Erasure-Resilient Sublinear-Time Graph Algorithms

We investigate sublinear-time algorithms that take partially erased graphs represented by adjacency lists as input. Our algorithms make degree and neighbor queries to the input graph and work with a specified fraction of adversarial erasures in adjacency entries. We focus on two computational tasks: testing if a graph is connected or ε-far from connected and estimating the average degree. For testing connectedness, we discover a threshold phenomenon: when the fraction of erasures is less than ε, this property can be tested efficiently (in time independent of the size of the graph); when the fraction of erasures is at least ε, then a number of queries linear in the size of the graph representation is required. Our erasure-resilient algorithm (for the special case with no erasures) is an improvement over the previously known algorithm for connectedness in the standard property testing model and has optimal dependence on the proximity parameter ε. For estimating the average degree, our results provide an “interpolation” between the query complexity for this computational task in the model with no erasures in two different settings: with only degree queries, investigated by Feige (SIAM J. Comput. ‘06), and with degree queries and neighbor queries, investigated by Goldreich and Ron (Random Struct. Algorithms ‘08) and Eden et al. (ICALP ‘17). We conclude with a discussion of our model and open questions raised by our work.

Threshold Phenomena for Random Cones

We consider an even probability distribution on the d-dimensional Euclidean space with the property that it assigns measure zero to any hyperplane through the origin. Given N independent random vectors with this distribution, under the condition that they do not positively span the whole space, the positive hull of these vectors is a random polyhedral cone (and its intersection with the unit sphere is a random spherical polytope). It was first studied by Cover and Efron. We consider the expected face numbers of these random cones and describe a threshold phenomenon when the dimension d and the number N of random vectors tend to infinity. In a similar way we treat the solid angle, and more generally the Grassmann angles. We further consider the expected numbers of k-faces and of Grassmann angles of index $$d-k$$ d - k when also k tends to infinity.

Investigating the Role of Green Infrastructure on Urban WaterLogging: Evidence from Metropolitan Coastal Cities

Urban green infrastructures (UGI) can effectively reduce surface runoff, thereby alleviating the pressure of urban waterlogging. Due to the shortage of land resources in metropolitan areas, it is necessary to understand how to utilize the limited UGI area to maximize the waterlogging mitigation function. Less attention, however, has been paid to investigating the threshold level of waterlogging mitigation capacity. Additionally, various studies mainly focused on the individual effects of UGI factors on waterlogging but neglected the interactive effects between these factors. To overcome this limitation, two waterlogging high-risk coastal cities—Guangzhou and Shenzhen, are selected to examine the effectiveness and stability of UGI in alleviating urban waterlogging. The results indicate that the impact of green infrastructure on urban waterlogging largely depends on its area and biophysical parameter. Healthier or denser vegetation (superior ecological environment) can more effectively intercept and store rainwater runoff. This suggests that while increasing the area of UGI, more attention should be paid to the biophysical parameter of vegetation. Hence, the mitigation effect of green infrastructure would be improved from the “size” and “health”. The interaction of composition and spatial configuration greatly enhances their individual effects on waterlogging. This result underscores the importance of the interactive enhancement effect between UGI composition and spatial configuration. Therefore, it is particularly important to optimize the UGI composition and spatial pattern under limited land resource conditions. Lastly, the effect of green infrastructure on waterlogging presents a threshold phenomenon. The excessive area proportions of UGI within the watershed unit or an oversized UGI patch may lead to a waste of its mitigation effect. Therefore, the area proportion of UGI and its mitigation effect should be considered comprehensively when planning UGI. It is recommended to control the proportion of green infrastructure at the watershed scale (24.4% and 72.1% for Guangzhou and Shenzhen) as well as the area of green infrastructure patches (1.9 ha and 2.8 ha for Guangzhou and Shenzhen) within the threshold level to maximize its mitigation effect. Given the growing concerns of global warming and continued rapid urbanization, these findings provide practical urban waterlogging prevention strategies toward practical implementations.

