Independence Attractor in Graph Dynamics

• Independence attractor is defined as the Hausdorff limit of zeros of independence polynomials from iterated lexicographic graph products, linking combinatorial structures to dynamical behavior.
• The reduced independence polynomial transforms the problem into studying preimages of -1 under iteration, revealing deep connections with Julia sets and fractal patterns.
• Rigidity results constrain the topology of attractors by ruling out circular forms and restricting line-segment possibilities to specific intervals determined by polynomial coefficients.

The independence attractor of a finite simple graph is the Hausdorff limit of the zero sets of the independence polynomials of the graph’s lexicographic powers. Through the reduced independence polynomial, it is simultaneously a graph-theoretic and a dynamical object: a limiting preimage set of $-1$ under polynomial iteration, often coinciding with a Julia set. For graphs with independence number three, its connectedness is classified explicitly by the cubic coefficients of the independence polynomial, while more general rigidity results show that independence attractors are never circles and that line-segment attractors can occur only as $\left[-\frac{4}{k},0\right]$ for $k\in\{1,2,3,4\}$ (Manna et al., 6 Aug 2025, Khetawat et al., 27 May 2025).

1. Definition and graph-theoretic construction

Let $G$ be a finite simple graph. An independent set in $G$ is a set of pairwise non-adjacent vertices. The independence polynomial of $G$ is

$I_G(z)=a_0+a_1 z+a_2 z^2+\cdots+a_\alpha z^\alpha,$

where $a_i$ is the number of independent sets of size $i$, $a_0=1$, and $\left[-\frac{4}{k},0\right]$0 is the independence number of $\left[-\frac{4}{k},0\right]$1, namely the cardinality of a largest independent set.

If $\left[-\frac{4}{k},0\right]$2 denotes the $\left[-\frac{4}{k},0\right]$3-times lexicographic product of $\left[-\frac{4}{k},0\right]$4 with itself, the independence attractor of $\left[-\frac{4}{k},0\right]$5 is defined by

$\left[-\frac{4}{k},0\right]$6

where the limit is taken in the Hausdorff metric on compact subsets of the plane (Manna et al., 6 Aug 2025).

The definition turns a sequence of algebraic zero sets into a single limiting compact subset of $\left[-\frac{4}{k},0\right]$7. In this sense, the independence attractor is not an attractor in the phase-space sense of an autonomous ODE; it is a limit set attached to iterated graph composition and polynomial zeros. Its basic input is entirely combinatorial—the independent sets of $\left[-\frac{4}{k},0\right]$8—but its asymptotic behavior is governed by polynomial iteration.

2. Reduced independence polynomials and complex dynamics

The natural dynamical polynomial is the reduced independence polynomial

$\left[-\frac{4}{k},0\right]$9

A key identity is

$k\in\{1,2,3,4\}$0

so that

$k\in\{1,2,3,4\}$1

Thus the zeros of the iterated independence polynomials are exactly the preimages of $k\in\{1,2,3,4\}$2 under iterates of $k\in\{1,2,3,4\}$3 (Manna et al., 6 Aug 2025).

This identity places independence attractors inside the framework of complex polynomial dynamics. The attractor is tied to the Julia set of $k\in\{1,2,3,4\}$4, and the relation is especially clean when $k\in\{1,2,3,4\}$5 is not a multiple root of $k\in\{1,2,3,4\}$6. In that case,

$k\in\{1,2,3,4\}$7

If $k\in\{1,2,3,4\}$8 is a multiple root, then $k\in\{1,2,3,4\}$9 is the disjoint union of $G$0 and the zeros of the iterates $G$1 (Manna et al., 6 Aug 2025).

A closely related object is the independence fractal $G$2. For $G$3, earlier work cited in the later literature gives

$G$4

and the relation between $G$5 and $G$6 is governed by the root structure at $G$7: if $G$8 is not a root or is a simple root of $G$9, then $G$0; if $G$1 is a multiple root, then $G$2 is $G$3 together with additional limit points coming from iterates of $G$4 (Khetawat et al., 27 May 2025).

For cubic independence polynomials,

$G$5

the condition that $G$6 be a multiple root is

$G$7

which is equivalent to

$G$8

This exceptional case is the source of the extra isolated or orbit-type pieces that can appear in $G$9.

3. Cubic parameter space for graphs with independence number three

When the independence number is three, the independence polynomial has the cubic form

$G$0

and the topology of $G$1 is controlled by the coefficients $G$2 (Manna et al., 6 Aug 2025).

