Abstract. We consider a Markov process on a connected graph, with edges labeled with transition rates between the adjacent vertices. The distribution of the Markov process converges to the uniform distribution at a rate determined by the second smallest eigenvalue λ 2 of the Laplacian of the weighted graph. In this paper we consider the problem of assigning transition rates to the edges so as to maximize λ 2 subject to a linear constraint on the rates. This is the problem of finding the fastest mixing Markov process (FMMP) on the graph. We show that the FMMP problem is a convex optimization problem, which can in turn be expressed as a semidefinite program, and therefore effectively solved numerically. We formulate a dual of the FMMP problem and show that it has a natural geometric interpretation as a maximum variance unfolding (MVU) problem, i.e., the problem of choosing a set of points to be as far apart as possible, measured by their variance, while respecting local distance constraints. This MVU problem is closely related to a problem recently proposed by Weinberger and Saul as a method for "unfolding" high-dimensional data that lies on a low-dimensional manifold. The duality between the FMMP and MVU problems sheds light on both problems, and allows us to characterize and, in some cases, find optimal solutions.

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BackgroundThe one-stage procedure for treatment of older developmental dislocation of the hip (DDH) is used widely. However, the best age group for this operation is still unknown. The aims of our study were to evaluate middle-term outcomes of one-stage surgical treatment of a large number of patients with late-diagnosed DDH, and to explore the best age group for treatment.MethodsWe retrospectively reviewed 652 patients with a total of 864 hip joints with DDH, all aged >18 months. All the hip joints were treated with one-stage procedures including open reduction, pelvic osteotomy, and femoral shortening. The patients were divided into three groups according to age at surgery: Group I: 1.5–2.5 years; Group II: 2.5–8 years; and Group III: >8 years. The latest clinical and radiographic outcomes, complications and avascular necrosis (AVN) of the femoral head were evaluated and compared among the three groups.ResultsThe mean age at surgery was 5.8 years (range: 1.5–13.2 years). The average time of follow-up was 6.2 years (range: 3.2–8.9 years). A total of 79.4% of good or excellent outcomes were obtained for clinical functional evaluation according to the McKay classification. For radiographic outcomes, 732 hips (84.7%) were classified as good or excellent according to the Severin classification. A total of 27.4% of all hips had a poor outcome according to the Kalamchi and MacEwen classification for AVN. The poorest outcomes were observed for clinical, radiographic and AVN results in Group III (p < 0.001). Compared with Group I, the better results for clinical and AVN outcomes were found in Group II (p < 0.001). However, similar clinical outcomes were observed between Groups I and II (p > 0.05). A significantly higher incidence of redislocation and residual acetabular dysplasia was observed in Tonnis grade II and III hip dislocation (p < 0.001).ConclusionsOne-stage treatment of late-diagnosed DDH had a good outcome in young and middle group. Younger patients achieved better results than older patients. However, the best age group was 2.5–8 years. Tonnis grade II and III DDH is a risk factor for redislocation and residual acetabular dysplasia after the one-stage operation.Electronic supplementary materialThe online version of this article (doi:10.1186/1471-2474-15-401) contains supplementary material, which is available to authorized users.

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