simulations are also carried out to validate the theoretical predictions. It is obviously shown that MMC method can be used to estimate the epidemic threshold; meanwhile, the inclusion of simplicial complexes in the virtual layer or introductory partial mapping relationship between layers can inhibit the spread of epidemics. Current results are conducive to understanding the coupling behaviors between epidemics and disease-related information.

The impact of the self-recognition ability and physical quality on coupled negative information-behavior-epidemic dynamics in multiplex networks

In recent years, as the COVID-19 global pandemic evolves, many unprecedented new patterns of epidemic transmission continue to emerge. Reducing the impact of negative information diffusion, calling for individuals to adopt immunization behaviors, and decreasing the infection risk are of great importance to maintain public health and safety. In this paper, we construct a coupled negative information-behavior-epidemic dynamics model by considering the influence of the individual's self-recognition ability and physical quality in multiplex networks. We introduce the Heaviside step function to explore the effect of decision-adoption process on the transmission for each layer, and assume the heterogeneity of the self-recognition ability and physical quality obey the Gaussian distribution. Then, we use the microscopic Markov chain approach (MMCA) to describe the dynamic process and derive the epidemic threshold. Our findings suggest that increasing the clarification strength of mass media and enhancing individuals' self-recognition ability can facilitate the control of the epidemic. And, increasing physical quality can delay the epidemic outbreak and leads to suppress the scale of epidemic transmission. Moreover, the heterogeneity of the individuals in the information diffusion layer leads to a two-stage phase transition, while it leads to a continuous phase transition in the epidemic layer. Our results can provide favorable references for managers in controlling negative information, urging immunization behaviors and suppressing epidemics.

Limited resource network modeling and its opinion diffusion dynamics

The preferential attachment of the Barabási-Albert model has been playing an important role in modeling practical complex networks. The preferential attachment mechanism describes the role of many real systems, which follows the characteristic "the rich get richer." However, there are some situations that are ignored by the preferential attachment mechanism, one of which is the existence of the limited resource. Vertices with the largest degree may not obtain new edges by the highest probability due to various factors, e.g., in social relationship networks, vertices with quite a lot of relationships may not connect to new vertices since their energy and resource are limited. Hence, the limit for degree growing is proposed in our new network model. We adjust the attachment rule in light of the population growth curve in biology, which considers both attraction and restriction of the degree. In addition, the unaware-aware-unaware opinion diffusion is studied on our proposed network. The celebrity effect is taken into consideration in the opinion diffusion process.

Role of Time Scales in the Coupled Epidemic-Opinion Dynamics on Multiplex Networks

Modelling the epidemic’s spread on multiplex networks, considering complex human behaviours, has recently gained the attention of many scientists. In this work, we study the interplay between epidemic spreading and opinion dynamics on multiplex networks. An agent in the epidemic layer could remain in one of five distinct states, resulting in the SIRQD model. The agent’s attitude towards respecting the restrictions of the pandemic plays a crucial role in its prevalence. In our model, the agent’s point of view could be altered by either conformism mechanism, social pressure, or independent actions. As the underlying opinion model, we leverage the q-voter model. The entire system constitutes a coupled opinion–dynamic model where two distinct processes occur. The question arises of how to properly align these dynamics, i.e., whether they should possess equal or disparate timescales. This paper highlights the impact of different timescales of opinion dynamics on epidemic spreading, focusing on the time and the infection’s peak.

Massive migration promotes the early spread of COVID-19 in China: a study based on a scale-free network

BACKGROUND

The coronavirus disease 2019 (COVID-19) epidemic met coincidentally with massive migration before Lunar New Year in China in early 2020. This study is to investigate the relationship between the massive migration and the coronavirus disease 2019 (COVID-19) epidemic in China.

METHODS

The epidemic data between January 25th and February 15th and migration data between Jan 1st and Jan 24th were collected from the official websites. Using the R package WGCNA, we established a scale-free network of the selected cities. Correlation analysis was applied to describe the correlation between the Spring Migration and COVID-19 epidemic.

RESULTS

The epidemic seriousness in Hubei (except the city of Wuhan) was closely correlated with the migration from Wuhan between January 10 and January 24, 2020. The epidemic seriousness in the other provinces, municipalities and autonomous regions was largely affected by the immigration from Wuhan. By establishing a scale-free network of the regions, we divided the regions into two modules. The regions in the brown module consisted of three municipalities, nine provincial capitals and other 12 cities. The COVID-19 epidemics in these regions were more likely to be aggravated by migration.

CONCLUSIONS

The migration from Wuhan could partly explain the epidemic seriousness in Hubei Province and other regions. The scale-free network we have established can better evaluate the epidemic. Three municipalities (Beijing, Shanghai and Tianjin), eight provincial capitals (including Nanjing, Changsha et al.) and 12 other cities (including Qingdao, Zhongshan, Shenzhen et al.) were hub cities in the spread of COVID-19 in China.

