Extended Newton Graphs

• Extended Newton Graphs are graph-based representations that encode Newton-type information by replacing dense analytic data with combinatorial structures.
• They extend classical Newton methods by incorporating nonsmooth derivatives, controlling graphs, and bifurcation models across diverse mathematical settings.
• Practical applications include classifying elliptic flows on tori, analyzing convergence in p-adic differential equations, and enabling efficient sparse computational techniques.

Searching arXiv for recent and foundational papers related to “Extended Newton Graphs,” including nonsmooth Newton flows, elliptic Newton graphs, pseudo Newton graphs, graphical Newton methods, and controlling graphs for convergence Newton polygons. Extended Newton graphs arise in several mathematically distinct research programs that extend classical Newtonian constructions by replacing dense local analytic data with graph-like objects. The literature uses closely related but nonidentical meanings, which suggests that the phrase does not denote a single universally standardized object. In one line of work, Newton graphs are cellularly embedded toroidal graphs that classify structurally stable elliptic Newton flows; in another, pseudo Newton graphs and nuclear Newton graphs encode degenerate or bifurcating elliptic flows; in $p$-adic analysis, locally finite controlling graphs and their refinements govern the variation of the convergence Newton polygon; and in nonsmooth optimization and sparse second-order computation, Newton structure is reformulated through graphical derivatives or computational graphs rather than classical Hessians (Helminck et al., 2016, Helminck et al., 2017, Poineau et al., 2024, Garrido et al., 2024, Srinivasan et al., 2015).

1. Terminological scope and common structure

The common feature across these usages is the transfer of Newton-type information from a purely local analytic object to a graph-based representation. Depending on context, the graph may encode a phase portrait on a torus, the locus where radii of convergence vary on a Berkovich curve, or the sparsity pattern of a structured optimization problem. This suggests a family resemblance among the constructions rather than a single formal definition.

Setting	Newton-related graph structure	Mathematical role
Elliptic Newton flows on a torus	Newton graph, dual graph	Encodes the phase portrait and classifies structurally stable flows
Degenerate elliptic Newton flows	Pseudo Newton graph, nuclear Newton graph	Encodes bifurcation and non-structurally stable configurations
$p$-adic differential equations	Controlling graph, cumulative and linearized graphs	Records where radii and partial heights of the convergence Newton polygon vary
Nonsmooth or sparse Newton frameworks	Graphical derivative inclusion, computational DAG	Generalizes Newton flow or exact Newton-step computation beyond smooth dense settings

A recurrent misconception is to treat all of these objects as variants of one combinatorial graph theory. The papers instead attach the Newton label to different structures: embedded toroidal multigraphs in elliptic dynamics, locally finite skeleton-like subsets in nonarchimedean geometry, and graph-organized second-order operators in optimization. The common denominator is methodological: graph structure is used to preserve Newton-type information in settings where the classical smooth Hessian picture is inadequate.

2. Generalized Newton flows in nonsmooth and nonconvex optimization

A continuous-time generalized Newton framework for nonsmooth and nonconvex optimization is developed for

$$\min \varphi(x), \qquad \varphi=\varphi_1+\varphi_2,$$

where $\varphi_1:\mathbb{R}^d\to\mathbb{R}\cup\{\infty\}$ is primal lower regular and $\varphi_2:\mathbb{R}^d\to\mathbb{R}$ is locally Lipschitz. The central evolution law is the differential inclusion

$$0\in \partial\varphi_1(x(t))+\partial\varphi_2(x(t))+DF(x(t))(\dot x(t)) \quad\text{for a.e. }t\in\mathbb{R}_+,\qquad x(0)=x_0,$$

where $F:\mathbb{R}^d\to\mathbb{R}^d$ is locally Lipschitz, $DF(x(t))(\dot x(t))$ is the graphical derivative, and $\partial\varphi_i$ denotes the Clarke subdifferential. The formulation is explicitly presented as the kind of framework one would describe as an “extended Newton graph”: the classical Newton flow

$\dot x(t)=-(\nabla^2 f(x(t)))^{-1}\nabla f(x(t))$

is replaced by a graph-based differential inclusion built from generalized derivatives (Garrido et al., 2024).

The graphical derivative is defined through the tangent geometry of the graph of a mapping. For a set-valued map $p$0,

$p$1

and

$p$2

with $p$3 the Bouligand tangent cone. When $p$4 is single-valued, the notation simplifies to $p$5. If $p$6 for smooth $p$7, then the graphical derivative reduces to Hessian action, so the inclusion becomes a genuine Newton-type flow. A chain rule for absolutely continuous curves,

$p$8

is the mechanism that links the generalized second-order object to the dynamics.

