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Arıkan’s exciting discovery of polar codes has provided an altogether new way to efficiently achieve Shannon capacity. Given a (constant-sized) invertible matrix M , a family of polar codes can be associated with this matrix and its ability to approach capacity follows from the polarization of an associated [0, 1]-bounded martingale, namely its convergence in the limit to either 0 or 1 with probability 1. Arıkan showed appropriate polarization of the martingale associated with the matrix ( G 2 = ( 1 1 0 1) to get capacity achieving codes. His analysis was later extended to all matrices M that satisfy an obvious necessary condition for polarization. While Arıkan’s theorem does not guarantee that the codes achieve capacity at small blocklengths (specifically in length, which is a polynomial in ( 1ε ) where (ε) is the difference between the capacity of a channel and the rate of the code), it turns out that a “strong” analysis of the polarization of the underlying martingale would lead to such constructions. Indeed for the martingale associated with ( G 2 ) such a strong polarization was shown in two independent works (Guruswami and Xia (IEEE IT’15) and Hassani et al. (IEEE IT’14)), thereby resolving a major theoretical challenge associated with the efficient attainment of Shannon capacity. In this work we extend the result above to cover martingales associated with all matrices that satisfy the necessary condition for (weak) polarization. In addition to being vastly more general, our proofs of strong polarization are (in our view) also much simpler and modular. Key to our proof is a notion of local polarization that only depends on the evolution of the martingale in a single time step. We show that local polarization always implies strong polarization. We then apply relatively simple reasoning about conditional entropies to prove local polarization in very general settings. Specifically, our result shows strong polarization over all prime fields and leads to efficient capacity-achieving source codes for compressing arbitrary i.i.d. sources, and capacity-achieving channel codes for arbitrary symmetric memoryless channels. We show how to use our analyses to achieve exponentially small error probabilities at lengths inverse polynomial in the gap to capacity. Indeed we show that we can essentially match any error probability while maintaining lengths that are only inverse polynomial in the gap to capacity. 

We propose and extend a qualitative, complex systems methodology from cognitive engineering, known as theabstraction hierarchy, to model how potential interventions that could be carried out by social media platforms might impact social equality. Social media platforms have come under considerable ire for their role in perpetuating social inequality. However, there is also significant evidence that platforms can play a role inreducingsocial inequality, e.g. through the promotion of social movements. Platforms’ role in producing or reducing social inequality is, moreover, not static; platforms can and often do take actions targeted at positive change. How can we develop tools to help us determine whether or not a potential platform change might actually work to increase social equality? Here, we present the abstraction hierarchy as a tool to help answer this question. Our primary contributions are two-fold. First, methodologically, we extend existing research on the abstraction hierarchy in cognitive engineering with principles from Network Science. Second, substantively, we illustrate the utility of this approach by using it to assess the potential effectiveness of a set of interventions, proposed in prior work, for how online dating websites can help mitigate social inequality.