Risk management for traffic safety control

This paper offers a range of modelling ideas and techniques from mathematical statistics appropriate for analysing traffic accident data for the East region operation of CLP Power Hong Kong Limited and for the Hong Kong population in general. We further make proposals for alternative ways to record and collect data, and discuss ways to identify the major contributing factors behind accidents. We hope that our findings will enable the design of effective accident prevention strategies for CLP

Rigidity percolation in a field

Rigidity Percolation with g degrees of freedom per site is analyzed on randomly diluted Erdos-Renyi graphs with average connectivity gamma, in the presence of a field h. In the (gamma,h) plane, the rigid and flexible phases are separated by a line of first-order transitions whose location is determined exactly. This line ends at a critical point with classical critical exponents. Analytic expressions are given for the densities n_f of uncanceled degrees of freedom and gamma_r of redundant bonds. Upon crossing the coexistence line, n_f and gamma_r are continuous, although their first derivatives are discontinuous. We extend, for the case of nonzero field, a recently proposed hypothesis, namely that the density of uncanceled degrees of freedom is a ``free energy'' for Rigidity Percolation. Analytic expressions are obtained for the energy, entropy, and specific heat. Some analogies with a liquid-vapor transition are discussed. Particularizing to zero field, we find that the existence of a (g+1)-core is a necessary condition for rigidity percolation with g degrees of freedom. At the transition point gamma_c, Maxwell counting of degrees of freedom is exact on the rigid cluster and on the (g+1)-rigid-core, i.e. the average coordination of these subgraphs is exactly 2g, although gamma_r, the average coordination of the whole system, is smaller than 2g. gamma_c is found to converge to 2g for large g, i.e. in this limit Maxwell counting is exact globally as well. This paper is dedicated to Dietrich Stauffer, on the occasion of his 60th birthday.Comment: RevTeX4, psfig, 16 pages. Equation numbering corrected. Minor typos correcte

The cut metric, random graphs, and branching processes

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 sequence of matrices of edge probabilities converges to an appropriate limit object (a kernel), but only in a very weak sense, namely in the cut metric. Our results thus generalize previous results on the phase transition in the already very general inhomogeneous random graph model we introduced recently, as well as related results of Bollob\'as, Borgs, Chayes and Riordan, all of which involve considerably stronger assumptions. We also prove corresponding results for random hypergraphs; these generalize our results on the phase transition in inhomogeneous random graphs with clustering.Comment: 53 pages; minor edits and references update

Cache as ca$h can

In this paper we consider the problem of placing proxy caches in a network to get a better performance of the net. We develop a heuristic method to decide in which nodes of the network proxies should be installed and what the sizes of these caches should be. The heuristic attempts to minimize a function of the waiting times in the network

Random subgraphs of the 2D Hamming graph: the supercritical phase

We study random subgraphs of the 2-dimensional Hamming graph H(2, n), which is the Cartesian product of two complete graphs on n vertices. Let p be the edge probability, and write p = (1 + ε)/(2(n − 1)) for some ε ∈ R. In Borgs et al. (Random Struct Alg 27:137–184, 2005; Ann Probab 33:1886–1944, 2005), the size of the largest connected component was estimated precisely for a large class of graphs including H(2, n) for ε ≤ �V−1/3, where � > 0 is a constant and V = n2 denotes the number of vertices in H(2, n). Until now, no matching lower bound on the size in the supercritical regime has been obtained. In this paper we prove that, when ε � (log V)1/3V−1/3, then the largest connected component has size close to 2εV with high probability.We thus obtain a law of large numbers for the largest connected component size, and show that the corresponding values of p are supercritical. Barring the factor (log V)1/3, this identifies the size of the largest connected component all the way down to the critical p window