Stochastic Topology Research Articles

Resilient Consensus via Weight Learning and Its Application in Fault-Tolerant Clock Synchronization

This paper addresses the distributed consensus problem in the presence of faulty nodes. A novel weight learning algorithm is introduced such that neither the network connectivity nor a sequence of history records is required to achieve resilient consensus. The critical idea is to dynamically update the interaction weights among neighbors learnt from their credibility measurement. Basically, we define a reward function that is inversely proportional to the state difference to its neighbor, and then adjust the credibility based on the reward derived at the present step and the previous credibility. In such a way, the interaction weights are updated at every step, which integrates the historic information and degrades the influences from faulty nodes. Four classes of faulty nodes are considered, and both fixed and stochastic topologies are involved in this paper. Furthermore, we apply this novel approach in clock synchronization problem. By updating the logical clock skew and offset via the corresponding weight learning algorithms, respectively, the logical clock synchronization is eventually achieved regardless of faulty nodes. Simulations are provided to illustrate the effectiveness of the strategy.

Research on Design Method of Multilayer Metamaterials Based on Stochastic Topology.

Metamaterials are usually designed using biomimetic technology based on natural biological characteristics or topology optimization based on prior knowledge. Although satisfactory results can be achieved to a certain extent, there are still many performance limitations. For overcoming the above limitations, this paper proposes a rapid metamaterials design method based on the generation of random topological patterns. This method realizes the combined big data simulation and structure optimization of structure-electromagnetic properties, which makes up for the shortcomings of traditional design methods. The electromagnetic properties of the proposed metamaterials are verified by experiments. The reflection coefficient of the designed absorbing metamaterial unit is all lower than -15 dB over 12-16 GHz. Compared with the metal floor, the radar cross section (RCS) of the designed metamaterial is reduced by a minimum of 14.5 dB and a maximum of 27.6 dB over the operating band. The performance parameters of metamaterial obtained based on the random topology design method are consistent with the simulation design results, which further verifies the reliability of the algorithm in this paper.

All-optical spiking neural network and optical spike-time-dependent plasticity based on the self-pulsing effect within a micro-ring resonator.

In this paper, we proposed an all-optical version of photonic spiking neurons and spike-time-dependent plasticity (STDP) based on the nonlinear optical effects within a micro-ring resonator. In this system, the self-pulsing effect was exploited to implement threshold control, and the equivalent pulse energy required for spiking, calculated by multiplying the input pulse power amplitude with its duration, was about 14.1 pJ. The positive performance of the neurons in the excitability and cascadability tests validated the feasibility of this scheme. Furthermore, two simulations were performed to demonstrate that such an all-optical spiking neural network incorporated with STDP could run stably on a stochastic topology. The essence of such an all-optical spiking neural network is a nonlinear spiking dynamical system that combines the advantages of photonics and spiking neural networks (SNNs), promising access to the high speed and lower consumption inherent to optical systems.

Bridge topology optimization considering stochastic moving traffic

Topology optimization of bridge structures is challenging due to the difficulty in accounting for the stochastic dynamic nature of the moving vehicle loads. As a result, optimization for this common type of structure and loading is limited in the existing literature. Existing topology optimization approaches for addressing stochastic dynamic loading can be classified as time domain, frequency domain, or random vibration methods. Herein, a new compact representation of random moving traffic loading as a filtered white noise is developed which enables stochastic topology optimization to be performed. This optimization utilizes an objective function which combines the mean and standard deviation of responses to minimize the extreme response to random traffic loading. Examples show a significant improvement of the bridge response, represented by a reduction in the standard deviation at a minimal cost to the mean response. Also, optimal topologies for different traffic parameters such as speed and arrival rate are relatively similar, indicating a robust solution. With this approach, bridge topology can be efficiently optimized for random moving traffic loading by enabling direct minimization of response extremes, which represents the probabilistic design intent to achieve adequate levels of performance under the loading uncertainties in typical bridges.

