# Evolution of Experts in Question Answering Communities

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6 BIC Criteria 2.5 x Number of Clusters Figure 6: BIC information criteria over the Stackoverflow dataset. this section, we systematically explore the different evolution patterns exhibited by experts and also explore models that can utilize expert evolution towards the task of expert identification. Expert Evolution Analysis In this section, we explore the different evolutional patterns exhibited by the experts. To do this, we consider the relative time series of the number of answers of an experts. Let the time series of i th expert be x i. Let X = {x 1,..., x N } for N experts to be independently and identically distributed (i.i.d). We tried to preprocess this data using the Haar and DB3 Wavelet transformations as these methods allows us to get a noise free version of the data, which might be more robust, but did not find any significant difference in comparison to the relative time series. We use Gaussian Mixture Model based clustering algorithm to find the clusters amongst N time series in X. We can write the likelihood of the observed data series as follows: N K P (X θ) = π ik P (x i θ k ) (4) i=1 k=1 where π ik is the probability of x i belonging to cluster k. The two sufficient conditions for this to be a probability distributions is that 0 π ik 1 and k π ik = 1. The likelihood of the data can then be written as, ln(p (X θ)) N i=1 k=1 K π ik ln(p (x i θ k )) (5) where we used Jensen s inequality to get the above lower bound. Now consider the data likelihood distribution to be a multivariate Gaussian distribution, defined as follows P (x i θ k ) 1 Σ k 1 2 exp{ 1 2 (x i µ k ) T Σ 1 k (x i µ k )} (6) where θ k = {µ k, Σ k } are the mean and covariance parameters of the k th cluster. The benefit of GMM clustering over a hard clustering algorithm such as KMeans is that in our setting there might be correlations between x b i and xb+1 i, i.e. number of answers during subsequent buckets for a user. These correlations are automatically captured by the Σ parameter in GMM, whereas KMeans considers them to be independent. To run KMeans, we need to orient the data using Singular Value Decomposition into a space where the correlations are minimized. An additional benefit of GMM is that it allows us to use Bayesian Information Criteria to automatically estimate the number of clusters in the dataset. This saves us from making arbitrary choice on the number of clusters. So we first estimate the number of clusters and then analyze the shapes and sizes of those clusters. Number of Expert Evolutional Patterns In order to estimate the number of evolutional patterns, we use the Bayesian Information Criteria (BIC). BIC has been shown to work successfully for large CQA datasets to automatically find the number of users that should be labeled as experts (Bouguessa, Dumoulin, and Wang 2008). In our setting, BIC is used to find how many different clusters exist in the observed temporal data. BIC(K) = 2 ln(p (X θ)) + K ln(n) (7) Without BIC criteria, we can see that setting K = N maximizes the data likelihood in equation 5 which leads to K D (D +3)/2 O(N) model parameters and every user lying in a different cluster. Figure 6 shows the BIC curve as a function of K for the Stackoverflow dataset. We pick K = 6 as it minimizes the BIC criteria in almost all the runs of the GMM algorithm. Expert Evolutional Patterns We run GMM with 6 clusters and aggregate the mean series of users in each cluster to compute the cluster aggregate series. Figure 7 shows the aggregate time series of number of answer for different clusters found by GMM. The 6 clusters contained roughly equal number of experts. We see three dominant patterns amongst the clusters. The clusters exhibiting these patterns are labeled as C, E, L in the figure 7. Cluster C consists of users who were consistently active in the community. On the other hand, users in cluster E were initially very active and later became dormant. Whereas users in cluster L were initially passive but later became very active in the community. The other three clusters were variant of the three dominant patterns with small amplitudes. The cluster output suggests that indeed there are three kinds of experts in the community even though they might look similar in terms of their overall contributions. We argue that for question routing schemes the experts in C are more valuable than in E, L. Between E and L it can depend on how much time has the user spent in the community. Additionally, we argue that the identification of these different kinds of experts can help in providing different measures to retain the experts.

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