In this paper we investigate the mutual influence of participants (agents) of a social network on each other using the framework of Markov models. The main objective of this study was to check several hypotheses concerning dependencies between the influence of agents and their impact on several computational models.
In this paper, we investigate the mutual influence of participants of a social network
(we will call them agents in accordance with the terminology used in [
Influence is understood as a process of changes in a subject caused by the behavior
of other entities, their settings, intentions, views, assessments and their actions during
cooperation with them [
Our analysis is based on data of three communities extracted from the Live Journal website. Live Journal (http://www.livejournal.com) consists of blogs, which contain sequences of messages called posts. Additionally we have available event logs, online diaries and other website content including images, multimedia, texts, etc.
The differences between a blog and a traditional diary are caused by the environment: blogs are usually public and involve third-party readers, who may enter into a public debate with the author, by commenting on blogs.
Authors of posts are called bloggers. The majority of posts are available for reading and commenting by other bloggers. Live Journal also provides an opportunity to bloggers to unite in a community, and subscribe to a community to read their blogs. In this case all the new posts in the selected blogs are displayed in a special news feed. The blogger can belong to several communities at the same time.
Information about communities, subscriptions and records themselves in most cases, are open and accessible to any Internet user. For each of the communities anyone can get the list of participants and the list of friends for each participant. Data consists of three tables: the list of communities, the list of bloggers and the list of links between bloggers. Further in this paper the terms "blogger" and "agent "will be used as synonyms. The number of entries in the list of participants who are members of one of the three communities of Live Journal are 964, 2960, 6587, and the number of links are 6359, 49504 and 190427 respectively. 2
There are numerous works about properties of Markov models [
Despite the high effectiveness of existing computational algorithms, the main disadvantage of a Markov model is the need to build the initial matrix of influence in the infinite or a high degree. In addition there is some uncertainty in the determination of the initial matrix of mutual trust agents and their relationships with other agents.
In this paper preliminary comparison of different methods for determining the influence of agents of the social network without taking into consideration the data of the messages exchanged between the agents, was conducted. The basis for the identification of the network was data of the agents about whose blogs they read. The format of available data and the sample data is presented in table 1.
The ID of the connection between agents
In literature, several approaches have been proposed to describe the interaction
between participants in a social network: a Markov model or model of De Groot [
In some models, ranking of agents is used, for example, by means of power
indices, index of Houde-Bakker [
Because of its wide popularity, the description of Markov models in this work for the
sake of brevity was not given. Details can be found, for example, in [
where aij is an element of the transitive closure of the matrix of direct influence, can also be computed for the original stochastic matrix of direct influence. In this case we will call it direct influence of i-th agent. The common method of agent identification is based on the direct influence matrix which is derived from the adjacency matrix by the formula where aij is a weight in the matrix of direct influence and bij is an element of the adjacency matrix.
wj aij ,
i aij
bij bij
i aij
fi bij fi bij
i
In some cases, one can try to take into account the impact of the authority of the agent on the strength of the influence. We consider the case when the impact of authority is proportional to the number of friends of the agent, where fj represents the credibility of the i-th agent. (1) (2) (3) 5
Hypotheses
fi x bij i (4) 1. Direct influence depends on the number of friends of an agent. 2. The number of friends is not correlated with transitive influence. 3. There is a correlation between transitive influences of agents, calculated by different methods taking into account the authority of agents. 4. The direct influence of the agent does not correlate with its transitive influence. 5. Implementation of hypotheses does not depend on the size of the network. 6
Testing hypothesis 1 reveals that direct influence depends on the number of friends of an agent, and the relationship between them is close to a power-law function as shown in figure 1. The coefficient of correlation is 0.85.
Testing hypothesis 2 revealed that the number of friends is not correlated with transitive influence. This is shown in figure 2. The coefficient of correlation is 0.72. In addition to a linear dependence we observe the almost vertical "tail". Its presence means that there are several agents who have a small number of friends, but a substantial influence. In particular, there are three agents for whom the transitive influence exceeds the transitive influence of the agent with the highest number of friends (whose influence can be assumed). The existence of this phenomenon has been theoretically predicted, but validation on real experimental data had not been performed yet. Note that we do not yet have an explanation for the presence of only two main lines in diagram and this issue should be investigated more thoroughly in the future. Fig. 3. The transitive influence of agents, calculated without taking into account their authority (horizontal axis) and calculated while taking authority into account (β=4) (vertical axis).
Testing hypothesis 3 revealed that there is no correlation between transitive influences of agents, neither without taking into account the authority, nor when taking authority into account. In figure 3 we see that there is no correlation. The coefficient of the correlation is 0.31. This is an important observation because by making assumptions about the impact of the number of friends of an agent on his credibility, you can get, generally speaking, different results. If we ignore some of the outlier observations, we can once again identify two "tails". The main tail shows a linear dependency, which is not equal to the constant β, and the second tail indicates a non-increasing transitive influence of the agents, despite of the increase in their transitive influence in the case of not taking into account their credibility. Moreover, the figure shows that there are a number of influential agents with low authority. This is consistent with the result that we received in the process of verification of hypothesis 2. So it can be argued that the correlation between transitive influences is complex in nature, and thus, hypothesis 3, cannot be affirmed without additional clarifications.
Testing hypothesis 4 revealed that the direct influence of an agent does not correlate with its transitive visibility. In figure 4 you can see that the linear correlation between direct and transitive influences is not clear. The coefficient of correlation is 0.78. This is also interesting, since we believe that not all agents can make decisions based on the computation of transitive influence, and therefore are forced to use direct influence measures. Fig. 5. The dependency of the transitive influence of the agent (vertical) from its direct influence (horizontal), calculated taking into account the authority of agents Then we come to a rather obvious conclusion, that in real social networks such agents can be mistaken. However, we conclude that there is no ground for hypothesis 4. In the case shown in figure 5, the linear correlation is noticeable. However, it is different for small values of direct influence than for higher values, where you can also identify the correlation. The coefficient of the correlation is 0.92.
Hypothesis 5 states the assumption that the validity of the hypotheses does not depend on the size of the network. This has not yet been verified and is a possible direction for future research. 7
The study showed the presence of a certain number of anomalies and effects that need to be taken into account while identifying optimal Markov model parameters for experimental data. It was shown that the credibility of agents has a significant impact on the influence of agents. It was shown that there is a specific dependence between transitive influence and direct influence. We identified an abnormal cluster of agents, which have a small number of friends, but which have a great transitive influence.
It may be interesting to continue our study by verifying hypothesis 5, as well as including in the analysis the possibility of taking into account the exchange of messages between agents. We also intend to investigate the ranking of agents using methods such as alpha-centrality, the PageRank algorithm, as well as other widely used methods based on direct and transitive influences of agents.
Acknowledgement. The research is supported by the grant 10-07-00129 of Russian Foundation for Basic Research. We would like to express our gratitude to Jonas Poelmans and Dmitry Ignatov for improving the language quality.