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From One Relay to Two: The Diamond Network

The relay channel has a single relay helping the source-destination pair. The natural next step is to ask: what happens with multiple relays? The diamond network is the simplest such model: a source communicates to a destination through two parallel relays, with no direct link between source and destination. This "diamond" topology (source → relay 1 and relay 2 → destination) is the building block for understanding larger relay networks and foreshadows the noisy network coding results of Chapter 23.

The diamond network reveals a key challenge: the two relays' signals must somehow be coordinated at the destination, even though the relays cannot communicate with each other. How to achieve this coordination — through decode-and-forward, compress-and-forward, or a hybrid — is the central question.

Definition:
The Diamond Network

The diamond network consists of:

A source $S$ with input $X \in \mathcal{X}$ ,
Two relays $R_1$ and $R_2$ with observations $Y_{r_1}, Y_{r_2}$ and inputs $X_{r_1}, X_{r_2}$ ,
A destination $D$ with observation $Y$ ,
Broadcast channel: $p(y_{r_1}, y_{r_2} | x)$ (source to relays),
Multiple-access channel: $p(y | x_{r_1}, x_{r_2})$ (relays to destination),
No direct link from source to destination.

The relays operate under the strictly causal constraint: $X_{r_k,i} = f_{k,i}(Y_{r_k}^{i-1})$ .

Diamond Network

A four-node relay network where a source communicates to a destination through two parallel relays with no direct source-destination link. The source broadcasts to both relays, which then transmit to the destination over a multiple-access channel.

Related: Relay Channel

Theorem: Cut-Set Bound for the Diamond Network

The capacity of the diamond network satisfies $C \leq \max_{p(x) p(x_{r_1}) p(x_{r_2})} \min\!\begin{cases} I(X; Y_{r_1}, Y_{r_2}) \\ I(X_{r_1}, X_{r_2}; Y) \\ I(X; Y_{r_1}) + I(X_{r_2}; Y | X_{r_1}) \\ I(X; Y_{r_2}) + I(X_{r_1}; Y | X_{r_2}) \end{cases}$ where the four terms correspond to: (1) the broadcast cut, (2) the MAC cut, (3) and (4) hybrid cuts isolating one relay with the source or destination.

The diamond network has four non-trivial cuts, compared to two for the single-relay channel. The broadcast cut limits how fast the source can send to both relays. The MAC cut limits how fast the relays can jointly send to the destination. The hybrid cuts capture scenarios where one relay is the bottleneck.

Proof

Enumerate all cuts

Partition the four nodes $\{S, R_1, R_2, D\}$ into a source-side set containing $S$ and a destination-side set containing $D$ . The non-trivial partitions are:

$\{S\}$ vs. $\{R_1, R_2, D\}$ : broadcast cut.
$\{S, R_1, R_2\}$ vs. $\{D\}$ : MAC cut.
$\{S, R_1\}$ vs. $\{R_2, D\}$ : hybrid cut isolating $R_2$ .
$\{S, R_2\}$ vs. $\{R_1, D\}$ : hybrid cut isolating $R_1$ .

Apply the cut-set bound

For each cut, the information flow is bounded by the mutual information across that cut. Since the relays have no direct link, the relay inputs are independent ( $p(x_{r_1}, x_{r_2}) = p(x_{r_1}) p(x_{r_2})$ ). Taking the minimum over all cuts and maximizing over the input distribution yields the bound.

,

Definition:
Quantize-Map-and-Forward

Quantize-map-and-forward (QMF) is a relay strategy for general networks:

Each relay quantizes its observation at the noise level (resolution $\sim \sqrt{\sigma^2}$ ).
Each relay maps the quantized observation to a channel input using a random mapping.
The destination collects all relay transmissions and jointly decodes.

QMF achieves within a constant gap of the cut-set bound for a wide class of Gaussian relay networks. The gap depends on the number of nodes but not on the channel gains or SNR — a universal approximation result.

Example: The Gaussian Diamond Network

Consider a Gaussian diamond network where:

Source to relays: $Y_{r_k} = h_k X + Z_{r_k}$ , $k = 1, 2$ , with $h_1 = 2$ , $h_2 = 1$ , noise variance $\sigma^2 = 1$ , and source power $P = 10$ .
Relays to destination: $Y = g_1 X_{r_1} + g_2 X_{r_2} + Z$ , with $g_1 = 1$ , $g_2 = 1.5$ , relay powers $P_{r_1} = P_{r_2} = 10$ , noise variance $1$ . Compute the broadcast cut and the MAC cut of the cut-set bound.