Stochastic Simulation of Flow Rate and Power Consumption Considering the Uncertainty of Pipeline Cracking Rate and Time-Dependent Topology of a Natural Gas Transmission Network

Various gas pipeline networks used for the transit of energy sources are some of the most important infrastructures. However, carrying gas from one point to another is not the only concern when planning the construction of a new network or expanding an already existing one. The reliability and environmental impact of the system are crucial when evaluating the network and risks posed by potential gas leaks, fires, explosions, etc. Even though everyone admits that reliability is a key aspect of any system, its constraints will still be most likely neglected in the assessment of the pipeline project. How much energy is wasted by pushing an additional amount of gas through the pipeline network, which will eventually gush out of the pipeline because of one crack or another? Moreover, if this additional power or fuel consumption and related environmental impact are significant, how could it be reduced? In this paper, an approach is introduced for the simulation and quantification of how much more power would be required if the pipelines are regarded as unreliable (i.e., by leaking, rupturing, or even exploding). By employing stochastic simulations and time-dependent topology (topology determined by the value of a variable representing time) of the pipeline network as a case study for the selected representative gas transmission network, the amount of additional power consumption in gas compressor stations due to uncertain cracking and the leaking rate was evaluated. Although the analysis of power consumption was performed for a hypothetical network, the estimates of the cracking rates, detection effectiveness, and leaking rates used were as close to the real cases as possible.

Approximation schemes for stochastic compliance-based topology optimization with many loading scenarios

In this paper, approximation schemes are proposed for handling load uncertainty in compliance-based topology optimization problems, where the uncertainty is described in the form of a set of finitely many loading scenarios. Efficient approximate methods are proposed to approximately evaluate and differentiate either (1) the mean compliance, or (2) a class of scalar-valued function of the individual load compliances such as the weighted sum of the mean and standard deviation. The computational time complexities of the proposed algorithms are analyzed, compared to the exact approaches and then experimentally verified. Finally, some mean compliance minimization problems and some risk-averse compliance minimization problems are solved for verification.

A stochastic topology optimization algorithm for improved fluid dynamics systems

Abstract The use of topology optimization in the design of fluid dynamics systems is still in its infancy. With the decreasing cost of additive manufacture, the application of topology optimization in the design of structural components has begun to increase. This paper provides a method for using topology optimization to reduce the power dissipation of fluid dynamics systems, with the novelty of it being the first application of stochastic mechanisms in the design of 3D fluid–solid geometrical interfaces. The optimization algorithm uses the continuous adjoint method for sensitivity analysis and is optimized against an objective function for fluid power dissipation. The paper details the methodology behind a vanilla gradient descent approach before introducing stochastic behavior through a minibatch-based system. Both algorithms are then applied to a novel case study for an internal combustion engine's piston cooling gallery before the performance of each algorithm's resulting geometry is analyzed and compared. The vanilla gradient descent algorithm achieves an 8.9% improvement in pressure loss through the case study, and this is surpassed by the stochastic descent algorithm which achieved a 9.9% improvement, however this improvement came with a large time cost. Both approaches produced similarly unintuitive geometry solutions to successfully improve the performance of the cooling gallery.

Invariant-set design of observer-based robust control for power systems under stochastic topology and parameters changes

Power systems are subject to stochastic fluctuations which can be classified as topological or parametric. This paper models the random topological changes due to a series of lightning strokes hitting a transmission line (associated with open/reclose of circuit breakers) in a power system as a Markov jump system. Whereas the line reactance (depending on the phase conductors spacing) changes due to the random wind speed fluctuations, are represented as a stochastic external disturbances. A novel excitation control for a stochastic power system is developed. The optimal two-controller and observer gains are designed based on the attracting ellipsoid method and the linear matrix inequalities (LMI) optimization. The effectiveness of the proposed control is tested on a multimachine system.

