We address the polling problem in social networks where individuals collaborate to choose the most favorite choice amongst some options, without divulging their vote and publicly exposing their potentially malicious actions. Given this social interaction model, Guerraoui et al. recently proposed polling protocols that do not rely on any central authority or cryptography system, using a simple secret sharing scheme along with verification procedures to accurately compute the poll's final result. However, these protocols can be deployed safely and efficiently provided that, inter alia, the social graph structure should be transformed into a ring structure-based overlay and the number of participating users is perfect square. Consequently, designing \emph{secure} and \emph{efficient} polling protocols regardless these constraints remains a challenging issue. In this paper, we present EPol, a simple decentralized polling protocol that relies on the current state of social graphs. More explicitly, we define one family of social graphs that satisfy what we call the $m$-broadcasting property (where $m$ is less than or equal to the minimum node degree) and show their structures enable low communication cost and constitute necessary and sufficient condition to ensure vote privacy and limit the impact of dishonest users on the accuracy of the polling output. Our protocol is effective to compute more precisely the final result. Furthermore, despite the use of richer social graph structures, the communication and spatial complexities of EPol are close to be linear.