[Paper Review] Existence of Small Separators Depends on Geometry for Geometric Inhomogeneous Random Graphs

Johannes Lengler, Lazar Todorović|arXiv (Cornell University)|Nov 10, 2017

Stochastic processes and statistical mechanics3 references3 citations

TL;DR

This paper demonstrates that the existence of linear-sized separators in Geometric Inhomogeneous Random Graphs (GIRGs) is fundamentally dependent on the underlying geometry: while Euclidean GIRGs exhibit sublinear separators, graphs with Minimum Component Distance (MCD) geometry do not. The authors establish this via a novel application of two-sided Azuma's inequality to handle edge dependencies, proving that MCD-based GIRGs have a constant clustering coefficient and thus lack small separators, contrasting sharply with Euclidean counterparts.

ABSTRACT

We show that Geometric Inhomogeneous Random Graphs (GIRGs) with power law weights may either have or not have separators of linear size, depending on the underlying geometry. While it was known that for Euclidean geometry it is possible to split the giant component into two linear size components by removing at most n^{1-\eps} edges, we show that this is impossible if the geometry is given by the minimum component distance.

Motivation & Objective

• To determine whether geometric inhomogeneous random graphs (GIRGs) with power-law weights admit small separators depending on the underlying geometry.
• To resolve the open question of whether sublinear separators exist in GIRGs when the geometry is non-Euclidean, particularly Minimum Component Distance (MCD).
• To overcome the challenge of strong edge dependencies in GIRGs, which prevent standard techniques like independent edge batching from being applied.
• To establish that MCD-based GIRGs have a constant clustering coefficient, implying the absence of small separators, in contrast to Euclidean GIRGs.

Proposed method

• The authors use a two-sided Azuma’s inequality with bounded differences to control concentration in the presence of dependent edges, a key innovation for handling GIRG dependencies.
• They condition on a high-probability event where vertex positions and weights are well-behaved, ensuring uniform distribution of neighbors within local neighborhoods.
• A random graph model is constructed by sampling vertex positions and weights independently, with edge probabilities depending on weights and geometric distance.
• The clustering coefficient is analyzed via conditional probability bounds, showing that neighbors of low-weight vertices are likely to be close in space and thus mutually connected.
• The proof leverages LeCam’s theorem to establish concentration of weighted sums of degrees and neighbor counts under the conditioning event.
• A martingale-based concentration argument is applied to the clustering coefficient function, using a carefully defined error event to exclude high-degree vertices that could disrupt concentration.

Experimental results

Research questions

• RQ1 Does the existence of small separators in GIRGs depend on the choice of geometric space, particularly between Euclidean and Minimum Component Distance (MCD) geometries?
• RQ2 Can standard techniques for proving sublinear separators—such as independent edge batching—be adapted to GIRGs with dependent edges?
• RQ3 What is the behavior of the clustering coefficient in MCD-based GIRGs, and how does it relate to the existence of small separators?
• RQ4 Is the giant component of MCD-based GIRGs stable under sublinear edge deletions, or are there inherent structural barriers to partitioning it?
• RQ5 How do edge dependencies in GIRGs affect the applicability of classical concentration inequalities like Azuma’s?

Key findings

• For Minimum Component Distance (MCD) geometry, GIRGs do not have sublinear separators; the giant component cannot be split into two linear-sized parts by removing o(n) edges.
• The clustering coefficient in MCD-based GIRGs is Ω(1) with high probability, indicating strong local clustering that prevents small separators.
• The authors prove that the expected clustering coefficient is Ω(1) for low-weight vertices, and concentration via Azuma’s inequality ensures it is Ω(1) w.h.p. across the graph.
• In contrast, Euclidean GIRGs do admit sublinear separators, showing that the geometric structure fundamentally alters graph partitioning properties.
• The proof overcomes edge dependence in GIRGs by using a two-sided Azuma’s inequality with a carefully defined error event, enabling concentration bounds despite dependencies.
• The result establishes a sharp dichotomy: sublinear separators exist in GIRGs if and only if the geometry is Euclidean (or similar), not for MCD geometry.