Reliable asynchronous control for delayed multi-agent systems with stochastic faults and switching topologies

The issue of exponential consensus for delayed multi-agent systems (MASs) with stochastic faults and switching topologies is addressed in this work. In contrast with the existing relative works on MASs with a fixed communication topology, the topological graph is governed by an uncertain Markov chain, where the restrictions on transition rates (TRs) of Markov switching topologies are loose, which may be partially known or completely unknown. Moreover, a reliable asynchronous control protocol with the stochastic failure behaviour of the actuator on account of the relative information of neighbouring agents is devised. Relaxation of criterions with respect to the consensus of delayed MASs based on the Lyapunov methodology, Wirtinger's inequality and an improved reciprocally convex combination strategy are derived. An unmanned aerial vehicle (UAV) is presented to demonstrate the superiority and applicability of the proposed method.

Interacting stochastic topology and Mott transition from light response

We develop a stochastic description of the topological properties in an interacting Chern insulator. We confirm the Mott transition's first-order nature in the interacting Haldane model on the honeycomb geometry from a mean-field variational approach supported by density matrix renormalization group results and Ginzburg-Landau arguments. From the Bloch sphere, we make predictions for circular dichroism of light related to the quantum Hall conductivity on the lattice and in the presence of interactions. This analysis shows that the topological number can be measured from the light response at the Dirac points. Electron-electron interactions can also produce a substantial number of particle-hole pairs above the band gap, which leads us to propose a stochastic Chern number as an interacting measure of the topology. The stochastic Chern number can describe disordered situations with a fluctuating staggered potential, and we build an analogy between interaction-induced particle-hole pairs and temperature effects. Our stochastic approach is physically intuitive, easy to implement, and leads the way to further studies of interaction effects.

A polynomial dimensional decomposition framework based on topology derivatives for stochastic topology sensitivity analysis of high-dimensional complex systems and a type of benchmark problems

In this paper, a new computational framework based on the topology derivative concept is presented for evaluating stochastic topological sensitivities of complex systems. The proposed framework, designed for dealing with high dimensional random inputs, dovetails a polynomial dimensional decomposition (PDD) of multivariate stochastic response functions and deterministic topology derivatives. On one hand, it provides analytical expressions to calculate topology sensitivities of the first three stochastic moments which are often required in robust topology optimization (RTO). On another hand, it offers embedded Monte Carlo Simulation (MCS) and finite difference formulations to estimate topology sensitivities of failure probability for reliability-based topology optimization (RBTO). For both cases, the quantification of uncertainties and their topology sensitivities are determined concurrently from a single stochastic analysis. Moreover, an original example of two random variables is developed for the first time to obtain analytical solutions for topology sensitivity of moments and failure probability. Another 53-dimension example is constructed for analytical solutions of topology sensitivity of moments and semi-analytical solutions of topology sensitivity of failure probabilities in order to verify the accuracy and efficiency of the proposed method for high-dimensional scenarios. Those examples are new and make it possible for researchers to benchmark stochastic topology sensitivities of existing or new algorithms. In addition, it is unveiled that under certain conditions the proposed method achieves better accuracies for stochastic topology sensitivities than for the stochastic quantities themselves.

Robust formation control of thrust-propelled vehicles under deterministic and stochastic topology

ABSTRACT Formation control of multiple thrust-propelled vehicles (TPVs) under deterministic and stochastic switching topologies and communication delay is addressed. Introducing a new version of variable structure control and based upon sliding mode technique, adaptive control and projection operator, we effectively handle the impact of uncertainties on the mass and inertia matrix and a set of time-varying disturbances affecting the translational and rotational dynamics. Global stability of the whole closed-loop system is guaranteed through Lyapunov stability theory. For the deterministic topology, sufficient condition in terms of LMIs is derived to achieve formation in the presence of jointly connected switching topology. In the case of stochastic topology, based on the concept of super-martingales, it is shown that if the probability of existing a connected topology is not zero, under some conditions, formation is almost surely solved in the network. Finally, numerical simulations verify the effectiveness of the proposed control framework.

