Shannon capacity of a graph¶
A graph invariant giving the asymptotic zero-error information rate of a confusability graph under repeated independent channel uses.
Core Idea¶
The Shannon capacity Θ(G) is the supremal exponential growth rate of the independence number of strong powers G^n. Vertices encode symbols and edges encode confusability; length-n codewords form vertices of the strong product, and a zero-error code is an independent set. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.
The load-bearing residual is not the broad topic of information theory. It is It is a zero-error combinatorial capacity and differs from probabilistic Shannon channel capacity; exact computation remains difficult for many graphs..
Scope of Application¶
Shannon capacity of a graph belongs to information theory and is useful where the analyst can specify a finite confusability graph, independent sets, strong graph powers, block length, logarithmic message count, and asymptotic growth rate, then evaluate the rate is derived from independence numbers of strong graph powers under the declared graph convention. The scope is broad within that domain but bounded by the need for the rate is derived from independence numbers of strong graph powers under the declared graph convention. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
Clarity¶
The abstraction clarifies a crowded vocabulary by making the rate is derived from independence numbers of strong graph powers under the declared graph convention the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Shannon capacity of a graph can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.
Manages Complexity¶
Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Shannon capacity of a graph. Shannon capacity of a graph compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.
Abstract Reasoning¶
- Identify the carrier. State what the elements, states, objects, or observations are: a finite confusability graph, independent sets, strong graph powers, block length, logarithmic message count, and asymptotic growth rate. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the rate is derived from independence numbers of strong graph powers under the declared graph convention independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of information theory because they reuse a finite confusability graph, independent sets, strong graph powers, block length, logarithmic message count, and asymptotic growth rate, Vertices encode symbols and edges encode confusability; length-n codewords form vertices of the strong product, and a zero-error code is an independent set., and type the carrier, state every parameter and convention in the definition, test that the rate is derived from independence numbers of strong graph powers under the declared graph convention, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Shannon capacity of a graph Domain-specific
Parents (1) — more general patterns this builds on
-
Shannon capacity of a graph is a kind of Channel Capacity Prime
The proposed strict upward parent is
prime:channel_capacity.
Hierarchy path (1) — routes to 1 parentless root
- Shannon capacity of a graph → Channel Capacity
Neighborhood in Abstraction Space¶
Shannon capacity of a graph sits in a crowded region of the domain-specific corpus (24th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Unclustered & Miscellaneous (1565 abstractions)
Nearest neighbors
- Independent set (graph theory) — 0.92
- Split graph — 0.92
- Strong product of graphs — 0.91
- Graph neural network — 0.90
- Triangle-free graph — 0.90
Computed from structural-signature embeddings · 2026-09-08