Authority estimation within social networks using regression analysis
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Abstract
This paper focuses on methods of machine learning, particularly on regression analysis to solve a problem of authority identification within social networks. Within this paper, linear, polynomial, and nonlinear regression types were considered. The aim was to find an approximation of dependency of the authority value on variables representing parameters of the structure and particularly the content of selected web discussions. The approximation function can be used at first for computation of the authority value of a given discussant, at second, for discrimination of an authoritative discussant from nonauthoritative contributors to the web discussion. This information is important for web users, who search for truthful and reliable information in the process of decision making about important things. The web users would like to be influenced by some credible professionals. The various regression methods were tested, particularly linear, polynomial, and nonlinear regression models. The best solution was implemented in the Application for the Machine Authority Identification.
Keywords
Authority identification Social networks Web mining Linear regression Nonlinear regression Web forums1 Introduction
We live in the information era. A volume of information, which is discovered each day, is too large and too timeconsuming to be processed by a human. Everybody from us needs sometimes an access to the relevant supporting information for our decisionmaking. To know the relevance of information we have found, we need information about sources of the obtained information and their credibility. In other words, it is important to know the sources, which are authoritative ones. A web forum discussion can be a repository of various kinds of useful information: facts, opinions, ideas, attitudes, and so on. However, useful information is mixed with nonuseful or misleading information. Every web user can join the web discussion, but many of them have not sufficient experiences or theoretical knowledge about the discussed themes. The web discussion often contains an opinion spam and an information trash. Therefore, it is the matter of principal to search for authoritative discussants to let them influence our important decisions. And just the searching for an authority and its machine identification among all discussants of web forum is our challenge.
 1.
To find such variables—parameters of the structure and content of the web discussion, which are the most related to the authoritative contributing.
 2.
To define a dependency of the variable “authority” of a web discussion on the independent variables selected in the first step. We tried to find an approximation of this dependency using the linear and nonlinear regression [1] based on the method of the ordinary least squares (OLSs) [2].
 3.
To use this approximation function for the discrimination of the authoritative from nonauthoritative contributors to the web discussion.
Finally, all learned results were tested using widely used measures of the efficiency—the precision and the recall. The best solution was implemented within the Application for the Machine Authority Identification (AMAI).
2 Authority and web discussion
2.1 Web discussion group

Discussion connected to a web article In this case, the discussion is only an additional function to the content of the article to enable feedback. The subject of such discussion can be the text of the article or the concerning theme or product.

Guestbook A place on a web site dedicated to reactions on the given web project, for example, a personal web page or a web page relevant to some theme.

Discussion forum A part of a more extensive discussion project. It enables to establish new discussion pages and to sectionalize them into groups according to themes. It is a place, where users can leave the contributions. These contributions (e.g., news) are often longer than one line as they are within chatrooms. They are temporarily archived. An approval of new contributions by moderator of the discussion can be desired before the contributions became visible for all users.

Questions and answers Some public institutions offer the public answering of question, suggestions or complaints on special web pages. In this case, a pronouncement of a responsible representative of the institution is expected.
2.2 Authority identification in general

Formal (functional) authority It represents a measure of influence of some person following from his formal position regardless of personal properties. It is leadership of the person, who is mandated to make decisions. It is obviously the result of a position, title or function of some person within an organization (an arbiter, teacher, politician, and so on). A leader could require submission, although this person is not honest or brave or predictable or able of a quick decisionmaking.

