In this paper, we investigate an energy-harvesting cooperative communication network, which comprises of a source, a destination, and multiple decode-and-forward (DF) relays in the presence of multiple passive eavesdropper (Es). Es attempt to intercept confidential information transmissions from the source to destination via DF relays. In this network, all the DF relays harvest energy from radio-frequency (RF) signals of a source through time-switching receivers. In order to improve the physical layer security of energyharvesting cooperative communication networks, we propose a best relay selection (BRS) scheme, where the ''best'' relay is chosen to assist the source-destination transmission. For the purpose of comparison, we consider the classic direct transmission (DT) and equal relay selection (ERS) as benchmark schemes. We derive the exact closed-form expressions of outage probability (OP) and intercept probability (IP) for the ERS and BRS schemes over Rayleigh fading channels. Besides, the security-reliability tradeoff (SRT) is analyzed as a metric to evaluate the tradeoff performance of the proposed BRS scheme. Numerical results show that the SRT of the BRS scheme consistently outperforms that of the ERS scheme, which demonstrates the advantage of our proposed scheme against eavesdroppers. Besides, it is verified that total error rate (TER) defined as the sum of OP and IP can be minimized for both the ERS and BRS schemes through changing the time allocation factor between information transmission and energy harvesting phases. Moreover, there is a best energy conversation efficiency to obtain a maximal SRT value of the ERS and BRS schemes. In addition, as the number of DF relays increases, the SRT of BRS scheme improves notably, while that of ERS scheme remains unchanged. And as the number of Es increases, the SRT of both the ERS and BRS schemes become worse. INDEX TERMS Cooperative communication, physical layer security, best relay selection (BRS), outage probability (OP), intercept probability (IP), security-reliability tradeoff (SRT).