Solution

Broadcast cut

The source broadcasts to both relays. The total capacity of this broadcast link is: $I(X; Y_{r_1}, Y_{r_2}) \leq \frac{1}{2}\log\!\left(1 + \frac{(|h_1|^2 + |h_2|^2) P}{\sigma^2}\right) = \frac{1}{2}\log(1 + 50) \approx 2.84 \text{ bits}.$ (This is an upper bound; the exact expression depends on the noise correlation structure.)

MAC cut

The two relays form a MAC to the destination: $I(X_{r_1}, X_{r_2}; Y) \leq \frac{1}{2}\log\!\left(1 + |g_1|^2 P_{r_1} + |g_2|^2 P_{r_2}\right) = \frac{1}{2}\log(1 + 10 + 22.5) = \frac{1}{2}\log(33.5) \approx 2.53 \text{ bits}.$

Cut-set bound

The cut-set bound is $\min\{2.84, 2.53, \ldots\} = 2.53$ bits (the MAC cut is the bottleneck in this example). The diamond network capacity is at most $2.53$ bits per channel use.

From the Diamond Network to Noisy Network Coding

The diamond network is the stepping stone to noisy network coding (Chapter 23). The key observation is that compress-and-forward generalizes naturally to networks with multiple relays: each relay compresses its observation and forwards the compressed version. The Lim-Kim-El Gamal-Chung noisy network coding scheme does exactly this, and achieves within a constant gap of the cut-set bound for any network topology.

The diamond network is the simplest test case for these general results, and it already illustrates the main ideas: compression at the relay level, joint decoding at the destination, and the interplay between the broadcast and MAC cuts.

The Diamond Network Topology — The diamond network: source $S$ broadcasts to relays $R_1, R_2$ , which transmit to destination $D$ over a MAC. No direct link exists between $S$ and $D$ .

🔧Engineering Note

Diamond Networks in Wireless Relay Deployment

The diamond network models a common wireless deployment scenario: a base station (source) communicates to a cell-edge user (destination) through two relay nodes. In 5G NR, this arises in integrated access and backhaul (IAB) architectures where multiple relay hops are used. The diamond network analysis reveals that the bottleneck is often the MAC cut (relay-to-destination), suggesting that relay placement should prioritize strong relay-destination links over strong source-relay links — a counterintuitive but information-theoretically grounded design principle.

Practical Constraints

•
Relay synchronization needed for coherent MAC transmission
•
Half-duplex constraint reduces effective capacity by ~50%

Common Mistake: Relay Coordination Does Not Require Message Sharing

Mistake:

Assuming that the two relays in a diamond network must share decoded messages to cooperate effectively.

Correction:

In the diamond network, the relays have no direct link and cannot share information. Cooperation emerges through code design: the source uses superposition coding to send different information to each relay, and the relays transmit correlated signals because they both heard (noisy versions of) the same source transmission. In compress-and-forward, each relay independently compresses and forwards, and all coordination happens at the destination through joint decoding.

Quick Check

In the diamond network, the capacity is generally limited by the minimum of the broadcast cut and the MAC cut. If we double the power of both relays (but not the source), which cut is affected?

Only the broadcast cut increases

Only the MAC cut increases

Both cuts increase equally

Neither cut changes

Correction:

Only the MAC cut increases

Correct. The MAC cut $I(X_{r_1}, X_{r_2}; Y)$ depends on the relay powers and relay-destination channels. Doubling relay power increases the MAC cut. The broadcast cut $I(X; Y_{r_1}, Y_{r_2})$ depends only on source power and source-relay channels, so it remains unchanged.

Key Takeaway

The diamond network — source to two parallel relays to destination — is the simplest multi-relay model. Its analysis reveals that capacity is limited by the minimum of broadcast and MAC cuts, and that compress-and-forward strategies (quantize-map-and-forward) can achieve within a constant gap of the cut-set bound without requiring relays to decode or coordinate with each other.

The Diamond Network

From One Relay to Two: The Diamond Network

Definition: The Diamond Network

Diamond Network

Theorem: Cut-Set Bound for the Diamond Network

Enumerate all cuts

Apply the cut-set bound

Definition: Quantize-Map-and-Forward

Example: The Gaussian Diamond Network

Broadcast cut

MAC cut

Cut-set bound

From the Diamond Network to Noisy Network Coding

The Diamond Network Topology

Diamond Networks in Wireless Relay Deployment

Common Mistake: Relay Coordination Does Not Require Message Sharing

Quick Check

Key Takeaway

Definition:
The Diamond Network

Definition:
Quantize-Map-and-Forward