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Epidemic model for information diffusion in web forums: experiments in marketing exchange and political dialog
 Jiyoung Woo^{1}Email authorView ORCID ID profile and
 Hsinchun Chen^{2}
 Received: 21 September 2015
 Accepted: 6 January 2016
 Published: 22 January 2016
Abstract
As social media has become more prevalent, its influence on business, politics, and society has become significant. Due to easy access and interaction between large numbers of users, information diffuses in an epidemic style on the web. Understanding the mechanisms of information diffusion through these new publication methods is important for political and marketing purposes. Among social media, web forums, where people in online communities disseminate and receive information, provide a good environment for examining information diffusion. In this paper, we model topic diffusion in web forums using the epidemiology model, the susceptibleinfectedrecovered (SIR) model, frequently used in previous research to analyze both disease outbreaks and knowledge diffusion. The model was evaluated on a large longitudinal dataset from the web forum of a major retail company and from a general political discussion forum. The fitting results showed that the SIR model is a plausible model to describe the diffusion process of a topic. This research shows that epidemic models can expand their application areas to topic discussion on the web, particularly social media such as web forums.
Keywords
 Information diffusion
 Epidemic model
 Contagion
 Web forum
 Social media
Background
Social media such as blogs, discussion forums, and social networking sites provide new channels for individuals to share information and express their opinions. The characteristics of social media, such as rich representation, low cost, easy accessibility, and rich user interaction, have created new opportunities for marketers and politicians to leverage social media for their businesses. The prevalence of social media enriches information that people share and accelerates its diffusion between them. The information diffusion process is a successive result by which people influence one another over a time period (Kleinberg 2008). The social interaction on the web has become a new source of information diffusion, which was only available to traditional mass media in the past.
As the influence of social media becomes more evident, understanding the mechanisms and properties of information diffusion through these new publication methods is important for political and marketing purposes. The wordofmouth propagation through blogs, email, and product review forums has been studied for marketing purposes. Researchers have also studied how political messages diffuse on the web through personal blogs or informationsharing websites. However, few studies have focused on the more restructured and spikey interactions expressed in public web forums. Web forums are important and popular for marketing exchange and political dialog. Unlike blog or email networks that are dominated by a few bloggers or known acquaintances, web forums allow opinions to be freely formed and spread in society. Anyone can begin a new thread of discussion and anyone can participate freely and equally. People who have common interests express and discuss their opinions and affect each other. Among all social media, web forums are promising for modeling information diffusion. In this article, we propose a new extension of the SIR model for information diffusion on web forums. Our design expands significantly from the baseline SIR epidemic model for information diffusion. This paper is organized as follows. “Related work” section summarizes previous research on diffusion modeling, focusing on information diffusion, and presents previous studies that support opinion contagiousness. In “Information diffusion model in web forums” section, we present the SIR model, develop the analogy between the epidemics and topic diffusion in the web forum and propose a new extension of the SIR model in the web forum. “System design: SIR for web forums (the SIRW system)” section presents the system design of diffusion modeling and elaborate the system components. Experiment results are reported in “Experiment result” section. Discussion including pros and cons of this research and conclusions and future directions are presented in “Discussions” and “Conclusions” sections.
