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

Switch chain mixing times through triangle counts

Sampling uniform simple graphs with power-law degree distributions with degree exponent τ∈(2,3) is a non-trivial problem. We propose a method to sample uniform simple graphs that uses a constrained version of the configuration model together with a Markov Chain switching method. We test the convergence of this algorithm numerically in the context of the presence of small subgraphs. We then compare the number of triangles in uniform random graphs with the number of triangles in the erased configuration model. Using simulations and heuristic arguments, we conjecture that the number of triangles in the erased configuration model is larger than the number of triangles in the uniform random graph, provided that the graph is sufficiently large

Optimal subgraph structures in scale-free configuration models

Subgraphs reveal information about the geometry and functionalities of complex networks. For scale-free networks with unbounded degree fluctuations, we count the number of times a small connected graph occurs as a subgraph (motif counting) or as an induced subgraph (graphlet counting). We obtain these results by analyzing the configuration model with degree exponent $\tau\in(2,3)$ and introducing a novel class of optimization problems. For any given subgraph, the unique optimizer describes the degrees of the nodes that together span the subgraph. We find that every subgraph occurs typically between vertices with specific degree ranges. In this way, we can count and characterize {\it all} subgraphs. We refrain from double counting in the case of multi-edges, essentially counting the subgraphs in the {\it erased} configuration model

Mesoscopic scales in hierarchical configuration models

To understand mesoscopic scaling in networks, we study the hierarchical configuration model (HCM), a random graph model with community structure. The connections between the communities are formed as in a configuration model. We study the component sizes of the hierarchical configuration model at criticality when the inter-community degrees have a finite third moment. We find the conditions on the community sizes such that the critical component sizes of the HCM behave similarly as in the configuration model. Furthermore, we study critical bond percolation on the HCM. We show that the ordered components of a critical HCM on $N$ vertices are of sizes $O(N^{2/3})$. More specifically, the rescaled component sizes converge to the excursions of a Brownian motion with parabolic drift, as for the scaling limit for the configuration model under a finite third moment condition

Triadic closure in configuration models with unbounded degree fluctuations

The configuration model generates random graphs with any given degree distribution, and thus serves as a null model for scale-free networks with power-law degrees and unbounded degree fluctuations. For this setting, we study the local clustering c(k) , i.e., the probability that two neighbors of a degree-k node are neighbors themselves. We show that c(k) progressively falls off with k and eventually for k=Ω(n − − √ ) settles on a power law c(k)∼k −2(3−τ) with τ∈(2,3) the power-law exponent of the degree distribution. This fall-off has been observed in the majority of real-world networks and signals the presence of modular or hierarchical structure. Our results agree with recent results for the hidden-variable model and also give the expected number of triangles in the configuration model when counting triangles only once despite the presence of multi-edges. We show that only triangles consisting of triplets with uniquely specified degrees contribute to the triangle counting

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