A Confidence-Based Approach for Balancing Fairness and Accuracy

We study three classical machine learning algorithms in the context of algorithmic fairness: adaptive boosting, support vector machines, and logistic regression. Our goal is to maintain the high accuracy of these learning algorithms while reducing the degree to which they discriminate against individuals because of their membership in a protected group. Our first contribution is a method for achieving fairness by shifting the decision boundary for the protected group. The method is based on the theory of margins for boosting. Our method performs comparably to or outperforms previous algorithms in the fairness literature in terms of accuracy and low discrimination, while simultaneously allowing for a fast and transparent quantification of the trade-off between bias and error. Our second contribution addresses the shortcomings of the bias-error trade-off studied in most of the algorithmic fairness literature. We demonstrate that even hopelessly naive modifications of a biased algorithm, which cannot be reasonably said to be fair, can still achieve low bias and high accuracy. To help to distinguish between these naive algorithms and more sensible algorithms we propose a new measure of fairness, called resilience to random bias (RRB). We demonstrate that RRB distinguishes well between our naive and sensible fairness algorithms. RRB together with bias and accuracy provides a more complete picture of the fairness of an algorithm

Disorder induced brittle to quasi-brittle transition in fiber bundles

We investigate the fracture process of a bundle of fibers with random Young modulus and a constant breaking strength. For two component systems we show that the strength of the mixture is always lower than the strength of the individual components. For continuously distributed Young modulus the tail of the distribution proved to play a decisive role since fibers break in the decreasing order of their stiffness. Using power law distributed stiffness values we demonstrate that the system exhibits a disorder induced brittle to quasi-brittle transition which occurs analogously to continuous phase transitions. Based on computer simulations we determine the critical exponents of the transition and construct the phase diagram of the system.Comment: 6 pages, 6 figure

Blending stiffness and strength disorder can stabilize fracture

Quasi-brittle behavior where macroscopic failure is preceded by stable damaging and intensive cracking activity is a desired feature of materials because it makes fracture predictable. Based on a fiber bundle model with global load sharing we show that blending strength and stiffness disorder of material elements leads to the stabilization of fracture, i.e. samples which are brittle when one source of disorder is present, become quasi-brittle as a consequence of blending. We derive a condition of quasi-brittle behavior in terms of the joint distribution of the two sources of disorder. Breaking bursts have a power law size distribution of exponent 5/2 without any crossover to a lower exponent when the amount of disorder is gradually decreased. The results have practical relevance for the design of materials to increase the safety of constructions.Comment: 9 pages, 6 figures, revte

Engineering graphene by oxidation: a first principles study

Graphene epoxide, with oxygen atoms lining up on pristine graphene sheets, is investigated theoretically in this Letter. Two distinct phases: metastable clamped and unzipped structures are unveiled in consistence with experiments. In the stable (unzipped) phase, epoxy group breaks underneath sp2 bond and modifies the mechanical and electronic properties of graphene remarkably. The foldable epoxy ring structure reduces its Young's modulus by 42.4%, while leaves the tensile strength almost unchanged. Epoxidation also perturbs the pi state and opens semiconducting gap for both phases, with dependence on the density of epoxidation. In the unzipped structures, localized states revealed near the Fermi level resembles the edge states in graphene nanoribbons. The study reported here paves the way for oxidation-based functionalization of graphene-related materials.Comment: 17 pages (4 figures, 1 table

Effect of disorder on temporal fluctuations in drying induced cracking

We investigate by means of computer simulations the effect of structural disorder on the statistics of cracking for a thin layer of material under uniform and isotropic drying. The layer is discretized into a triangular lattice of springs. The drying process is captured by reducing the natural length of all springs by the same factor, and the amount of quenched disorder is controlled by varying the width {\xi} of the distribution of the random breaking thresholds for the springs. Once a spring breaks, redistribution of the load may trigger an avalanche of breaks. Our simulations revealed that the system exhibits a phase transition with the amount of disorder as control parameter: at low disorders, the breaking process is dominated by a macroscopic crack at the beginning, and the size distribution of the subsequent breaking avalanches shows an exponential form. At high disorders, the fracturing proceeds in small-sized avalanches with an exponential distribution, generating a large number of micro-cracks which eventually merge and break the layer. Between both phases a sharp transition occurs at a critical amount of disorder {\xi}_c = 0.40 \pm 0.01, where the avalanche size distribution becomes a power law with exponent {\tau} = 2.6 \pm 0.08, in agreement with the mean-field value {\tau} = 5/2 of the fiber bundle model. Good quality data collapses from the finite-size scaling analysis show that the average value of the largest burst can be identified as the order parameter, with {\beta}/{\nu} = 1.4 and 1/{\nu} = 1.0, and that the average ratio of the second m2 and first moments m1 of the avalanche size distribution shows similar behaviour to the susceptibility of a continuous transition, with {\gamma}/{\nu} = 1., 1/{\nu} = 0.9. These suggest that the disorder induced transition of the breakup of thin layers is analogous to a continuous phase transition.Comment: 9 figure

Disorder driven collapse of the mobility gap and transition to an insulator in fractional quantum Hall effect

We study the nu=1/3 quantum Hall state in presence of the random disorder. We calculate the topologically invariant Chern number, which is the only quantity known at present to unambiguously distinguish between insulating and current carrying states in an interacting system. The mobility gap can be determined numerically this way, which is found to agree with experimental value semiquantitatively. As the disorder strength increases towards a critical value, both the mobility gap and plateau width narrow continuously and ultimately collapse leading to an insulating phase.Comment: 4 pages with 4 figure