Before revision:  We prove that the Triggering Probability Modulated (TPM) condition for CMAB still holds , and then utilize the property of competitive diffusion to introduce a new offline oracle, and discuss how to implement this new oracle in various cases. We propose an OCIM-OIFU algorithm with such an oracle that achieves logarithmic regret .
After revision:  We prove that the Triggering Probability Modulated (TPM) condition for CMAB still holds in OCIM, which is instrumental for our proposed algorithms OCIM-TS and OCIM-OFU to achieve logarithmic Bayesian regret and frequentist regret, respectively .