Budgeted Broadcast: Constrained Information Spread

• Budgeted Broadcast is a framework that imposes explicit resource limits on broadcast power, traffic, or neural connectivity across various systems.
• It unifies graph-theoretic, wireless, and neural approaches using constrained optimization techniques such as Lagrangian duality and entropy maximization.
• Empirical studies demonstrate BB’s practicality by improving throughput in wireless systems, reducing error rates in ASR, and enhancing performance in neural network pruning.

Budgeted Broadcast (BB) encompasses a collection of principled frameworks and algorithms that control, limit, or optimize information dissemination—be it in combinatorial networks, wireless systems, or neural architectures—by introducing explicit constraints or budgets on broadcast “power,” “traffic,” or resource allocation. These frameworks can arise in graph theory (dominating broadcasts), communication networks (bandwidth- or time-limited content multicasting), or as homeostatic rules in artificial neural networks that regulate unit connectivity and activity. This article synthesizes the core definitions, analytical foundations, algorithmic solutions, and quantitative results across the principal BB research tracks.

1. Theoretical Foundations: Budget Constraints in Networked Systems

Budgeted Broadcast structures impose upper bounds on the aggregate or per-node resources used to “cover,” “dominate,” or serve network elements via broadcast:

• In graph theory, the dominating $k$-broadcast bounds the signal strength at each vertex by $k$, restricting the “radius” of nodes permitted to broadcast to other nodes and defining the minimal cost to dominate a network (Cáceres et al., 2016).
• In wireless networks, BB formalizes the split between broadcast and unicast, minimizing a cost (latency, bandwidth) under a hard “broadcast time” or bandwidth constraint (Chatzinotas et al., 2015).
• In neural networks, each unit is assigned a traffic budget (product of activity and fan-out); global entropy is maximized under this aggregate constraint, yielding a selectivity-audience balance (Meirovitch et al., 26 Sep 2025).

These formalisms are united by the analytic machinery of constrained optimization (Lagrangian duality, combinatorial bounds, entropy maximization).

2. Graph-Theoretic Budgeted Broadcast: Dominating $k$-Broadcast

Let $G = (V, E)$ be a connected graph and $k \ge 1$. A function $f: V \rightarrow \{0, 1, \ldots, k\}$ is called a dominating $k$-broadcast if $\forall u \in V$, there exists $v$ with $f(v) > 0$ and $k$0. The cost is $k$1. The dominating $k$2-broadcast number is

$k$3

Key properties:

• For $k$4, recovers the domination number: $k$5.
• For $k$6, recovers the classical broadcast number: $k$7.

A unified upper bound holds:

$k$8

where $k$9 and $k$0 (Cáceres et al., 2016). For $k$1, this interpolates smoothly between classical domination and unconstrained broadcast.

The decision problem—given $k$2, is $k$3?—is NP-complete for every fixed $k$4. The reduction constructs gadgets corresponding to logical variables and clauses such that satisfying assignments map to minimum-cost broadcasts.

3. Budgeted Broadcast in Communication Networks: Proactive Caching and Scheduling

In cooperative multi-point (CoMP) cloud-radio access networks (C-RANs), BB enables efficient joint delivery of multimedia by dividing data into “proactive broadcast” (to all cached users) and “on-demand unicast” streams (Chatzinotas et al., 2015). The network seeks to:

• Partition the file library at index $k$5: files $k$6 are broadcast to all users; the remainder are unicasted.
• Minimize the total frame time $k$7 under broadcast time or bandwidth constraints.

Assuming Zipf popularity ($k$8), the optimal threshold is

$k$9

where $G = (V, E)$0 and $G = (V, E)$1 are the per-user spectral efficiencies of broadcast and unicast, $G = (V, E)$2 is the user count (Chatzinotas et al., 2015).

Key empirical outcomes:

• Integrated BB can yield up to 80% throughput gain over UC-only systems when $G = (V, E)$3, with the convex cost function $G = (V, E)$4 minimized at the analytical $G = (V, E)$5.
• Adjustment of the BC budget $G = (V, E)$6 trades off push-period length with on-demand unicast latency.

4. Budgeted Broadcast in Neural Network Pruning

BB in neural computation implements activity-dependent structural plasticity by maximizing total coding entropy $G = (V, E)$7, where $G = (V, E)$8 and $G = (V, E)$9 is the long-term on-rate of unit $k \ge 1$0 (Meirovitch et al., 26 Sep 2025). The key constraint is a global traffic budget: $k \ge 1$1, with $k \ge 1$2 the fan-out of unit $k \ge 1$3.

Constrained entropy maximization yields the selectivity-audience equilibrium:

$k \ge 1$4

or $k \ge 1$5.

