Self-Stabilizing Repeated Balls-into-Bins

We study the following synchronous process that we call "repeated balls-into-bins". The process is started by assigning $n$ balls to $n$ bins in an arbitrary way. In every subsequent round, from each non-empty bin one ball is chosen according to some fixed strategy (random, FIFO, etc), and re-assigned to one of the $n$ bins uniformly at random. We define a configuration "legitimate" if its maximum load is $\mathcal{O}(\log n)$. We prove that, starting from any configuration, the process will converge to a legitimate configuration in linear time and then it will only take on legitimate configurations over a period of length bounded by any polynomial in $n$, with high probability (w.h.p.). This implies that the process is self-stabilizing and that every ball traverses all bins in $\mathcal{O}(n \log^2 n)$ rounds, w.h.p

Self-Stabilization in the Distributed Systems of Finite State Machines

The notion of self-stabilization was first proposed by Dijkstra in 1974 in his classic paper. The paper defines a system as self-stabilizing if, starting at any, possibly illegitimate, state the system can automatically adjust itself to eventually converge to a legitimate state in finite amount of time and once in a legitimate state it will remain so unless it incurs a subsequent transient fault. Dijkstra limited his attention to a ring of finite-state machines and provided its solution for self-stabilization. In the years following his introduction, very few papers were published in this area. Once his proposal was recognized as a milestone in work on fault tolerance, the notion propagated among the researchers rapidly and many researchers in the distributed systems diverted their attention to it. The investigation and use of self-stabilization as an approach to fault-tolerant behavior under a model of transient failures for distributed systems is now undergoing a renaissance. A good number of works pertaining to self-stabilization in the distributed systems were proposed in the yesteryears most of which are very recent. This report surveys all previous works available in the literature of self-stabilizing systems

Memory requirements for silent stabilization

A self-stabilizing algorithm is silent if it converges to a glc)bal state after which the values stored in the com-munication registers are fixed. The silence property of self-stabilizing algorithms is a desirable property in terms of simplicity and communication overhead. In this work we show that no constant memory silent self-stabilizing algorithms exist for identification of the centers of a graph, leader election, and spanning tree construction. Lower bounds of Cl(log n) bits per communication register are obtained for each of the above tasks. The existence of a silent legitimate global state that uses less than log n bits per register is assumed. This legitimate global state is used to construct a silent global state that is illegitimate.

Entanglement in a fermion chain under continuous monitoring

We study the entanglement entropy of the quantum trajectories of a free fermion chain under continuous monitoring of local occupation numbers. We propose a simple theory for entanglement entropy evolution from disentangled and highly excited initial states. It is based on generalized hydrodynamics and the quasi-particle pair approach to entanglement in integrable systems. We test several quantitative predictions of the theory against extensive numerics and find good agreement. In particular, the volume law entanglement is destroyed by the presence of arbitrarily weak measurement.Comment: 18 pages, 8 figures, 2 new figure

The structure of Herpesvirus Fusion Glycoprotein B-Bilayer Complex reveals the protein-membrane and lateral protein-protein interaction

Glycoprotein B (gB) is a key component of the complex herpesvirus fusion machinery. We studied membrane interaction of two gB ectodomain forms and present an electron cryotomography structure of the gB-bilayer complex. The two forms differed in presence or absence of the membrane proximal region (MPR) but showed an overall similar trimeric shape. The presence of the MPR impeded interaction with liposomes. In contrast, the MPR-lacking form interacted efficiently with liposomes. Lateral interaction resulted in coat formation on the membranes. The structure revealed that interaction of gB with membranes was mediated by the fusion loops and limited to the outer membrane leaflet. The observed intrinsic propensity of gB to cluster on membranes indicates an additional role of gB in driving the fusion process forward beyond the transient fusion pore opening and subsequently leading to fusion pore expansion

Performance Evaluation of Self-stabilizing Algorithms by Probabilistic Model Checking

