Variational principle for scale-free network motifs

For scale-free networks with degrees following a power law with an exponent $\tau\in(2,3)$, the structures of motifs (small subgraphs) are not yet well understood. We introduce a method designed to identify the dominant structure of any given motif as the solution of an optimization problem. The unique optimizer describes the degrees of the vertices that together span the most likely motif, resulting in explicit asymptotic formulas for the motif count and its fluctuations. We then classify all motifs into two categories: motifs with small and large fluctuations

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 obtain the asymptotics of the number of times a small connected graph occurs as a subgraph or as an induced subgraph. 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 vertices that together span the subgraph. We find that subgraphs typically occur 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.Comment: 50 pages, 2 figure

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 $k=\Omega(\sqrt{n})$ settles on a power law $c(k)\sim k^{-2(3-\tau)}$ with $\tau\in(2,3)$ 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

Cluster tails for critical power-law inhomogeneous random graphs

Recently, the scaling limit of cluster sizes for critical inhomogeneous random graphs of rank-1 type having finite variance but infinite third moment degrees was obtained (see previous work by Bhamidi, van der Hofstad and van Leeuwaarden). It was proved that when the degrees obey a power law with exponent in the interval (3,4), the sequence of clusters ordered in decreasing size and scaled appropriately converges as n goes to infinity to a sequence of decreasing non-degenerate random variables. Here, we study the tails of the limit of the rescaled largest cluster, i.e., the probability that the scaling limit of the largest cluster takes a large value u, as a function of u. This extends a related result of Pittel for the Erd\H{o}s-R\'enyi random graph to the setting of rank-1 inhomogeneous random graphs with infinite third moment degrees. We make use of delicate large deviations and weak convergence arguments.Comment: corrected and updated first referenc

Novel scaling limits for critical inhomogeneous random graphs

We find scaling limits for the sizes of the largest components at criticality for rank-1 inhomogeneous random graphs with power-law degrees with power-law exponent \tau. We investigate the case where $\tau\in(3,4)$, so that the degrees have finite variance but infinite third moment. The sizes of the largest clusters, rescaled by $n^{-(\tau-2)/(\tau-1)}$, converge to hitting times of a "thinned" L\'{e}vy process, a special case of the general multiplicative coalescents studied by Aldous [Ann. Probab. 25 (1997) 812-854] and Aldous and Limic [Electron. J. Probab. 3 (1998) 1-59]. Our results should be contrasted to the case \tau>4, so that the third moment is finite. There, instead, the sizes of the components rescaled by $n^{-2/3}$ converge to the excursion lengths of an inhomogeneous Brownian motion, as proved in Aldous [Ann. Probab. 25 (1997) 812-854] for the Erd\H{o}s-R\'{e}nyi random graph and extended to the present setting in Bhamidi, van der Hofstad and van Leeuwaarden [Electron. J. Probab. 15 (2010) 1682-1703] and Turova [(2009) Preprint].Comment: Published in at http://dx.doi.org/10.1214/11-AOP680 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

From satisfaction to expectation: The patient's perspective in lower limb prosthetic care

Neck pain is a common musculoskeletal complaint and a relationship with reduced work-related functional capacity is assumed. A validated instrument to test functional capacity of patients with neck pain is unavailable. The objective of this study was to develop a Functional Capacity Evaluation (FCE), which is content valid for determining functional capacity in patients with work related neck disorders (WRND). A review of epidemiological review literature was conducted to identify physical risk factors for WRND. Evidence was found that physical risk factors contribute in development of WRND. Physical risk factors were related to repetitive movements, forceful movements, awkward positions and static contractions of the neck or the neck/shoulder region. An FCE was designed based on the risk factors identified. Eight tests were selected to cover all risk factors: repetitive side reaching, repetitive reaching overhead, static overhead work, front carry, forward static bend neck, overhead lift and the neck strength test. Content validity of this FCE was established by providing the rationale, specific objectives and operational definitions of the FCE. Further research is needed to establish reliability and other aspects of validity of the neck-FCE Aim Worldwide, family- centred and co- ordinated care are seen as the two most desirable and effective methods of paediatric care delivery. This study outlines current views on how team collaboration comprising professionals in paediatric rehabilitation and special education and the parents of children with disabilities should be organized, and analyses the policies of five paediatric rehabilitation settings associated with the care of 44 children with cerebral palsy ( CP) in the Netherlands. Methods For an overview of current ideas on collaboration, written statements of professional associations in Dutch paediatric rehabilitation were examined. The policy statements of the five participating settings were derived from their institutional files. Documents detailing the collaborative arrangements involving the various professionals and parents were evaluated at the institutional level and at the child level. Involvement of the stakeholders was analysed based on team conferences. Results Also in the Netherlands collaboration between rehabilitation and education professionals and parents is endorsed as the key principle in paediatric rehabilitation, with at its core the team conference in which the various priorities and goals are formulated and integrated into a personalized treatment plan. As to their collaborative approaches between rehabilitation centre and school, the five paediatric settings rarely differed, but at the child level approaches varied. Teams were large ( averaging 10.5 members), and all three stakeholder groups were represented, but involvement differed per setting, as did the roles and contributions of the individual team members. Conclusion Collaboration between rehabilitation and education professionals and parents is supported and encouraged nationwide. Views on collaboration have been formulated, and general guidelines on family- centred and co- ordinated care are available. Yet, collaborative practices in Dutch paediatric care are still developing. Protocols that carefully delineate the commitments to collaborate and that translate the policies into practical, detailed guidelines are needed, as they are a prerequisite for successful teamwork

Pattern theorems, ratio limit theorems and Gumbel maximal clusters for random fields

We study occurrences of patterns on clusters of size n in random fields on Z^d. We prove that for a given pattern, there is a constant a>0 such that the probability that this pattern occurs at most an times on a cluster of size n is exponentially small. Moreover, for random fields obeying a certain Markov property, we show that the ratio between the numbers of occurrences of two distinct patterns on a cluster is concentrated around a constant value. This leads to an elegant and simple proof of the ratio limit theorem for these random fields, which states that the ratio of the probabilities that the cluster of the origin has sizes n+1 and n converges as n tends to infinity. Implications for the maximal cluster in a finite box are discussed.Comment: 23 pages, 2 figure

Clustering Spectrum of scale-free networks

Real-world networks often have power-law degrees and scale-free properties such as ultra-small distances and ultra-fast information spreading. In this paper, we study a third universal property: three-point correlations that suppress the creation of triangles and signal the presence of hierarchy. We quantify this property in terms of $\bar c(k)$, the probability that two neighbors of a degree-$k$ node are neighbors themselves. We investigate how the clustering spectrum $k\mapsto\bar c(k)$ scales with $k$ in the hidden variable model and show that $c(k)$ follows a {\it universal curve} that consists of three $k$-ranges where $\bar c(k)$ remains flat, starts declining, and eventually settles on a power law $\bar c(k)\sim k^{-\alpha}$ with $\alpha$ depending on the power law of the degree distribution. We test these results against ten contemporary real-world networks and explain analytically why the universal curve properties only reveal themselves in large networks