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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.

Version
v1 · 2026-09-28 · History
Domain-specific #
9736
Domain group
Natural Sciences
Origin domain
Physics
Subdomains
Network Science, Complex Networks → Physics

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. Tie policy is part of the construction and must be stated before structural claims are compared.

Structural Signature

Sig role-phrases:

  • substrate graph. Defines allowed local adjacency. Constitutive base. If altered: Without substrate edges the neighborhood rule changes.
  • scalar potential. Assigns a comparable value to each node. Constitutive field. If altered: Vector or incomparable labels lack a simple extremum.
  • closed neighborhood. Combines each node with its immediate neighbors. Identity-bearing choice set. If altered: Excluding self can force an edge where a local optimum should loop.
  • extremum rule. Selects local minimum or maximum under a declared convention. Constitutive transformation. If altered: A random neighbor produces another directed graph.
  • one-out-link result. Creates a functional directed edge from every node. Diagnostic structure. If altered: Tie handling must preserve a specified rule.

What It Is Not

  • Continuous gradient. Is a derivative vector or discrete edge meant?
  • Edge orientation. Are all edges directed rather than one selected?
  • Nearest-neighbor graph. Is geometric distance instead of potential used?
  • Flow network. Are capacities and conservation present?

Scope of Application

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.

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.

Abstract Reasoning

  1. Specify the undirected substrate graph.
  2. Assign and validate one scalar potential per node.
  3. Form each closed one-step neighborhood.
  4. Apply a consistent argmin or argmax and tie rule.
  5. Analyze the derived directed graph separately from the substrate.

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.

Examples

Canonical

On a substrate graph, each node compares its own height with adjacent heights and points to the lowest under a fixed tie rule; a local minimum receives a self-loop.

Mapped back: substrate graph → allowed adjacencies; scalar potential → node heights; closed neighborhood → self plus neighbors; extremum rule → argmin; one-out-link result → directed edge or self-loop.

Applied / In Practice

A routing scheme chooses the lowest-potential node anywhere in the network, not only a neighbor; it is excluded because it violates the substrate-neighborhood constraint.

Mapped back: substrate graph → present but ignored; scalar potential → node score; closed neighborhood → replaced by global set; extremum rule → global minimum; one-out-link result → nonlocal edge.

Structural Tensions

T1: local descent vs. global optimum. Each edge follows a neighborhood extremum but can terminate at a nonglobal basin. Diagnostic: Which minima are reachable?

T2: deterministic rule vs. tie ambiguity. Equal potentials can create different derived topology. Diagnostic: How are ties resolved?

Structural–Framed Character

Description turns on substrate graph, scalar potential, closed neighborhood, extremum rule, one-out-link result. Skeletal core. A local ordered choice rule transforms a relation graph into directed functional flow. Domain-bound accent. Substrate nodes, neighbors, scalar potentials, extrema, self-links, and in-degree define the network. Transfer remains bounded because Why not prime. Local extremum routing is portable; this is a network-science construction. The negative boundary is concrete: Any directed network, gradient field, nearest-neighbor graph, flow network, or orientation of edges is not automatically a gradient network. Gradient network is structural: graph adjacency, scalar order, and local selection fully determine the object once ties are fixed. Its character: a one-out-link directed graph induced by local potential extrema.

Structural Core vs. Domain Accent

Skeletal core. A local ordered choice rule transforms a relation graph into directed functional flow.

Domain-bound accent. Substrate nodes, neighbors, scalar potentials, extrema, self-links, and in-degree define the network.

Why not prime. Local extremum routing is portable; this is a network-science construction.

This entry is a kind of Network.

  • Graph transformation. Substrate plus field produces a derived digraph.
  • Argmax/argmin. Local ordering selects one destination.
  • No strict parent is asserted.

Relationships to Other Abstractions

Local relationship map for Gradient NetworkParents appear above the current abstraction, mutual partners to the right, and children below. Node labels state whether each abstraction is prime or domain-specific; colors identify relation types.Gradient NetworkDOMAINPrime abstraction: Network — is a kind ofNetworkPRIME

Current abstraction Gradient Network Domain-specific

Parents (1) — more general patterns this builds on

  • 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

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

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With

  • Continuous gradient. Tell: Is a derivative vector or discrete edge meant?
  • Edge orientation. Tell: Are all edges directed rather than one selected?
  • Nearest-neighbor graph. Tell: Is geometric distance instead of potential used?
  • Flow network. Tell: Are capacities and conservation present?

References

  • Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Gradient_network (revision 1348378333).
  • Preserved source candidate: https://www.sciencedirect.com/science/article/pii/S0577907316301654
  • Preserved source candidate: http://cnls.lanl.gov/External/people/highlights/Toroczkai_net.pdf
  • Preserved source candidate: https://web.archive.org/web/20061004090327/https://cnls.lanl.gov/External/people/highlights/Toroczkai_net.pdf
  • Preserved source candidate: https://www.sciencedirect.com/science/article/pii/S0378437111002317

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.