David Smith, Joseph Samuel Myers, Craig S. Kaplan, Chaim Goodman-Strauss

A longstanding open problem asks for an aperiodic monotile, also known as an “einstein”: a shape that admits tilings of the plane, but never periodic tilings. We answer this problem for topological disk tiles by exhibiting a continuum of combinatorially equivalent aperiodic polygons. We first show that a representative example, the “hat” polykite, can form clusters called “metatiles”, for which substitution rules can be defined. Because the metatiles admit tilings of the plane, so too does the hat. We then prove that generic members of our continuum of polygons are aperiodic, through a new kind of geometric incommensurability argument. Separately, we give a combinatorial, computer-assisted proof that the hat must form hierarchical — and hence aperiodic — tilings.

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Sandeep Chowdhary, Elsa Andres, Adriana Manna, Luka Blagojević, Leonardo Di Gaetano & Gerardo Iñiguez 

EPJ Data Science volume 12, Article number: 7 (2023)

Human communication, the essence of collective social phenomena ranging from small-scale organizations to worldwide online platforms, features intense reciprocal interactions between members in order to achieve stability, cohesion, and cooperation in social networks. While high levels of reciprocity are well known in aggregated communication data, temporal patterns of reciprocal information exchange have received far less attention. Here we propose measures of reciprocity based on the time ordering of interactions and explore them in data from multiple communication channels, including calls, messaging and social media. By separating each channel into reciprocal and non-reciprocal temporal networks, we find persistent trends that point to the distinct roles of one-to-one exchange versus information broadcast. We implement several null models of communication activity, which identify memory, a higher tendency to repeat interactions with past contacts, as a key source of temporal reciprocity. When adding memory to a model of activity-driven, time-varying networks, we reproduce the levels of temporal reciprocity seen in empirical data. Our work adds to the theoretical understanding of the emergence of reciprocity in human communication systems, hinting at the mechanisms behind the formation of norms in social exchange and large-scale cooperation.

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Fabiano L. Ribeiro, Diego Rybski

Physics Reports

Volume 1012, 23 April 2023, Pages 1-39

The quest for a theory of cities that could offer a quantitative and systematic approach to managing cities represents a top priority. If such a theory is feasible, then its formulation must be in a mathematical way. As a contribution to organizing the mathematical ideas that deal with such a systematic way of understanding urban phenomena, we review the main theoretical models present in the literature that aim at explaining the origin and emergence of urban scaling. We intend to present the models, identify similarities and connections between them, and find situations in which different models lead to the same output. In addition, we report situations where some ideas initially introduced in a particular model can also be introduced in another one, generating more diversification and increasing the scope of the original works. The models treated in this paper explain urban scaling from different premises, i.e. from gravity ideas, densification and cites’ geometry to a hierarchical organization and social network properties. We also investigate scenarios in which these different fundamental ideas could be interpreted as similar — where the similarity is likely but not obvious. Furthermore, concerning the gravity model, we propose a general framework that includes all analyzed models as particular cases. We conclude the paper by discussing perspectives of this field and how future research designs and schools of thought can build on the ideas treated here.

Read the full article at: www.sciencedirect.com

Read the full article at: www.pnas.org