Is star complexity a proxy for information based complexity of graphs?

TL;DR

The paper investigates whether star complexity serves as a proxy for the information-based complexity (IBC) of graphs. It defines ${\cal C}$ via edge-list encodings with a graph-automorphism classifier and introduces ${\rm Star}(\mathcal{G})$ and ${\cal C}^*$ as an IBC analogue derived from star constructions, along with a computable upper bound ${\overline{\mathrm{Star}}({\cal G})}$. Through experiments on 10- and 22-vertex graphs and large Erdős–Rényi graphs, it finds weak correlation between ${\rm Star}$ and ${\cal C}$, but strong coupling between ${\cal C}^*$ and ${\rm Star}$, with ${\overline{\mathrm{Star}}}$ tightly tracking Star. The results provide empirical evidence that practical star-based measures and Standish's ${\cal C}$ capture related information content, especially for larger graphs, while highlighting computational challenges in exact star calculations. This supports the conjecture that practical IBC measures are asymptotically interchangeable under universal encodings, with a computable upper bound offering scalable insight into graph complexity.

Abstract

Information-based complexity (IBC) is a well-defined complexity measure of any object given a description in a language and a classifier that identifies those descriptions with the object. Of course, the exact numerical value will vary according to the descriptive language and classifier, but under certain universality conditions (eg the classifier identifies programs of a universal Turning machine that halt and output the same value), asymptotically, the complexity measure is independent of the classifier up to a constant of O(1). The hypothesis being investigated in this work that any practical IBC measure will similarly be asymptotically equivalent to any other practical IBC measure. Standish presented an IBC measure for graphs ${\cal C}$ that encoded graphs by their links, and identifies graphs as those that are automorphic to each other. An interesting alternate graph measure is {\em star complexity}, which is defined as the number of union and intersection operations of basic stars that can generate the original graph. Whilst not an IBC itself, it can be related to an IBC (called ${\cal C}^*$) that is strongly correlated with star complexity. In this paper, 10 and 22 vertex graphs are constructed up to a star complexity of 8, and the ${\cal C}^*$ compared emprically with ${\cal C}$. Finally, an easily computable upper bound of star complexity is found to be strongly related to ${\cal C}$.

Figures (5)

• Figure 1: Plot of Star vs ${\cal C}$ for (left) all 10 vertex graphs and (right) all 22 vertex graphs with star complexity 8 or less. The line is a linear fit, with correlation coefficient $\rho=0.082$ for the 10 vertex graphs, and $-0.041$ for the 22 vertex graphs.
• Figure 2: Plot of ${\cal C}^*$ vs ${\cal C}$ for (left) all 10 vertex graphs and (right) all 22 vertex graphs with star complexity 8 or less. The line is a linear fit, with correlation coefficient $\rho=0.143$ for the 10 vertex graphs, and $-1.28\times10^{-3}$ for the 22 vertex graphs
• Figure 3: Plot of $\mathrm{Star}$ vs ${\cal C}^*$ for (left) all 10 vertex graphs and (right) all 22 vertex graphs with star complexity 8 or less. The line is a linear fit, with correlation coefficient $\rho=0.638$ for the 10 vertex graphs, and $0.611$ for the 22 vertex graphs
• Figure 4: Plot of $\mathrm{Star}$ vs $\overline{\mathrm{Star}}$ for (left) all 10 vertex graphs and (right) all 22 vertex graphs with star complexity 8 or less. The red line is a linear fit, with correlation coefficient $\rho=0.863$ for the 10 vertex graphs, and $0.850$ for the 22 vertex graphs. The green line is the identity $\mathrm{Star}=\overline{\mathrm{Star}}$.
• Figure 5: Plot of ${\cal C}$ against $\overline{\mathrm{Star}}$ for 1000 Erdös-Renyi randomly generated graphs of 1000 vertices each.