BB is realized via two sparse pruning actuators:

• SP-out (axonal pruning): For each source unit $k \ge 1$6, hold outgoing edges at target $k \ge 1$7 (Top-$k \ge 1$8 by magnitude).
• SP-in (dendritic pruning): For each target unit $k \ge 1$9, hold incoming edges at target $f: V \rightarrow \{0, 1, \ldots, k\}$0.

This process is repeated periodically (mask refresh) and can include natural regrowth (sampling from the full dense matrix). The system self-organizes toward entropy maximization under global traffic constraint.

5. Empirical Results Across Domains

Experimental studies demonstrate the quantitative efficacy of BB:

Automatic Speech Recognition (ASR):

• Model: Transformer; Dataset: LibriSpeech.
• Under 0.85 density, SP-in achieves Word Error Rate Reduction (WERR) of +3.2% (test_clean), outperforming magnitude pruning (WERR $f: V \rightarrow \{0, 1, \ldots, k\}$1 0) (Meirovitch et al., 26 Sep 2025).

Face Identification:

• Model: ResNet-101; Dataset: VGGFace2-7k.
• At density 0.5, SP-in attains Top-1 accuracy 88.4% and verification accuracy 0.883, exceeding the dense baseline (87.9%, 0.870) and magnitude pruning (87.1%, 0.860).

Change Detection and Synapse Prediction:

• On LEVIR-CD (change detection), BB improves mean IoU by +10.8% over dense.
• On SmartEM (3D EM synapse segmentation), SP-in achieves PR-AUC 0.7407 vs. 0.6952 for dense, and F1=0.6752 vs. 0.6578.

Wireless Scheduling (CoMP Cloud-RAN):

• For $f: V \rightarrow \{0, 1, \ldots, k\}$2 base stations, $f: V \rightarrow \{0, 1, \ldots, k\}$3 users, and $f: V \rightarrow \{0, 1, \ldots, k\}$4, $f: V \rightarrow \{0, 1, \ldots, k\}$5 b/s/Hz, the optimal threshold is at $f: V \rightarrow \{0, 1, \ldots, k\}$6 (Zipf $f: V \rightarrow \{0, 1, \ldots, k\}$7), matching the convex optimum (Chatzinotas et al., 2015).

Domain	Model/Arch	Metric	Best BB Result	Baseline
ASR	Transformer	WERR (test_clean)	+3.2% (SP-in)	~0 (Mag)
Face ID	ResNet-101	Top-1 @0.5 density	88.4%	87.9% (dense)
Change Detect	FC-Siam U-Net	IoU (mean)	0.678 (SP-in)	0.611 (dense)
Synapse Pred.	3D U-Net, SmartEM	PR-AUC (GT2)	0.7407 (SP-in)	0.6952

6. Implementation, Overhead, and Limitations

BB methods involve minimal additional state:

• Hardware & compute: One per-channel float for EMA $f: V \rightarrow \{0, 1, \ldots, k\}$8; $f: V \rightarrow \{0, 1, \ldots, k\}$9 operations per mask refresh. No per-weight penalty or counter is needed.
• Integration: Insert channel-wise EMA accumulators, periodic mask (binary) refresh after warm-up, and mask-weight multiplication.
• Hyperparameters: Mask refresh interval $k$0, Lagrange multiplier $k$1 (per layer), minimum degree $k$2 = 1.
• Limitations: Current implementations produce unstructured sparsity; hardware acceleration requires downstream mapping. BB does not yet regulate lateral or attention-head connections; “Budgeted Attention” is a proposed extension (Meirovitch et al., 26 Sep 2025).

In graph algorithms, combinatorial reductions (e.g., from 3-SAT) demonstrate NP-completeness across fixed $k$3. In wireless BB, implementation uses proactive caching, centralized request statistics, and periodical optimization of the BC/UC split (Chatzinotas et al., 2015).

7. Synthesis and Emerging Directions

Budgeted Broadcast provides a unified perspective on constrained coverage and dissemination across discrete, physical, and computational networks. BB formalizes how local or global resource budgets shape the optimal topologies, broadcast radii, or neural connectivity, and achieves state-of-the-art results in neural pruning, wireless scheduling, and combinatorial covering (Cáceres et al., 2016, Chatzinotas et al., 2015, Meirovitch et al., 26 Sep 2025). Ongoing developments include structured sparsity, adaptive budget allocation (e.g., input-conditional), and application to richer network forms such as Transformer attention or recurrent architectures. A plausible implication is that BB schemes, by maximizing entropy under resource constraints, naturally promote diverse, efficient encoding with minimal redundancy—a property relevant across information, communication, and learning systems.