Related papers: On the evolution of scale-free graphs
We study the height of a spanning tree $T$ of a graph $G$ obtained by starting with a single vertex of $G$ and repeatedly selecting, uniformly at random, an edge of $G$ with exactly one endpoint in $T$ and adding this edge to $T$.
We analyse graphs in which each vertex is assigned random coordinates in a geometric space of arbitrary dimensionality and only edges between adjacent points are present. The critical connectivity is found numerically by examining the size…
We investigate the dynamic formation of regular random graphs. In our model, we pick a pair of nodes at random and connect them with a link if both of their degrees are smaller than d. Starting with a set of isolated nodes, we repeat this…
This article explores and analyzes the unsupervised clustering of large partially observed graphs. We propose a scalable and provable randomized framework for clustering graphs generated from the stochastic block model. The clustering is…
We study the growth of random networks under a constraint that the diameter, defined as the average shortest path length between all nodes, remains approximately constant. We show that if the graph maintains the form of its degree…
The determination of cluster centers generally depends on the scale that we use to analyze the data to be clustered. Inappropriate scale usually leads to unreasonable cluster centers and thus unreasonable results. In this study, we first…
We study a one parameter family of random graph models that spans a continuum between traditional random graphs of the Erd\H{o}s-R\'enyi type, where there is no underlying structure, and percolation models, where the possible edges are…
Motivated by the analysis of social networks, we study a model of random networks that has both a given degree distribution and a tunable clustering coefficient. We consider two types of growth processes on these graphs: diffusion and…
We introduce a very general model of an inhomogenous random graph with independence between the edges, which scales so that the number of edges is linear in the number of vertices. This scaling corresponds to the p=c/n scaling for G(n,p)…
The unsupervised learning of community structure, in particular the partitioning vertices into clusters or communities, is a canonical and well-studied problem in exploratory graph analysis. However, like most graph analyses the…
Probabilistic networks display a wide range of high average clustering coefficients independent of the number of nodes in the network. In particular, the local clustering coefficient decreases with the degree of the subtending node in a…
We investigate the contact process on four different types of scale-free inhomogeneous random graphs evolving according to a stationary dynamics, where each potential edge is updated with a rate depending on the strength of the adjacent…
Given a weighted graph, we introduce a partition of its vertex set such that the distance between any two clusters is bounded from below by a power of the minimum weight of both clusters. This partition is obtained by recursively merging…
Scaling behavior of scale-free evolving networks arising in communications, citations, collaborations, etc. areas is studied. We derive universal scaling relations describing properties of such networks and indicate limits of their…
This paper investigates whether nonlinear gravitational instability can account for the clustering of galaxies on large and small scales, and for the evolution of clustering with epoch. No CDM-like spectrum is consistent with the shape of…
We develop tail estimates for the number of edges in a Chung-Lu random graph with regularly varying weight distribution. Our results show that the most likely way to have an unusually large number of edges is through the presence of one or…
A scale-free network is grown in the Euclidean space with a global directional bias. On a vertical plane, nodes are introduced at unit rate at randomly selected points and a node is allowed to be connected only to the subset of nodes which…
Many real-world networks exhibit scale-free feature, have a small diameter and a high clustering tendency. We have studied the properties of a growing network, which has all these features, in which an incoming node is connected to its…
For random percolation at p_c, the probability distribution P(n) of the number of spanning clusters (n) has been studied in large scale simulations. The results are compatible with $P(n) \sim \exp(-an^2)$ for all dimensions. We also study…
In this paper we study the component structure of random graphs with independence between the edges. Under mild assumptions, we determine whether there is a giant component, and find its asymptotic size when it exists. We assume that the…