Related work
Diffusion refers the process whereby a phenomenon of interest (e.g., information, innovation, or disease) spreads from one to another (Cliff and Haggett 2005). Related to human, diffusion is a long history of the research topic in epidemiology and sociology domains. Rich literature deals with information diffusion between people. As the social media became an everyday communication channel between people, various online social networks are formed, and they reflect the realworld social network. As the social network draws much attention from researchers, the diffusion process on the online social network became an ongoing research topic. Due to similar patterns in the spread of epidemics and social contagion processes, most research adopts the same theoretical principles for epidemics in describing the information diffusion. The mainstream theory that explains the epidemic defines the disease diffusion as the spread of memes of infection (Blackmore 2000). Thus, most epidemic models are mainly based on the contagion through the interaction between people. Microlevel epidemic diffusion models firstly set the population structure and build nonlinear differential equations that describe the status of change in the population class. These microlevel models are named as equationbased models. Equationbased models (EBMs) operate on global laws defined by the equations and applied to all members of the compartment. The underlying assumption of EBMs is that the population is homogeneous and is governed by holistic rules. They assume that people have a constant contact rate and are infected by a disease that has a unique transmission rate, and recover at a certain rate. The stochastic model uses the concept of independent and identically distributed objects, but it still describes the diffusion process with holistic rules (Bobashev et al. 2007). Using diffusion models, we can understand how new diseases, information, or products spreads, to predict their success or failure in the early stages, and to increase or reduce the chances of diffusion. Early stage models do not reflect the social network underlying in the population. In some disease, this assumption fits well since the disease can diffuse through indirect interaction as well as direct contact. The model that assumes the homogeneous mixing between individuals, in other words, random contact is named the population model. The population model divides a population into classes that reflect the status of individuals in the population. The networkbased model considers the network in which diffusion happens and focuses on the effects of network properties in the diffusion process. Diffusion studies have been supported by recent findings from realworld networks, including social networks and their topological features (Barabási and Albert 1999). The SIR model (Kermack and McKendrick 1927), a representative epidemic model, has three compartments of susceptible, infective, and recovered. The model expresses the status change of three compartments using differential equations. The independent cascade model (ICM) (Goldenberg et al. 2001), which is widely adopted in describing the information diffusion on online social networks, is a special case of the SIR model reflecting the network structure of the population. The first study on information diffusion modeling using epidemic models has been made using the study on the spread of scientific ideas. Goffman and Newill (1964) developed the analogy between the adoption of scientific information and the spread of infectious disease. Bettencourt et al. (2006, 2008) developed Goffman and Newill’s (1964) work by proposing the competency model that describes the diffusion process of two competing theories. Epidemic models were also tested to rumor propagation (Kawachi 2008). Epidemic models have also been used for modeling financial information diffusion. Fan (1985) proposed ideodynamics model that embeds people’s contact and content characteristics. Fan and Cook (2003) added the sentiment of mass media content to model consumer sentiment about the economy. Shive (2010) modified the SIR model to predict the buying and selling of a stock by adding situational determinants such as total trade amount, return on investment, and income level to the social interaction. Shtatland and Shtatland (2008) approximated the SIR model into a firstorder autoregressive model for the financial outbreak. From a marketing perspective, online wordofmouth has become a new topic of diffusion modeling. Goldenberg et al. (2001) studied the underlying process of wordofmouth on the network. Song et al. (2007) proposed the ratebased information flow model using the Markov chain and applied it to recommendation systems. Bampo et al. (2008) applied the SIR model to various ideal networks to measure the efficiency of email marketing campaigns. As online information diffusion has become a major topic for diffusion studies due to the growth of email, the web, and social media, epidemic models have been applied to modeling of information diffusion on the web. New attempts to apply diffusion models to email, blogs, and forums have emerged. Many studies examine information diffusion in the blogsphere. Gruhl et al. (2004) defined the characteristics of diffusing topics in the blog and proposed a method for estimating the transmission probability for ICMs. Saito et al. (2008) used expectation maximization to estimate the transmission probability for ICMs. Leskovec et al. (2007) suggested the cascade generation model under the SIS framework with fixed transmission probability.
The research on web forums differs from that on blogs in that it focuses on the diffusion vector such as topics, news, and documents. Kubo et al. (2007) showed the analogy between the disease propagation model, the SIR model, and posting data in web forums. Woo et al. (2011) adopted the SIR model to model violent topic diffusion in the Jihadi forum. Woo and Chen (2012) extended Kubo’s work incorporating the new media effects. They focused on how new media effects can be reflected in the SIR model. Kubo et al. (2007), Woo et al. (2011) and Woo and Chen (2012) performed modeling the post dynamics with the SIR model and without the sound analogy between information diffusion and epidemic, logical arguments and interpretation of the proposed model. These works viewed the information diffusion in postlevel not authorlevel that is more appropriate for modeling the authors interaction and thereby contagion of a topic. The postlevel works consider the post as the carrier and itself as the result of diffusion, this results that susceptible class does not explain clearly. To overcome these shortcomings, we develop the sound analogy between information diffusion in the web forum and epidemic model, and propose the system design to examine the information diffusion in the web forum.
