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This paper considers the Byzantine consensus problem for nodes with binary inputs. The nodes are interconnected by a network represented as an undirected graph, and the system is assumed to be synchronous. Under the classical point-to-point communication model, it is well-known that the following two conditions are both necessary and sufficient to achieve Byzantine consensus among n nodes in the presence of up to ƒ Byzantine faulty nodes: n & 3 #8805; 3 ≥ ƒ+ 1 and vertex connectivity at least 2 ƒ + 1. In the classical point-to-point communication model, it is possible for a faulty node to equivocate, i.e., transmit conflicting information to different neighbors. Such equivocation is possible because messages sent by a node to one of its neighbors are not overheard by other neighbors. This paper considers the local broadcast model. In contrast to the point-to-point communication model, in the local broadcast model, messages sent by a node are received identically by all of its neighbors. Thus, under the local broadcast model, attempts by a node to send conflicting information can be detected by its neighbors. Under this model, we show that the following two conditions are both necessary and sufficient for Byzantine consensus: vertex connectivity at least ⌋ 3 fƒ / 2 ⌊ + 1 and minimum node degree at least 2 ƒ. Observe that the local broadcast model results in a lower requirement for connectivity and the number of nodes n, as compared to the point-to-point communication model. We extend the above results to a hybrid model that allows some of the Byzantine faulty nodes to equivocate. The hybrid model bridges the gap between the point-to-point and local broadcast models, and helps to precisely characterize the trade-off between equivocation and network requirements. 

We study downlink transmission in a multi-band heterogeneous network comprising unmanned aerial vehicle (UAV) small base stations and ground-based dual mode mmWave small cells within the coverage area of a microwave (μW) macro base station. We formulate a two-layer optimization framework to simultaneously find efficient coverage radius for the UAVs and energy efficient radio resource management for the network, subject to minimum quality-of-service (QoS) and maximum transmission power constraints. The outer layer derives an optimal coverage radius/height for each UAV as a function of the maximum allowed path loss. The inner layer formulates an optimization problem to maximize the system energy efficiency (EE), defined as the ratio between the aggregate user data rate delivered by the system and its aggregate energy consumption (downlink transmission and circuit power). We demonstrate that at certain values of the target SINR τ introducing the UAV base stations doubles the EE. We also show that an increase in τ beyond an optimal EE point decreases the EE.