## 5 Relationship between Connectivity and Capacity
### Capacity Constraints imposed by Connectivity  
As mentioned previously, the work of [PERSON] and [PERSON] [26] pointed out that the minimum density required for connectivity limits the possible throughput per node. They considered both a SINR model and model based directly on distances, for both optimal and random placement of nodes. For systems with \(n\) nodes, they established an upper bound on the capacity under the SINR model of \(\Omega(1/\sqrt{n})\) in each case, and lower bounds of \(\Omega(1/\sqrt{n})\) for optimal placement and \(\Omega(1/\sqrt{n\log n})\) for random placement. [PERSON] _et al._[65] showed that the lower bound for random placement is also \(\Omega(1/\sqrt{n})\), giving matching asymptotic upper and lower bounds.  
The fact that the capacity scales less than linearly with the number of nodes has prompted the study of hybrid networks containing some wired base stations. However, it has been shown that a linear scaling of capacity with the number of nodes is possible if the modeling assumptions are relaxed.  
As already mentioned, [PERSON], [PERSON], and [PERSON] [27] showed that hierarchical cooperative beamforming can, in principle, achieve capacity proportional to the number of nodes, albeit at the expense of requiring an unrealistic degree of cooperation.  
[PERSON] _et al._[66] demonstrated that the linear scaling is possible if a fixed (but arbitrarily small) fraction of the nodes are allowed not to be connected. Naturally, the achievable rate per node is a function of the fraction of nodes which are allowed to remain disconnected from the main connected component. In particular, if we want to impose a minimum data rate of transmission between nodes, then as the number of nodes goes to infinity, a fraction of nodes, which is a function of the required data rate, are automatically disconnected. This result was proved both for a Boolean (disk) model, and fora novel capacity-based model. In the latter model, the network is considered connected if and only any pair of nodes can communicate with a predefined minimum data rate, using other nodes as relays.  
Note that the connectivity requirement is only an issue if the designer has control over the transmission range, but cannot add wireless relay nodes. An often overlooked consideration in these studies is that the _total_ network capacity increases as the number of nodes increases. The fact that the capacity per node decreases as more nodes are added arises because it is assumed that additional nodes also have data to carry. If additional relay nodes are introduced purely to provide connectivity, then these nodes have no data of their own to send, and so do not reduce the capacity available for the original users. To see this, consider a network of a finite density of user nodes. Superimpose on this (a) a very much higher density of ad hoc relay nodes and (b) a planar wired network. In the ad hoc network, users can communicate by relaying data over the relay nodes lying on the paths of the wired network. As the density of the ad hoc network increases, the required transmission ranges vanish and so the interference between \"virtual wires\" also vanishes. This allows the users to communicate with arbitrarily high capacities.  
To investigate the relationship between connectivity and capacity, [PERSON] and [PERSON] [67] proved that the attenuation function is one of the major factors affecting the properties of dense ad hoc networks. Under the assumption of power law attenuation, the network connectivity probability becomes higher as the density of nodes increases, even though the interference also increases. The increase of network total capacity is bounded by \(\sqrt{n}\). However, the total network capacity remains constant with attenuation functions that are bounded above, and the network maintains connectivity at the expense of capacity.