Previous research on information diffusion
Key papers  Model specification  Applications  Contributions 

Goffman and Newill (1964)  SIR, SIS  Scientific theory  The first analogy development between information and disease diffusion 
Kawachi (2008)  SIRvariants  Rumor  The novel model with offsetting effect 
Fan (1985)  SIR  Financial information  The novel model with content characteristics ideodynamics model 
Shive (2010)  SIR  WOM of stock  Novel model with corporate financial information 
Shtatland and Shtatland (2008)  SIR  Financial information  Outbreak detection using the diffusion model 
Goldenberg et al. (2001)  SIR  Word of mouth (WOM)  The network effects on WOM 
Bampo et al. (2008)  ICM  WOM  The network effects on WOM 
Gruhl et al. (2004)  ICM  Blog  The empirical test 
Saito et al. (2008)  ICM  The method to estimate infection rate  
Leskovec et al. (2007)  Network SIS  Blog  The empirical test 
Kubo et al. (2007)  SIR  Web forum  The analogy development between topic diffusion in the web forum and disease spread 
Toole et al. (2012)  Network SIS  The novel model with geolocation information, the empirical test  
Myers et al. (2012)  ICM  The novel model with external effect  
Tang et al. (2014)  Network SI  Chinese Twitter  The empirical test 
Liu and Zhang (2014)  ICM  Syntactic data  The novel model with rewiring friendship 
Wang et al. (2015)  Network SI  The novel model, emotionbased spreaderignorantstifler (ESIS) model 
Information diffusion model in web forums
SIR model in web forums
The analogy between epidemics and topic diffusion in the web forum
Elements of SIR model  Epidemics  Topic diffusion in web forums 

What flows  Disease  Idea/topic (keywords) 
Susceptible: S(t)  People who can have contact with an infective and possibly will become infected  Possible authors (including commenters) who might read posts on a topic 
Infective: I(t)  People who have a disease and possibly will infect others  Current authors who write posts on a topic 
Recovered: R(t)  People who recover from a disease and lose the power to infect others  Past authors whose posts lose influence toward others 
Infection rate: α  The probability of transmission in a contact between an infective and a susceptible  The probability of writing a comment or thread after reading posts on the topic 
Recovery rate: β  The probability that the infective becomes recovered  The probability that posts lose infectivity 
Recruitment rate: μ  The proportional increase rate of the population  The proportional increase rate of author pools 
Carrying capacity: K  The maximum population that the environment can support  The highest value of the total authors that a topic can recruit 
Mathematical formulation
Equation (5) is the objective function for the iterative parameter estimation. I(t) is an observation variable, in our case, it is the number of authors who participated in the discussion on a topic. The parameter set is composed of \(\alpha\), \(\beta\), and \(\mu\), and they are optimized to minimize the above objective function. The parameters should be nonnegative and be <1 except \(\mu\). The initial condition of S(0), the author pool, K, are estimated. S(t), I(t), and R(t) are iteratively updated following Eqs. (2)–(4).
System design: SIR for web forums (the SIRW system)
Data collection
Topic extraction
To derive keywords, we performed topic clustering using a probabilistic topic model, especially latent Dirichlet allocation (LDA) (Blei et al. 2003). A topic modeling technique is designed to automatically uncover thematic structure in a large collection of unstructured text (McCallum 2002). According to topic modeling, a document is defined as a mixture of various topics. A topic is defined as a set of words that frequently occurred together. The probabilistic model assumes that documents are mixtures of topics, where a topic is a probability distribution over words. The probabilistic topic model works to find out the best set of words and to explain the shownup words in documents. The probabilistic topic model specifies a simple probabilistic procedure by which documents can be generated. To make a new document, one chooses a distribution over topics. Then, for each word in that document, one chooses a topic at random according to this distribution and draws a word from that topic. This process is inverted using statistical methods inferring the set of topics that were responsible for generating a collection of documents (Blei et al. 2003). The adopted probabilistic model clusters messages according to probable keywords. In LDA, the topic distribution is assumed to have a Dirichlet prior that is often used as prior distributions in Bayesian statistics. Dirichlet priors make the statistical inference simple and output more reasonable mixtures of topics in a document (Steyvers and Griffiths 2007).
