Related papers: Stable graphs: distributions and line-breaking con…
We prove a metric space scaling limit for a critical random graph with independent and identically distributed degrees having power-law tail behaviour with exponent $\alpha+1$, where $\alpha \in (1,2)$. The limiting components are…
We introduce a new, relatively simple, line-breaking construction of the $\alpha$-stable tree which realises its random finite-dimensional distributions. This is a direct analogue of Aldous' line-breaking construction of the Brownian…
We give a new, simple construction of the $\alpha$-stable tree for $\alpha \in (1,2]$. We obtain it as the closure of an increasing sequence of $\mathbb{R}$-trees inductively built by gluing together line-segments one by one. The lengths of…
We study the structure of trees minimizing their number of stable sets for given order $n$ and stability number $\alpha$. Our main result is that the edges of a non-trivial extremal tree can be partitioned into $n-\alpha$ stars, each of…
The stability number of a graph G is the cardinality of a stability system of G (that is of a stable set of maximum size of G). A graph is alpha-stable if its stability number remains the same upon both the deletion and the addition of any…
We investigate random connected graphs from a block-stable class whose distribution is weighted based on the number of $2$-connected components, or blocks. This includes the class of planar graphs. For this, we develop a notion of a…
The global structure of the minimal spanning tree (MST) is expected to be universal for a large class of underlying random discrete structures. However, very little is known about the intrinsic geometry of MSTs of most standard models, and…
For a uniform random labelled tree, we find the limiting distribution of tree parameters which are stable (in some sense) with respect to local perturbations of the tree structure. The proof is based on the martingale central limit theorem…
We define decorated $\alpha$-stable trees which are informally obtained from an $\alpha$-stable tree by blowing up its branchpoints into random metric spaces. This generalizes the $\alpha$-stable looptrees of Curien and Kortchemski, where…
We present a linear stability analysis of stationary states (or fixed points) in large dynamical systems defined on random directed graphs with a prescribed distribution of indegrees and outdegrees. We obtain two remarkable results for such…
A growing family of random graphs is called robust if it retains a giant component after percolation with arbitrary positive retention probability. We study robustness for graphs, in which new vertices are given a spatial position on the…
In recent decades, it has been emphasized that the evolving structure of networks may be shaped by interaction principles that yield sparse graphs with a vertex degree distribution exhibiting an algebraic tail, and other structural traits…
We study both numerically and analytically what happens to a random graph of average connectivity "alpha" when its leaves and their neighbors are removed iteratively up to the point when no leaf remains. The remnant is made of isolated…
This paper introduces and studies the stability of the strong domination number of a graph, denoted $\operatorname{st}_{\gamma_{st}}(G)$, defined as the minimum number of vertices whose removal changes the strong domination number…
Fractal scaling--a power-law behavior of the number of boxes needed to tile a given network with respect to the lateral size of the box--is studied. We introduce a new box-covering algorithm that is a modified version of the original…
We introduce a class of random compact metric spaces L(\alpha) indexed by \alpha \in (1,2) and which we call stable looptrees. They are made of a collection of random loops glued together along a tree structure, and can be informally be…
We investigate a class of stochastic fragmentation processes involving stable and unstable fragments. We solve analytically for the fragment length density and find that a generic algebraic divergence characterizes its small-size tail.…
We investigate the asymptotic structure of a random perfect graph $P_n$ sampled uniformly from the perfect graphs on vertex set $\{1,\ldots,n\}$. Our approach is based on the result of Pr\"omel and Steger that almost all perfect graphs are…
We investigate fragmentation processes with a steady input of fragments. We find that the size distribution approaches a stationary form which exhibits a power law divergence in the small size limit, P(x) ~ x^{-3}. This algebraic behavior…
We study a random fiber bundle model with tips of the fibers placed on a graph having co-ordination number 3. These fibers follow local load sharing with uniformly distributed threshold strengths of the fibers. We have studied the critical…