FCFG-N: Extension of Fractal-Continued-Fraction Geometry to Network Sciences

 Author: Zhang Suhang, Luoyang, Henan

Abstract

Within the framework of Fractal-Continued-Fraction Geometry (FCFG), this paper extends the isomorphism relation between finite continued fractions and finite self-similar fractals to a class of recursive fractal networks. We define recursive fractal networks, the global similarity ratio of networks, and continued-fraction-type network invariants, establish the isomorphism axioms for FCFG-N, and prove that the values of continued fractions are strictly equal to the global similarity ratios of networks under specific recursive constructions. We further present the correspondence theorems for convergence, periodicity and boundedness, together with a dimensional formula.

Keywords: fractal-continued-fraction geometry; recursive fractal network; continued fraction; global similarity ratio; network invariant

1. Introduction

Fractal-Continued-Fraction Geometry (FCFG) establishes a formal isomorphism:

r_n = S_n

where r_n denotes the n-th order finite continued fraction, and S_n denotes the global similarity ratio of an n-th order finite self-similar fractal. This isomorphism indicates that arithmetic recursion and geometric recursion represent the same structure in finite-order cases.

Networks are mathematical constructs for applications. Graph-theoretic tools, fractal networks and box-counting methods in network science are essentially implementations of mathematical structures in concrete systems. Therefore, if FCFG can describe fractal geometry, and fractal geometry can describe fractal networks, FCFG can naturally be extended to the field of networks.

2. Preliminaries: Review of Core FCFG

2.1 Finite continued fractions

Let a_1,a_2,\dots,a_n be a sequence of positive integers. The n-th order finite continued fraction is defined as:

r_n = \cfrac{1}{a_1+\cfrac{1}{a_2+\cdots+\cfrac{1}{a_n}}}

We have r_n\in\mathbb{Q}.

2.2 Finite self-similar fractals

Let F_0 be the initial figure. We recursively copy a_k copies scaled by the factor 1/a_k to obtain F_k. The global similarity ratio of F_n is denoted S_n.

2.3 FCFG isomorphism axiom

r_n = S_n

3. Recursive Fractal Networks

Definition 3.1 (Recursive fractal network)

Let G_0=(V_0,E_0) be a finite connected seed graph. Given a positive integer sequence a_1,\dots,a_n, we define recursively:

G_k = \mathcal{R}(G_{k-1},a_k)

where \mathcal{R} denotes the recursive construction operator satisfying:

1. Replication: take a_k copies of G_{k-1}, denoted G_{k-1}^{(1)},\dots,G_{k-1}^{(a_k)};

2. Scaling: each copy is scaled by the factor 1/a_k;

3. Connection: connect the copies following fixed rules such that G_k remains connected.

G_n is called an n-th order recursive fractal network.

Definition 3.2 (Global similarity ratio of a network)

Let \ell(G) denote the characteristic length of network G. The global similarity ratio of G_n is defined as:

S_n(G) = \frac{\ell(G_n)}{\ell(G_{n-1})}

We require that S_n(G) is uniquely determined by a_1,\dots,a_n.

Definition 3.3 (Continued-fraction-type network invariant)

Let f be a network invariant. If f satisfies the recursion:

f(G_k) = a_k + \frac{1}{f(G_{k-1})}

then f is termed a continued-fraction-type invariant.

4. FCFG-N Isomorphism Axiom

Axiom 4.1 (FCFG-N isomorphism axiom)

For any recursive fractal network G_n generated by the sequence a_1,\dots,a_n, if a continued-fraction-type invariant f exists:

\boxed{\frac{1}{f(G_n)} = r_n}

In particular, if we set S_n(G)=1/f(G_n), then:

S_n(G) = r_n

5. Existence Theorem

Theorem 5.1 (Isomorphism for recursive path networks)

There exists a class of recursive fractal networks G_n such that:

S_n(G) = r_n

Proof:

Construct a recursive path network:

- G_0: one edge with two vertices;

- G_k: take a_k copies of G_{k-1} connected in series, with the equivalent length of each segment scaled by 1/a_k.

Let L_k be the equivalent path length of G_k. Then:

L_k = a_k + \frac{1}{L_{k-1}}

Thus

\frac{1}{L_k} = \cfrac{1}{a_k+\cfrac{1}{L_{k-1}}}

Recursive expansion yields:

\frac{1}{L_n} = r_n

Set S_n(G)=1/L_n, then S_n(G)=r_n.

\square

6. Fundamental Property Theorems

Theorem 6.1 (Convergence correspondence)

r_n converges if and only if S_n(G) converges.

Proof: By Axiom 4.1, S_n(G)=r_n, so their convergence is equivalent.

\square

Theorem 6.2 (Periodicity correspondence)

If a_k is a periodic sequence, then S_n(G) is a quadratic irrational number, and the network possesses self-similar periodicity.

Proof: Periodic continued fractions converge to quadratic irrationals. By Axiom 4.1, S_n(G) is also a quadratic irrational. The periodicity of the recursive network construction is directly determined by the periodicity of a_k.

\square

Theorem 6.3 (Boundedness correspondence)

If a_k is bounded, the degree distribution of the network is bounded and the fractal dimension is finite.

Proof: Bounded a_k implies bounded replication numbers and scaling ratios at each layer, hence bounded vertex degrees of the network. The fractal dimension is given by \log N/\log a, which remains finite under the boundedness condition.

\square

7. Dimensional Formula

Theorem 7.1 (Fractal dimension of the network)

Let the replication number at each layer be N and the scaling ratio be a. The fractal dimension of the network reads:

D = \frac{\log N}{\log a}

Proof: From the box-counting definition, N boxes of radius 1/a cover the network. Thus

D = \lim_{\epsilon\to 0} \frac{\log N(\epsilon)}{\log(1/\epsilon)} = \frac{\log N}{\log a}

\square

8. Discussion

8.1 Relation to FCFG

FCFG-N is the extension of FCFG to network domains. FCFG deals with continuous geometric objects, while FCFG-N addresses discrete network objects. Both share the same isomorphism axiom.

8.2 Relation to network science

Networks are products of applied mathematics. FCFG-N does not supersede existing methods in network science; instead, it provides a continued-fraction description for recursive fractal networks.

8.3 Limitations

- Valid only for recursive, self-similar, multi-scale networks;

- Random networks, scale-free networks and small-world networks generally do not satisfy the conditions;

- Only finite-order theory is established at present.

8. Conclusion

This paper establishes a rigorous mathematical connection between recursive fractal networks and finite continued fractions within the FCFG framework. The main results are as follows:

1. FCFG-N isomorphism axiom: S_n(G)=r_n;

2. Existence theorem: recursive path networks satisfy the isomorphism;

3. Correspondence theorems for convergence, periodicity and boundedness;

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4. Fractal dimensional formula for networks.

References

[1] Original materials of the FCFG framework.

[2] Research on fractal geometry and continued fractions.

[3] Fractals and box-counting methods in complex networks.

[4] Recursive structures and multiscale analysis in network science.