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Register a new userAn Analytical Solution to the $k$-core Pruning Process
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$k$-core decomposition is widely used to identify the center of a large network, it is a pruning process in which the nodes with degrees less than $k$ are recursively removed. Although the simplicity and effectiveness of this method facilitate its implementation on broad applications across many scientific fields, it produces few analytical results. We here simplify the existing theoretical framework to a simple iterative relationship and obtain the exact analytical solutions of the $k$-core pruning process on large uncorrelated networks. From these solutions we obtain such statistical properties as the degree distribution and the size of the remaining subgraph in each of the pruning steps. Our theoretical results resolve the long-lasting puzzle of the $k$-core pruning dynamics and provide an intuitive description of the dynamic process.
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Multi-layer networks or multiplex networks are generally considered as the networks that have the same set of vertices but different types of edges. Multi-layer networks are especially useful when describing the systems with several kinds of interactions. In this paper we study the analytical solution of $textbf{k}$-core pruning process on multi-layer networks. $k$-core decomposition is a widely used method to find the dense core of the network. Previously the Nonbacktracking Expand Branch (NBEB) is found to be able to easily derive the exact analytical results in the $k$-core pruning process. Here we further extend this method to solve the $textbf{k}$-core pruning process on multi-layer networks by designing a variation of the method called Multicolor Nonbacktracking Expand Branch (MNEB). Our results show that, given any initial multi-layer network, Multicolor Nonbacktracking Expand Branch can offer the exact solution for each intermediate state of the pruning process, these results do not only apply to uncorrelated network, but also apply to networks with either interlayer correlations or in-layer correlations.
We induce the NonBacktracking Expansion Branch method to analyze the k-core pruning process on the monopartite graph G which does not contain any self-loop or multi-edge. Different from the traditional approaches like the generating functions or the degree distribution evolution equations which are mathematically difficult to solve, this method provides a simple and intuitive solution of the k-core pruning process. Besides, this method can be naturally extended to study the k-core pruning process on correlated networks, which is among the few attempts to analytically solve the problem.
We apply moment methods to obtaining an approximate analytical solution to Knudsen layers. Based on the hyperbolic regularized moment system for the Boltzmann equation with the Shakhov collision model, we derive a linearized hyperbolic moment system to model the scenario with the Knudsen layer vicinity to a solid wall with Maxwell boundary condition. We find that the reduced system is in an even-odd parity form that the reduced system proves to be well-posed under all accommodation coefficients. We show that the system may capture the temperature jump coefficient and the thermal Knudsen layer well with only a few moments. With the increasing number of moments used, qualitative convergence of the approximate solution is observed.
In this paper we present a framework which provides an analytical (i.e., infinitely differentiable) transformation between spatial coordinates and orbital elements for the solution of the gravitational two-body problem. The formalism omits all singular variables which otherwise would yield discontinuities. This method is based on two simple real functions for which the derivative rules are only required to be known, all other applications -- e.g., calculating the orbital velocities, obtaining the partial derivatives of radial velocity curves with respect to the orbital elements -- are thereafter straightforward. As it is shown, the presented formalism can be applied to find optimal instants for radial velocity measurements in transiting exoplanetary systems to constrain the orbital eccentricity as well as to detect secular variations in the eccentricity or in the longitude of periastron.
Multiplex networks are convenient mathematical representations for many real-world -- biological, social, and technological -- systems of interacting elements, where pairwise interactions among elements have different flavors. Previous studies pointed out that real-world multiplex networks display significant inter-layer correlations -- degree-degree correlation, edge overlap, node similarities -- able to make them robust against random and targeted failures of their individual components. Here, we show that inter-layer correlations are important also in the characterization of their $mathbf{k}$-core structure, namely the organization in shells of nodes with increasingly high degree. Understanding $k$-core structures is important in the study of spreading processes taking place on networks, as for example in the identification of influential spreaders and the emergence of localization phenomena. We find that, if the degree distribution of the network is heterogeneous, then a strong $mathbf{k}$-core structure is well predicted by significantly positive degree-degree correlations. However, if the network degree distribution is homogeneous, then strong $mathbf{k}$-core structure is due to positive correlations at the level of node similarities. We reach our conclusions by analyzing different real-world multiplex networks, introducing novel techniques for controlling inter-layer correlations of networks without changing their structure, and taking advantage of synthetic network models with tunable levels of inter-layer correlations.
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