The algorithm works in an iterative way as follows. First, we set the number of topics. Then algorithm assigns the every word to a temporary topic according to a probability distribution. Since each word is assigned in a somewhat random manner, a word shown up more than once in different documents can be assigned to different topics. Then the algorithm takes an iterative topic assignment as follows. A word is assigned to the topic where this word is the most prevalent and a document is assigned to a topic where the words in the document are mostly assigned. After the iteration converges, topic modeling is done. For topic clustering, we used Mallet (McCallum 2002) that implements LDA for largescaled unstructured data. In Mallet, we need to address the semantic meaning of topics by considering the words in the topic. When the number of topics is set to too low, words in a topic are general and a topic includes the words that are not close semantically. When the number of topics is set to too high, words in a topic become too specific, and words distribute to various topics. This causes semantic overlapping of topics. Thus, we varied the number of topics and examined whether topic modeling generates proper semantic clusters. We incrementally set the number of topics by ten and observed the semantic generation.
We defined key topic as the topic cluster that has a significant volume of posts and authors, is discussed actively at current and is composed of meaningful keywords. We cluster messages in a large number of classes and select meaningful topic clusters with keywords that reflect user needs. We selected bigrams from extracted keywords to derive topics that are meaningful and have significant volume.
Timeseries patterns
In the timeseries pattern derivation module, the numbers of distinct authors are derived by aggregating postings that include a topic in a time frame. According to Gruhl et al. (2004), topics are classified into chatter topics and spikey topics. The chatter topic is defined the topic of ongoing discussions whose subtopic flow is largely determined by decisions of the authors. The spikey topics are highintensity discussions of realworld events that are relevant to the topic (Gruhl et al. 2004). The spikey topic induces sharp rises in postings. They looked at topic occurrences and defined a spike as an area where the posts on a given day exceeded \(\mu + 2\sigma\) (Gruhl et al. 2004). We followed the same manner. We analyzed the timeseries patterns of key topics to exclude chatter topics. Chatter topics with ongoing patterns were also excluded from the analysis because topics without epidemic patterns were not considered to be contagious and cause contagion between users. Spiky topics, which are topics that draw interest, i.e., are infectious, are then selected. In the next step, interaction rules, observation variable, estimation variable, objective function, and parameters to be estimated are defined as mentioned in “Information diffusion model in web forums” section.
Model fitting
In the model fitting step, data are tested for the model using a userdefined optimization algorithm. The Genetic Algorithm (GA) is employed as the optimization algorithm for parameter estimation. The wide range of methods to implement GA has been developed. We outlined the major procedures and selected algorithm in each procedure. First, a fitness function that indicates how well the current population fits the objective function is determined using the linearranking algorithm of Baker (1987). The fitness function affects the population selection. To reproduce the population in each generation, the selection method that extracts chromosomes from population should be fixed. In this work, we used roulette wheel selection (Golberg 1989). The crossover routines recombine pairs of individuals with given probability to produce offspring. Singlepoint (Booker 1987) is used. For mutation operation, realvalue mutation (Mühlenbein and SchlierkampVoosen 1993) is adopted.
Experiment results

Yahoo! Finance Walmart message board (January 1999–June 2008; 139,062 threads; 441,954 messages; 25,500 authors).

US Politics Online Breaking News in Politics (May 2005–March 2010; 2192 threads; 130,850 messages; 1124 authors).
