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A Counterintuitive Result

Readers new to scaling-law analysis often expect that stacking two mechanisms ("spatial reuse" + "coded multicasting") multiplies gains. The Ji-Caire-Molisch 2015 result shows this expectation is wrong at the scaling-order level. This section provides the rigorous statement and intuition for why.

Theorem: Coded D2D Scaling Law

In a D2D caching network with random uniform placement, random uniform demands, and protocol-model interference, both the uncoded delivery scheme (Ch 10) and the MAN-style coded delivery scheme achieve per-user throughput $T_n(M) \;=\; \Theta(\mu).$ The coded scheme achieves a larger constant: $T_{\text{coded}}(M) = c_{\text{coded}} \cdot \mu \; \text{vs} \; T_{\text{uncoded}}(M) = c_{\text{uncoded}} \cdot \mu,$ with $c_{\text{coded}} > c_{\text{uncoded}}$ but both constants finite. The scaling order is the same; only the constant differs.

The coded scheme's per-cluster MAN gain $(1 + K_g M/N)$ is offset by the per-cluster airtime overhead $\binom{K_g}{t+1}$ . These two effects cancel at the scaling-law order, yielding the same $\Theta(M/N)$ scaling. The coded scheme's edge shows up only in the constant.

Proof

Achievability — coded scheme

Partition into clusters of size $K_g = \Theta(\log n)$ . Per cluster: MAN delivery with gain $1 + K_g M/N$ . Airtime per cluster: $\binom{K_g}{t+1}$ XOR messages. Number of clusters: $\Theta(n/K_g)$ . With spatial reuse: $\Theta(n/(K_g r^2))$ concurrent clusters. Aggregate: $\Theta(n \cdot (1 + K_g M/N)/(K_g \binom{K_g}{t+1}))$ $\approx \Theta(n \cdot M/N)$ . Per-user: $\Theta(M/N)$ .

Achievability — uncoded scheme

From Theorem 10.1: $T_n = \Theta(M/N)$ .

Same order

Both schemes achieve $\Theta(M/N)$ — the order is the same. The coded scheme has a better constant (explicit analysis gives a factor of roughly $1 + K_g M/N$ per cluster, divided by the cluster size normalization).

Converse

The cut-set bound caps per-user at $O(M/N + 1/\sqrt{n})$ (Chapter 10 Exercise). The coded scheme doesn't escape this cap, so no scaling-order gain is possible. $\blacksquare$

Key insight

The reason two gains don't compound: they're competing for the same resource (transmit opportunities per unit time). Spatial reuse gives you many concurrent links; coded multicast gives you more content per link. Both reduce total airtime, but you can only benefit from one at a time. The bottleneck shifts from "concurrent" to "content per link" but the per-user bound is unchanged.

🎓CommIT Contribution(2016)

Spatial Reuse and Coded Multicasting Do Not Cumulate

M. Ji, G. Caire, A. F. Molisch — IEEE Journal on Selected Areas in Communications, vol. 34, no. 1

The Ji-Caire-Molisch 2015 paper shows that spatial reuse and coded multicasting — two of the main gain mechanisms in wireless caching — do not cumulate in scaling-law terms. Both achieve $\Theta(M/N)$ per-user throughput; coding improves only the constant factor.

Key insights.

Shared resource constraint. Spatial reuse and coded multicasting both reduce airtime. They compete for the same "time budget" and cannot both multiplicatively improve scaling.
Constant factor matters. Although not asymptotically dominant, the constant improvement is substantial (factor of ~2-3 in practical regimes) and worth implementing.
Design guidance. Don't expect compound asymptotic gains from stacking mechanisms. Instead, optimize cluster size and coordination overhead for best constant factor.

This result has been foundational for understanding 6G D2D designs: architects should not over-invest in one mechanism at the expense of another.

coded-cachingd2dcommitscaling-lawView Paper →

Coded vs Uncoded D2D Scaling

Both coded and uncoded D2D schemes exhibit $\Theta(M/N)$ scaling (flat in $n$ , log-log). Only the constants differ. Compare with Gupta-Kumar (no caching, decreasing) baseline.

Parameters

Memory ratio μ0.3

Max network size1000

Uncoded constant1

Coded constant1.3

Example: Where Does the Gain 'Disappear'?

Suppose $K_g = 10$ , $\mu = 0.2$ , $n = 100$ . The MAN coded gain per cluster is $1 + K_g M/N = 3$ . Why doesn't this triple the overall per-user throughput?

Solution

Per-cluster rate

Within a cluster of 10 users, MAN delivers 10 files using $\binom{10}{t+1} = \binom{10}{3} = 120$ subfile-sized XOR messages instead of 10 files. The airtime per delivered file is $120/10 = 12$ subfiles. Uncoded: 10 unicasts × 1 file each = 10 file-units.

Per-cluster airtime comparison

Coded per-cluster airtime: $120 \cdot (1/\binom{10}{2}) = 120/45 \approx 2.67$ file-units. Uncoded: 10 file-units. Coded is about 3.75× faster per cluster — matches the MAN gain factor.

Cluster-level parallelism

Across clusters, spatial reuse lets them run in parallel. Number of concurrent clusters: $\Theta(n/(K_g r^2))$ . Aggregate rate: cluster-level rate × concurrent clusters. Under uncoded: $10/n$ per user × concurrent = $\Theta(M/N)$ . Under coded: $2.67 \cdot 3.75/n$ × concurrent = $\Theta(M/N)$ . The MAN gain gets offset by the airtime savings already baked into the scaling.

Net effect

Same scaling. Better constant. The coded gain is "real" — just not asymptotically new.

What the Result Teaches Us

The non-cumulating result has deep implications for how we think about caching architectures:

Do not stack mechanisms naively. Adding coded multicasting to a D2D network does not give a multiplicative asymptotic gain. It gives a constant-factor gain. This is still worth having, but less than naive intuition suggests.
Distinguish order vs constant. Some design decisions affect scaling order (e.g., caching itself — from $1/\sqrt{n}$ to $\Theta(1)$ ); others affect only constants (e.g., coding on top of caching). Be clear about which.
The scaling is set by the weakest link. In coded D2D, spatial reuse and caching both contribute to the $\Theta(M/N)$ bound. Adding coding doesn't escape it.

Designers should therefore prioritize: first, ensure caching (which gives an order of magnitude improvement in scaling); second, add coding (for a constant factor improvement on top). The relative priorities are clear when viewed through the scaling lens.

Common Mistake: Reject the 'Stacked Gains' Intuition

Mistake:

Stating "D2D + caching + coded multicasting combines to give $\Theta((1 + KM/N) \cdot M/N)$ per-user throughput."

Correction:

This naive product is wrong. The correct statement is $\Theta(M/N)$ — same order as D2D + caching alone. The MAN coded multicasting gain is absorbed by the per-cluster airtime cost; it doesn't multiply the D2D scaling.

If you want higher scaling orders, you need to add fundamentally different resources: multi-antenna transmitters (Chapter 5), server cooperation (Chapter 9), or infrastructure offloading (Chapter 8). Those mechanisms do affect the scaling order when combined with caching.

Why the Gains Do Not Cumulate