Characterization of topic-based online communities by combining network data and user generated content

Abstract

This study proposes a model for characterizing online communities by combining two types of data: network data and user-generated-content (UGC). The existing models for detecting the community structure of a network employ only network information. However, not all people connected in a network share the same interests. For instance, even if students belong to the same community of “school,” they may have various hobbies such as music, books, or sports. Hence, it is more realistic and beneficial for companies to identify communities according to their interests uncovered by their communications on social media. In addition, people may belong to multiple communities such as family, work, and online friends. Our model explores multiple overlapping communities according to their topics identified using two types of data jointly. By way of validating the main features of the proposed model, our simulation study shows that the model correctly identifies the community structure that could not be found without considering both network data and UGC. Furthermore, an empirical analysis using Twitter data clarifies that our model can find realistic and meaningful community structures from large online networks and has a good predictive performance.

Appendix

1.1 Appendix 1: Derivation of the collapsed Gibbs sampler for the proposed model

In Sect. 3.2, we derived the conditional posterior distributions of latent variables (Equations (4) and (5)). To derive these posteriors, we need the full conditional posterior distributions for model parameters, and these are given as follows:

$$\begin{aligned}&P(\eta _i | S, R, X, \gamma ) \nonumber \\&\quad = \frac{\varGamma \left( \sum _k N_{ik} + M_{ik} + \gamma _k \right) }{\prod _k \varGamma (N_{ik} + M_{ik} + \gamma _k)} \prod _{k=1}^{K} \eta _{ik}^{N_{ik} + M_{ik} + \gamma _k} \end{aligned}$$

(7)

$$\begin{aligned}&P(\psi _{kk'} | A, S, R, \delta , \epsilon ) \nonumber \\&\quad = \frac{\varGamma (n_{kk'}^{(+)} + n_{kk'}^{(-)} + \delta _{kk'} + \epsilon _{kk'})}{\varGamma (n_{kk'}^{(+)} + \delta _{kk'}) \varGamma (n_{kk'}^{(-)} + \epsilon _{kk'})} \nonumber \\&\qquad \times \psi _{kk'}^{{\mathbb {I}}(a_{ij}=1)} (1 - \psi _{kk'})^{{\mathbb {I}}(a_{ij}=0)} \end{aligned}$$

(8)

$$\begin{aligned}&P(\theta _k | X, Z, \alpha ) = \frac{\varGamma \left( \sum _l M_{kl} + \alpha _l \right) }{\varPi _l \varGamma (M_{kl} + \alpha _l)} \prod _{l=1}^{L} \theta _{kl}^{M_{kl} + \alpha _l} \end{aligned}$$

(9)

$$\begin{aligned}&P(\phi _l | W, Z, \beta ) = \frac{\varGamma \left( \sum _v M_{lv} + \beta _v \right) }{\varPi _v \varGamma (M_{lv} + \beta _v)} \prod _{v=1}^{V} \phi _{lv}^{M_{lv} + \beta _v}, \end{aligned}$$

(10)

where $N_{ik}$ is the count number of when node i is assigned community k on the edges from node i to other nodes and from other nodes to node i. $M_{ik}$ is the count number of when words in node i’s document are assigned to community k. $n_{kk'}^{(+)}$$(n_{kk'}^{(-)})$ is the number of links (non-links) from nodes in community k to nodes in community $k'$. $M_{kl}$ is the count number of when words are assigned to community k and topic l. $M_{lv}$ is the count number of when word v is assigned to topic l. $\varGamma$ is the gamma function, and ${\mathbb {I}}$ is the indicator function that returns 1, if the condition is satisfied, and 0 otherwise.

Collapsed Gibbs sampling repeats the sampling procedure according to Equations (4) and (5). The pseudo algorithm for the proposed model is provided in algorithm 1.

1.2 Appendix 2: Definition of WAIC for proposed model

The definition of WAIC for our model is as follows:

$$\begin{aligned} lpd^{(i)}&= \log \left( \frac{1}{G} \sum _{g=b+1}^{G} \prod _{j=1}^{D} P\left( a_{ij} | H^{(g)}, \varPsi ^{(g)}\right) \right. \nonumber \\&\quad \left. \prod _{m=1}^{M_i} P\left( w_{im} | H^{(g)}, \varTheta ^{(g)}, \varPhi ^{(g)}\right) \right) \end{aligned}$$

(11)

$$\begin{aligned} P_{waic}^{(i)}&= \frac{G}{G-1} \left( \frac{1}{G} \sum _{g=b+1}^{G} \left( \sum _{j=1}^{D} \log P\left( a_{ij} | H^{(g)}, \varPsi ^{(g)}\right) ^2 \right. \right. \nonumber \\&\quad \left. + \sum _{m=1}^{M_i} \log P\left( w_{im} | H^{(g)}, \varTheta ^{(g)}, \varPhi ^{(g)}\right) ^2\right) \nonumber \\&\quad - \left( \frac{1}{G} \sum _{g=b+1}^{G} \left( \sum _{j=1}^{D} \log P\left( a_{ij} | H^{(g)}, \varPsi ^{(g)}\right) \right. \right. \nonumber \\&\quad \left. \left. \left. + \sum _{m=1}^{M_i} \log P\left( w_{im} | H^{(g)}, \varTheta ^{(g)}, \varPhi ^{(g)}\right) \right) \right) ^2 \right) \end{aligned}$$

(12)

$$\begin{aligned} WAIC&= -2 \sum _{i=1}^{D} \left( lpd^{(i)} - P_{waic}^{(i)} \right) , \end{aligned}$$

(13)

where $P\left( a_{ij} | H^{(g)}, \varPsi ^{(g)}\right)$ and $P\left( w_{im} | H^{(g)}, \varTheta ^{(g)}, \varPhi ^{(g)}\right)$ are the model likelihood conditioned with the parameters estimated using samples at sth iteration

$$\begin{aligned}&P\left( a_{ij} | H^{(g)}, \varPsi ^{(g)}\right) \nonumber \\&\quad = \sum _{k=1}^{K} \sum _{k'=1}^{K} \eta _{ik} \cdot \eta _{jk'}^{(g)} \cdot \psi _{kk'}^{(g) {\mathbb {I}}(a_{ij}=1)} \cdot \left( 1 - \psi _{kk'}\right) ^{(g) {\mathbb {I}}(a_{ij}=0)} \end{aligned}$$

(14)

$$\begin{aligned}&P\left( w_{im} | H^{(g)}, \varTheta ^{(g)}, \varPhi ^{(g)}\right) = \sum _{k=1}^{K} \sum _{l=1}^{L} \eta _{ik}^{(g)} \cdot \theta _{kl}^{(g)} \cdot \phi _{lw_{im}}^{(g)}. \end{aligned}$$

(15)