Marketing exchange: Walmart forum
The major topics and keywords in the Walmart forum
Topic group  Topic  Keywords 

Investor  Stock price  Growth, share, earnings, price, stock, market 
Sales  Sales, percent, quarter, increase, fiscal, earnings, expected, results  
Customer  Low price  Prices, low, economy, consumer, cost, market 
Shopping convenience  Shopping, items, manager, shoppers, service, line, door, experience  
Employee  Healthcare  Healthcare, employees, insurance, medical, plan 
Labor law  Labor, illegal, federal, laws, violations, rights  
Wage  Pay, wages, benefits, employees, hour, working paid average hours, minimum, poverty, paying 
Parameter estimation results on the Walmart forum
Topic  MSE  \(R^{2}\)  S(0)  \(\alpha\)  \(\beta\)  \(\mu\)  K 

Stock price  5.28E+03  0.6198  163  0.0045  0.6798  0.1226  1384 
Sales  2.72E+03  0.6320  100  0.0081  0.7270  0.1388  997 
Low price  3.64E+03  0.7262  122  0.0059  0.7506  0.1419  1401 
Shopping convenience  1.98E+03  0.6433  116  0.0078  0.7914  0.1230  1000 
Healthcare  3.83E+03  0.7190  116  0.0065  0.7677  0.1361  1200 
Labor law  1.16E+03  0.7510  89  0.0088  0.7433  0.1324  800 
Wage  6.55E+03  0.5209  100  0.0053  0.6000  0.1524  950 
Political dialog: political web forum
The major topics and keywords in the US Politics Online forum
Topic group  Topic  Keywords 

International issue  Nuclear weapon  Iran, nuclear, weapons, United States, Ahmadinejad, Russia 
Iraq war  Iraq, war, troops, Iraqi, military, forces, security, government  
Domestic issue  Healthcare bill  Tax, healthcare, plan, pay, cost, insurance, income, program 
Election issue  McCain  McCain, campaign, Palin, John, Governor, Presidential, Sarah 
Obama  Obama, president, Barack, presidential 
Parameter estimation results on the political forum
Topic  MSE  \(R^{2}\)  S(0)  \(\alpha\)  \(\beta\)  \(\mu\)  K 

Nuclear weapon  9.90E+03  0.4379  142.9  0.0076  0.9500  0.2670  931.8 
Iraq war  9.27E+03  0.4739  166.4  0.0062  0.9115  0.2565  861.4 
Healthcare bill  6.81E+02  0.5761  21.7  0.0180  0.5696  0.2995  208.7 
Barack Obama  8.65E+03  0.7929  67.2  0.0039  0.2212  0.0937  1022.7 
John McCain  8.20E+03  0.7190  140.4  0.0034  0.8078  0.2232  709.3 
Discussions
In this work, we tried to find out the underlying mechanism in the occurrence of the spiky discussion on a specific topic. Previous research that addressed the idea and opinion contagiousness formed a basis of this research. In a web forum, we could derive following observations. Users in the web forum react to others posts. It implies that users interact with each other through posts. Thus, we adopt the disease diffusion model that explains the disease outbreak through the contact between people. The aim of the diffusion model is to understand the mechanisms of the spread of new diseases, ideas and products, to predict success or failure of diffusion in the early stages, and to design strategy to increase or reduce the chances of diffusion. The timeseries model also has an aim to forecasting future trend with current data; the timeseries model is mainly based on the identification of common patterns of consecutive data points. While the timeseries model is the absence of reasoning on diffusion process except examining data occurrence patterns, the diffusion model is based on the fact that diffusion happens mainly due to user interaction. In this work, we tested the feasibility of a baseline epidemic model to describing the topic diffusion in the web forum. Through the empirical tests, we presented the coefficient value, Rsquared that indicates how well data fit a model and this is widely adopted in timeseries modeling. Since the mathematical model, especially the deterministic model, simplifies the diffusion process, it does not provide a complete analysis on that. The purpose of the mathematical model is the description of the diffusion process and not a complete analysis. We also simplified the topic diffusion process without consideration forum characteristics except the contagion between people. This simplification enables us to build a theoretical model. The mathematical model does not provide individuallevel knowledge such as who will be infected by a topic or when a user will be infected. However, the usefulness of a mathematical model is to obtain systemlevel measurements and test hypotheses using them. According to the estimation results, we found that the SIR model is a plausible model for the topic diffusion in the web forum. For major topics, we can say that 43 % of variances in timeseries patterns are explained by the diffusion model at least because the lowest Rsquare value is 0.43. All topics that we tested exceed the lower bound of the moderate range. Thus, we claim that the topic diffuses among authors mainly by the interaction between them and thereby, it causes a peak of author participations. Heeler and Hustad (1980) addressed that the soundness of structural test guarantees the forecasting validity of the model. Colbaugh and Glass (2009) proved that very early dispersion of a diffusion process across network communities is a reliable early indicator that the diffusion will ultimately involve a substantial number of individuals with case studies involving emergence of the Swedish Social Democratic Party at the turn of the twentieth century, the spread of SARS in 2002–2003, and blogging dynamics.
