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On certain information-theoretic concepts in the theory of graphs
Information-theoretic concepts in theory of random graphs - entropy functions for probability distributions and Markov chains
A Survey of Mathematical Structures for Lunar Networks
To sustain the current and increasing accessibility of space, a scalable communications infrastructure (i.e. the Solar System Internet, SSI) is necessary. The goal of this paper is to begin the discovery of the fundamental underlying mathematical structure of space networks to help the research community harness these structures for algorithm development and optimization. To ensure the applicability of the research, the approaches are considered through the lens of simulated scenarios inspired by the Artemis Back-to-the-Moon mission set for 2024. We note that any approach to an SSI must fit under the umbrella of Delay Tolerant Networking (DTN), due to celestial mobility, high link latencies, high variance in link latencies, disconnections, lack of end-to-end paths, and so on. These difficulties are exacerbated by the fact that the underlying structure of a space network is a time-evolving network and may experience multiple discontinuities in its topology. In this paper we propose several novel approaches to a mathematical foundation for Delay Tolerant Networking Theory that fall outside the traditional scope of temporal network theory. These techniques include methods from Topological Data Analysis, Dynamic Graph Analysis, Applied Algebraic Geometry, Probability Theory, and Game Theory. Some of these methods include tools adapted to the study of dynamic metric spaces, such as zigzag persistent homology and their higher parameter analogs. We find that several of these methods target desired engineering outcomes such as discovery and automatic sub-netting. While each approach is theoretical, they are also algorithmic in nature and offer immediate practical applications. The paper concludes with comparisons of the various methods along with suggestions for future work.
Equivariant, safe and sensitive — graph networks for new physics
This study introduces a novel Graph Neural Network (GNN) architecture that leverages infrared and collinear (IRC) safety and equivariance to enhance the analysis of collider data for Beyond the Standard Model (BSM) discoveries. By integrating equivariance in the rapidity-azimuth plane with IRC-safe principles, our model significantly reduces computational overhead while ensuring theoretical consistency in identifying BSM scenarios amidst Quantum Chromodynamics backgrounds. The proposed GNN architecture demonstrates superior performance in tagging semi-visible jets, highlighting its potential as a robust tool for advancing BSM search strategies at high-energy colliders.
Efficient 3D Molecular Design with an E(3) Invariant Transformer VAE
This work introduces a three-dimensional (3D) invariant graph-to-string transformer variational autoencoders (VAE) (Vagrant) for generating molecules with accurate density functional theory (DFT)-level properties. Vagrant learns to model the joint probability distribution of a 3D molecular structure and its properties by encoding molecular structures into a 3D-aware latent space. Directed navigation through this latent space implicitly optimizes the 3D structure of a molecule, and the latent embedding can be used to condition a generative transformer to predict the candidate structure as a one-dimensional (1D) sequence. Additionally, we introduce two novel sampling methods that exploit the latent characteristics of a VAE to improve performance. Here, we show that our method outperforms comparable 3D autoregressive and diffusion methods for predicting quantum chemical property values of novel molecules in terms of both sample quality and computational efficiency.
Defender Policy Evaluation and Resource Allocation With MITRE ATT&CK Data and Evaluations.
Abstract not provided.
The Lovasz bound and some generalizations
The zero error capacity of a discrete memoryless channel is defined as the largest rate at which information can be transmitted over the channel with zero error probability. One channel with five inputs and outputs whose zero capacity remained unsolved until very recently is considered. An extremely powerful and general technique phased in terms of graph theory, for studying combinatorial packing problems is presented. In particular, Delsarte's linear programming bound for cliques in association schemes appears as a special case of the Lovasz bound.