For

$G$3

the critical points are the zeros of $G$4: $G$5 Accordingly, the sign of

$G$6

determines whether the polynomial is bicritically non-real, unicritical, or bicritically real: $G$7

A second discriminant governs the nonzero fixed points of $G$8. Since

$G$9

the nonzero fixed points are

$I_G(z)=a_0+a_1 z+a_2 z^2+\cdots+a_\alpha z^\alpha,$0

Hence the sign of

$I_G(z)=a_0+a_1 z+a_2 z^2+\cdots+a_\alpha z^\alpha,$1

determines whether the nonzero fixed points are non-real, repeated, or real.

These two discriminants provide the basic geometry of the parameter space. The first determines the critical-point configuration, and the second determines the fixed-point configuration. The classification of connectedness for independence number three is obtained by combining them with several finer threshold values: $I_G(z)=a_0+a_1 z+a_2 z^2+\cdots+a_\alpha z^\alpha,$2

4. Connectedness classification for independence number three

The case $I_G(z)=a_0+a_1 z+a_2 z^2+\cdots+a_\alpha z^\alpha,$3 is exceptional. There is only one graph with independence number three and $I_G(z)=a_0+a_1 z+a_2 z^2+\cdots+a_\alpha z^\alpha,$4, namely

$I_G(z)=a_0+a_1 z+a_2 z^2+\cdots+a_\alpha z^\alpha,$5

Then

$I_G(z)=a_0+a_1 z+a_2 z^2+\cdots+a_\alpha z^\alpha,$6

and $I_G(z)=a_0+a_1 z+a_2 z^2+\cdots+a_\alpha z^\alpha,$7 is conformally conjugate to $I_G(z)=a_0+a_1 z+a_2 z^2+\cdots+a_\alpha z^\alpha,$8 via $I_G(z)=a_0+a_1 z+a_2 z^2+\cdots+a_\alpha z^\alpha,$9. Its Julia set is the circle centered at $a_i$0 with radius $a_i$1, and

$a_i$2

This is the unique “circle plus isolated point” example in the cubic theory (Manna et al., 6 Aug 2025).

For $a_i$3, the connectedness picture is coefficient-driven.

Bicritically non-real and unicritical regime: if $a_i$4, then $a_i$5 is totally disconnected. If $a_i$6, then the only non-totally-disconnected case is the exceptional $a_i$7 example above; for $a_i$8, $a_i$9 is again totally disconnected. In the bicritically non-real case, the proof uses the fact that the critical points are complex conjugates and that their forward orbits escape, with estimates involving the critical disk

$i$0

Real critical points but non-real nonzero fixed points: if

$i$1

then $i$2 is totally disconnected.

Repeated nonzero fixed point: if

$i$3

then $i$4 is connected when $i$5, and disconnected but not totally disconnected for all other values of $i$6.

Real fixed-point regime below $i$7: if

$i$8

then $i$9 is disconnected but not totally disconnected. If

$a_0=1$0

then $a_0=1$1 is connected for $a_0=1$2 and disconnected but not totally disconnected for $a_0=1$3. If

$a_0=1$4

then $a_0=1$5 is disconnected but not totally disconnected. If

$a_0=1$6

then $a_0=1$7 is connected for $a_0=1$8 and disconnected but not totally disconnected for $a_0=1$9. If

$\left[-\frac{4}{k},0\right]$00

then $\left[-\frac{4}{k},0\right]$01 is generally disconnected, except for the two special connected cases

$\left[-\frac{4}{k},0\right]$02

Boundary $\left[-\frac{4}{k},0\right]$03: in this case,

$\left[-\frac{4}{k},0\right]$04

and

$\left[-\frac{4}{k},0\right]$05

Moreover, when $\left[-\frac{4}{k},0\right]$06, the Julia set is actually a line segment.

Beyond $\left[-\frac{4}{k},0\right]$07: if

$\left[-\frac{4}{k},0\right]$08

then $\left[-\frac{4}{k},0\right]$09 is disconnected. The paper notes that it can be totally disconnected when both critical orbits escape; an explicit example is

$\left[-\frac{4}{k},0\right]$10

which has totally disconnected attractor (Manna et al., 6 Aug 2025).

