Private blockchain platforms tend to apply deterministic consensus mechanisms as a more efficient alternative to the proof-based consensus. Deterministic mechanisms tolerate two types of failures, Byzantine, and Crash-Fault. Byzantine-Fault tolerant consensus assumes restrictive assumptions of time and number of failures to guarantee the validity, while the termination depends on node message broadcasting. Crash-Fault tolerant consensus mechanisms induce lower overhead and faster termination than Byzantine-Fault tolerant mechanisms at the cost of not tolerating malicious behaviors. This paper explores different overlay topologies to assess a lightweight consensus mechanism based on vicinity voting with reliable message broadcasting. The paper proposes different ways to compose the consensus quorum according to the vicinity models. Vicinity models applied in the overlay network allow for relaxing the trade-off between agreement and termination. Analytical and experimental results for different physical topologies show that certain vicinity models guarantee higher number of nodes reached at the time of reaching the consensus threshold (higher than 90% of all nodes) with a lower cost at the exchange of lower fault tolerance. Other models induce a lower agreement while increase fault tolerance (higher than 50%).