Suppression of epidemic spreading process on multiplex networks via active immunization

Spatial epidemic spreading, a fundamental dynamical process upon complex networks, attracts huge research interest during the past few decades. To suppress the spreading of epidemic, a couple of effective methods have been proposed, including node vaccination. Under such a scenario, nodes are immunized passively and fail to reveal the mechanisms of active activity. Here, we suggest one novel model of an observer node, which can identify infection through interacting with infected neighbors and inform the other neighbors for vaccination, on multiplex networks, consisting of epidemic spreading layer and information spreading layer. In detail, the epidemic spreading layer supports susceptible-infected-recovered process, while observer nodes will be selected according to several algorithms derived from percolation theory. Numerical simulation results show that the algorithm based on large degree performs better than random placement, while the algorithm based on nodes' degree in the information spreading layer performs the best (i.e., the best suppression efficacy is guaranteed when placing observer nodes based on nodes' degree in the information spreading layer). With the help of state probability transition equation, the above phenomena can be validated accurately. Our work thus may shed new light into understanding control of empirical epidemic control.

Global stability for epidemic models on multiplex networks

In this work, we consider an epidemic model in a two-layer network in which the dynamics of susceptible-infected-susceptible process in the physical layer coexists with that of a cyclic process of unaware-aware-unaware in the virtual layer. For such multiplex network, we shall define the basic reproduction number [Formula: see text] in the virtual layer, which is similar to the basic reproduction number [Formula: see text] defined in the physical layer. We show analytically that if [Formula: see text] and [Formula: see text], then the disease and information free equilibrium is globally stable and if [Formula: see text] and [Formula: see text], then the disease free and information saturated equilibrium is globally stable for all initial conditions except at the origin. In the case of [Formula: see text], whether the disease dies out or not depends on the competition between how well the information is transmitted in the virtual layer and how contagious the disease is in the physical layer. In particular, it is numerically demonstrated that if the difference in [Formula: see text] and [Formula: see text] is greater than the product of [Formula: see text], the deviation of [Formula: see text] from 1 and the relative infection rate for an aware susceptible individual, then the disease dies out. Otherwise, the disease breaks out.

Dynamics of epidemic diseases on a growing adaptive network

The study of epidemics on static networks has revealed important effects on disease prevalence of network topological features such as the variance of the degree distribution, i.e. the distribution of the number of neighbors of nodes, and the maximum degree. Here, we analyze an adaptive network where the degree distribution is not independent of epidemics but is shaped through disease-induced dynamics and mortality in a complex interplay. We study the dynamics of a network that grows according to a preferential attachment rule, while nodes are simultaneously removed from the network due to disease-induced mortality. We investigate the prevalence of the disease using individual-based simulations and a heterogeneous node approximation. Our results suggest that in this system in the thermodynamic limit no epidemic thresholds exist, while the interplay between network growth and epidemic spreading leads to exponential networks for any finite rate of infectiousness when the disease persists.

Reconstructing direct and indirect interactions in networked public goods game

Network reconstruction is a fundamental problem for understanding many complex systems with unknown interaction structures. In many complex systems, there are indirect interactions between two individuals without immediate connection but with common neighbors. Despite recent advances in network reconstruction, we continue to lack an approach for reconstructing complex networks with indirect interactions. Here we introduce a two-step strategy to resolve the reconstruction problem, where in the first step, we recover both direct and indirect interactions by employing the Lasso to solve a sparse signal reconstruction problem, and in the second step, we use matrix transformation and optimization to distinguish between direct and indirect interactions. The network structure corresponding to direct interactions can be fully uncovered. We exploit the public goods game occurring on complex networks as a paradigm for characterizing indirect interactions and test our reconstruction approach. We find that high reconstruction accuracy can be achieved for both homogeneous and heterogeneous networks, and a number of empirical networks in spite of insufficient data measurement contaminated by noise. Although a general framework for reconstructing complex networks with arbitrary types of indirect interactions is yet lacking, our approach opens new routes to separate direct and indirect interactions in a representative complex system.