Informal (natural) authority It is based on human and personal properties and professional assumptions of a person. It is the result of a personal profile, capability, adequate selfconfidence, and social activities. Such person has natural, spontaneous influence on others, because of his/her persuasiveness and good experiences with his/her advices and decisions. The people, who let an authority to lead them, enforce the weight of this authority.
2.3 Authority of a web discussion
The virtual web authority has different characteristics as the authority in real life. It is related to the structure of the web, which is based on hyperlinks among web pages. The Google has discovered very complicated relations among web pages and references. Wellknown tool for the web page authority calculation is PageRank [6]. Other known approaches to the web page authority calculating are hyperlinkinduced topic search (HITS) algorithm [7] and stochastic approach for linkstructure analysis (SALSA) [8]. These approaches are also based on an input and output hyperlinks of the evaluated web page. There are also tools of the respected portal “Seomoz,” for example, MozTrust(Moz’s global link trust score) [9] and Open Site Explorer [10]. All these tools cannot be easily used for calculating of an authority of the web discussion forum. There is also interesting work [11] concerned on a qualitative analysis of discussion forum. However, this work has not the aim to estimate the value of Authority of web discussant.
The authority identification from web discussion forums is a similar problem as web page authority calculation, because authority identification from web discussion is concentrated on web page, the discussion runs on. On the other hand, it is also a different problem, because no input or output references between this page and other pages are considered. Only references inside this page between various discussants are considered. These references are represented by reactions on contributions. All mentioned methods (PageRank, HITS, SALSA, MozTrust, and Open Site Explorer) calculate authority of each web page separately. One page leads to one measure of authority. Within the authority mining from the web conversation, not only one but all contributions of the given discussant are evaluated. All information about all contributions related to one discussant has to be concentrated and used for the authority estimation. Nevertheless, we can inspire ourselves by these techniques and take into account the number of references as reactions on an actual contribution.
In our previous work [12], we have taken into account mentioned number of reactions on all contributions of evaluated discussant, but also the number of all contributions of this discussant, the number of reaction of the discussant on the bottom level of the conversation tree (Fig. 4), the polarity matching between opinion of the discussant and opinion of all discussion, the positions of contributions in the conversation tree and the length of his/her contributions. Some of these variables have appeared to be not so important for the precise estimation of the authority. Another problem of this approach was in way of the estimation function generation. For these reasons, we decided to modify the set of variables—arguments of the conversational structure and to use the regression methods for training the authority estimation function.
3 Used methods
3.1 Linear and nonlinear regression

\(y = b_{0} + b_{1}x\) (basic linear regression)

\(y = b_{0} + b_{1}x + b_{2}x^{2}\) (quadratic regression)

\(y = b_{0} + b_{1}x + {\cdots } + b_{r}x^{r}\) (polynomial regression)
3.2 Ordinary least squares method
3.3 Specification of the variables of a discussion structure

AE Average evaluation of the contribution.

K Value of the karma of the user, which is the contribution author.

NCH Number of characters within his/her contributions.

AL Average layer in the conversation tree (see Fig. 4).

ANR Average number of reactions on his/her contributions.

NC Number of contributions of given discussant.
Average evaluation of the contribution (AE) is represented by the ratio of the sum of all reactions [agree (\(+\)) and disagree (−)] on the contributions of given discussant to the number of all his contributions. This average evaluation is available on the web discussion page. The range of the AE is the number from 0 to 80.
Value of karma (K) of the discussant is also available on the discussion web page. The karma is a number from 0 to 200, which represents activity of the discussant from last 3 months (within the portal “http://www.sme.sk”).
Number of characters (NCH) represents the average length of discussant contributions. It penalized authors with too short and so less informative contributions. We assume that an authoritative contributor does not insert extremely short contributions.
Average layer (AL) in the conversation tree (see Fig. 4) is the average number of all layers, which the contributions of the discussant are situated in. The conversation tree is a graphical representation of the web discussion. The AL represents the information, when the discussant joined the discussion, from the beginning or at the end.
Average number of reactions (ANR) on the all contributions of the given discussant is the number of reactions per one his contribution.
Number of contributions (NC) is simply the whole number of contributions of the given discussant. The parameter NC penalizes authors, which join a discussion rarely.
All these parameters, taking separately, indicate rather chatty contributors than authoritative ones. However, taking them together as one entity, the emergency phenomenon arises. This phenomenon can indicate the authoritative contributors.
It may happen that a good contribution of already wellknown authority finishes the discussion on the Web. It is truth that in such a case, there is no reaction on this contribution. It does not disturb the measure of the authority, because of high probability that there were more previous contributions of this contributor with many reactions within the given discussion. These reactions can balance the lack of reactions on the finishing contribution.
 1.
Evaluation of each discussant by “human expert”;
 2.
Evaluation of each discussant by other discussants and it represents “wisdoms of the crowd.”
Average deviation of four versions of authority estimation function
Version  Average deviation 

