Several consequences of adding or removing an edge from an electric network

Identifiability of Linear Compartmental Models: The Impact of Removing Leaks and Edges

A mathematical model is identifiable if its parameters can be recovered from data. Here, we focus on a particular class of model, linear compartmental models, which are used to represent the transfer of substances in a system. We analyze what happens to identifiability when operations are performed on a model, specifically, adding or deleting a leak or an edge. We first consider the conjecture of Gross et al. that states that removing a leak from an identifiable model yields a model that is again identifiable. We prove a special case of this conjecture, and also show that the conjecture is equ…

几乎距离正则图的扰动

On perturbations of almost distance-regular graphs

本文证明了一类几乎距离正则图，即所谓的$h$ -点行走正则图，可以通过其扰动图的共谱性来刻画.对于给定的$h\le D$，如果距离为$h$的一对顶点$u,v$之间的长度为$\ell$的路的个数只依赖于$\ell$，则称直径为$D$的图$G$为$h$ -点行走正则图.这里考虑的图扰动是删除一个顶点，添加一个环，添加一个悬垂边，添加/删除一个边，合并顶点，并添加一个桥接顶点。我们表明，步行-正则图这些操作是等价的，在这个意义上说，一个扰动产生共谱图当且仅当其他的。我们的研究是基于Cvetković，Godsil，McKay，Rowlinson，Schwenk等人开发的图扰动理论。从而得到了距离正则图的一些新的刻画。

Local rewiring algorithms to increase clustering and grow a small world

Many real-world networks have high clustering among vertices: vertices that share neighbors are often also directly connected to each other. A network's clustering can be a useful indicator of its connectedness and community structure. Algorithms for generating networks with high clustering have been developed, but typically rely on adding or removing edges and nodes, sometimes from a completely empty network. Here, we introduce algorithms that create a highly clustered network by starting with an existing network and rearranging edges, without adding or removing them; these algorithms can pre…