The function class is broad. Primal lower regularity is expressed by

$p$9

for nearby $$\min \varphi(x), \qquad \varphi=\varphi_1+\varphi_2,$$0 and $$\min \varphi(x), \qquad \varphi=\varphi_1+\varphi_2,$$1. The class contains proper lsc convex functions, weakly convex functions, lower-$$\min \varphi(x), \qquad \varphi=\varphi_1+\varphi_2,$$2 functions, and qualified convexly composite functions. The paper also introduces the notion of a $$\min \varphi(x), \qquad \varphi=\varphi_1+\varphi_2,$$3-lower-definite mapping,

$$\min \varphi(x), \qquad \varphi=\varphi_1+\varphi_2,$$4

as the generalized analogue of positive definiteness; if $$\min \varphi(x), \qquad \varphi=\varphi_1+\varphi_2,$$5 is locally strongly monotone, then $$\min \varphi(x), \qquad \varphi=\varphi_1+\varphi_2,$$6 is $$\min \varphi(x), \qquad \varphi=\varphi_1+\varphi_2,$$7-lower-definite.

Well-posedness is formulated through energetic solutions. A trajectory $$\min \varphi(x), \qquad \varphi=\varphi_1+\varphi_2,$$8 is energetic if

$$\min \varphi(x), \qquad \varphi=\varphi_1+\varphi_2,$$9

is nonincreasing, equivalently

$\varphi_1:\mathbb{R}^d\to\mathbb{R}\cup\{\infty\}$0

Under hypotheses $\varphi_1:\mathbb{R}^d\to\mathbb{R}\cup\{\infty\}$1, there exists a $\varphi_1:\mathbb{R}^d\to\mathbb{R}\cup\{\infty\}$2-energetic solution and $\varphi_1:\mathbb{R}^d\to\mathbb{R}\cup\{\infty\}$3 is absolutely continuous on compact intervals. Uniqueness is obtained only in special cases, namely when $\varphi_1:\mathbb{R}^d\to\mathbb{R}\cup\{\infty\}$4 is linear and $\varphi_1:\mathbb{R}^d\to\mathbb{R}\cup\{\infty\}$5, or when $\varphi_1:\mathbb{R}^d\to\mathbb{R}\cup\{\infty\}$6 and $\varphi_1:\mathbb{R}^d\to\mathbb{R}\cup\{\infty\}$7. In general, uniqueness can fail.

The asymptotic theory is organized around stationarity and metric-analytic regularity. For any bounded energetic trajectory, every accumulation point $\varphi_1:\mathbb{R}^d\to\mathbb{R}\cup\{\infty\}$8 is stationary: $\varphi_1:\mathbb{R}^d\to\mathbb{R}\cup\{\infty\}$9 If $\varphi_2:\mathbb{R}^d\to\mathbb{R}$0 is strongly metrically subregular at $\varphi_2:\mathbb{R}^d\to\mathbb{R}$1, then every energetic solution converges to $\varphi_2:\mathbb{R}^d\to\mathbb{R}$2. Under additional weak convexity assumptions, the function gap decays exponentially, and in one case the trajectory itself satisfies an exponential estimate. An alternative convergence mechanism uses a Kurdyka–Łojasiewicz inequality of the form

$\varphi_2:\mathbb{R}^d\to\mathbb{R}$3

which yields $\varphi_2:\mathbb{R}^d\to\mathbb{R}$4, $\varphi_2:\mathbb{R}^d\to\mathbb{R}$5, and rates ranging from finite-time to exponential and polynomial depending on $\varphi_2:\mathbb{R}^d\to\mathbb{R}$6. In this setting, the “extended Newton graph” idea is not an embedded combinatorial graph, but a generalized second-order evolution encoded through the graph of a locally Lipschitz mapping.

3. Newton graphs for structurally stable elliptic Newton flows

In the elliptic-flow literature, a Newton graph is a precise combinatorial-topological object attached to the phase portrait of an elliptic Newton flow on a torus. For a non-constant elliptic function $\varphi_2:\mathbb{R}^d\to\mathbb{R}$7 of order $\varphi_2:\mathbb{R}^d\to\mathbb{R}$8 with period lattice $\varphi_2:\mathbb{R}^d\to\mathbb{R}$9, the planar Newton flow

$$0\in \partial\varphi_1(x(t))+\partial\varphi_2(x(t))+DF(x(t))(\dot x(t)) \quad\text{for a.e. }t\in\mathbb{R}_+,\qquad x(0)=x_0,$$0