Cross Layer Interaction for Strong Route Selection in A O D V for MA N E T

Wireless communication could be a requirement of this era as all industries are approaching for the internet of things and automation. Even supposing multiple choices are ready for wireless communication, Mobile ad hoc Network (M A N E T) gives wireless communication while not having any infrastructure. M A N E T typically adopts a preferred and well fitted routing protocol that is Ad hoc On Demand Distance Vector (A O D V). Stochastic topology inside M A N E T antecedent link breaks various times in network because of that feeble link quality that appears as packet extinction within the network. In the modern wireless communication study, it is noted that the performance of system deteriorates because of feeble link creation or link wreckage in the network. In this paper, link quality improvement in AODV routing protocol is focused. Link quality concern is proposed to resolve through Cross layer design (C L D) interaction in the OSI communication model. C L D enforced by interacting Physical initial layer and network middle layer of the OSI model, C L D facilitate in robust route formation in AODV named as C L D A O D V. The result exhibits growth within the system Performance in terms of factors like throughput, Packet Delivery Ratio (P D R) and decline in the packet losses and delays in the network. Network simulator used for performance analysis are N S 2 and N S 3.

Distributed formation control for multiple quadrotor UAVs under Markovian switching topologies with partially unknown transition rates

This paper addresses distributed formation control for a group of quadrotor unmanned aerial vehicles (UAVs) under Markovian switching topologies with partially unknown transition rates. Instead of the general stochastic topology, the graph is governed by a set of Markov chains to the edges, which can recover the traditional Markovian switching topologies in line with the practical communication network. Extended high gain observers (EHGOs) are constructed with a two-time-scale format to deal with the issue of nonlinear input coefficients, so that there could be a higher estimation precision of the system uncertainties. To impel multiple quadrotor UAVs to achieve a predesigned formation shape, a modified integral sliding mode (ISM) control protocol is proposed here with a multi-time-scale structure, which allows independent analysis of the dynamics in each time scale. The stability proof for the system state space origin is derived from the singular perturbation method and Lyapunov stability theory. In addition, the introduced ISM controller can deal with the time varying desired references with the bounded accelerations and is robust to the disturbances. Finally, simulations on six quadrotor UAVs are given to verify the effectiveness of the theoretical results.

Spectral Gaps of Random Graphs and Applications

AbstractWe study the spectral gap of the Erdős–Rényi random graph through the connectivity threshold. In particular, we show that for any fixed $\delta> 0$ if $$\begin{equation*} p \geq \frac{(1/2 + \delta) \log n}{n}, \end{equation*}$$then the normalized graph Laplacian of an Erdős–Rényi graph has all of its nonzero eigenvalues tightly concentrated around $1$. This is a strong expander property. We estimate both the decay rate of the spectral gap to $1$ and the failure probability, up to a constant factor. We also show that the $1/2$ in the above is optimal, and that if $p = \frac{c \log n}{n}$ for $c < 1/2,$ then there are eigenvalues of the Laplacian restricted to the giant component that are separated from $1.$ We then describe several applications of our spectral gap results to stochastic topology and geometric group theory. These all depend on Garland’s method [24], a kind of spectral geometry for simplicial complexes. The following can all be considered to be higher-dimensional expander properties. First, we exhibit a sharp threshold for the fundamental group of the Bernoulli random $2$-complex to have Kazhdan’s property (T). We also obtain slightly more information and can describe the large-scale structure of the group just before the (T) threshold. In this regime, the random fundamental group is with high probability the free product of a (T) group with a free group, where the free group has one generator for every isolated edge. The (T) group plays a role analogous to that of a “giant component” in percolation theory. Next we give a new, short, self-contained proof of the Linial–Meshulam–Wallach theorem [35, 39], identifying the cohomology-vanishing threshold of Bernoulli random $d$-complexes. Since we use spectral techniques, it only holds for $\mathbb{Q}$ or $\mathbb{R}$ coefficients rather than finite field coefficients, as in [35] and [39]. However, it is sharp from a probabilistic point of view, providing for example, hitting time type results and limiting Poisson distributions inside the critical window. It is also a new method of proof, circumventing the combinatorial complications of cocycle counting. Similarly, results in an earlier preprint version of this article were already applied in [33] to obtain sharp cohomology-vanishing thresholds in every dimension for the random flag complex model.

