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Social systems consists of individuals and the social system in which they interact. Thoughts feelings and actions of individuals affect depend on their social context. On the other side individual behavior creates the social system of which they are parts. Social group have emergent properties that cannot be trivially derived from the properties of their members. Very simple rules concerning individual behavior can result in very complex properties on the group level. Emergent group level phenomena like group norms and values influence in turn individuals and their interactions. This bi-directional causal relationship is at the essence of complexity in the social domain. Understanding the nature of the relationship between different levels at which social phenomena exist has been made possible due to the tools and insights generated in the complexity approach. Explanation of this relationship calls for examining the types of interactions that link individuals in social groups.

Types of social interactions

At least three different classes of interactions assemble individuals into groups and societies.

Passing information .First individuals pass information to other individuals. An individual who knows an item of information will with a certain probability share this information with another individual. A form of this process may be described as gossip. This process underlies the spread of information in social groups and societies. Once an individual learns the information from a certain source, hearing it from other sources is irrelevant to the process. The most fundamental question is under what conditions the information will penetrate the society. In the complexity approach this line of inquiry often is termed the percolation problem. Dynamics of information transmission, at least in simpler cases is well understood. Whether the information will penetrate a social group or a society depends primarily on the probability that it will be passed from the individual who knows on to the interaction partner and on the number of interaction partners. Dynamics of information transmission is a threshold phenomenon, where below certain probability the information will spread in a limited group; above the threshold it will penetrate the whole society.

Social influence. Research in social sciences suggests that interaction between individuals, does not just involve sharing information but, in large part, its function is to construct a shared reality that consists not only of shared information but also of agreed upon opinions. In this process, individuals do not simply transmit information, but more importantly, they influence one another to arrive at a common interpretation of information. In contrast with a simple spread of information hearing the same opinion from a number of sources results in a higher probability of adopting this opinion, than hearing it form a single source. It also matters whom individuals hear this information form. Social influence is usually defined as a change in individual's thoughts feelings or actions resulting from the real or imagined presence of others. Social influence concerns not only formation of opinions but also a variety of other social phenomena such as learning from observation i.e. social modeling, attitude changes and norm formation. Empirical research has revealed that three critical factors determine the magnitude of social influence: (1) the number of sources exerting the influence, (2) the immediacy of the source(s) to the target(s), and (3) the strength of the source(s). Greatest influence is exerted by individuals who are strongest and nearest to the recipient. With respect to locality of influence, research has shown that the probability of interaction, decreases with the square of the distance between individuals and also that. the impact of information decreases with distance between source and recipient.

Multi agent modeling of social influence in the formation of public opinion has shown that individual variation in strength and locality of the influence function are both critical factors in shaping dynamics of influence in a social network. Models show that as a result of social influence initial majority opinion gains in popularity and all the opinions cluster in space. Important class of applications of this class of models, developed by Nowak and Vallacher, is in the approach of complexity to social change. Locality of the influence function also shapes the dynamics of social change. Computer models of social change based on the social influence formula indicate that social transitions occur as clusters of 'new' that appear and grow in the sea of 'old.' In the process of social influence, individuals interact to define and shape their shared social reality. Because this happens on a local level, it promotes different social realities that are separated in space. Social transitions occur as the 'new' reality gains at the expense of the 'old' reality. Form this perspective a number of social and economic phenomena co- evolve in processes of social transitions. Clustering in the process of change is common across a variety of social and economic phenomena. Since the social influence process gets stronger with the increase in the number of sources of influence at a higher level of abstraction it can be described as an autocatalytic process. As such it becomes a specific example of the general class of autocatalytic local growth AB models developed by Solomon and his collaborators. . The complexity model of social change predicts well the empirical data concerning the economic and political transformations in Poland following the collapse of communism in the late 1980s. Especially the AB model predicts the non-linear time course of the socio-economic system in Poland, where the pro-market economic reform has resulted in the initial rapid decline in social-economic activity to be followed by fast and strong but localized growth.

