  In our recent works, we developed a probabilistic framework for structural
analysis in undirected networks and directed networks. The key idea of that
framework is to sample a network by a symmetric and asymmetric bivariate
distribution and then use that bivariate distribution to formerly defining
various notions, including centrality, relative centrality, community, and
modularity. The main objective of this paper is to extend the probabilistic
definition to attributed networks, where sampling bivariate distributions by
exponentially twisted sampling. Our main finding is that we find a way to deal
with the sampling of the attributed network including signed network. By using
the sampling method, we define the various centralities in attributed networks.
The influence centralities and trust centralities correctly show that how to
identify centralities in signed network. The advertisement-specific influence
centralities also perfectly define centralities when the attributed networks
that have node attribute. Experimental results on real-world dataset
demonstrate the different centralities with changing the temperature. Further
experiments are conducted to gain a deeper understanding of the importance of
the temperature.