A self-stabilizing protocol is one that starting from any arbitrary initial state recovers to legitimate states in a finite number of steps, and once it stabilizes to a set of legitimate states, it remains there unless it is perturbed by transient faults. The traditional methods existing for performance evaluation of a self-stabilizing algorithm usually work based on the analysis of worst case computational complexity. Another method that has been commonly used in evaluating these algorithms is simulation, which assumes the system starts from an initial state. Here, it is argued that the traditional methods have shortcomings and do not give enough insight about the behavior of the system. Moreover, they do not provide a decent method of comparison. We propose a novel method for evaluation of self-stabilizing algorithms. This method works based on probabilistic model checking and computation of the expected number of recovery steps. We execute some experiments on the case studies, and the results indicate that we can gain insight about the faults and their structure in the protocol. Next, we explain the difficulty of designing a self-stabilizing algorithm for a system and show how it is impossible to do so for some classes of protocols. This resulted in some relaxation in the definition of self-stabilization. One of the relaxations made in the definition of self-stabilization is weak-stabilization. A weak-stabilizing protocol ensures the existence of a recovery path from an arbitrary initial configuration. Thus, some paths may contain connected components or cycles. Since a weak-stabilizing algorithm may get stuck in connected components forever, we cannot evaluate weak-stabilizing protocols by traditional and existing methods. We calculate the expected number of recovery steps for evaluating weak-stabilization. However, since it does not give us enough intuition about the structure of faults, we apply a graph-theoretic formula for estimating the weak-stabilizing algorithm's performance. This formula is based on the number of cycles and their reachability. Based on the observations we made by performance evaluation of these protocols, we suggest algorithms called state encoding for modifying the performance of the algorithms. State encoding works based on changing the bit mapping of the states of the system. The aim is to make the states with faster recovery steps more probable to occur. There are three algorithms, one of which works based on betweenness centrality which is a measure of centrality of a node within a graph. The other one works based on feedback arc set which is a set of arcs whose removal makes a graph acyclic. The third algorithm works based on the length of the shortest recovery path for the states. The other problem investigated here is the problem of state space explosion in model checking. Similar to traditional methods of model checking, probabilistic model checking also suffers from the problem of state space explosion, i.e., the number of states grows exponentially in terms of the number of components in the distributed system. Abstraction methods, which are described briefly here, are designed to combat this problem. We argue that they are not effcient enough, and there is still the lack of a suffcient abstraction method that works for systems with an arbitrary number of processes. We also propose a new approach for evaluation of an abstraction function. Then, based on the intuition gained, a new abstraction algorithm is proposed that is exclusively designed for verification of reachability properties. After executing experiments on a case study, we compare the result of our algorithm with the results obtained by existing methods. The results support our claim that our method is more effcient and precise

Translucent windows: How uncertainty in competitive interactions impacts detection of community pattern

Trait variation and similarity among coexisting species can provide a window into the mechanisms that maintain their coexistence. Recent theoretical explorations suggest that competitive interactions will lead to groups, or clusters, of species with similar traits. However, theoretical predictions typically assume complete knowledge of the map between competition and measured traits. These assumptions limit the plausible application of these patterns for inferring competitive interactions in nature. Here we relax these restrictions and find that the clustering pattern is robust to contributions of unknown or unobserved niche axes. However, it may not be visible unless measured traits are close proxies for niche strategies. We conclude that patterns along single niche axes may reveal properties of interspecific competition in nature, but detecting these patterns requires natural history expertise firmly tying traits to niches.Comment: Main text: 18 pages, 6 figures. Appendices: A-G, 6 supplementary figures. This is the peer reviewed version of the article of the same title which has been accepted for publication at Ecology Letters. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archivin

Modelling protein localisation and positional information in subcellular systems

Cells and their component structures are highly organised. The correct function of many biological systems relies upon not only temporal control of protein levels but also spatial control of protein localisation within cells. Mathematical modelling allows us to quantitatively test potential mechanisms for protein localisation and spatial organisation. Here we present models of three examples of spatial organisation within individual cells. In the bacterium E. coli, the site of cell division is partly determined by the Min proteins. The Min proteins oscillate between the cell poles and suppress formation of the division ring here, thereby restricting division to midcell. We present a stochastic model of the Min protein dynamics, and use this model to investigate partitioning of the Min proteins between the daughter cells during cell division. The Min proteins determine the correct position for cell division by forming a timeaveraged concentration gradient which is minimal at midcell. Concentration gradients are involved in a range of subcellular processes, and are particularly important for obtaining positional information. By analysing the low copy number spatiotemporal uctuations in protein concentrations for a single polar gradient and two oppositelydirected gradients, we estimate the positional precision that can be achieved in vivo. We nd that time-averaging is vital for high precision. The embryo of the nematode C. elegans has become a model system for the study of cell polarity. At the one-cell stage, the PAR proteins form anterior and posterior domains in a dynamic process driven by contraction of cortical actomyosin. We present a continuum model for this system, including a highly simpli ed model of the actomyosin dynamics. Our model suggests that the known PAR protein interactions 5 are insu cient to explain the experimentally observed cytoplasmic polarity. We discuss a number of modi cations to the model which reproduce the correct cytoplasmic distributions