LEXPERT  17.3489 
LCROWD  3.2998 
PLEXPERT  24.0123 
PLCROWD  8.7912 
NLEXPERT  18.1131 
NLCROWD  6.5618 
Values of precision and recall of six versions of regression functions were obtained in the threetime cross validation
Precision  Recall  

Test  Version  Expert  Crowd  Expert  Crowd 
Cross val. 12_3  Linear regression  0.78  0.99  0.69  0.99 
Polynomial regression  0.77  0.84  0.65  0.97  
Nonlinear regression  0.72  0.99  0.66  0.88  
Cross val. 13_2  Linear regression  0.65  0.98  0.65  0.93 
Polynomial regression  0.63  0.77  0.60  0.91  
Nonlinear regression  0.67  0.97  0.67  0.86  
Cross val. 23_1  Linear regression  0.68  0.97  0.67  0.67 
Polynomial regression  0.62  0.72  0.58  0.95  
Nonlinear regression  0.69  0.97  0.69  0.67  
Average  Linear regression  0.70  0.98  0.67  0.80 
Polynomial regression  0.67  0.78  0.61  0.94  
Nonlinear regression  0.67  0.97  0.67  0.80 
4 Implementation and testing
 1.Linear function learned from the “human expert” (LEXPERT) is represented by:$$\begin{aligned} A= & {} 0.4383\mathrm{AE} + 0.0746K + 0.0281\mathrm{NCH}  2.1932\mathrm{AL}\nonumber \\&\, 3.4386\mathrm{ANR} + 8.0102\mathrm{NC} \end{aligned}$$(7)
 2.Linear function learned from the “wisdoms of the crowd” (LCROWD) is represented by:$$\begin{aligned} A= & {} 0.4385\mathrm{AE} + 0.325K + 0.002\mathrm{NCH}  0.2928\mathrm{AL}\nonumber \\&\, 0.0853\mathrm{ANR} + 1.0728\mathrm{NC} \end{aligned}$$(8)
 3.Polynomial function learned from the “human expert” (PLEXPERT) is represented by:$$\begin{aligned} A= & {} 0.0001\mathrm{AE}^{3}{ 0.0004K}^{2} + 0.0303\mathrm{NCH}  1.5539\mathrm{AL}\nonumber \\&\,2.0557\mathrm{ANR} + 12.1589\mathrm{NC} \end{aligned}$$(9)
 4.Polynomial function learned from the “wisdoms of the crowd” (PLCROWD) is represented by:$$\begin{aligned} A= & {} 0.0001\mathrm{AE}^{3}{ 0.0009K}^{2}+ 0.0043\mathrm{NCH} + 0.7473\mathrm{AL}\nonumber \\&+\,1.9875\mathrm{ANR} + 6.7507\mathrm{NC} \end{aligned}$$(10)
 5.Nonlinear function learned from the “human expert” (NLEXPERT) is represented by:$$\begin{aligned} A= & {} 0.0382\mathrm{AE}^{1,7192}{ 0.3295K}^{0,959}{ + 0.4470\mathrm{NCH}}^{0,681}\nonumber \\&{+\,0.1825\mathrm{AL}}^{0,0001}{ 0.6269\mathrm{ANR}}^{3,2394}{ + 20.2509\mathrm{NC}}^{0,2977}\nonumber \\ \end{aligned}$$(11)
 6.Nonlinear function learned from the “wisdoms of the crowd” (NLCROWD) is represented by:$$\begin{aligned} A= & {} 0.0185\mathrm{AE}^{1,8135}{+141.5704K}^{78,39}{ + 0.0018\mathrm{NCH}}^{1,0457}\nonumber \\&{\, 0.0011\mathrm{AL}}^{3,7717}{ 0.5562\mathrm{ANR}}^{0,0001}{+ 37.6642\mathrm{NC}}^{0,0038}\nonumber \\ \end{aligned}$$(12)
All the versions of the regression function for authority estimation (from (7) to (12)) were tested. The concise results of these tests are shown in Tables 1 and 2.
Some key and the most important achieved results of tests are presented in Table 2.
The linear regression learned from the “crowd,” with the best test results, was implemented in the Application for the Machine Authority Identification (AMAI). This application provides the list of all discussants with the actual value of their authority. The AMAI also displays the value of the authority of the discussant, which was selected by a user. This value is from the interval \(\left<0, 100\right>\). The application provides not only the binary decision whether the discussant is or is not the authority, but also it provides a precise numeric value of its authority.
5 Conclusions
