The favoured antero-posterior lip position was affected by not only the antero-posterior facial disproportion but also by the vertical dimensions. The favoured lip positions differed between orthodontists and laypersons. These results might be helpful in deciding between extraction and non-extraction treatment in borderline cases.
In this paper, we investigate asymptotic properties of a consensus protocol taking place in a class of temporal (i.e., time-varying) networks called the activity driven network. We first show that a standard methodology provides us with an estimate of the convergence rate toward the consensus, in terms of the eigenvalues of a matrix whose computational cost grows exponentially fast in the number of nodes in the network. To overcome this difficulty, we then derive alternative bounds involving the eigenvalues of a matrix that is easy to compute. Our analysis covers the regimes of 1) sparse networks and 2) fast-switching networks. We numerically confirm our theoretical results by numerical simulations.
A. Mathematical PreliminariesWe let I n denote the identity matrix of dimension n and O n,m denote the n × m zero matrix. We denote by 1 n the n-dimensional vector whose elements are all one. We let J n denote the n × n matrix whose elements are all one. For a matrix M, let M denote the transpose of M. If M is symmetric, we denote the (real) eigenvalues of M by λ 1 (M) ≤ · · · ≤ λ n (M). For a vector x, its Euclidean norm is denoted by x . The probability of an event is denoted by P(·). For an integrable random variable X, we let E[X] denote its expectation.An undirected network is a pair G = (V, E ), where V = {1, . . . , n} is the set of nodes, and E is the set of
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