obtained by coherent transmission is dominant. In order to allow this, relays should be able to receive sufficient information from their previous nodes. Thus, they tend to be clustered near the source.

In Figure \[fig:G\_vs\_N\] we plot the relaying gain $G(N,\lambda)$ in dB vs. the number of relays $N$, for various values of $\lambda$. As proved in Theorem \[theorem:capacity\_increasing\_with\_N\], we see that $G(N,\lambda)$ increases with $N$ for fixed $\lambda$. On the other hand, $G(N,\lambda)$ increases with $\lambda$ for fixed $N$, as proved in Theorem \[theorem:G\_increasing\_in\_lambda\].

![$G (N,\lambda)$ vs $N$ for total power constraint.[]{data-label="fig:G_vs_N"}](optimal-placement-fixed-line.pdf){height="3cm" width="8cm"}

Uniformly Placed Relays, Large $N$
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When the relays are uniformly placed, the behaviour of $R^{opt}_{P_T}(y_1,\cdots,y_N)$ (called $R_{N}$ in the next theorem) for large number of relays is captured by the following:

\[theorem:large\_nodes\_uniform\] For exponential path-loss and sum power constraint, if $N$ relay nodes are placed uniformly between the source and the sink, resulting in $R_{N}$ achievable rate, then $ \lim_{N \rightarrow \infty} R_N= C \left(\frac{P_{T}}{\sigma^{2}×}\right)$.

See Appendix \[appendix:proof\_of\_large\_nodes\_uniform\].

[*Remark:*]{} From Theorem \[theorem:large\_nodes\_uniform\], it is clear that we can achieve a rate arbitrarily close to $C(\frac{P_{T}}{ \sigma ^{2}})$ (i.e., the effect of path-loss can completely be mitigated) by placing a large enough number of relay nodes. 