The design of solving the problem of the authority identification from conversational content using the linear and nonlinear regression was presented. The measure of the authority A was estimated as dependency on variables (AE, K, NCH, AL, ANR, and NC)—parameters of the structure and content of given web discussions. Another parameter which could be relevant is the vocabulary of the discussant—the literary language with scientific concepts can determine the level of the discussant and thus indicate the authority of the writer. In addition, the type of emoticons used in the discussion could be helpful. On the contrary, the dirty language may reflect a low level of the discussant. This language can be identified using some prepared special dictionary of dirty words. We would like to involve these parameters in the future.
The six generated estimation functions were tested. According to the values of average deviations (see Table 1), the best solution is the linear function learned from crowd (LCROWD). The second one is the nonlinear function learned for crowd (NLCROWD). Linear and nonlinear functions learned from a single human evaluator—expert—seem to be worse. The same conclusions can be deduced from the resulting average values of precision and recall in Table 2. It is surprising that the linear model is better than the higher order model. It can be caused by a character of input data—parameters of the web discussion. Together with an increasing of values of these parameters, also the value of authority increases. Therefore, linear model is sufficient for the authority estimation.
It can be hardly said who is the expert on the authority identification. In addition, an opinion of a psychologist may be also subjective. On the other hand, combined opinion of many discussants can be objective.
There are other existing authority identification methods, as Klout, TwentyFeet, My Web Carrer [14], and our previous work [12]. All these methods use formulas for authority estimation, but these formulas were generated more experimentally without considering a theoretically based way. For this reason, we tried to generate the relation between the authority and the structure of web discussion using the classic mathematical approach based on the linear and nonlinear regressions. For the future, we plan to elicit the constants of linear and nonlinear equations using evolutionary algorithms [15, 16] to calculate not only constant values but the form of a nonlinear regression function as well.
The presented approach can be used also in weighted opinion analysis of some discussions on social networks. Within the classic opinion analysis, the whole discussion is recognized to be positive (or negative) when there are more positive (or negative) contributions within it and each contribution has the same weight within determination of the summarized opinion. The weighted opinion analysis could multiply the measure of positivity of a given contribution with the weight represented by the estimated authority value of the contributor, who is the author of the given contribution. Thus, the opinions of authoritative contributors would have greater influence on the summarized opinion. We would like to apply the weighted opinion analysis in a domain of recognizing personality aberration from written text [17]. The designed approach and its implementation can be used to solve the problem of the decreasing of a web user cognitive load [18].
Notes
Acknowledgments
The work presented in this paper was supported by the Slovak Grant Agency of the Ministry of Education and Academy of Science of the Slovak Republic under VEGA Grant No. 1/0493/16.
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