Realizability and uniqueness in graphs

AbstractConsider a finite graph G(V,E). Let us associate to G a finite list P(G) of invariants. To any P the following two natural problems arise: (R) Realizability. Given P, when is P=P(G) for some graph G?, (U) Uniqueness. Suppose P(G)=P(H) for graphs G and H. When does this imply G ≅ H? The best studied questions in this context are the degree realization problem for (R) and the reconstruction conjecture for (U). We discuss the problems (R) and (U) for the degree sequence and the size sequence of induced subgraphs for undirected and directed graphs, concentrating on the complexity of the corresponding decision problems and their connection to a natural search problem on graphs

The Architecture of Tell Jalul

The Problem. Buildings excavated in Tell Jalul have been studied separately for the last twenty years. However, until the present moment, no comprehensive study has been conducted. Therefore, this thesis will discuss the architecture of the main edifications found in Tell Jalul from 1992 to 2012. -- The Method. For this project, the ruins found in fields A, B, C, D, G and W were considered. First, a literature review was done to find similar structures in Jordan and the neighboring areas and their basic features. Then, a three-dimensional reconstruction of the ruins was suggested. The reconstruction was made in AutoCad 2013, software widely used for engineering, based on the two-dimensional drawings previously prepared. -- The Results. Parallels to the majority of the constructions were found, making it possible to compare them and suggest a reconstruction and possible implications. In addition, a three-dimensional model was made of each individual building and also of the entire tell with the studied constructions. -- Conclusions. It was possible to conclude that Jalul was a significant site in different periods of history. The size and complexity of the constructions point to a centralized power. However, the tell does not have large enough structures to be a royal city, nor is it small enough to fit in the village description

Robust Network Reconstruction in Polynomial Time

This paper presents an efficient algorithm for robust network reconstruction of Linear Time-Invariant (LTI) systems in the presence of noise, estimation errors and unmodelled nonlinearities. The method here builds on previous work on robust reconstruction to provide a practical implementation with polynomial computational complexity. Following the same experimental protocol, the algorithm obtains a set of structurally-related candidate solutions spanning every level of sparsity. We prove the existence of a magnitude bound on the noise, which if satisfied, guarantees that one of these structures is the correct solution. A problem-specific model-selection procedure then selects a single solution from this set and provides a measure of confidence in that solution. Extensive simulations quantify the expected performance for different levels of noise and show that significantly more noise can be tolerated in comparison to the original method.Comment: 8 pages, to appear in 51st IEEE Conference on Decision and Contro

Power-Constrained Sparse Gaussian Linear Dimensionality Reduction over Noisy Channels

In this paper, we investigate power-constrained sensing matrix design in a sparse Gaussian linear dimensionality reduction framework. Our study is carried out in a single--terminal setup as well as in a multi--terminal setup consisting of orthogonal or coherent multiple access channels (MAC). We adopt the mean square error (MSE) performance criterion for sparse source reconstruction in a system where source-to-sensor channel(s) and sensor-to-decoder communication channel(s) are noisy. Our proposed sensing matrix design procedure relies upon minimizing a lower-bound on the MSE in single-- and multiple--terminal setups. We propose a three-stage sensing matrix optimization scheme that combines semi-definite relaxation (SDR) programming, a low-rank approximation problem and power-rescaling. Under certain conditions, we derive closed-form solutions to the proposed optimization procedure. Through numerical experiments, by applying practical sparse reconstruction algorithms, we show the superiority of the proposed scheme by comparing it with other relevant methods. This performance improvement is achieved at the price of higher computational complexity. Hence, in order to address the complexity burden, we present an equivalent stochastic optimization method to the problem of interest that can be solved approximately, while still providing a superior performance over the popular methods.Comment: Accepted for publication in IEEE Transactions on Signal Processing (16 pages