Epidemic spreading with activity-driven awareness diffusion on multiplex network

There has been growing interest in exploring the interplay between epidemic spreading with human response, since it is natural for people to take various measures when they become aware of epidemics. As a proper way to describe the multiple connections among people in reality, multiplex network, a set of nodes interacting through multiple sets of edges, has attracted much attention. In this paper, to explore the coupled dynamical processes, a multiplex network with two layers is built. Specifically, the information spreading layer is a time varying network generated by the activity driven model, while the contagion layer is a static network. We extend the microscopic Markov chain approach to derive the epidemic threshold of the model. Compared with extensive Monte Carlo simulations, the method shows high accuracy for the prediction of the epidemic threshold. Besides, taking different spreading models of awareness into consideration, we explored the interplay between epidemic spreading with awareness spreading. The results show that the awareness spreading can not only enhance the epidemic threshold but also reduce the prevalence of epidemics. When the spreading of awareness is defined as susceptible-infected-susceptible model, there exists a critical value where the dynamical process on the awareness layer can control the onset of epidemics; while if it is a threshold model, the epidemic threshold emerges an abrupt transition with the local awareness ratio α approximating 0.5. Moreover, we also find that temporal changes in the topology hinder the spread of awareness which directly affect the epidemic threshold, especially when the awareness layer is threshold model. Given that the threshold model is a widely used model for social contagion, this is an important and meaningful result. Our results could also lead to interesting future research about the different time-scales of structural changes in multiplex networks.

Two-stage effects of awareness cascade on epidemic spreading in multiplex networks

Human awareness plays an important role in the spread of infectious diseases and the control of propagation patterns. The dynamic process with human awareness is called awareness cascade, during which individuals exhibit herd-like behavior because they are making decisions based on the actions of other individuals [Borge-Holthoefer et al., J. Complex Networks 1, 3 (2013)]. In this paper, to investigate the epidemic spreading with awareness cascade, we propose a local awareness controlled contagion spreading model on multiplex networks. By theoretical analysis using a microscopic Markov chain approach and numerical simulations, we find the emergence of an abrupt transition of epidemic threshold β(c) with the local awareness ratio α approximating 0.5, which induces two-stage effects on epidemic threshold and the final epidemic size. These findings indicate that the increase of α can accelerate the outbreak of epidemics. Furthermore, a simple 1D lattice model is investigated to illustrate the two-stage-like sharp transition at α(c)≈0.5. The results can give us a better understanding of why some epidemics cannot break out in reality and also provide a potential access to suppressing and controlling the awareness cascading systems.

Epidemic variability in hierarchical geographical networks with human activity patterns

Recently, some studies have revealed that non-Poissonian statistics of human behaviors stem from the hierarchical geographical network structure. On this view, we focus on epidemic spreading in the hierarchical geographical networks and study how two distinct contact patterns (i.e., homogeneous time delay (HOTD) and heterogeneous time delay (HETD) associated with geographical distance) influence the spreading speed and the variability of outbreaks. We find that, compared with HOTD and null model, correlations between time delay and network hierarchy in HETD remarkably slow down epidemic spreading and result in an upward cascading multi-modal phenomenon. Proportionately, the variability of outbreaks in HETD has the lower value, but several comparable peaks for a long time, which makes the long-term prediction of epidemic spreading hard. When a seed (i.e., the initial infected node) is from the high layers of networks, epidemic spreading is remarkably promoted. Interestingly, distinct trends of variabilities in two contact patterns emerge: high-layer seeds in HOTD result in the lower variabilities, the case of HETD is opposite. More importantly, the variabilities of high-layer seeds in HETD are much greater than that in HOTD, which implies the unpredictability of epidemic spreading in hierarchical geographical networks.

Exact solution of a stochastic susceptible-infectious-recovered model

The susceptible-infectious-recovered (SIR) model describes the evolution of three species of individuals which are subject to an infection and recovery mechanism. A susceptible S can become infectious with an infection rate beta by an infectious I type provided that both are in contact. The I type may recover with a rate gamma and from then on stay immune. Due to the coupling between the different individuals, the model is nonlinear and out of equilibrium. We adopt a stochastic individual-based description where individuals are represented by nodes of a graph and contact is defined by the links of the graph. Mapping the underlying master equation onto a quantum formulation in terms of spin operators, the hierarchy of evolution equations can be solved exactly for arbitrary initial conditions on a linear chain. In the case of uncorrelated random initial conditions, the exact time evolution for all three individuals of the SIR model is given analytically. Depending on the initial conditions and reaction rates beta and gamma , the I population may increase initially before decaying to zero. Due to fluctuations, isolated regions of susceptible individuals evolve, and unlike in the standard mean-field SIR model, one observes a finite stationary distribution of the S type even for large population size. The exact results for the ensemble-averaged population size are compared with simulations for single realizations of the process and also with standard mean-field theory, which is expected to be valid on large fully connected graphs.

Dynamic phenomena and human activity in an artificial society

We study dynamic phenomena in a large social network of nearly 3x10;{4} individuals who interact in the large virtual world of a massive multiplayer online role playing game. On the basis of a database received from the online game server, we examine the structure of the friendship network and human dynamics. To investigate the relation between networks of acquaintances in virtual and real worlds, we carried out a survey among the players. We show that, even though the virtual network did not develop as a growing graph of an underlying network of social acquaintances in the real world, it influences it. Furthermore we find very interesting scaling laws concerning human dynamics. Our research shows how long people are interested in a single task and how much time they devote to it. Surprisingly, exponent values in both cases are close to -1 . We calculate the activity of individuals, i.e., the relative time daily devoted to interactions with others in the artificial society. Our research shows that the distribution of activity is not uniform and is highly correlated with the degree of the node, and that such human activity has a significant influence on dynamic phenomena, e.g., epidemic spreading and rumor propagation, in complex networks. We find that spreading is accelerated (an epidemic) or decelerated (a rumor) as a result of superspreaders' various behavior.

