Distributive Network Utility Maximization (NUM) over Time-Varying Fading Channels

Distributed network utility maximization (NUM) has received an increasing intensity of interest over the past few years. Distributed solutions (e.g., the primal-dual gradient method) have been intensively investigated under fading channels. As such distributed solutions involve iterative updating and explicit message passing, it is unrealistic to assume that the wireless channel remains unchanged during the iterations. Unfortunately, the behavior of those distributed solutions under time-varying channels is in general unknown. In this paper, we shall investigate the convergence behavior and tracking errors of the iterative primal-dual scaled gradient algorithm (PDSGA) with dynamic scaling matrices (DSC) for solving distributive NUM problems under time-varying fading channels. We shall also study a specific application example, namely the multi-commodity flow control and multi-carrier power allocation problem in multi-hop ad hoc networks. Our analysis shows that the PDSGA converges to a limit region rather than a single point under the finite state Markov chain (FSMC) fading channels. We also show that the order of growth of the tracking errors is given by O(T/N), where T and N are the update interval and the average sojourn time of the FSMC, respectively. Based on this analysis, we derive a low complexity distributive adaptation algorithm for determining the adaptive scaling matrices, which can be implemented distributively at each transmitter. The numerical results show the superior performance of the proposed dynamic scaling matrix algorithm over several baseline schemes, such as the regular primal-dual gradient algorithm

Resolving the Berezinskii-Kosterlitz-Thouless transition in the 2D XY model with tensor-network based level spectroscopy

Berezinskii-Kosterlitz-Thouless transition of the classical XY model is re-investigated, combining the Tensor Network Renormalization (TNR) and the Level Spectroscopy method based on the finite-size scaling of the Conformal Field Theory. By systematically analyzing the spectrum of the transfer matrix of the systems of various moderate sizes which can be accurately handled with a finite bond dimension, we determine the critical point removing the logarithmic corrections. This improves the accuracy by an order of magnitude over previous studies including those utilizing TNR. Our analysis also gives a visualization of the celebrated Kosterlitz Renormalization Group flow based on the numerical data

Convergence of Tomlin's HOTS algorithm

The HOTS algorithm uses the hyperlink structure of the web to compute a vector of scores with which one can rank web pages. The HOTS vector is the vector of the exponentials of the dual variables of an optimal flow problem (the "temperature" of each page). The flow represents an optimal distribution of web surfers on the web graph in the sense of entropy maximization. In this paper, we prove the convergence of Tomlin's HOTS algorithm. We first study a simplified version of the algorithm, which is a fixed point scaling algorithm designed to solve the matrix balancing problem for nonnegative irreducible matrices. The proof of convergence is general (nonlinear Perron-Frobenius theory) and applies to a family of deformations of HOTS. Then, we address the effective HOTS algorithm, designed by Tomlin for the ranking of web pages. The model is a network entropy maximization problem generalizing matrix balancing. We show that, under mild assumptions, the HOTS algorithm converges with a linear convergence rate. The proof relies on a uniqueness property of the fixed point and on the existence of a Lyapunov function. We also show that the coordinate descent algorithm can be used to find the ideal and effective HOTS vectors and we compare HOTS and coordinate descent on fragments of the web graph. Our numerical experiments suggest that the convergence rate of the HOTS algorithm may deteriorate when the size of the input increases. We thus give a normalized version of HOTS with an experimentally better convergence rate.Comment: 21 page

Scaling Behaviors of Weighted Food Webs as Energy Transportation Networks

Food webs can be regarded as energy transporting networks in which the weight of each edge denotes the energy flux between two species. By investigating 21 empirical weighted food webs as energy flow networks, we found several ubiquitous scaling behaviors. Two random variables $A_i$ and $C_i$ defined for each vertex $i$, representing the total flux (also called vertex intensity) and total indirect effect or energy store of $i$, were found to follow power law distributions with the exponents $\alpha\approx 1.32$ and $\beta\approx 1.33$, respectively. Another scaling behavior is the power law relationship, $C_i\sim A_i^\eta$, where $\eta\approx 1.02$. This is known as the allometric scaling power law relationship because $A_i$ can be treated as metabolism and $C_i$ as the body mass of the sub-network rooted from the vertex $i$, according to the algorithm presented in this paper. Finally, a simple relationship among these power law exponents, $\eta=(\alpha-1)/(\beta-1)$, was mathematically derived and tested by the empirical food webs

Finite-size and finite bond dimension effects of tensor network renormalization

We propose a general procedure for extracting the running coupling constants of the underlying field theory of a given classical statistical model on a two-dimensional lattice, combining tensor network renormalization (TNR) and the finite-size scaling theory of conformal field theory. By tracking the coupling constants at each scale, we are able to visualize the renormalization group (RG) flow and demonstrate it with the classical Ising and 3-state Potts models. Furthermore, utilizing the new methodology, we reveal the limitations due to finite bond dimension D on TNR applied to critical systems. We find that a finite correlation length is imposed by the finite bond dimension in TNR, and it can be attributed to an emergent relevant perturbation that respects the symmetries of the system. The correlation length shows the same power-law dependence on D as the "finite entanglement scaling" of the Matrix Product States

Scaling of transmission capacities in coarse-grained renewable electricity networks

Network models of large-scale electricity systems feature only a limited spatial resolution, either due to lack of data or in order to reduce the complexity of the problem with respect to numerical calculations. In such cases, both the network topology, the load and the generation patterns below a given spatial scale are aggregated into representative nodes. This coarse-graining affects power flows and thus the resulting transmission needs of the system. We derive analytical scaling laws for measures of network transmission capacity and cost in coarse-grained renewable electricity networks. For the cost measure only a very weak scaling with the spatial resolution of the system is found. The analytical results are shown to describe the scaling of the transmission infrastructure measures for a simplified, but data-driven and spatially detailed model of the European electricity system with a high share of fluctuating renewable generation.Comment: to be published in EP