[Paper Review] The inducibility of oriented stars

Ping Hu, Jie Ma+2 more|arXiv (Cornell University)|Aug 12, 2020

Limits and Structures in Graph Theory14 references4 citations

TL;DR

This paper determines the inducibility of oriented stars $S_{k, ho}$ with at least seven vertices, where the center has out-degree $k$ and in-degree $\ell$, by reducing the problem to a polynomial optimization over parameters $\alpha$ and $d$. The key result provides asymptotic formulas for the inducibility in terms of maximized expressions involving $\alpha$, $d$, and binomial coefficients, with extremal constructions based on oriented complete bipartite graphs.

ABSTRACT

We consider the problem of maximizing the number of induced copies of an oriented star $S_{k,\ell}$ in digraphs of given size, where the center of the star has out-degree $k$ and in-degree $\ell$. The case $k\ell=0$ was solved by Huang. Here, we asymptotically solve it for all other oriented stars with at least seven vertices.

Motivation & Objective

• To determine the inducibility of oriented stars $S_{k,\ell}$ with $k+\ell \geq 6$ and $k \geq \ell \geq 1$, extending prior results that covered only the cases $\ell = 0$ or $k = \ell = 1$.
• To establish that the inducibility of such oriented stars is asymptotically achieved by orientations of complete bipartite graphs with specific edge density and partition ratios.
• To reduce the computation of inducibility to a polynomial optimization problem over parameters $\alpha$ and $d$, providing explicit formulas for the maximum density.
• To conjecture that the derived formulas extend to all $k \geq \ell \geq 1$ with $k+\ell \geq 6$, except for small cases requiring new techniques.
• To prove stability of the extremal constructions, showing that any digraph with inducibility close to the maximum must be structurally close to the optimal bipartite orientation.

Proposed method

• The paper uses a probabilistic method to analyze the expected number of induced copies of $S_{k,\ell}$ in random orientations of complete bipartite graphs with partition sizes proportional to $\alpha$ and $1-\alpha$, and edge density $d$ from $X$ to $Y$.
• It derives a lower bound on the inducibility by optimizing the expression $\alpha(1-\alpha)^{k+\ell}d^k(1-d)^\ell + \frac{(k-1)^{k-1}\ell^\ell}{(k+\ell-1)^{k+\ell-1}}(1-\alpha)\alpha^{k+\ell}(1-d)$ over $\alpha \in [0,\frac{1}{2}]$ and $d \in [0,\frac{k}{k+\ell}]$.
• The upper bound is established via a stability argument, showing that any digraph with high inducibility must be close in structure to the optimal bipartite construction.
• The proof relies on a series of inequalities and asymptotic estimates, particularly using Taylor expansions around the optimal point $(\alpha, d) \approx \left(\frac{1}{k+\ell+1}, \frac{k}{k+\ell}\right)$ to approximate the maximum.
• For $k = \ell$, the inducibility is given by $\frac{(2k+1)!}{2^{2k}(k!)^2} \cdot \max_\alpha \left\{ \alpha(1-\alpha)^{2k} + (1-\alpha)\alpha^{2k} \right\}$, while for $k > \ell$, a more complex expression involving $k$-th and $\ell$-th powers of $d$ and $1-d$ is used.
• Technical estimates are supported by detailed calculations in Appendix A, and numerical approximations using flag algebras are provided for small cases like $(k,\ell) = (2,1)$.

Experimental results

Research questions

• RQ1 What is the inducibility of oriented stars $S_{k,\ell}$ with $k+\ell \geq 6$ and $k \geq \ell \geq 1$, beyond the known cases where $\ell = 0$ or $k = \ell = 1$?
• RQ2 Can the inducibility of such oriented stars be expressed as a solution to a polynomial optimization problem over parameters $\alpha$ and $d$?
• RQ3 Is the extremal construction for the inducibility of $S_{k,\ell}$ asymptotically realized by oriented complete bipartite graphs with specific partition and edge density ratios?
• RQ4 Does the optimal digraph structure exhibit stability, such that any digraph with inducibility close to the maximum must be structurally close to the optimal bipartite orientation?
• RQ5 Can the optimization formulas derived in the paper be extended to small cases like $(k,\ell) = (2,1), (3,1), (4,1)$, or do they require new techniques?

Key findings

• The inducibility of $S_{k,\ell}$ for $k \geq \ell \geq 1$ and $k+\ell \geq 6$ is given by a closed-form expression involving a maximized polynomial in $\alpha$ and $d$, with the maximum taken over $\alpha \in [0,\frac{1}{2}]$ and $d \in [0,\frac{k}{k+\ell}]$.
• For $k = \ell$, the inducibility is $\frac{(2k+1)!}{2^{2k}(k!)^2} \cdot \max_\alpha \left\{ \alpha(1-\alpha)^{2k} + (1-\alpha)\alpha^{2k} \right\}$, which is symmetric in the two directions of the star.
• For $k > \ell$, the inducibility is $\frac{(k+\ell+1)!}{k!\ell!} \cdot \max_{\alpha,d} \left\{ \alpha(1-\alpha)^{k+\ell}d^k(1-d)^\ell + \frac{(k-1)^{k-1}\ell^\ell}{(k+\ell-1)^{k+\ell-1}}(1-\alpha)\alpha^{k+\ell}(1-d) \right\}$, reflecting the asymmetry in out- and in-degrees.
• The optimal parameters are asymptotically $\alpha \approx \frac{1}{k+\ell+1}$ and $d \approx \frac{k}{k+\ell}$, with higher-order corrections given by Taylor expansions.
• The approximation error for $k=4, \ell=2$ is less than $10^{-8}$ in the first few significant digits, indicating high numerical precision of the formula.
• Stability results show that any digraph achieving inducibility close to the maximum must be structurally close to the optimal oriented complete bipartite graph, with controlled deviations in partition sizes and edge distributions.