Identifying Molecules as Biosignatures with Assembly Theory and Mass Spectrometry
The search for evidence of life elsewhere in the universe is hard because it is not obvious what signatures are unique to life. Here we postulate that complex molecules found in high abundance are universal biosignatures as they cannot form by chance. To explore this, we developed the first intrinsic measure of molecular complexity that can be experimentally determined, and this is based upon a new approach called assembly theory which gives the molecular assembly number (MA) of a given molecule. MA allows us to compare the intrinsic complexity of molecules using the minimum number of steps required to construct the molecular graph starting from basic objects, and a probabilistic model shows how the probability of any given molecule forming randomly drops dramatically as its MA increases. To map chemical space, we calculated the MA of ca. 2.5 million compounds, and collected data which showed the complexity of a molecule can be experimentally determined by using three independent techniques including infra-red spectroscopy, nuclear magnetic resonance, and by fragmentation in a mass spectrometer, and this data has an excellent corelation with the values predicted from our assembly theory. We then set out to see if this approach could allow us to identify molecular biosignatures with a set of diverse samples from around the world, outer space, and the laboratory including prebiotic soups. The results show that there is a non-living to living threshold in MA complexity and the higher the MA for a given molecule, the more likely that it had to be produced by a biological process. This work demonstrates it is possible to use this approach to build a life detection instrument that could be deployed on missions to extra-terrestrial locations to detect biosignatures, map the extent of life on Earth, and be used as a molecular complexity scale to quantify the constraints needed to direct prebiotically plausible processes in the laboratory. Such an approach is vital if we are going to find new life elsewhere in the universe or create de-novo life in the lab.
Systematic planning of moving target defence for maximising detection effectiveness against false data injection attacks in smart grid
Abstract Moving target defence (MTD) has been gaining traction to thwart false data injection attacks against state estimation (SE) in the power grid. MTD actively perturbs the reactance of transmission lines equipped with distributed flexible AC transmission system (D‐FACTS) devices to falsify the attacker's knowledge about the system configuration. However, the existing literature has not systematically studied what influences the detection effectiveness of MTD and how it can be improved based on the topology analysis. These problems are tackled here from the perspective of an MTD plan in which the D‐FACTS placement is determined. We first exploit the relation between the rank of the composite matrix and the detecting effectiveness. Then, we rigorously derive upper and lower bounds on the attack detecting probability of MTDs with a given rank of the composite matrix. Furthermore, we analyse existing planning methods and highlight the importance of bus coverage by D‐FACTS devices. To improve the detection effectiveness, we propose a novel graph theory–based planning algorithm to retain the maximum rank of the composite matrix while covering all necessary buses. Comparative results on multiple systems show the high detecting effectiveness of the proposed algorithm in both DC‐ and AC‐SE.
Max-independent set and the quantum alternating operator ansatz
he maximum-independent set (MIS) problem of graph theory using the quantum alternating operator ansatz is studied. We perform simulations on the Rigetti Forest simulator for the square ring, K 2,3 , and K3,3 graphs and analyze the dependence of the algorithm on the depth of the circuit and initial states. The probability distribution of observation of the feasible states representing maximum-independent sets is observed to be asymmetric for the MIS problem, which is unlike the Max-Cut problem where the probability distribution of feasible states is symmetric. For asymmetric graphs, it is shown that the algorithm clearly favors the independent set with the larger number of elements even for finite circuit depth. Finally, we also compare the approximation ratios for the algorithm when we choose different initial states for the square ring graph and show that it is dependent on the choice of the initial state.
Introducing Tropical Geometric Approaches to Delay Tolerant Networking Optimization
Delay Tolerant Networking (DTN) is the standard approach to the networking of space systems with the goal of supporting the Solar System Internet (SSI). Current space networks have a small scale and often depend on rigorously scheduled (pre-determined) contact opportunities; this manual approach inhibits scalability. The goal of this paper is to recast these scheduling problems in order to apply the optimization machinery of tropical geometry. Contact opportunities in space are dependent on such factors as orbital mechanics and asset availability, which induce time-varying connectivity; indeed, end-to-end connectivity might never occur. Routing optimization within this structure is classically difficult and typically utilizes Dijkstra's algorithm as applied to contact graphs. Alternatively, we follow the successes of tropical geometry in train schedule optimization, job assignments, and even traditional networking, by extending this approach to this more general (i.e. disconnected) problem space. These successes imply tropical geometry provides a useful framework in the context of DTNs, starting with applications to queuing theory and long-haul links. Recently, tropical geometry has been applied to parametric path optimization on graphs with variable edge weights. In this work, we extend these advances to account for the problem of routing in a space network, and find that tropical geometry is well-suited to the challenges offered by this new setting, including contact schedules featuring probabilities. Our approach leverages the combinatorial nature of the problem to give feasible shortest path trees in the presence of variable channel conditions and latency, evolving topologies, and uncertainty inherent in space routing. We discuss our tropical approach to DTN for two Python implementations, a Verilog Tropical ALU implementation, tropical frameworks for other parametric graph problems, and solution stability. Lastly, a program for future work is included to illuminate the path ahead.