is desingularized to a $$0\in \partial\varphi_1(x(t))+\partial\varphi_2(x(t))+DF(x(t))(\dot x(t)) \quad\text{for a.e. }t\in\mathbb{R}_+,\qquad x(0)=x_0,$$1-vector field and descends to a globally defined flow on the torus $$0\in \partial\varphi_1(x(t))+\partial\varphi_2(x(t))+DF(x(t))(\dot x(t)) \quad\text{for a.e. }t\in\mathbb{R}_+,\qquad x(0)=x_0,$$2. Zeros of $$0\in \partial\varphi_1(x(t))+\partial\varphi_2(x(t))+DF(x(t))(\dot x(t)) \quad\text{for a.e. }t\in\mathbb{R}_+,\qquad x(0)=x_0,$$3 are attractors, poles are repellors, and critical points are saddles. Along every non-singular trajectory, $$0\in \partial\varphi_1(x(t))+\partial\varphi_2(x(t))+DF(x(t))(\dot x(t)) \quad\text{for a.e. }t\in\mathbb{R}_+,\qquad x(0)=x_0,$$4 is constant and $$0\in \partial\varphi_1(x(t))+\partial\varphi_2(x(t))+DF(x(t))(\dot x(t)) \quad\text{for a.e. }t\in\mathbb{R}_+,\qquad x(0)=x_0,$$5 is strictly decreasing (Helminck et al., 2016).

Structural stability is characterized by non-degeneracy: all zeros, poles, and critical points are simple, and no critical points are connected by Newton-flow trajectories. For such an $$0\in \partial\varphi_1(x(t))+\partial\varphi_2(x(t))+DF(x(t))(\dot x(t)) \quad\text{for a.e. }t\in\mathbb{R}_+,\qquad x(0)=x_0,$$6, the phase portrait determines an embedded graph $$0\in \partial\varphi_1(x(t))+\partial\varphi_2(x(t))+DF(x(t))(\dot x(t)) \quad\text{for a.e. }t\in\mathbb{R}_+,\qquad x(0)=x_0,$$7 on the torus with vertices equal to the $$0\in \partial\varphi_1(x(t))+\partial\varphi_2(x(t))+DF(x(t))(\dot x(t)) \quad\text{for a.e. }t\in\mathbb{R}_+,\qquad x(0)=x_0,$$8 zeros, edges equal to the $$0\in \partial\varphi_1(x(t))+\partial\varphi_2(x(t))+DF(x(t))(\dot x(t)) \quad\text{for a.e. }t\in\mathbb{R}_+,\qquad x(0)=x_0,$$9 unstable manifolds at the $F:\mathbb{R}^d\to\mathbb{R}^d$0 critical points, and faces equal to the $F:\mathbb{R}^d\to\mathbb{R}^d$1 basins of repulsion of the poles. The graph is a multigraph with no loops, each edge lies in the boundaries of two different faces, the facial walk around each face is Eulerian, and the embedding is connected and cellular. Consequently, $F:\mathbb{R}^d\to\mathbb{R}^d$2 has exactly $F:\mathbb{R}^d\to\mathbb{R}^d$3 vertices, $F:\mathbb{R}^d\to\mathbb{R}^d$4 edges, and $F:\mathbb{R}^d\to\mathbb{R}^d$5 faces.

Two combinatorial properties are decisive. The E-property requires that every facial walk be Eulerian; concretely, each edge is adjacent to two different faces, and every vertex on a facial boundary has even degree. The A-property is expressed in terms of positive angles satisfying local normalization conditions around vertices and faces, and is equivalent to a Hall-type condition. If $F:\mathbb{R}^d\to\mathbb{R}^d$6 is nonempty and $F:\mathbb{R}^d\to\mathbb{R}^d$7 is the subgraph generated by the faces indexed by $F:\mathbb{R}^d\to\mathbb{R}^d$8, then

$F:\mathbb{R}^d\to\mathbb{R}^d$9

The same paper relates this to a transversal condition and to solvability of a linear system derived from the incidence matrix of angles versus vertices and faces.

A Newton graph of rank $DF(x(t))(\dot x(t))$0 is therefore a cellularly embedded toroidal graph with $DF(x(t))(\dot x(t))$1 vertices, $DF(x(t))(\dot x(t))$2 edges, and $DF(x(t))(\dot x(t))$3 faces that satisfies both the A-property and the E-property. This definition is exact and restrictive. A toroidal graph is not a Newton graph merely because it has the correct counts; the Hall/Angle and Eulerian conditions are essential.

