Flooding and Diameter in General Weighted Random Graphs

Flooding in weighted sparse random graphs of active and passive nodes

This paper discusses first passage percolation and flooding on large weighted sparse random graphs with two types of nodes: active and passive nodes. In mathematical physics passive nodes can be interpreted as closed gates where fluid flow or water cannot pass through and active nodes can be interpreted as open gates where water may keep flowing further. The model of this paper has many applications in real life, for example, information spreading, where passive nodes are interpreted as passive receivers who may read messages but do not respond to them. In the epidemic context passive nodes ma…

Flooding in Weighted Random Graphs

In this paper, we study the impact of edge weights on distances in diluted random graphs. We interpret these weights as delays, and take them as i.i.d exponential random variables. We analyze the weighted flooding time defined as the minimum time needed to reach all nodes from one uniformly chosen node, and the weighted diameter corresponding to the largest distance between any pair of vertices. Under some regularity conditions on the degree sequence of the random graph, we show that these quantities grow as the logarithm of $n$, when the size of the graph $n$ tends to infinity. We also derive…

随机加权图上均匀生成树的直径

Diameter of uniform spanning trees on random weighted graphs

本文研究随机加权图上均匀生成树的直径，证明在有界度扩展图和有限盒子中，其直径以高概率为 n^{1/2+o(1)} 阶。

The diameter of weighted random graphs

In this paper we study the impact of random exponential edge weights on the distances in a random graph and, in particular, on its diameter. Our main result consists of a precise asymptotic expression for the maximal weight of the shortest weight paths between all vertices (the weighted diameter) of sparse random graphs, when the edge weights are i.i.d. exponential random variables.

Far-out Vertices In Weighted Repeated Configuration Model

We consider an edge-weighted uniform random graph with a given degree sequence (Repeated Configuration Model) which is a useful approximation for many real-world networks. It has been observed that the vertices which are separated from the rest of the graph by a distance exceeding certain threshold play an important role in determining some global properties of the graph like diameter, flooding time etc., in spite of being statistically rare. We give a convergence result for the distribution of the number of such far-out vertices. We also make a conjecture about how this relates to the longest…

External diffusion limited aggregation on a spanning-tree-weighted random planar map

Let $M$ be the infinite spanning-tree-weighted random planar map, which is the local limit of finite random planar maps sampled with probability proportional to the number of spanning trees they admit. We show that a.s. the $M$-graph-distance diameter of the external diffusion-limited aggregation (DLA) cluster on $M$ run for $m$ steps is of order $m^{2/d + o_m(1)}$, where $d$ is the metric ball volume growth exponent for $M$ (which was shown to exist by Ding-Gwynne, 2018). By known bounds for $d$, one has $0.55051\ldots \leq 2/d \leq 0.563315\ldots$. Along the way, we also prove that loop-eras…