On the Degree Distribution of Haros Graphs

Abstract

Haros graphs are a graph-theoretical representation of real numbers in the unit interval. The degree distribution of the Haros graphs provides information regarding the topological structure and the associated real number. This article provides a comprehensive demonstration of a conjecture concerning the analytical formulation of the degree distribution. Specifically, a theorem outlines the relationship between Haros graphs, the corresponding continued fraction of its associated real number, and the subsequent symbolic paths in the Farey binary tree. Moreover, an expression that is continuous and piecewise linear in subintervals defined by Farey fractions can be derived from an additional conclusion for the degree distribution of Haros graphs.

1. Introduction

The study of the structure of real numbers has been approached from a variety of perspectives [1,2,3,4]. The representation by continued fractions and the representation through the Farey tree are examples of canonical representations [5,6,7]. Recent graph-theoretical research has provided a new representation of real numbers using Haros graphs [8]. These graphs are swayed by the approach of Horizontal Visibility Graphs to the quasiperiodic route [9,10,11,12]. Furthermore, Haros graphs are similar to other structures, such as Farey graphs [13,14,15]. Haros graphs provide a graph description of the unit interval $[0,1]$ establishing a one-to-one correspondence with the well-known Farey sequences ${\mathcal{F}}_n$, where

$${\mathcal{F}}_n=\left\{\frac{p}{q}\in [0,1]:0\le p\le q\le n,(p,q)=1\right\}.$$

The Haros graph set $\mathcal{G}$ is recursively generated from an initial graph (defined as two nodes joined by an edge) and the concatenation graph-operator ⊕ shown in Figure 1. Hence, the set $\mathcal{G}$ may be represented as a binary tree (see Figure 1). Since ${lim}_n{\mathcal{F}}_n=[0,1]$, the bijection can be extended to the unit interval, where rational numbers are associated with finite Haros graphs, and irrational numbers correspond to infinite Haros graphs.

Consequently, a one-to-one correspondence $\tau$ exists between real numbers $x\in [0,1]$ and Haros graphs $\tau \left(x\right)=G_x\in \mathcal{G}$. One of the main features investigated in [8] is the degree distribution $P(k,x)$ of Haros graphs $G_x$, which is the probability that a randomly selected node in $G_x$ has degree k. The degree distribution was deemed a fruitful tool because Haros graphs are uniquely determined by the degree sequence [16], whereas the degree distribution is a marginal distribution of the degree sequence. Indeed, the degree distribution for the three initial values of k confirms:

$$P(k,x<1/2)=\left\{\begin{array}{cc}x,\hfill & k=2\hfill \\ 1-2x,\hfill & k=3\hfill \\ 0,\hfill & k=4;\hfill \end{array}\right.P(k,x>1/2)=\left\{\begin{array}{cc}1-x,\hfill & k=2\hfill \\ 2x-1,\hfill & k=3\hfill \\ 0,\hfill & k=4.\hfill \end{array}\right.$$

(1)

In contrast to the initial values, which are related to the real number x associated with $G_x$, the closed form of the degree distribution $P(k,x)$ has only been drawn for degrees $k\ge 5$. Taking the above fact into account, this paper outlines two theorems to complete the degree distribution expression, based on two distinct approaches. Initially, a complete description of $P(k,x)$ is provided only in terms of the continued fraction of x or, alternatively, in terms of the Haros graph creation process codified along the symbolic binary path. The second result demonstrates the properties of $P(k,x)$ as a continuous real-valued piecewise linear function.

This paper is separated into four sections. Section 2 gives a brief overview of Haros graphs and their connections to Farey sequences, continued fractions, and the Farey binary tree. Section 3 provides the two main results: the first theorem states the closed form of $P(k,p/q)$ with respect to truncations of the continued fraction of $p/q$, whereas a second theorem rewrites the preceding result using the position of $p/q$ in the Farey binary tree. Section 4 concludes the work. In addition, Appendix A includes detailed proofs for the assertions in Section 3.