The main theorems are classification and representation. Structurally stable elliptic Newton flows are classified up to conjugacy by their Newton graphs: $DF(x(t))(\dot x(t))$4 Conversely, every Newton graph can be realized as $DF(x(t))(\dot x(t))$5 for some structurally stable elliptic function $DF(x(t))(\dot x(t))$6. Duality is induced by $DF(x(t))(\dot x(t))$7: the flow for $DF(x(t))(\dot x(t))$8 is the reverse of the flow for $DF(x(t))(\dot x(t))$9, and the associated graph is the negative dual graph. The theory also yields a polynomial-time recognition procedure: the E-property reduces to checking even vertex degrees, while the A-property becomes a Hall-type condition on an associated bipartite graph.

4. Pseudo Newton graphs, nuclear Newton graphs, and bifurcation

The Newton-graph classification applies only to structurally stable elliptic Newton flows. A further extension addresses non-structurally stable or degenerate configurations through pseudo Newton graphs and nuclear Newton graphs. Starting from a Newton graph $\partial\varphi_i$0, one deletes an edge shared by two faces, merges those faces, and then deletes degree-1 vertices and their incident edges as long as possible. The resulting connected cellularly embedded toroidal multigraph is a pseudo Newton graph of order $\partial\varphi_i$1. These graphs generally fail the E-property and need not satisfy the exact Newton-graph counts (Helminck et al., 2017).

The paper distinguishes one-face pseudo Newton graphs, denoted $\partial\varphi_i$2 or $\partial\varphi_i$3, and, for $\partial\varphi_i$4, additional two-face pseudo Newton graphs $\partial\varphi_i$5. The construction is explicitly intended to capture how structurally stable configurations degenerate. This is an extension of the earlier Newton-graph program not by enlarging the class of structurally stable graphs, but by introducing a graph language for bifurcation and loss of structural stability.

The extreme degenerate case is the nuclear Newton graph. It is a cellularly embedded graph on the torus with one vertex and two edges. Such graphs are connected, have one face, have two loops, have trivial rotation system, satisfy the A-property, and do not satisfy the E-property. They are therefore not Newton graphs. Dynamically, nuclear graphs arise from elliptic functions with one zero and one pole, both of order $\partial\varphi_i$6; the associated flow has exactly two critical points, hence two saddles, and all nuclear flows of the same order are mutually conjugate.

The bifurcation picture proceeds by splitting multiple zeros and poles and by breaking saddle connections through perturbation. In this way, the nuclear case acts as a combinatorial seed for more elaborate structurally stable flows. For $\partial\varphi_i$7 and $\partial\varphi_i$8, two realization results are proved: the graph $\partial\varphi_i$9 associated with a structurally stable flow in the relevant class is a pseudo Newton graph, and every pseudo Newton graph of the admissible types can be represented by some elliptic Newton flow. The main theorem states that any pseudo Newton graph of order $\dot x(t)=-(\nabla^2 f(x(t)))^{-1}\nabla f(x(t))$0, $\dot x(t)=-(\nabla^2 f(x(t)))^{-1}\nabla f(x(t))$1, represents an elliptic Newton flow; in particular, the third-order nuclear Newton flow creates, by splitting up zeros and poles, all structurally stable elliptic Newton flows of order $\dot x(t)=-(\nabla^2 f(x(t)))^{-1}\nabla f(x(t))$2, up to duality and topological equivalency.

This extension clarifies another frequent confusion. Newton graphs, pseudo Newton graphs, and nuclear Newton graphs are not interchangeable. The first classify structurally stable flows; the latter two describe degenerate or bifurcating situations in which the original Newton-graph axioms no longer hold, but a graph-theoretic phase-portrait model still survives.

5. Controlling graphs for convergence Newton polygons in $\dot x(t)=-(\nabla^2 f(x(t)))^{-1}\nabla f(x(t))$3-adic analysis

A different use of the phrase concerns the convergence Newton polygon of a $\dot x(t)=-(\nabla^2 f(x(t)))^{-1}\nabla f(x(t))$4-adic differential equation on a Berkovich curve. Here the relevant graph is not a phase portrait and not an optimization graph. The geometric setting consists of a complete valued field $\dot x(t)=-(\nabla^2 f(x(t)))^{-1}\nabla f(x(t))$5 of characteristic $\dot x(t)=-(\nabla^2 f(x(t)))^{-1}\nabla f(x(t))$6, a quasi-smooth Berkovich $\dot x(t)=-(\nabla^2 f(x(t)))^{-1}\nabla f(x(t))$7-analytic curve $\dot x(t)=-(\nabla^2 f(x(t)))^{-1}\nabla f(x(t))$8, and a locally free $\dot x(t)=-(\nabla^2 f(x(t)))^{-1}\nabla f(x(t))$9-module with connection $p$00 of finite rank $p$01. A pseudo-triangulation $p$02 yields a skeleton $p$03 together with a proper retraction

$p$04

This skeleton organizes the geometry on which radii of convergence are studied (Poineau et al., 2024).