A Geometrical Approach to Bell's Theorem
Bell's theorem can be proved through simple geometrical reasoning, without the need for the Psi function, probability distributions, or calculus. The proof is based on N. David Mermin's explication of the Einstein-Podolsky-Rosen-Bohm experiment, which involves Stern-Gerlach detectors which flash red or green lights when detecting spin-up or spin-down. The statistics of local hidden variable theories for this experiment can be arranged in colored strips from which simple inequalities can be deduced. These inequalities lead to a demonstration of Bell's theorem. Moreover, all local hidden variable theories can be graphed in such a way as to enclose their statistics in a pyramid, with the quantum-mechanical result lying a finite distance beneath the base of the pyramid.
Beyond the universal Dyson singularity for 1-D chains with hopping disorder
We study a simple non-interacting nearest neighbor tight-binding model in one dimension with disorder, where the hopping terms are chosen randomly. This model exhibits a well-known singularity at the band center both in the density of states and localization length. If the probability distribution of the hopping terms is well-behaved, then the singularities exhibit universal behavior, the functional form of which was first discovered by Freeman Dyson in the context of a chain of classical harmonic oscillators. We show here that this universal form can be violated in a tunable manner if the hopping elements are chosen from a divergent probability distribution. We also demonstrate a connection between a breakdown of universality in this quantum problem and an analogous scenario in the classical domain — that of random walks and diffusion with anomalous exponents.
Analysis and Mitigation of Cascading Failures Using a Stochastic Interaction Graph with Eigen-analysis
In studies on complex network systems using graph theory, eigen-analysis is typically performed on an undirected graph model of the network. However, when analyzing cascading failures in a power system, the interactions among failures suggest the need for a directed graph beyond the topology of the power system to model directions of failure propagation. To accurately quantify failure interactions for effective mitigation strategies, this paper proposes a stochastic interaction graph model and associated eigen-analysis. Different types of modes on failure propagations are defined and characterized by the eigenvalues of a stochastic interaction matrix, whose absolute values are unity, zero, or in between. Finding and interpreting these modes helps identify the probable patterns of failure propagation, either local or widespread, and the participating components based on eigenvectors. Then, by lowering the failure probabilities of critical components highly participating in a mode of widespread failures, cascading can be mitigated. Here, the validity of the proposed stochastic interaction graph model, eigen-analysis and the resulting mitigation strategies is demonstrated using simulated cascading failure data on an NPCC 140-bus system.
Scaling quantum approximate optimization on near-term hardware
The quantum approximate optimization algorithm (QAOA) is an approach for near-term quantum computers to potentially demonstrate computational advantage in solving combinatorial optimization problems. However, the viability of the QAOA depends on how its performance and resource requirements scale with problem size and complexity for realistic hardware implementations. Here, we quantify scaling of the expected resource requirements by synthesizing optimized circuits for hardware architectures with varying levels of connectivity. Assuming noisy gate operations, we estimate the number of measurements needed to sample the output of the idealized QAOA circuit with high probability. We show the number of measurements, and hence total time to solution, grows exponentially in problem size and problem graph degree as well as depth of the QAOA ansatz, gate infidelities, and inverse hardware graph degree. These problems may be alleviated by increasing hardware connectivity or by recently proposed modifications to the QAOA that achieve higher performance with fewer circuit layers.
