SAME DEGREES
Same neighbors counted. Different worlds.
Keep three neighbors each. Change the connections. How far can a pulse reach?
Six links join the two rings.
One hop = one edge · the same source in every case
Twelve identities. Three neighbors each.
The degree sequence never changes. Inspect the actual neighbors for the selected construction.
A local count cannot describe the whole network.
Degree counts immediate neighbors. A route depends on how those connections continue. Here, matching every local count still permits two separate components.
Try source A at hop 2, then compare all three cases. At the final hop, switch to Apart: the other ring remains unreachable however long you wait.
The same budget, carefully stated
Every construction has 12 × 3 / 2 = 18 edges. That preserves the number of connections, not their physical length, construction cost, capacity, or travel time.
Spread is a comparison specimen, not an optimum. No claim is made that its layout is best for every source, hop, or network task.
Read the study
What stays the same
The labeled vertices A–L, their screen positions, and all twelve degrees. These are simple, undirected, unweighted graphs: each edge joins two different vertices, once. A–F occupy the outer ring; G–L occupy the inner ring.
What changes
Only the pairs of vertices connected by edges. Selecting another source changes the question for all three constructions together. Switching constructions preserves the source and selected hop.
How the three graphs are built
Spread is a hexagonal prism: a six-cycle on each ring, plus six corresponding cross-ring edges. Narrow starts from two triangular prisms, deletes C–F and I–L, and adds C–I and F–L. Apart consists of two separate triangular prisms.
The displayed ring partition has six, two, and zero crossing edges respectively. The global minimum edge cut has size three, two, and zero. These values describe these exact specimens.
What a pulse means
The source is reached at hop 0. Breadth-first search computes the fewest edges needed to reach each vertex. At hop h, all vertices with distance at most h are reached. The warm ring marks those reached for the first time at that hop.
Play advances those discrete layers for inspection. Its pacing is a presentation choice, not a simulation of elapsed physical time, traffic, infection, or electrical current. Edges illuminate after both ends are reached; that does not designate a unique route.
Exact network, finite drawing
Adjacency, degrees, components, and hop distances are computed from integer vertex IDs. The diagram uses ordinary floating-point drawing coordinates. Intersecting lines do not create extra connections; letters identify the only vertices.
No automatic motion starts on page load. Playback stops when the page is hidden. With reduced motion enabled, Send pulse shows the completed reachable set immediately; the hop slider and Next hop still inspect each layer.