2. Preliminaries

The Farey binary tree is a canonical way of representing the set of rational numbers in $[0,1]$ as a binary tree starting with the fractions $0/1,1/1$—the elements of the Farey sequence ${\mathcal{F}}_1$—and creating new irreducible fractions by the mediant sum of two consecutive fractions in ${\mathcal{F}}_n$:

$$\frac{p}{q}\oplus \frac{r}{s}=\frac{p+r}{q+s}.$$

(2)

The binary tree representation allocates each rational number to a level k of the tree, denoted ${\ell }_k$. For instance, the three first levels consist of:

$${\ell }_1=\left\{\frac{0}{1},\frac{1}{1}\right\};{\ell }_2=\left\{\frac{1}{2}\right\};{\ell }_3=\left\{\frac{1}{3},\frac{2}{3}\right\}.$$

This representation is closely related to the continued fraction, a powerful technique for representing a real number in the interval $[0,1]$ as

$$x=\frac{1}{a_1+\frac{1}{a_2+\frac{1}{a_3+\dots }}}=[a_1,a_2,a_3,\dots ].$$

The relationship was presented in [7], and it has been established that a number with a continued fraction expression $[a_1,a_2,a_3,\dots ]$ is associated with a symbolic binary path in the Farey binary tree $L^{a_1}{R}^{a_2}{L}^{a_3}\dots$, where $L^q$ is interpreted as a sequence of q symbols L (if the symbolic path is finite, the last symbol has an index $a_n-1$). Therefore, since every irrational number has an infinite continued fraction, the irrational numbers are reachable through an infinite path in the Farey binary tree. Moreover, the continued fraction allows a sequence of rational so-called convergents [6] defined as:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{p}_k=\hfill & a_k·p_{k-1}+p_{k-2}\hfill \\ q_k=\hfill & a_k·q_{k-1}+q_{k-2},\hfill \end{array}\right.\end{array}$$

(3)

with initial values $p_{-2}=0,q_{-2}=1,p_{-1}=1,$ and $q_{-1}=0$, where

$$\underset{k\to \infty }{lim}\frac{p_k}{q_k}=\underset{k\to \infty }{lim}[a_1,\dots ,a_k]=[a_1,a_2,\cdots ]=x.$$

As stated in the preceding section, the Haros graphs set $\mathcal{G}$ provides a graph-based representation of the unit interval $[0,1]$. The primary objective is to reproduce, in a graph scenario, the mediant sum—described in Equation (2)—that was utilized to construct ${\mathcal{F}}_n$ [1]. The concatenation graph is depicted in Figure 1, and Flanagan et al. [17] provide a comprehensive definition. Every Haros graph G (except for the initial graph) is therefore described as $G=G_L\oplus G_R$, where $G_L,G_R$ are also Haros graphs. However, just as the mediant sum only takes to two nearby fractions in Farey sequences, Haros graphs only can be concatenated if $G_L$ and $G_R$ are also adjacent.

In order to analyze the topological structure of a Haros graph, the probability distribution degree $P(k,x)$ proves to be a useful instrument. Equation (1) identifies the three first values of k, although by construction $P(k,0)=P(k,1)=0$. In addition, for degrees $k\ge 5$, Theorem 2, in [8], determines that the degree values for which $P(k,x)=0$ rely on the symbol repetition—$RR$ or $LL$—in the symbolic path of the Haros graph tree to reach $G_x$. Moreover, the same research conjectures follow for the closed form of $P(k,x)$. The aim is to provide a formal proof of this claim.

Prior to this, a brief explanation of the emergence of degrees $k\ge 5$ is provided: the emergence of degrees $k\ge 5$ occurs if there is a change in symbol $L\to R$ or $R\to L$ in the path of the Haros graph tree. Consequently, the degree it emerges, or not, is related to the level at which this symbolic change occurs. Suppose that we have covered the path $L^{a_1}{R}^{a_2}\dots L^{a_{k-1}}$. Next, the downstream of $R^{a_k}$ generates a new degree, which had previously appeared as a boundary node before the shift in direction (the node resulting from the identification of the extreme nodes). Specifically, in the first downstream R, the degree appears in the merging node, and the number of nodes with this degree increases by one with each descent. Therefore, when we reach $R^{a_k-1}$, there will be $0+(a_k-1)=a_k-1$ nodes of that degree. In addition, the Haros graph achieved is associated with $p_k/q_k=[a_1,\dots ,a_k]$.