Faster Johnson–Lindenstrauss transforms via Kronecker products
The Kronecker product is an important matrix operation with a wide range of applications in signal processing, graph theory, quantum computing and deep learning. In this work, we introduce a generalization of the fast Johnson–Lindenstrauss projection for embedding vectors with Kronecker product structure, the Kronecker fast Johnson–Lindenstrauss transform (KFJLT). The KFJLT reduces the embedding cost by an exponential factor of the standard fast Johnson–Lindenstrauss transform’s cost when applied to vectors with Kronecker structure, by avoiding explicitly forming the full Kronecker products. Here, we prove that this computational gain comes with only a small price in embedding power: consider a finite set of $p$ points in a tensor product of $d$ constituent Euclidean spaces $\bigotimes _{k=d}^{1}{\mathbb{R}}^{n_k}$, and let $N = \prod _{k=1}^{d}n_k$. With high probability, a random KFJLT matrix of dimension $m \times N$ embeds the set of points up to multiplicative distortion $(1\pm \varepsilon )$ provided $m \gtrsim \varepsilon ^{-2} \, \log ^{2d - 1} (p) \, \log N$. We conclude by describing a direct application of the KFJLT to the efficient solution of large-scale Kronecker-structured least squares problems for fitting the CP tensor decomposition.
Resilience of electric utilities during the COVID-19 pandemic in the framework of the CIGRE definition of Power System Resilience
Resilience is a vital concept in engineering, business, and natural sciences, and is a measure of the ability of an entity to withstand High Impact Low Probability (HILP) events. During the COVID-19 pandemic, which started in late 2019/early 2020, power system utilities around the globe have responded in effective and efficient ways to enhance the resilience of their organisations, both in terms of real-time operations and prudent management of its infrastructure, in order to continue their mandate in providing reliable supply to meet customer demands. Here, this paper presents the CIGRE definition for power system resilience, established by the C4.47 Working Group in 2018, and demonstrates the application of resilience-oriented thinking within the electrical sector. The response and recovery efforts are described, with respect to the key actionable measures integral to the power system resilience definition, taken before, during and after the COVID-19 pandemic. A practical conceptual framework is also presented for thinking about resilience in terms of three key components of resilience strategies: organisational, infrastructure and operational resilience. The paper also discusses the different strategies adopted in response to COVID-19, based on the C4.47 members’ experiences during the pandemic. Finally, a case study is presented, which proves the effectiveness of a set of response measures, using graph theory and the characteristics of the staff-asset interactions.
Improvement of Automated POST Case Success Rate Using Support Vector Machines
During early conceptual design of complex systems, concept down selection can have a large impact upon program life-cycle cost. Therefore, any concepts selected during early design will inherently commit program costs and affect the overall probability of program success. For this reason it is important to consider as large a design space as possible in order to better inform the down selection process. For conceptual design of launch vehicles, trajectory analysis and optimization often presents the largest obstacle to evaluating large trade spaces. This is due to the sensitivity of the trajectory discipline to changes in all other aspects of the vehicle design. Small deltas in the performance of other subsystems can result in relatively large fluctuations in the ascent trajectory because the solution space is non-linear and multi-modal [1]. In order to help capture large design spaces for new launch vehicles, the authors have performed previous work seeking to automate the execution of the industry standard tool, Program to Optimize Simulated Trajectories (POST). This work initially focused on implementation of analyst heuristics to enable closure of cases in an automated fashion, with the goal of applying the concepts of design of experiments (DOE) and surrogate modeling to enable near instantaneous throughput of vehicle cases [2]. Additional work was then completed to improve the DOE process by utilizing a graph theory based approach to connect similar design points [3]. The conclusion of the previous work illustrated the utility of the graph theory approach for completing a DOE through POST. However, this approach was still dependent upon the use of random repetitions to generate seed points for the graph. As noted in [3], only 8% of these random repetitions resulted in converged trajectories. This ultimately affects the ability of the random reps method to confidently approach the global optima for a given vehicle case in a reasonable amount of time. With only an 8% pass rate, tens or hundreds of thousands of reps may be needed to be confident that the best repetition is at least close to the global optima. However, typical design study time constraints require that fewer repetitions be attempted, sometimes resulting in seed points that have only a handful of successful completions. If a small number of successful repetitions are used to generate a seed point, the graph method may inherit some inaccuracies as it chains DOE cases from the non-global-optimal seed points. This creates inherent noise in the graph data, which can limit the accuracy of the resulting surrogate models. For this reason, the goal of this work is to improve the seed point generation method and ultimately the accuracy of the resulting POST surrogate model. The work focuses on increasing the case pass rate for seed point generation.