核心拥塞是双曲网络中固有的
Core congestion is inherent in hyperbolic networks
摘要：
我们研究了负曲率对大规模网络中的流量拥塞的影响。我们证明了每个Gromov双曲网络$G$都有一个核心，从而肯定地回答了Jonckheere、Lou、Bonahon和Baryshnikov在Internet Mathematics，7（2011）中提出的一个猜想，该猜想是基于Narayan和Saniee在Physical Review E，84（2011）中的实验观察，即具有小双曲性的真实世界网络具有核心拥塞。也就是说，我们证明了对于$δ$ -双曲图$G$的每个顶点子集$X$，存在$G$的一个顶点$m$，使得以$m$为中心的半径为$4 δ$的圆盘$D(m,4 δ)$至少截取$X$的所有顶点对之间的总流的一半，其中两个顶点$x,y\in X$之间的流由测地（或准测地）$(x,y)$ -路承载.如果$S$与$x$和$y$之间的每个最短路径相交，则集合$S$拦截两个节点$x$和$y$之间的流。根据Jonckheere等人的研究，我们证明了$m$不是（也不可能是）$X$的质心，而是靠近$X$的所谓内射壳中的$X$的中值的节点。在$X$的节点之间的非均匀业务的情况下（在这种情况下，单位流仅存在于由商品图$R$定义的$X$的某些节点对之间），我们证明了原始-对偶结果，该结果表明对于任何$ρ>5δ$，$ρ$ -多核的大小（即，截取所有$R$对的半径为$ρ$）的盘的数目的上限为$R$的成对分开的$(ρ-3δ)$对的最大数目。
英文摘要：
We investigate the impact the negative curvature has on the traffic congestion in large-scale networks. We prove that every Gromov hyperbolic network $G$ admits a core, thus answering in the positive a conjecture by Jonckheere, Lou, Bonahon, and Baryshnikov, Internet Mathematics, 7 (2011) which is based on the experimental observation by Narayan and Saniee, Physical Review E, 84 (2011) that real-world networks with small hyperbolicity have a core congestion. Namely, we prove that for every subset $X$ of vertices of a $δ$-hyperbolic graph $G$ there exists a vertex $m$ of $G$ such that the disk $D(m,4 δ)$ of radius $4 δ$ centered at $m$ intercepts at least one half of the total flow between all pairs of vertices of $X$, where the flow between two vertices $x,y\in X$ is carried by geodesic (or quasi-geodesic) $(x,y)$-paths. A set $S$ intercepts the flow between two nodes $x$ and $y$ if $S$ intersect every shortest path between $x$ and $y$. Differently from what was conjectured by Jonckheere et al., we show that $m$ is not (and cannot be) the center of mass of $X$ but is a node close to the median of $X$ in the so-called injective hull of $X$. In case of non-uniform traffic between nodes of $X$ (in this case, the unit flow exists only between certain pairs of nodes of $X$ defined by a commodity graph $R$), we prove a primal-dual result showing that for any $ρ>5δ$ the size of a $ρ$-multi-core (i.e., the number of disks of radius $ρ$) intercepting all pairs of $R$ is upper bounded by the maximum number of pairwise $(ρ-3δ)$-apart pairs of $R$.
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