21211
Random Minimum Length Spanning Trees in Regular Graphs
r-regular n-vertex graph G with random independent edge lengths, each uniformly distributed on (0, 1). Let mst(G) be the expected length of a minimum spanning tree. We show that mst(G) can be estimated quite accurately under two distinct circumstances. Firstly, if r is large and G has a modest edge expansion property then , where . Secondly, if G has large girth then there exists an explicitly defined constant such that . We find in particular that .
12475992
On Markov Chains for Randomly H-Coloring a Graph
Let H=(W,F) be a graph without multiple edges, but with the possibility of having loops. Let G=(V,E) be a simple graph. A homomorphism c is a map c:V?W with the property that (v,w)?E implies that (c(v),c(w))?F. We will often refer to c(v) as the color of v and c as an H-coloring of G. We consider the problem of choosing a random H-coloring of G by Markov chain Monte Carlo. The probabilistic model we consider includes random proper colorings, random independent sets, and the Widom?Rowlinson and Beach models of statistical physics. We prove negative results for uniform sampling and a positive result for weighted sampling when H is a tree.
6378032
Adding random edges to dense graphs
This paper investigates the addition of random edges to arbitrary dense graphs; in particular, we determine the number of random edges required to ensure various monotone properties including the appearance of a fixed size clique, small diameter and k-connectivity.
6616465
Multi-Coloured Hamilton Cycles In Random Edge-Coloured Graphs
We define a space of random edge-coloured graphs Gn,m,κ which correspond naturally to edge κ-colourings of Gn,m. We show that there exist constants K0, K1 ≤ 21 such that, provided m ≥ K0n log n and κ ≥ K1n, then a random edge-coloured graph contains a multi-coloured Hamilton cycle with probability tending to 1 as the number of vertices n tends to infinity.
448902
A general model of web graphs
We describe a very general model of a random graph process whose proportional degree sequence obeys a power law. Such laws have recently been observed in graphs associated with the world wide web.
2168830
Generating and Counting Hamilton Cycles in Random Regular Graphs
LetGbe chosen uniformly at random from the set G(r,n) ofr-regular graphs with vertex set n. We describe polynomial time algorithms thatwhp(i) find a Hamilton cycle inG, and (ii) approximately count the number of Hamilton cycles inG.
5329008
Clustering in large graphs and matrices
We consider the problem of dividing a set of m points in Euclidean n-space into k clusters (m, n are variable while k is fixed), so as to minimize the sum of distances squared of each point to its “cluster center”. This formulation differs in two ways from the most frequently considered clustering problems in the literature, namely, here we have k fixed and m, n variable, and we use the sum of squared distances as our measure; we will argue that our problem is natural in many contexts. We consider a relaxation of the discrete problem: find the k-dimensional subspace V so that the sum of distances squared to V (of the m points) is minimized. We show: (i) The relaxation can be solved by the Singular Value Decomposition (SVD) of Linear Algebra. (ii) The solution of the relaxation can be used to get a 2-approximation algorithm for the original problem. More importantly, (iii) we argue that in fact the relaxation provides a generalized clustering which is useful in its own right. Finally, (iv) we show that the SVD of a randomly chosen submatrix (according to a suitable probability distribution) of the matrix provides an approximation to the SVD of the whole matrix, thus yielding a very fast randomized algorithm. This cam be applied to problems of very large size which typically arise in modern applications.