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Register a new userDirectional Statistics of Preferential Orientations of Two Shapes in Their Aggregate and Its Application to Study Preferential Attachment of Nanoparticles
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Added by
Chiwoo Park
Publication date
2016
fields
Mathematical Statistics
and research's language is
English
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Nanoscientists have long conjectured that adjacent nanoparticles aggregate with one another in certain preferential directions during a chemical synthesis of nanoparticles, which is referred to the oriented attachment. For the study of the oriented attachment, the microscopy and nanoscience communities have used dynamic electron microscopy for direct observations of nanoparticle aggregation and have been so far relying on manual and qualitative analysis of the observations. We propose a statistical approach for studying the oriented attachment quantitatively with multiple aggregation examples in imagery observations. We abstract an aggregation by an event of two primary geometric objects merging into a secondary geometric object. We use a point set representation to describe the geometric features of the primary objects and the secondary object, and formulated the alignment of two point sets to one point set to estimate the orientation angles of the primary objects in the secondary object. The estimated angles are used as data to estimate the probability distribution of the orientation angles and test important hypotheses statistically. The proposed approach was applied for our motivating example, which demonstrated that nanoparticles of certain geometries have indeed preferential orientations in their aggregates.
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We consider the degree distributions of preferential attachment random graph models with choice similar to those considered in recent work by Malyshkin and Paquette and Krapivsky and Redner. In these models a new vertex chooses $r$ vertices according to a preferential rule and connects to the vertex in the selection with the $s$th highest degree. For meek choice, where $s>1$, we show that both double exponential decay of the degree distribution and condensation-like behaviour are possible, and provide a criterion to distinguish between them. For greedy choice, where $s=1$, we confirm that the degree distribution asympotically follows a power law with logarithmic correction when $r=2$ and shows condensation-like behaviour when $r>2$.
Generalized preferential attachment is defined as the tendency of a vertex to acquire new links in the future with respect to a particular vertex property. Understanding which properties influence link acquisition tendency (LAT) gives us a predictive power to estimate the future growth of network and insight about the actual dynamics governing the complex networks. In this study, we explore the effect of age and degree on LAT by analyzing data collected from a new complex-network growth dataset. We found that LAT and degree of a vertex are linearly correlated in accordance with previous studies. Interestingly, the relation between LAT and age of a vertex is found to be in conflict with the known models of network growth. We identified three different periods in the networks lifetime where the relation between age and LAT is strongly positive, almost stationary and negative correspondingly.
In the Yule-Simon process, selection of words follows the preferential attachment mechanism, resulting in the power-law growth in the cumulative number of individual word occurrences. This is derived using mean-field approximation, assuming a continuum limit of both the time and number of word occurrences. However, time and word occurrences are inherently discrete in the process, and it is natural to assume that the cumulative number of word occurrences has a certain fluctuation around the average behavior predicted by the mean-field approximation. We derive the exact and approximate forms of the probability distribution of such fluctuation analytically and confirm that those probability distributions are well supported by the numerical experiments.
We give an explicit construction of the weak local limit of a class of preferential attachment graphs. This limit contains all local information and allows several computations that are otherwise hard, for example, joint degree distributions and, more generally, the limiting distribution of subgraphs in balls of any given radius $k$ around a random vertex in the preferential attachment graph. We also establish the finite-volume corrections which give the approach to the limit.
Global degree/strength based preferential attachment is widely used as an evolution mechanism of networks. But it is hard to believe that any individual can get global information and shape the network architecture based on it. In this paper, it is found that the global preferential attachment emerges from the local interaction models, including distance-dependent preferential attachment (DDPA) evolving model of weighted networks(M. Li et al, New Journal of Physics 8 (2006) 72), acquaintance network model(J. Davidsen et al, Phys. Rev. Lett. 88 (2002) 128701) and connecting nearest-neighbor(CNN) model(A. Vazquez, Phys. Rev. E 67 (2003) 056104). For DDPA model and CNN model, the attachment rate depends linearly on the degree or strength, while for acquaintance network model, the dependence follows a sublinear power law. It implies that for the evolution of social networks, local contact could be more fundamental than the presumed global preferential attachment. This is onsistent with the result observed in the evolution of empirical email networks.
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