Epidemic spreading in complex networks

The study of epidemic spreading in complex networks is currently a hot topic and a large body of results have been achieved. In this paper, we briefly review our contributions to this field, which includes the underlying mechanism of rumor propagation, the epidemic spreading in community networks, the influence of varying topology, and the influence of mobility of agents. Also, some future directions are pointed out.

Fluctuating epidemics on adaptive networks

A model for epidemics on an adaptive network is considered. Nodes follow a susceptible-infective-recovered-susceptible pattern. Connections are rewired to break links from noninfected nodes to infected nodes and are reformed to connect to other noninfected nodes, as the nodes that are not infected try to avoid the infection. Monte Carlo simulation and numerical solution of a mean field model are employed. The introduction of rewiring affects both the network structure and the epidemic dynamics. Degree distributions are altered, and the average distance from a node to the nearest infective increases. The rewiring leads to regions of bistability where either an endemic or a disease-free steady state can exist. Fluctuations around the endemic state and the lifetime of the endemic state are considered. The fluctuations are found to exhibit power law behavior.

The epidemics of donations: logistic growth and power-laws

This paper demonstrates that collective social dynamics resulting from individual donations can be well described by an epidemic model. It captures the herding behavior in donations as a non-local interaction between individual via a time-dependent mean field representing the mass media. Our study is based on the statistical analysis of a unique dataset obtained before and after the tsunami disaster of 2004. We find a power-law behavior for the distributions of donations with similar exponents for different countries. Even more remarkably, we show that these exponents are the same before and after the tsunami, which accounts for some kind of universal behavior in donations independent of the actual event. We further show that the time-dependent change of both the number and the total amount of donations after the tsunami follows a logistic growth equation. As a new element, a time-dependent scaling factor appears in this equation which accounts for the growing lack of public interest after the disaster. The results of the model are underpinned by the data analysis and thus also allow for a quantification of the media influence.

Partially mixed household epidemiological model with clustered resistant individuals

We study the dynamics of the spread of an infectious disease within a population partitioned into households, and stratified into resistant and nonresistant individuals. Variability in the level of resistance between households increases the initial rate of spread of the infection, as well as the infection level at the endemic equilibrium. This phenomenon is seen even when all individuals in the population are equally likely to be resistant, and can also be predicted by including spatial clustering of resistant individuals within an improved mean-field approximation.

Modelling disease spread and control in networks: implications for plant sciences

Networks are ubiquitous in natural, technological and social systems. They are of increasing relevance for improved understanding and control of infectious diseases of plants, animals and humans, given the interconnectedness of today's world. Recent modelling work on disease development in complex networks shows: the relative rapidity of pathogen spread in scale-free compared with random networks, unless there is high local clustering; the theoretical absence of an epidemic threshold in scale-free networks of infinite size, which implies that diseases with low infection rates can spread in them, but the emergence of a threshold when realistic features are added to networks (e.g. finite size, household structure or deactivation of links); and the influence on epidemic dynamics of asymmetrical interactions. Models suggest that control of pathogens spreading in scale-free networks should focus on highly connected individuals rather than on mass random immunization. A growing number of empirical applications of network theory in human medicine and animal disease ecology confirm the potential of the approach, and suggest that network thinking could also benefit plant epidemiology and forest pathology, particularly in human-modified pathosystems linked by commercial transport of plant and disease propagules. Potential consequences for the study and management of plant and tree diseases are discussed.

Evolution of a social network: the role of cultural diversity

We present a simple deterministic and based on local rules model of evolving social network, which leads to a network with the properties of a real social system, e.g., small-world topology and assortative mixing. The state of an individual Si is characterized by the values of Q cultural features, drawn from Gaussian distribution with variance sigma. The other control parameter is sociability Ti, which describes the maximal number of connections of an individual. The state of individuals and connections between them evolve in time. As results from numerical computations, an initial diversity of cultural features in a community has an essential influence on an evolution of social network. It was found that for a critical value of control parameter sigma c(Q) there is a structural transition and a hierarchical network with small-world topology of connections and a high clustering coefficient emerges. The emergence of small-world properties can be related to the creation of subculture groups in a community. The power-law relation between the clustering coefficient of a node and its connectivity C(k) approximately k-beta was observed in the case of a scale-free distribution of sociability Ti and a high enough cultural diversity in a population.