In this work, we just showed the structural soundness of the baseline epidemic model over the topic diffusion in the web forum. However, after testing structural soundness, we can perform forecasting. Additionally, we can estimate how many authors have latent interest on each topic at the initial phase of the diffusion process. We can also estimate the expected duration and the intensity of diffusion process at an initial stage. Even though these estimation measurements become reliable when it closes to the peak, we can estimate them in an adaptive way. We can compare those values for topics that lead the peak. We also can predict the outbreak of topics through the examination on initial stage of diffusion process using the following broad principal of the mathematical epidemic model. If the average number of secondary infections caused by an average infective is <1 a disease will die out, while if it exceeds one there will be an epidemic (Diekmann et al. 1990). In the case when the diffusion process follows an epidemic model, this threshold behavior enables us to estimate the likelihood of an outbreak, which is a peak of the diffusion process. The most important usefulness of building a mathematical model is that we can incorporate the forum characteristics into the diffusion model. We can perform further investigation on other factors that may affect the diffusion dynamics by incorporating those factors into the model. For instance, the sentiment of posts may determine the infectivity of a topic. Then, we can design the model with the infective rate varying depending on the cumulated sentiment score of posts. Another possible hypothesis is that there exists the preydecay mechanism in the web forum. It is a common sense that when newcoming influencing news come out, old news die out. We expect that same phenomena may occur in the web forum. When an influencing topic emerges, people’s interest may stop participate in a previous topic. The competing relationship between a new emerging topic and a previous topic can be incorporated into the proposed baseline model to test the above hypotheses. For a marketing purpose of this research, the marketer can use this model by identifying key words related to the marketing campaign. Using this model, the marketer can predict the outbreak and dieout of a marketing campaign and how long it lasts when it diffuses by examining the initial patterns of diffusion process. In a case when a new marketing campaign comes out, the market can find a similar one from previous campaigns and apply the parameters of the similar one for forecasting success or failure of a new marketing campaign. For a political perspective, the politician can examine which politicalrelated topics are diffused contagiously. For example, during an election, based on the current diffusion patterns of key words related to candidates, we can infer who will win the election. Our work has following limitations. This system level model does not provide specific information about who will be infected. It just gives aggregated information how many users will react to a topic. Second, even though, we can use this model for forecasting for an emerging topic based on diffusion process of similar topics in past, but the reliability of forecasting becomes reliable when the point of forecasting reaches to the peak. Furthermore, identifying similar topics based on semantic similarity is not trivial and a new topic can generate totally different diffusion process from past topics even if it is similar to them semantically.
Conclusions
We proposed an integrated and novel methodology to model opinion/idea diffusion in web forums. The SIR model, frequently used in previous research to analyze both disease outbreaks and knowledge diffusion, is adopted for the web forum. The model was evaluated on a large longitudinal dataset from the web forum of a major retail company and a dataset from a general political discussion forum. The experiment results revealed that the SIR model performed well in modeling topic diffusion in web forums. This research has two contributions. We extended the information diffusion research to a new domain: web forums. We also examined the possibility of applying the epidemic model to topic diffusion in web forums. For future research, we plan to apply an epidemic model to sentiment diffusion in web forums. The epidemic model with the two classes of positive and negative opinion would be fit to the sentiment diffusion. Consideration of the competency and interaction between positive opinion and negative opinion on a topic would also improve the modeling accuracy.
Declarations
Authors’ contributions
Conceived and designed the experiments: JW and HC. Performed the experiments: JW. Analyzed the data: JW. Wrote the paper: JW, HC. Both authors read and approved the final manuscript.
Acknowledgements
This work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF2009352D00329 and NRF2013R1A1A3011816).
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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