Now, in order to reach the Haros graph associated with $[a_1,\dots ,a_k,a_{k+1}]$, we must descend to R and then $a_{k+1}-1$ times to L. If there were $a_k-1$ nodes of that degree previously, a new descent will result in $a_k$ nodes. Then, performing a descent $L^{a_{k+1}-1}$ requires concatenating the Haros graph reached by $R^{a_k}$ with $a_{k+1}-1$ copies of the Haros graph reached by $R^{a_k-1}$, so that the resultant Haros graph has $(a_k-1)·(a_{k+1}-1)+a_k=(a_k-1)·\left(a_{k+1}\right)+1$ nodes of that degree.

In other words, the emergence of the degree occurs, according to the recursive equation $q_n=q_{n-2}+a_n·q_{n-1}$, where the terms $a_i$ correspond to the continued fraction $[a_k-1,a_{k+1},\dots ,a_m]$, with initial conditions $q_{-1}=0,q_0=1$. Hence, this recursive equation converges to the denominator of the continued fraction $[a_k-1,a_{k+1},\dots ,a_m]$.

3. Main Results

The first presented result provides an explicit description of the degree distribution related to the truncations of the continued fraction of $p/q$, the rational number associated with Haros graph $G_{p/q}$:

Theorem 1. 

Let $p/q\in [0,1/2]$, with $p/q=[a_1,\dots ,a_m]$. Then, the degree distribution $P(k,p/q)$ of the Haros graph $G_{p/q}$ is:

$$P\left(k,\frac{p}{q}\right)=\left\{\begin{array}{cc}p/q,\hfill & k=2\hfill \\ (q-2p)/q,\hfill & k=3\hfill \\ 0,\hfill & k=4\hfill \\ s^{\left(l\right)}/q,\hfill & for values k=\sum _{i=1}^l{a}_i+3, with \forall l=1,\dots ,m-1,\hfill \\ \hfill & where r^{\left(l\right)}/s^{\left(l\right)}=[a_{l+1}-1,\dots ,a_m]\hfill \\ 1/q,\hfill & if k=\sum _{i=1}^m{a}_i+2\hfill \\ 0,\hfill & otherwhise.\hfill \end{array}\right.$$

(4)

Proof. 

See Appendix A.1 for a complete proof. □

The theorem has several consequences: first, it unveils the whole expression for $P(k,x)$ providing a large amount of topological information. Moreover, as stated in the previous section, the values $k\ge 5$ are related to the continued fraction and, consequently, with the symbolic path reached in the Haros graph tree. In addition, the result reduces the computational cost for obtaining the degree distribution $P(k,p/q)$ of the Haros graph $G_{p/q}$. Initially, we observe that the denominator of the n-th convergent $q_n$ of $p/q$ verifies $q_n\ge {\varphi }^{n-1}$, where $\varphi$ is the Golden number [18]. Hence, as the Haros graph $G_{p_n/q_n}$ has $q_n+1$ nodes, its growth is exponential, but Theorem 1 uses only continued fractions [3,19].

Let us illustrate an example: consider the Haros graph $G_{10/23}$, where $10/23=[a_1,a_2,a_3]=[2,3,3]$. Hence, the symbolic path in the Haros graph tree is $L^2{R}^3{L}^2$ (see Figure 2 for an illustration). Numerically, its degree distribution is as follows:

$$P\left(k,\frac{10}{23}\right)=\{\begin{array}{ll}\frac{10}{23},& k=2,\\ \frac{3}{23},& k=3,\\ 0,& k=4,\\ \frac{7}{23},& k=5,\\ \frac{2}{23},& k=8,\\ \frac{1}{23},& k=10,\\ 0,& \mathrm{otherwise}.\end{array}$$

(5)

Then, we can verify that there are $m_{5,10/23}=7$ nodes of degree $k=5$ and that this number corresponds with the denominator of the continued fraction

$$[a_2-1,a_3]=[2,3]=\frac{3}{7};$$

moreover, there are $m_{8,10/23}=2$ nodes of degree $k=8$, which is the denominator of the continued fraction

$$[a_3-1]=\left[2\right]=\frac{1}{2}.$$

Lastly, $k=10$ corresponds with the degree of the boundary node; therefore, it only appears once.

With Theorem 1, we are able to provide a proof for the conjecture presented in [8]. Contrary to the previous finding, the subsequent theorem results in a formulation of the degree distribution relating to the number x, but the computation of $P(k,x)$ for every k depends on the location of x in the subintervals given by the levels in the Farey binary tree ${\ell }_k$.

Theorem 2. 

