Gradient Network¶
A directed network derived from an undirected substrate graph and a scalar potential, with each node pointing to the extremal-potential node in its closed neighborhood.
Core Idea¶
A gradient network converts an undirected substrate graph with scalar node values into a directed functional graph. For each node i, its closed neighborhood contains i and all adjacent nodes; the gradient rule points i toward the member with minimum or maximum potential, depending on convention. Because every node has one selected out-link, local extrema can point to themselves and other nodes flow toward basins or cycles shaped by the potential and graph.
Scope of Application¶
Use gradient network for network-science constructions with substrate, potentials, neighborhood, extremum orientation, and ties explicit. Use gradient network for network-science constructions with substrate, potentials, neighborhood, extremum orientation, and ties explicit.
- Transport models. Routes toward local potential extrema.
- Congestion studies. Examines in-degree concentration.
- Network landscapes. Identifies basins and local extrema.
- Optimization analogies. Studies local greedy motion.
- Graph transformation. Compares derived and substrate topology.
Clarity¶
Gradient here is a discrete selected edge, not a derivative vector. Including the node itself is decisive because it represents a local extremum as a self-link rather than forced motion. The closest near miss sets the boundary: A steepest-descent graph is closest: it may choose only strictly lower neighbors and stop at minima, whereas this definition includes self and yields one selected out-link under its convention.
Manages Complexity¶
The construction compresses graph and scalar field into a one-out-degree digraph. This reveals flow concentration while discarding nonselected substrate edges and sensitivity to ties or small potential changes. The central local descent–global optimum tradeoff is this: Each edge follows a neighborhood extremum but can terminate at a nonglobal basin. A second deterministic rule–tie ambiguity tension matters because Equal potentials can create different derived topology.
Abstract Reasoning¶
Use three linked moves: specify the undirected substrate graph; assign and validate one scalar potential per node; form each closed one-step neighborhood. As a collapse test, the case exits when links ignore substrate adjacency, potentials, or the local extremum rule. A fourth check is to apply a consistent argmin or argmax and tie rule.
Knowledge Transfer¶
Local-extremum routing transfers to energy landscapes and greedy dynamics. The exact graph definition stops at scalar potentials and closed neighborhoods; continuous gradients or arbitrary routing are only analogies. The nearest stopping boundary is explicit: A steepest-descent graph is closest: it may choose only strictly lower neighbors and stop at minima, whereas this definition includes self and yields one selected out-link under its convention. The inclusion test remains: A network qualifies when it is derived from a substrate graph by sending every node to an extremal-potential member of its closed one-step neighborhood. The structure no longer applies when the case exits when links ignore substrate adjacency, potentials, or the local extremum rule. No canonical parent prime is currently asserted; broader structural comparisons remain related-prime analogies until separately adjudicated in the DAG. Substrate plus field produces a derived digraph.
Relationships to Other Abstractions¶
Current abstraction Gradient Network Domain-specific
Parents (1) — more general patterns this builds on
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Gradient Network is a kind of Network Prime
Gradient Network is a domain-specific kind of network under the frozen identity and differentia.
Hierarchy path (1) — routes to 1 parentless root
- Gradient Network → Network → Reservoir-Flux Network → Conservation Laws → Invariance
Neighborhood in Abstraction Space¶
Gradient Network sits in a sparse region of the domain-specific corpus (60th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Dynamical Systems & Differential Structures (37 abstractions)
Nearest neighbors
- Twin-width — 0.86
- Edge Covering Number — 0.85
- Closed Linear Operator — 0.85
- Filtration (algebra) — 0.84
- Double-Pushout Graph Rewriting — 0.84
Computed from structural-signature embeddings · 2026-10-08