Social interdependence. Social interdependence defines the third type of interaction between individuals in a society. Social interdependence can be understood as a dependence of outcomes of one individual on another individual behavior. Such a relationship between payoffs for choices of different individuals is usually described with the formalism of the game theory. In this formalism choices of one agent change the payoff structure of other agents. Prisoner's dilemma is a mathematical model of conflict between individual and group interests. In prisoner's dilemma each individual can either compete or collaborate with a partner. The payoff for each individual is always higher if the individual chooses to compete. On the group level, however, the payoff is always higher if the individuals cooperate. In multi-agent models agent's decisions are dictated by agent's strategies. .Since on the individual level competition is the dominating strategy, the main question is how can cooperation between individuals emerge? To answer this question one can equip the agents with pre-specified strategies, behavior rules and properties and observe to consequences of their interactions as they play the game, or allow for the evolution of strategies by random mutations of strategies in genetic algorithms The famous result obtained by Axelrod is that cooperation will eventually emerge even in a society of egoistic individuals, as these pairs of players that will be able to converge on cooperation will gain more that those that will be locked in competition. Simulation results concerning conditions that facilitate the emergence of cooperation have important practical implications. An important field for applications of these models are social dilemmas. Social dilemmas can be defined as situations where choices that are beneficial for individuals have negative consequences for the group. An example of social dilemma is smog in the city. If everyone communicates by car (which is the easer choice for an individual, than taking public transportation) then the smog level is high and everybody suffers. Combinations of empirical results and computer simulations has suggested many practical advices on how to increase the probability of solving social dilemmas, for example to help solve such dilemmas one can: enable communication, make group size smaller, facilitate the creation of trust etc.

Since interdependence defines one of the most fundamental ways to link individuals many different issues have been addressed in this paradigm. These results concern not only strategies, individuals features and interaction patterns that facilitate vs. inhibit cooperation in the group, but also the formation of social structures, like the emergence of solidarity and mutual help networks, power structures, patterns of settlements , trust networks etc.

Structural reaction to extreme events, the relations change.

Patterns of social interactions

For the emergence of group level phenomena the pattern of social interactions is critical Global interactions. Some social processes and some simulation models assume global pattern of interaction. Everyone interacts with a certain probability with everyone else in the social group. Such interaction corresponds to communication trough mass media, milling crowd, densely connected social network, posts on the message boards etc. In this approach no local phenomena are possible because there is no locality. Epidemic models of such phenomena as gossip, smoking or violence belong to this class. Epidemic models have autocatalytic properties since everybody who becomes infected becomes the source of infection. The dynamics of infection can be described by logistic equation. This equation is widely recognized to underlay most of social processes.

Local interactions

Individuals in most social processes do not interact with everyone with equal probability. Space imposes important constrains on interaction. Individuals interact with the highest probability with others who are nearby. Empirical research has shown, for example, that the probability of interactions drops down as a square of the distance between the places where each individual lives.

Cellular automata are the models of choice in the social sciences for investigating the emergence of patterns from local interactions among elements. A cellular automaton is a computer simulation model composed of elements adopting discrete states. These elements are arranges in a discrete spatial arrangement such as a 2D lattice, and the time proceeds in discrete states. Each cell, is characterized by its location and its state. In cellular automaton the interaction rules are local, in that the state of each cell depends on the state of neighboring states in a way specified by a specific rule.

.In the social interpretation each cell corresponds to an individual. It is assumed in the classical approach to cellular automata, that all the individual are identical. Neighborhood structures created by spatial proximity can capture locality of human interactions. Each individual can thus react to the social context created by other nearby individuals. The updating rules can specify principles governing changes of states of individuals (such as attitude change) or their location in space.

Two general classes of cellular automata are used to simulate social processes. In the first class, elements change their state, they do not, however change their spatial location. Most models of opinion change and the spread of information belong to this class. In the second class, elements migrate, they do not, however, change their state. Models of racial segregation, acculturation and migration belong to this class.

In cellular automata even very simple rules can produce amazingly complex dynamics, and no direct relationship exists between the complexity of the rules and the complexity of the resultant dynamics. Cellular automata are most useful for discovering spatial and spatio-temporal patterns

Individualized societies

The distinction between anonymous and individualized societies provides a base for distinguishing between societies of ants and humans. Individuals have different properties perceived by others. They occupy different social roles and social positions. Human societies can be characterized in terms of power and trust relations. The structure of human societies consists both of groups such as local communities and of social networks that link individuals, groups and organizations.

Individual properties by which individuals differ provide the basis on which social structure is built. Even meaningless features acquire meaning in social interactions if they are perceived by interaction partners. Any distinctive feature that can be perceived by others may influence the probability of specific behaviors as shown by Macy. He conducted multi-agent simulations in which the agents played Prisoners Dilemma game. The agents were randomly tagged by an initially meaningless symbol. In a random process agents with a particular tag behaved somewhat more cooperatively, with another tag somewhat more competitively. Other players recognized this tendency and started acting cooperatively when they met an individual with the tag corresponding to cooperative tendency and competitively when interacting with someone who had a tag that happened to be associated with competition. The experience of the agents with the first tag was that other agents are cooperative which resulted in more cooperation on the part of these agents, and the experience of the agents with the other tag was that others were competitive, what resulted in more competition on the part of those agents. Initially random tag with time become really predictive of cooperation.

