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1. A degree theorem for the simplicial closure of Auter Space

   https://doi.org/10.4310/HHA.2024.v26.n1.a13  

   Aygun, Juliet; Miller, Jeremy (January 2024, Homology, Homotopy and Applications)

   Sinha, Dev (Ed.)

   The degree of a based graph is the number of essential non-basepoint vertices after generic perturbation. Hatcher–Vogtmann’s degree theorem states that the subcomplex of Auter Space of graphs of degree at most is (d-1)-connected. We extend the definition of degree to the simplicial closure of Auter Space and prove a version of Hatcher–Vogtmann’s result in this context. 

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2. Short-range-order degree dominated physical and mechanical properties of refractory multi-principal element alloys: a first-principles study

   https://doi.org/10.1088/1402-4896/ad3a49  

   Yu, Libo; Li, Jia; Liaw, Peter K; Fang, Qihong (May 2024, Physica Scripta)

   Abstract The degree of short-range order (SRO) can influence the physical and mechanical properties of refractory multi-principal element alloys (RMPEAs). Here, the effect of SRO degree on the atomic configuration and properties of the equiatomic TiTaZr RMPEA is investigated using the first-principles calculations. Their key roles on the lattice parameters, binding energy, elastic properties, electronic structure, and stacking fault energy (SFE) are analyzed. The results show the degree of SRO has a significant effect on the physical and mechanical properties of TiTaZr. During the SRO degree increasing in TiTaZr lattice, the low SRO degree exacerbates the lattice distortion and the high SRO degree reduces the lattice distortion. The high degree of SRO improves the binding energy and elastic stiffness of the TiTaZr. By analyzing the change in charge density, this change is caused by the atomic bias generated during the formation of the SRO, which leading to a change in charge-density thereby affecting the metal bond polarity and inter-atomic forces. The high SRO degree also reduces SFE, which means the capability of plastic deformation of the TiTaZr is enhanced. 

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3. On the Rational Degree of Boolean Functions and Applications

   Iyer, Vishnu; Jain, Siddhartha; Kovacs-Deak, Matt; Kumar, Vinayak M; Schaeffer, Luke; Wang, Daochen; Whitmeyer, Michael (October 2023, arXivorg)

   On the Rational Degree of Boolean Functions and Applications Vishnu Iyer, Siddhartha Jain, Matt Kovacs-Deak, Vinayak M. Kumar, Luke Schaeffer, Daochen Wang, Michael Whitmeyer We study a natural complexity measure of Boolean functions known as the (exact) rational degree. For total functions f, it is conjectured that rdeg(f) is polynomially related to deg(f), where deg(f) is the Fourier degree. Towards this conjecture, we show that symmetric functions have rational degree at least deg(f)/2 and monotone functions have rational degree at least sqrt(deg(f)). We observe that both of these lower bounds are tight. In addition, we show that all read-once depth-d Boolean formulae have rational degree at least Ω(deg(f)1/d). Furthermore, we show that almost every Boolean function on n variables has rational degree at least n/2−O(sqrt(n)). In contrast to total functions, we exhibit partial functions that witness unbounded separations between rational and approximate degree, in both directions. As a consequence, we show that for quantum computers, post-selection and bounded-error are incomparable resources in the black-box model. 

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4. Personalized Degrees: Effects on Link Formation in Dynamic Networks from an Egocentric Perspective

   https://doi.org/10.1145/3308560.3316699  

   Arastuie, Makan; S. Xu, Kevin (January 2019, Companion Proceedings of the World Wide Web Conference)

   Understanding mechanisms driving link formation in dynamic social networks is a long-standing problem that has implications to understanding social structure as well as link prediction and recommendation. Social networks exhibit a high degree of transitivity, which explains the successes of common neighbor-based methods for link prediction. In this paper, we examine mechanisms behind link formation from the perspective of an ego node. We introduce the notion of personalized degree for each neighbor node of the ego, which is the number of other neighbors a particular neighbor is connected to. From empirical analyses on four on-line social network datasets, we find that neighbors with higher personalized degree are more likely to lead to new link formations when they serve as common neighbors with other nodes, both in undirected and directed settings. This is complementary to the finding of Adamic and Adar that neighbor nodes with higher (global) degree are less likely to lead to new link formations. Furthermore, on directed networks, we find that personalized out-degree has a stronger effect on link formation than personalized in-degree, whereas global in-degree has a stronger effect than global out-degree. We validate our empirical findings through several link recommendation experiments and observe that incorporating both personalized and global degree into link recommendation greatly improves accuracy. 

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5. Scale-free degree distributions, homophily and the glass ceiling effect in directed networks

   https://doi.org/10.1093/comnet/cnac007  

   Nettasinghe, Buddhika; Alipourfard, Nazanin; Iota, Stephen; Krishnamurthy, Vikram; Lerman, Kristina; Estrada, ed., Ernesto (April 2022, Journal of Complex Networks)

   Abstract Preferential attachment, homophily, and their consequences such as scale-free (i.e. power-law) degree distributions, the glass ceiling effect (the unseen, yet unbreakable barrier that keeps minorities and women from rising to the upper rungs of the corporate ladder, regardless of their qualifications or achievements) and perception bias are well-studied in undirected networks. However, such consequences and the factors that lead to their emergence in directed networks (e.g. author–citation graphs, Twitter) are yet to be coherently explained in an intuitive, theoretically tractable manner using a single dynamical model. To this end, we present a theoretical and numerical analysis of the novel Directed Mixed Preferential Attachment model in order to explain the emergence of scale-free degree distributions and the glass ceiling effect in directed networks with two groups (minority and majority). Specifically, we first derive closed-form expressions for the power-law exponents of the in-degree and out-degree distributions of each of the two groups and then compare the derived exponents with each other to obtain useful insights. These insights include answers to questions such as: when does the minority group have an out-degree (or in-degree) distribution with a heavier tail compared to the majority group? what factors cause the tail of the out-degree distribution of a group to be heavier than the tail of its own in-degree distribution? what effect does frequent addition of edges between existing nodes have on the in-degree and out-degree distributions of the majority and minority groups? Answers to these questions shed light on the interplay between structure (i.e. the in-degree and out-degree distributions of the two groups) and dynamics (characterized collectively by the homophily, preferential attachment, group sizes and growth dynamics) of various real-world directed networks. We also provide a novel definition of the glass ceiling faced by a group via the number of individuals with large out-degree (i.e. those with many followers) normalized by the number of individuals with large in-degree (i.e. those who follow many others) and then use it to characterize the conditions that cause the glass ceiling effect to emerge in a directed network. Our analytical results are supported by detailed numerical experiments. The DMPA model and its theoretical and numerical analysis provided in this article are useful for analysing various phenomena on directed networks in fields such as network science and computational social science. 

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