Let $p/q\in [0,1/2]$ and the associated Haros graph $G_{p/q}$. Let us consider ${\ell }_{k-3}={\left\{\frac{a_i}{b_i}\right\}}_{i=1}^{2^{k-5}}$, and ${\ell }_{k-2}={\left\{\frac{p_j}{q_j}\right\}}_{j=1}^{2^{k-4}}$; clearly, we have $\forall i=1,\dots ,2^{k-5}$:

$$\frac{p_{2i-1}}{q_{2i-1}}<\frac{a_i}{b_i}<\frac{p_{2i}}{q_{2i}}.$$

(6)

Therefore, the degree distribution of the Haros graph $G_{p/q}$ for degrees $k\ge 5$ is:

$$P\left(k,\frac{p}{q}\right)=\{\begin{array}{l}{q}_{2i-1}·(p/q)-p_{2i-1},\mathrm{if}p/q\in \left(\frac{p_{2i-1}}{q_{2i-1}},\frac{a_i}{b_i}\right),\forall i=1,\dots ,2^{k-5},\\ -q_{2i}·(p/q)+p_{2i},\mathrm{if}p/q\in \left(\frac{a_i}{b_i},\frac{p_{2i}}{q_{2i}}\right),\forall i=1,\dots ,2^{k-5},\\ 1/q_j,\mathrm{if}\frac{p}{q}=\frac{p_j}{q_j}\in {\ell }_{k-2},\\ 0,\mathrm{otherwise}.\end{array}$$

(7)

Proof. 

See Appendix A.2 for a complete proof. □

The theorem can be extended from rational numbers to all real numbers. Figure 3 depicts a numerical computation of $P(k,x)$ for the first values of $k\ge 5$ verifying the statement of Theorem 2. Moreover, this new formulation allows the following statement to emphasize some aspects of the degree distribution $P(k,x)$ as a real function over the variable x:

Corollary 1. 

The degree distribution $P(k,x)$ over the variable x is a piecewise linear and continuous function, with the exception of the measure null set.

Now, let us illustrate the theorem by applying the result to the case when $k=5$. Then, it is a simple matter to confirm that if $p/q\in (1/3,1/2)$, then

$$P\left(k=5,\frac{p}{q}\right)=3·\frac{p}{q}-1=\frac{3p-q}{q}.$$

In comparison, the continued fractions of rational numbers $p/q\in (1/3,1/2)$ start with the term $a_1=2$. In virtue of Theorem 1, the degree $k=a_1+3=5$ would have a frequency of $s^{\left(1\right)}/q$, where $s^{\left(1\right)}$ is the denominator of $[a_2-1,a_3,\dots ,a_m]$. Let us examine how the two expressions coincide:

$$\begin{array}{c}\hfill \frac{p}{q}=\frac{1}{2+\frac{1}{a_2+\ddots }}\Rightarrow \frac{q}{p}-2=\frac{q-2p}{p}=\frac{1}{a_2+\frac{1}{a_3+\ddots }}\Rightarrow \\ \hfill \frac{p}{q-2p}-1=\frac{3p-q}{q-2p}=(a_2-1)+\frac{1}{a_3+\frac{1}{a_4+\ddots }}\Rightarrow \\ \hfill \frac{q-2p}{3p-q}=\frac{1}{(a_2-1)+\frac{1}{a_3+\frac{1}{a_4+\ddots }}}.\end{array}$$

Hence, $3p-q$ is the denominator of the continued fraction $[a_2-1,a_3,\dots ,a_m]$ according to Theorem 1. This finding may be generalizable to all degree values k, requiring the partition of $[0,1]$ by the levels ${\ell }_{k-3}$ and ${\ell }_{k-2}$.

4. Conclusions

The topological properties of rational Haros graphs were investigated in detail. The full formulation of the degree distribution $P(k,x)$ of a Haros graph $G_x$ was demonstrated. Different methods were used to prove two theorems. The first theorem unveiled the closed form of $P(k,x)$ obtained by truncations of the continued fraction of x. With this result, not only did we obtain the complete expression of the degree distribution, but we also computed it more efficiently. The second theorem provided a piecewise linear and continuous expression of $P(k,x)$, except for a measure null set. This result confirmed a conjecture presented in [8] revealing that the degree distribution exhibited a self-similar behavior related to the subintervals defined by Farey fractions.