arXiv subjects
Publications and source records attributed to P. Van Mieghem.
The replacement of the classical derivative in the governing equation of a continuous-time Markov process with finite state space by a fractional derivative of arbitrary order $α>1$ produces solutions that are mathematically incompatible with the theory of stochastic processes.
The current density in a rectangle in the $D=2$ dimensional Euclidean space, where opposite point charges are placed in a source $s$ and a destination $d$, is computed as the gradient of a potential that obeys the Poisson equation (first law of Maxwell). Although the computation of the current density in a 2D rectangle is a classical problem, several barriers (e.g. very slow convergence of double Fourier series) were eventually alleviated by a $q$-analysis, related to Gaussian polynomials and Jacobi's theta-functions, that form the basic building blocks for any elliptic function. We present an exact and computationally very efficient analytic formula of the potential and the magnitude of the current density in a rectangle.
We investigate a generalization of Binet's factorial series in the parameter $α$ \[ μ\left( z\right) =\sum_{m=1}^{\infty}\frac{b_{m}\left( α\right) }{\prod_{k=0}^{m-1}(z+α+k)}% \] due to Gilbert, for the Binet function \[ μ\left( z\right) =\logΓ\left( z\right) -\left( z-\frac{1} {2}\right) \log z+z-\frac{1}{2}\log\left( 2π\right) \] After a review of the Binet function $μ\left( z\right) $ and Gilbert's investigations of $μ\left( z\right) $, several properties of the Binet polynomials $b_{m}\left( α\right) $ are presented. We compare Gilbert's generalized factorial series with Stirling's asymptotic expansion and demonstrate by a numerical example that, with a same number of terms evaluated, the Gilbert generalized factorial series with an optimized value of $α$ can beat the best possible accuracy of Stirling's expansion. Finally, we extend Binet's method to factorial series of Laplace transforms.
Heterogeneous Markovian Susceptible-Infected-Susceptible (SIS) epidemics with a general infection rate matrix $\widetilde{A}$ are considered. Using a non-negative matrix factorization to approximate $\widetilde{A}$, we are able to identify when a metastable state can be expected, and that the metastable distribution, under certain conditions, will feature a normal distribution with known expectation and covariance. Furthermore, we model a heterogeneous Markovian SIS epidemic, that starts with a fraction of initially infected nodes different from that in the metastable state, by approximating its behaviour by a standard linear stochastic differential equation (SDE) in sufficiently high dimensions. By exploiting the knowledge of the covariance matrix from the SDE, we demonstrate significant accuracy improvements over the first-order mean-field approximation NIMFA.
It is well-known that the synchronization of diffusively-coupled systems on networks strongly depends on the network topology. In particular, the so-called algebraic connectivity $μ_{N-1}$, or the smallest non-zero eigenvalue of the discrete Laplacian operator plays a crucial role on synchronization, graph partitioning, and network robustness. In our study, synchronization is placed in the general context of networks-of-networks, where single network models are replaced by a more realistic hierarchy of interdependent networks. The present work shows, analytically and numerically, how the algebraic connectivity experiences sharp transitions after the addition of sufficient links among interdependent networks.
We propose that hypergraphs can be used to model social networks with overlapping communities. The nodes of the hypergraphs represent the communities. The hyperlinks of the hypergraphs denote the individuals who may participate in multiple communities. The hypergraphs are not easy to analyze, however, the line graphs of hypergraphs are simple graphs or weighted graphs, so that the network theory can be applied. We define the overlapping depth $k$ of an individual by the number of communities that overlap in that individual, and we prove that the minimum adjacency eigenvalue of the corresponding line graph is not smaller than $-k_{max}$, which is the maximum overlapping depth of the whole network. Based on hypergraphs with preferential attachment, we establish a network model which incorporates overlapping communities with tunable overlapping parameters $k$ and $w$. By comparing with the Hyves social network, we show that our social network model possesses high clustering, assortative mixing, power-law degree distribution and short average path length.
We propose a new algorithm MARINLINGA for reverse line graph computation, i.e., constructing the original graph from a given line graph. Based on the completely new and simpler principle of link relabeling and endnode recognition, MARINLINGA does not rely on Whitney's theorem while all previous algorithms do. MARINLINGA has a worst case complexity of O(N^2), where N denotes the number of nodes of the line graph. We demonstrate that MARINLINGA is more time-efficient compared to Roussopoulos's algorithm, which is well-known for its efficiency.
In recent years, many network perturbation techniques, such as topological perturbations and service perturbations, were employed to study and improve the robustness of complex networks. However, there is no general way to evaluate the network robustness. In this paper, we propose a new global measure for a network, the reconstructability coefficient θ, defined as the maximum number of eigenvalues that can be removed, subject to the condition that the adjacency matrix can be reconstructed exactly. Our main finding is that a linear scaling law, E[θ]=aN, seems universal, in that it holds for all networks that we have studied.