Universal diameter bounds for random graphs with given degrees

TL;DR

The paper establishes universal, non-asymptotic diameter bounds for random graphs with prescribed degree sequences by decomposing graphs into forest, core, and kernel components. It develops a probabilistic framework built on a simple kernel encoding, plus a switching technique and composition-biased-tree tails, to bound the height of attachable forests and the lengths of core paths, culminating in a high-probability O(log m) kernel diameter and an overall E[diam(G)] bound of O(√(n/ε)+log n/ε) when a positive fraction of vertices have degree different from 2. For minimum degree at least 3, the results imply logarithmic diameters, matching best possible behavior in many regimes. The work advances the understanding of how fixed-degree constraints shape distances in random graphs, with implications for sparse networks and configuration-model-type ensembles, and it introduces robust, non-asymptotic techniques (forest-core-kernel decomposition, path switching, and composition-biased tree analysis).

Abstract

Given a graph $G$, let $\mathrm{diam}(G)$ be the greatest distance between any two vertices of $G$ which lie in the same connected component, and let $\mathrm{diam}^+(G)$ be the greatest distance between any two vertices of $G$; so $\mathrm{diam}^+(G)=\infty$ if $G$ is not connected. Fix a sequence $(d_1,\ldots,d_n)$ of positive integers, and let $\mathbf{G}$ be a uniformly random connected simple graph with $V(\mathbf{G})=[n]:=\{1,\ldots,n\}$ such that $\mathrm{deg}_{\mathbf{G}}(v)=d_v$ for all $v \in [n]$. We show that, unless a $1-o(1)$ proportion of vertices have degree $2$, then $\mathbb{E}[\mathrm{diam}(\mathbf{G})]=O(\sqrt{n})$. It is not hard to see that this bound is best possible for general degree sequences (and in particular in the case of trees, in which $\sum_{v=1}^n d_v = 2(n-1)$). We also prove that this bound holds without the connectivity constraint. As a key input to the proofs, we show that graphs with minimum degree $3$ are with high probability connected and have logarithmic diameter: if $\min(d_1,\ldots,d_n) \ge 3$ and $\mathbf{G}$ is a uniformly random simple graph with $V(\mathbf{G})=[n]$ such that $\mathrm{deg}_{\mathbf{G}}(v)=d_v$ for all $v \in [n]$, then $\mathrm{diam}^+(\mathbf{G})=$ $O_{\mathbb{P}}(\log n)$; this bound is also best possible.

Figures (2)

• Figure 1: Left, a connected graph $\mathbf{G}$, with its kernel vertices circled in bold. Only labels of vertices in the core are shown, for readability. Right: The tree $\mathbf{T}$ constructed from (the trees hanging from the core paths of) $\mathbf{G}$, with root $9$. Listing the kernel edges as $e_1,\ldots,e_6$ in lexicographic order, then the numbers of internal vertices of the paths $C(e_1),\ldots,C(e_6)$ are $(2,3,0,2,0,0)$, which is a partition of $7=\operatorname{ht}_{\mathbf{T}}(0)$ with $6=e(K(\mathbf{G}))$ parts. The graph $\mathbf{G}$ can be recovered from $\mathbf{T}$ together with this partition.
• Figure 2: Left, a portion of $C(A)$ for an augmented core $A$, corresponding to pairs $(\{ui,vj\},(p_1,\ldots,p_a))$ and $(\{xk,y\ell\},(q_1,\ldots,q_b))$. If $ui\prec_{\operatorname{lex}}vj$ then $P$ is $(p_1,\ldots,p_a)$, and otherwise $P$ is $(p_a,\ldots,p_1)$, and likewise $Q$ is either $(q_1,\ldots,q_b)$ or $(q_b,\ldots,q_1)$ depending on whether or not $xk\prec_{\operatorname{lex}}y\ell$. Right, the same portion of $C(A)$ after switching on $e=(ui,vj)$ and $f=(xk,y\ell)$. The curved arrows in the left figure indicate how the paths $P$ and $Q$ "pivot" as a result of the switching operation. The red dotted edge attached to $v$ corresponds to half-edge $vj$, which is "detached from the head of $P$ and reattached to the tail of $Q$"; likewise, the edge corresponding to half-edge $xk$ is detached from the tail of $Q$ and reattached to the head of $P$.

Theorems & Definitions (74)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 1.5
• Corollary 1.6
• Conjecture 1
• Conjecture 2
• Conjecture 3
• Theorem 1.7