For each $p$05, one has Baldassarri’s radii of convergence

$p$06

Each radius factors through a locally finite graph

$p$07

called the controlling graph of the $p$08-th radius. Outside $p$09, the radius is locally constant on virtual open disks. The cumulative graphs

$p$10

control the first $p$11 radii simultaneously and are closely tied to the convergence Newton polygon.

The partial heights

$p$12

are the main Newton-polygon invariants. The paper then defines a refined or linearized graph,

$p$13

by adding vertices at points where one of the radii fails to be $p$14-affine along an edge. In this sense, the “extended” graph is the subdivision that captures both the locus where the polygon changes and the precise places where slope breaks occur. This is a different notion of extended Newton graph from the toroidal-flow literature: it is a locally finite combinatorial support for the variation of radii and partial heights, not an encoding of a Newton iteration or a phase portrait.

The analytic control comes from Laplacian estimates. For a function on a curve,

$p$15

For every $p$16 and every $p$17,

$p$18

Outside an exceptional set $p$19, one has super-harmonicity,

$p$20

At interior points of the skeleton, under condition $p$21, the paper proves a local inequality involving $p$22, $p$23, $p$24, and the number $p$25 of spectral non-solvable radii.

The main new contribution is quantitative control of graph size. Constants

$p$26

govern estimates on the numbers of weighted vertices and edges of the linearized graph on virtual open disks, open pseudo-annuli, and finite curves. The paper also establishes a Grothendieck–Ogg–Shafarevich-type formula for the total height $p$27. Under suitable hypotheses, the de Rham cohomology is finite-dimensional and

$p$28

where $p$29 is defined from boundary slopes and local Laplacians. Under the same assumptions, the size of the graph controlling the total height is bounded in terms of $p$30, $p$31, and $p$32. In this setting, extended Newton graphs are finite or quasi-finite combinatorial objects that measure the complexity of variation of the convergence Newton polygon.

6. Computational graphs, polynomiography, and conceptual distinctions

Other Newton-related graph formalisms use graph structure in still different ways. In graphical second-order optimization, a function $p$33 is represented by a directed acyclic computational graph $p$34 with input nodes, intermediate states, node maps $p$35, and local costs $p$36. The structured objective is written as a sum of local terms over the graph, while state variables satisfy recursive equations $p$37. The Newton step for the original unconstrained objective is recovered from the Lagrange-Newton step of a KKT system for the constrained graph formulation, provided the current state is feasible and the multipliers are chosen as reverse-mode gradients $p$38. The resulting linear system can be solved by message passing on a tree decomposition in time linear in graph size and cubic in the tree-width of a related graph (Srinivasan et al., 2015).

This use of graph structure is algorithmic rather than topological. The graph is a computational DAG, not a toroidal embedding and not a locally finite control locus. The advantage is structural efficiency: Newton’s method is executed on the graph before substitution destroys sparsity. The paper’s central insight is that the cost of an exact Newton step can be governed by tree-width rather than ambient dimension.

A separate but adjacent line of work studies an extended Newton-type iteration for scalar equations $p$39,

$p$40

where $p$41. Under $p$42, $p$43, $p$44 on $p$45, and $p$46, the iteration converges to the unique root for every $p$47. The same idea is then used in the complex plane to generate polynomiographs: the color records which root attracts the initial point, and the shade records the number of iterations, with tolerance $p$48 and maximal iteration count $p$49 in the reported experiments (Karaca et al., 2017).

These polynomiographs are graphical outputs of Newton-type dynamics, but they are not Newton graphs in the toroidal sense and not controlling graphs in the $p$50-adic sense. Taken together, the various literatures indicate that “extended Newton graphs” is best understood as a family of graph-based extensions of Newtonian structure. In elliptic dynamics it means an embedded graph encoding the global phase portrait; in $p$51-adic analysis it means a graph controlling the convergence Newton polygon; in nonsmooth optimization it means a graphical-derivative formulation of Newton flow; and in sparse computation it means a computational graph that makes exact Newton steps tractable. This suggests that the phrase names a methodological pattern—encoding Newton-type information by graph-like structures—rather than a single canonical object.