Beyond the clustering coefficient: A topological analysis of node neighbourhoods in complex networks

In Network Science, node neighbourhoods, also called ego-centered networks, have attracted significant attention. In particular the clustering coefficient has been extensively used to measure their local cohesiveness. In this paper, we show how, given two nodes with the same clustering coefficient, the topology of their neighbourhoods can be significantly different, which demonstrates the need to go beyond this simple characterization. We perform a large scale statistical analysis of the topology of node neighbourhoods of real networks by first constructing their clique complexes, and then computing their Betti numbers. We are able to show significant differences between the topology of node neighbourhoods of real networks and the stochastic topology of null models of random simplicial complexes revealing local organisation principles of the node neighbourhoods. Moreover we observe that a large scale statistical analysis of the topological properties of node neighbourhoods is able to clearly discriminate between power-law networks, and planar road networks.

ABOUT THE DEFINITIONS OF THE STOCHASTIC TOPOLOGICAL SPACE

In this paper, we put forword to the definition of stochastic normed space, and study its some properties, from there we obtain the definition of stochastic topology space.

Large-scale stochastic topology optimization using adaptive mesh refinement and coarsening through a two-level parallelization scheme

Topology optimization under uncertainty of large-scale continuum structures is a computational challenge due to the combination of large finite element models and uncertainty propagation methods. The former aims to address the ever-increasing complexity of more and more realistic models, whereas the latter is required to estimate the statistical metrics of the formulation. In this work, the computational burden of the problem is addressed using a sparse grid stochastic collocation method, to calculate the statistical metrics of the topology optimization under uncertainty formulation, and a parallel adaptive mesh refinement method, to efficiently solve each of the stochastic collocation nodes. A two-level parallel processing scheme (TOUU-PS2) is proposed to profit from parallel computation on distributed memory systems: the stochastic nodes are distributed through the distributed memory system, and the efficient computation of each stochastic node is performed partitioning the problem using a domain decomposition strategy and solving each subdomain using an adaptive mesh refinement method. A dynamic load-balancing strategy is used to balance the workload between subdomains, and thus increasing the parallel performance by reducing processor idle time. The topology optimization problem is addressed using the topological derivative concept in combination with a level-set method. The performance and scalability of the proposed methodology are evaluated using several numerical benchmarks and real-world applications, showing good performance and scalability up to thousands of processors.

Distributed Event‐Triggered Consensus for Multiple Underactuated Systems Under Markovian Switching Topologies

AbstractThis paper presents a distributed consensus algorithm that employs event‐triggered communication for multiple underactuated systems under Markovian switching topologies. Instead of the general stochastic topology, the graph of the entire system is governed by a set of Markov chains to the edges, which can recover the general Markovian switching topologies in line with the practical communication network. By utilizing integral sliding mode control strategy, rigorous analysis of the asymptotic convergence results has been performed through graph theory and Lyapunov stability theory. An event‐triggered communication law is provided for each agent and Zeno behavior of triggering time sequences is excluded. It will yield to the very first application of the multiple underactuated systems, in which the system states could be enforced to track the leader. Finally, the illustrative simulations on six underactuated two‐link manipulators are given to demonstrate the effectiveness of theoretical results.

Robust topology optimization of skeletal structures with imperfect structural members

A topology optimization framework is proposed for robust design of skeletal structures with stochastically imperfect structural members. Imperfections are modeled as uncertain members’ out-of-straightness using curved frame elements in the form of predefined functions with random magnitudes throughout the structure. The stochastic perturbation method is used for propagating the imperfection uncertainty up to the structural response level, and the expected value of performance measure or constraint is used to form the stochastic topology optimization problem. Sensitivities are derived explicitly using the adjoint method and are used in conjunction with an efficient gradient-based optimizer in search for robust optimal topologies. Topological designs for three representative examples are investigated with the proposed algorithm and the resulting topologies are compared with the deterministic designs. It is observed that the new designs primarily feature load path diversification, which is pronounced with increasing level of uncertainty, and occasionally member thickening to mitigate the impact of the uncertainty in members’ out-of-straightness on structural performance.