Inferring Social Ties In some papers, they proposed a model in which the joint posterior probability was modeled as[1]:
$$ P(Y|\textbf{X},G) = \frac{P(\textbf{X},G|Y)P(Y)}{P(X,G)} \propto P(\textbf{X}|Y) \cdot P(Y|G)$$
where $\textbf{X}$ is attribute vectors set and $Y$ is the labels of all the edges in the network. 

I know the meaning of the first equal sign which can be applied with Bayesian Theorem. But I feel confused about the third term, $P(\textbf{X}|Y) \cdot P(Y|G)$. How this term comes out?

[1]: J. Tang, T. Lou, and J. Kleinberg, “Inferring social ties across heterogenous networks,” in Proceedings of the fifth ACM international conference on Web search and data mining - WSDM ’12, 2012, p. 743.
This model seems a little confusing due to the role of $G$ and I didn't find their definition of $P(Y|G)$ very sound.

You can consider it the following way:

$$P(Y|\textbf{X},G) \propto P(\textbf{X}|Y) \cdot P(G|Y) \cdot P(Y) = P(\textbf{X}|Y) \cdot [P(G|Y) \cdot P(Y)] \propto P(\textbf{X}|Y) \cdot P(Y|G)$$