Weighted distances in scale-free preferential attachment models

We study three preferential attachment models where the parameters are such that the asymptotic degree distribution has infinite variance. Every edge is equipped with a non-negative i.i.d. weight. We study the weighted distance between two vertices chosen uniformly at random, the typical weighted distance, and the number of edges on this path, the typical hopcount. We prove that there are precisely two universality classes of weight distributions, called the explosive and conservative class. In the explosive class, we show that the typical weighted distance converges in distribution to the sum of two i.i.d. finite random variables. In the conservative class, we prove that the typical weighted distance tends to infinity, and we give an explicit expression for the main growth term, as well as for the hopcount. Under a mild assumption on the weight distribution the fluctuations around the main term are tight.Comment: Revised version, results are unchanged. 30 pages, 1 figure. To appear in Random Structures and Algorithm

Fluctuations in a general preferential attachment model via Stein's method

We consider a general preferential attachment model, where the probability that a newly arriving vertex connects to an older vertex is proportional to a sublinear function of the indegree of the older vertex at that time. It is well known that the distribution of a uniformly chosen vertex converges to a limiting distribution. Depending on the parameters, this model can show power law, but also stretched exponential behaviour. Using Stein's method we provide rates of convergence for the total variation distance. Our proof uses the fact that the limiting distribution is the stationary distribution of a Markov chain together with the generator method of Barbour

Prescription of prostheric ankle-foot mechanisms after lower limb amputation

BACKGROUND: A prosthesis can be divided into several components: the prosthetic socket; the prosthetic ankle‐foot mechanism; and for higher levels of amputation, the prosthetic knee. This review focuses on the prosthetic ankle‐foot mechanism, which forms an important part of the prosthesis in terms of mobility. A correct prosthetic prescription can be derived by matching the functional abilities of the individual with a lower limb amputation with the technical and functional aspects of the various prosthetic ankle‐foot mechanisms. However, there seems to be no clear clinical consensus on the precise prescription criteria for the various prosthetic ankle‐foot mechanisms in relation to the functional abilities of individuals with a lower limb amputation. OBJECTIVES: To obtain information about aspects of prosthetic ankle‐foot mechanisms and daily functioning of individuals with a lower limb prosthesis, for appropriate prosthetic prescription criteria. SEARCH METHODS: We searched the Cochrane Bone, Joint and Muscle Trauma Group Specialised Register (April 2006), the Cochrane Central Register of Controlled Trials (The Cochrane Library 2006, Issue 2), MEDLINE (1966 to April 2006), EMBASE (1983 to April 2006), CINAHL (1982 to April 2006), AMED (Allied and Complimentary Medicine) (1985 to April 2006), and reference lists of articles. No language restrictions were applied. SELECTION CRITERIA: All randomised controlled trials and quasi‐randomised controlled trials comparing different ankle foot mechanisms for lower limb amputation in adults. No language restrictions were applied. DATA COLLECTION AND ANALYSIS: Two review authors independently identified potential articles from the literature search. Methodological quality was assessed using a checklist comprising 13 criteria. The reviewers extracted data using pre‐defined extraction forms. MAIN RESULTS: Twenty‐six trials were included, with a total of 245 participants. The numbers of participants in the included trials ranged from three to sixteen. The methodological quality was moderate. Only one study was of high quality. All included studies used cross‐over designs allowing sufficient control for confounding. In individuals with a transtibial amputation, there seems to be a small tendency towards a greater stride length when walking with the Flex‐foot in comparison to the SACH (solid‐ankle cushioned heel) foot. When walking speed was increased, the energy cost was lower. In high activity individuals with a transfemoral amputation, there is limited evidence for the superiority of the Flex foot during level walking compared with the SACH foot in respect of energy cost and gait efficiency. AUTHORS' CONCLUSIONS: There is insufficient evidence from high quality comparative studies for the overall superiority of any individual type of prosthetic ankle‐foot mechanism, although there is a small trend towards the Flex‐foot in comparison with the SACH foot for greater stride length and lower energy cost in individuals with a transtibial amputation, and improved gait efficiency and lower energy cost in high activity individuals with a transfemoral amputation. In prescribing prosthetic‐ankle foot mechanisms for individuals with a lower limb amputation, practitioners should take into account availability, patient functional needs, the type of knee mechanism to be prescribed and the inter‐relationship with ankle‐foot mechanisms, and cost