Groups

Social groups reflect the basic organization of individuals. Family groups, local communities, religious groups all share important features. It is relatively well defined who belongs to the group. All the individuals in the group interact more likely with in-group members than with out-group members. Social groups are the sources of identity, they also provide for creation and maintenance of social bonds between their members. Interaction of individuals in social groups leads to the emergence of group norms, where different groups are likely to have different norms. Individual characteristics such as attitudes and behaviors may depend on the social groups they belong to in two ways. First, they may directly result from interactions with other individuals in the group. Second, they may reflect some emergent group level properties such as group norms. Since emergent group properties and individual beliefs and opinions depend on the pattern of interactions between individuals, groups structure of the society is reflected in such characteristics of group members as opinions, beliefs, attitudes, norms, values, behaviors etc. Group structure also allows for mutual monitoring in local communities, so social groups impose social control. Group structure provides for coherence among group members on may dimensions..

Social groups often have hierarchical structure and patterns of social interaction and follow the hierarchical structure of groups. Multi agent simulations has shown that patterns of individual characteristics shaped by social interactions follow such a structure.

Usually people belong to many social groups, where each group governs the pattern of interactions with respect to different topics. The pattern of opinions will follow the structure of this group which governs the interactions with respect to this topic. The structure of a traditional society may be well captured in terms of social groups. Modern societies often cannot be adequately characterized as a set of distinct groups because they resemble networks more than groups.

Social networks

In societies individuals are recognized, they have identity. Human interactions are governed by formal and informal networks. The approach of social networks formalizes the description of networks of relations in a society. In social networks individuals or at a different level of description social groups can be represented as nodes, and relations or flows between individuals or groups can be portrayed as links. These links can represent different types of relations between individuals such as friendship, trust, exchange of information, collaboration or flows e.g. transfers of knowledge. In a network of institutions links may represent flow of money, products or services, organizational relations etc. Affiliation networks are composed by two types of nodes: institutions and individuals. All the links in the network connect nodes of one type to the nodes of another type i.e. individuals and institutions. Nodes of the same type are connected together, if both of them are connected to the same node of the other type. For example one type of nodes may correspond to a supervisory board and another to an individual. Two individuals are connected of they both belong to the same supervisory board. Two supervisory boards are connected if they there is an individuals that belongs to both of them. In multimodal networks nodes may be connected by more than one type of connection where each type of connection corresponds to a different type of relation. Multimodal networks (in graphs theory hyper graphs) are one of the most powerful for practical applications.

Although the formalism of social networks is relatively old, in recent years there has been a rapid growth of interest in networks in general and social networks in particular. Different types of networks occurring in nature and societies have been shown to have universal properties such as:

• short paths: the path linking any two nodes have to go though only a small (on the average 6) number of other nodes. It means that in principle messages can spread in very few steps through the entire society.
• scale free: networks properties do not depend on their size
• power law distribution: there is a small number of nodes with a high number of connections (super-hubs)and a high number of nodes with low number of connections.

Social networks are also auto-affiliative in the sense that nodes that are similar with respect to the degree (i.e. number of other nodes connected to it) or other characteristic are more likely to be connected than other nodes.

Structural properties of network can be derived from the dynamics by which the networks are created. The process by which new social ties are created is far from random formation of connections. Formation of new links for example may happen by individuals introducing others to their friends what results in the most connected nodes acquiring most new connections,. This process produces power distribution of the number of connections. In real societies networks are created by social processes , in which individual create and maintain social relationships, and they in turn influence the dynamics of these (and other) processes. This process results in the self organization of social structures, where social relations depend on and in turn they shape the relation between individuals and between groups in societies.

With respect to social networks a lot is known about measuring their structural properties. Lately the work has been concentrated on the rules of networks formation and on the relationships between the process of network creation and the structure of the resulting network. Little in known, however about the dynamics of processes occurring in networks, and how these processes depend on network properties. Among the simulation research that has been conducted on the dynamics of processes in social networks transmission of information was studied the most. The direction of the research to come is to understand how the structure of social networks determines the dynamics of various types of social processes occurring in the networks. In this respect the most promising models from the perspective of complexity are such, in which the structure of the nets determine the processes which in turn shape the structure.

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