Proposal of a mathematical model for measuring the impact of focus on network and behaviour dynamics of social networks

Social networks are social systems composed of individuals who interact directly or indirectly. An individual has various kinds of relationship in his/her social life and he/she is located in different type of network structures depending on this relationship. Each social interaction, which is a component of the social life, has its own internal dynamics and structures with different characteristics. Therefore, it is necessary to know the characteristics of social networks in order to analyze the structural features of the relations in the network. This study focuses on modeling of social network structures in their dynamic environments. It is assumed that, there are two types of changes occurred in a social network; one is the network change addressing the change of actor's network structure, the other is the behavior change identifying the change of attitudes of social actors. In general, the idea behind the study is to understand (1) how the links between actors (nodes) of a social network change over time based on their social interactions, and (2) how their behavior changes due to these social interactions. With the help of stochastic actor-based modeling suggested by Snijders et al (1997) these type of changes can be observed and the changes paired with selection and influence process explained based on homophily. In the model, the probability of links depends on the actor's links and actor-pairs. In the study, it is claimed that each variable of social interaction/network foci has a different effect on the behavior or attitude of the actor and a specific focus is the primary mechanism to explain the changes in the network. The network structure may change not only based on individual characteristics and interactions of pairs of actors as it is focused in stochastic actor based model, but it may also varies depending upon the property of the network. In this context, the stochastic actor based model is reconsidered using systematic perspective of structuration theory based on focus theory. The aim of this study is to understand the reason of change of actors' political leaning based on social interaction variables/foci. Within this context, a mathematical model is proposed in the study to indicate the effects of network foci using the proximity of clusters of political leaning. It can be inferred from the results of the research on two different types of network foci that, the model for the political discussion network explains the changes in political leaning better than the model do for the friendship network. Based on this result, the claim which states that "a specific focus has the primary effect regarding the explanation of the change of individual's behavior or attitude compared to other network focus" is supported. As the actor's undertaken different mission in the network, in order to explain the changes in social network one should have to look at the overall picture to determine the network foci, rather than focusing on the individual. Additionally, proposed model reveals that time-dependent influences of friends (current friend, new friend, lost friend) has different impact on exposure and adoption of opinions in social networks. The influence of lost friend is smaller than the influence of current friends while the influence of the current friends is smaller than the influence of new friends. Due to the dynamic nature of social networks, it is observed that time-dependent relationship has different impact on the behavior of actors. With the proposed model, time-dependent direct and indirect effects of the interactions are also measured. Consequently, having focused on individual characteristics and properties of the pairs of actors', stochastic actor-based model is insufficient in explaining the changes in the network. For this reason, the proposed model in this study provides a new perspective for explaining the changes in the network. Keywords: Social Network, Social Network Analysis, Stochastic Actor Based Model, Focus Theory, Proximity.