The cubic theory therefore distinguishes all three topological outcomes—connected, disconnected but not totally disconnected, and totally disconnected—within a single coefficient space. It also shows that these outcomes are not merely abstract possibilities: the paper provides examples exhibiting each one.

5. Simple topological forms: circles, line segments, and rigidity

A later structural analysis addresses the possibility that an independence attractor might be topologically simple in a global sense. The strongest negative result is that no graph has an independence attractor that is a circle (Khetawat et al., 27 May 2025).

This does not conflict with the cubic example

$\left[-\frac{4}{k},0\right]$11

because that attractor is not a circle; it is a disjoint union of a circle and an isolated point. The obstruction to circularity is dynamical. If $\left[-\frac{4}{k},0\right]$12 is not a root or is a simple root of $\left[-\frac{4}{k},0\right]$13, then $\left[-\frac{4}{k},0\right]$14, hence the attractor equals the Julia set of $\left[-\frac{4}{k},0\right]$15. A key lemma shows that if the Julia set of a reduced independence polynomial is a circle, then the polynomial must be of the form

$\left[-\frac{4}{k},0\right]$16

which corresponds only to a complete graph $\left[-\frac{4}{k},0\right]$17. Complete graphs, however, do not produce circular independence attractors. If $\left[-\frac{4}{k},0\right]$18 is a multiple root, then the attractor contains extra pieces coming from the orbit of $\left[-\frac{4}{k},0\right]$19, so assuming that the whole attractor is a circle again leads to a contradiction.

The line-segment case is more rigid still. If $\left[-\frac{4}{k},0\right]$20 is a line segment, then

$\left[-\frac{4}{k},0\right]$21

This extends an earlier result for graphs with independence number three to arbitrary independence number. The proof uses the fact that if the Julia set of $\left[-\frac{4}{k},0\right]$22 is a line segment, then it is affinely conjugate to a Chebyshev polynomial, while the positivity and integrality of the coefficients of $\left[-\frac{4}{k},0\right]$23 force the segment to lie on the real axis and restrict the affine scaling to four possibilities.

The four admissible intervals are therefore

$\left[-\frac{4}{k},0\right]$24

The same paper constructs graphs with independence number four realizing each of these possibilities. In particular, if

$\left[-\frac{4}{k},0\right]$25

then

$\left[-\frac{4}{k},0\right]$26

and for each $\left[-\frac{4}{k},0\right]$27 there exists at least one graph with independence number $\left[-\frac{4}{k},0\right]$28 whose independence attractor is $\left[-\frac{4}{k},0\right]$29 (Khetawat et al., 27 May 2025).

These results show that topological simplicity is highly constrained. A circle never occurs, and a line segment, when it occurs, is confined to a short explicit list.

6. Scope of the term and adjacent usages of “independence”

In current arXiv usage, “independence attractor” is a specific graph-theoretic term referring to the limit set

$\left[-\frac{4}{k},0\right]$30

It should be distinguished from several nearby but different uses of “independence” in attractor-related literature.

In inflationary cosmology, “renormalization group independence of cosmological attractors” refers to the fact that for $\left[-\frac{4}{k},0\right]$31- and $\left[-\frac{4}{k},0\right]$32-attractors the leading-order predictions

$\left[-\frac{4}{k},0\right]$33

remain robust under RG improvement when the RG scale satisfies a regularity condition; the object under discussion is a family of inflationary models, not a set-valued attractor in the complex plane (Fumagalli, 2016).

In ELKO cosmology, the relevant statement is that a fast-roll inflation attractor is independent of the form of the potential. There the fixed-point condition

$\left[-\frac{4}{k},0\right]$34

is expressed entirely in terms of kinetic and matter fractions, and the perturbed equations on that branch do not explicitly depend on the potential (Basak et al., 2012).

In finite-time basin stability, “independence time” is the recovery time after which successive large perturbations can be treated as approximately independent. It is defined by

$\left[-\frac{4}{k},0\right]$35

and quantifies when the system has effectively returned close enough to the attractor that later kicks have the same basin-exit probability as kicks applied from the attractor itself (Schultz et al., 2017).

These usages share the vocabulary of “attractor” and “independence,” but they concern different structures: RG robustness, potential-independence of fixed points, recovery-time separation of perturbations, and, in graph theory, a Hausdorff limit of polynomial zero sets. The graph-theoretic independence attractor is therefore best understood as a specialized object at the interface of combinatorics, polynomial iteration, and complex dynamics.