Search in Complex Networks : a New Method of Naming

We suggest a method for routing when the source does not posses full information about the shortest path to the destination. The method is particularly useful for scale-free networks, and exploits its unique characteristics. By assigning new (short) names to nodes (aka labelling) we are able to reduce significantly the memory requirement at the routers, yet we succeed in routing with high probability through paths very close in distance to the shortest ones.Comment: 5 pages, 4 figure

Diameters in preferential attachment models

In this paper, we investigate the diameter in preferential attachment (PA-) models, thus quantifying the statement that these models are small worlds. The models studied here are such that edges are attached to older vertices proportional to the degree plus a constant, i.e., we consider affine PA-models. There is a substantial amount of literature proving that, quite generally, PA-graphs possess power-law degree sequences with a power-law exponent \tau>2. We prove that the diameter of the PA-model is bounded above by a constant times \log{t}, where t is the size of the graph. When the power-law exponent \tau exceeds 3, then we prove that \log{t} is the right order, by proving a lower bound of this order, both for the diameter as well as for the typical distance. This shows that, for \tau>3, distances are of the order \log{t}. For \tau\in (2,3), we improve the upper bound to a constant times \log\log{t}, and prove a lower bound of the same order for the diameter. Unfortunately, this proof does not extend to typical distances. These results do show that the diameter is of order \log\log{t}. These bounds partially prove predictions by physicists that the typical distance in PA-graphs are similar to the ones in other scale-free random graphs, such as the configuration model and various inhomogeneous random graph models, where typical distances have been shown to be of order \log\log{t} when \tau\in (2,3), and of order \log{t} when \tau>3

Mean-field driven first-order phase transitions in systems with long-range interactions

We consider a class of spin systems on $\Z^d$ with vector valued spins (\bS_x) that interact via the pair-potentials J_{x,y} \bS_x\cdot\bS_y. The interactions are generally spread-out in the sense that the $J_{x,y}$'s exhibit either exponential or power-law fall-off. Under the technical condition of reflection positivity and for sufficiently spread out interactions, we prove that the model exhibits a first-order phase transition whenever the associated mean-field theory signals such a transition. As a consequence, e.g., in dimensions $d\ge3$, we can finally provide examples of the 3-state Potts model with spread-out, exponentially decaying interactions, which undergoes a first-order phase transition as the temperature varies. Similar transitions are established in dimensions $d=1,2$ for power-law decaying interactions and in high dimensions for next-nearest neighbor couplings. In addition, we also investigate the limit of infinitely spread-out interactions. Specifically, we show that once the mean-field theory is in a unique ``state,'' then in any sequence of translation-invariant Gibbs states various observables converge to their mean-field values and the states themselves converge to a product measure.Comment: 57 pages; uses a (modified) jstatphys class fil

Random graph asymptotics on high-dimensional tori. II. Volume, diameter and mixing time

For critical bond-percolation on high-dimensional torus, this paper proves sharp lower bounds on the size of the largest cluster, removing a logarithmic correction in the lower bound in Heydenreich and van der Hofstad (2007). This improvement finally settles a conjecture by Aizenman (1997) about the role of boundary conditions in critical high-dimensional percolation, and it is a key step in deriving further properties of critical percolation on the torus. Indeed, a criterion of Nachmias and Peres (2008) implies appropriate bounds on diameter and mixing time of the largest clusters. We further prove that the volume bounds apply also to any finite number of the largest clusters. The main conclusion of the paper is that the behavior of critical percolation on the high-dimensional torus is the same as for critical Erdos-Renyi random graphs. In this updated version we incorporate an erratum to be published in a forthcoming issue of Probab. Theory Relat. Fields. This results in a modification of Theorem 1.2 as well as Proposition 3.1.Comment: 16 pages. v4 incorporates an erratum to be published in a forthcoming issue of Probab. Theory Relat. Field