UNIVERSAL METHOD FOR COMPUTATIONAL MODELING OF THRESHOLD PHENOMENON IN THE NONSTEADY BIOLOGICAL PROCESSES

Context. In modern conditions occur abrupt changes in ecosystems. The species composition of Caspian Sea is changing rapidly. The dynamics of populations acquires an extreme character with the development of rapid invasions. The mathematical description of scale transformations requires new modeling methods. Complicated population regimes of changes have features of the threshold phenomenon in process of its development. Objective. We set the goal of computational modeling of practically important scenarios – groups of situations that relate to extreme and transitional dynamics of ecosystems, like outbreaks at the onset of dangerous invasions. We are developing a method that, on the basis of the survival model of generations, will conduct a description of sudden transitions to rapid but limited outbreak of numbers or, on contrary, a collapse of stocks like Atlantic cod in 1992 or Peruan anchovy Engraulis ringens in 1985. The purpose of our modeling is to improve the accuracy of forecasts of the population size when experts are estimates a rational strategy for the exploitation of biological resources. Method. Situations of abrupt but short-term changes in population processes cannot be calculated by traditional mathematical models and expressed in terms of asymptotic dynamics – closed limit trajectory sets. The basis of the idea of the method proposed by us is the formalization of nonlinear efficiency of reproduction, which changes in a threshold manner only in strictly defined environmental conditions. We use continuous-discrete time in the model for early ontognosis of the cod fish and insect pests. The method with triggers allows us to take into account in simulation experiments logic and motivation of making decisions by experts, people who manage the strategy of exploiting biological resources. Models assess variability for development of situations Results. We have implemented new method of bounded trigger functionals into hybrid system of the equations, that acting in selected specific states of biosystems. Analysis of new model scenarios with modifications of functionals in the basic hybrid system for extreme situations in fish and insect pests is carried out. Conclusions. We consider the method to be universal, since selection of the functional can be adapted to a wide class of models using differential equations on a fixed interval.

A Sharp Threshold Phenomenon in String Graphs

A string graph is the intersection graph of curves in the plane. We prove that for every $$\epsilon >0$$ ϵ > 0 , if G is a string graph with n vertices such that the edge density of G is below $${1}/{4}-\epsilon $$ 1 / 4 - ϵ , then V(G) contains two linear sized subsets A and B with no edges between them. The constant 1/4 is a sharp threshold for this phenomenon as there are string graphs with edge density less than $${1}/{4}+\epsilon $$ 1 / 4 + ϵ such that there is an edge connecting any two logarithmic sized subsets of the vertices. The existence of linear sized sets A and B with no edges between them in sufficiently sparse string graphs is a direct consequence of a recent result of Lee about separators. Our main theorem finds the largest possible density for which this still holds. In the special case when the curves are x-monotone, the same result was proved by Pach and the author of this paper, who also proposed the conjecture for the general case.

An Approach to Size Sub-Wavelength Surface Crack Measurements Using Rayleigh Waves Based on Laser Ultrasounds

In this paper, the interaction of a broadband Rayleigh wave generated by a laser and an artificial rectangular notch is analyzed theoretically and experimentally. For the theoretical analysis, a Gaussian function is adopted to analyze the modulation of notch depth on the frequency spectrum via reflection and transmission coefficients. By the finite element method, the Rayleigh wave generated by pulsed laser beam irradiation and its scattering waves at cracks are calculated. A curve with a slope close to 4 fitted by crack depth and critical wavelength of the threshold phenomenon is obtained by the wavelet transform and Parseval’s theorem according to simulated and experimental results. Based on this relationship, the critical frequency at which the threshold phenomenon happens due to energy transformation of transmission/reflection Rayleigh waves is adopted to determine the size of sub-wavelength surface crack. The experimental results of artificial notch depth estimation on aluminum alloy specimens consistent with theoretical analysis validates the usefulness of the critical frequency method based on a broadband Rayleigh wave generated by laser ultrasonic.