The Accounting Failure in Neighborhood Definitions
Standard neighborhood definitions in directed graphs suffer from a fundamental accounting failure: they cannot be partitioned without destroying ancestral data.
Loss of Ancestry
When a node's influence passes to its first neighbors, the connection to its second neighbors is reduced to a simple count—stripping distance-two neighbors of their lineage.
Degree Disconnect
This loss creates a severe disconnect, making it impossible to track the precise distribution of out-degree influence across multi-step hops.
Definition: The Seymour Diamond
This counting problem is canonically demonstrated in directed graph structures through the presence of shared descendants:
Let $u_i$ be a parent node, and let $v_{i+1}, w_{i+1} \in N^+(u_i)$ be distinct children of $u_i$.
Then the quadruple (u_i, v_{i+1}, w_{i+1}, y_{i+2}) forms a Seymour diamond.
Visualizing the Seymour Diamond
An interactive canvas representation of the quadruple showing how arcs converge on the second-hop descendant node.
The Double-Counting Problem
Seymour diamonds double-count second neighbors because summation treats distinct outgoing arcs as distinct target nodes.
When two children at distance one ($v_{i+1}$ and $w_{i+1}$) both point to the same node $y_{i+2}$, a sum of their out-degrees fails by counting $y_{i+2}$ twice.
The actual contribution to the second out-neighborhood is only one, as both directed arcs terminate at the exact same target node, $y_{i+2}$.
Remark: Why New Definitions Are Needed
Standard neighborhoods $N^+(v)$ and $N^{++}(v)$ discard layer-ancestry information, causing the Seymour diamond double-counting problem.
The canonical partition preserves positional data across layers, making double-counting resolvable.
The second neighborhood must be analyzed as a union of forward neighbor sets rather than an arithmetic sum.