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Graphs and networks are a crucial tool to model various phenomena in almost every branch of science. This book studies several different problems that are related to networks; the common theme among these problems is that they all describe some kind of an entity or effect that propagates though the network. The first part of the book focuses on majority/minority processes, when the actors in the network prefer to switch to the most/least common state in their neighborhood. These processes can model various situations in practice: they can represent the spreading of political opinions in a social network, the adoption of different social media platforms, the production strategies of rival companies, the frequency selection of devices in a wireless network, and many more. The book present several upper and lower bounds on the stabilization time of these processes in a general network topology. The second part studies financial networks: a system of banks that are interconnected by different kinds of debt contracts. If one of the banks goes bankrupt and cannot pay its liabilities, then this might cause other banks to go bankrupt as well, leading to a ripple effect through the whole system. There are countless interesting questions in this setting that are related to the properties of the underlying network, such as how regulators can enforce certain properties in these systems, how banks can execute different actions to improve their situation, or how the system behaves if default announcements are made in a step-by-step fashion. Finally, we conclude with a brief discussion of pebble games, where the propagation effect that we study is computation itself: the network represents the interdependencies of variables in a complex computational task, and a pebble game models our progress of computing more and more values in this graph until we arrive at the final result. The book assumes that readers have a strong background in theoretical computer science, but it requires no prior knowledge of the discussed network problems or the corresponding application areas.

This is an open access article distributed under the terms of the Creative Commons CC-BY license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ; Here we present the results of visible range light curve observations of ten Centaurs using the Kepler Space Telescope in the framework of the K2 mission. Well defined periodic light curves are obtained in six cases allowing us to derive rotational periods, a notable increase in the number of Centaurs with known rotational properties. The low amplitude light curves of (471931) 2013 PH44 and (250112) 2002 KY14 can be explained either by albedo variegations, binarity or elongated shape. (353222) 2009 YD7 and (514312) 2016 AE193 could be rotating elongated objects, while 2017 CX33 and 2012 VU85 are the most promising binary candidates due to their slow rotations and higher light curve amplitudes. (463368) 2012 VU85 has the longest rotation period, P = 56.2 h observed among Centaurs. The P > 20 h rotation periods obtained for the two potential binaries underlines the importance of long, uninterrupted time series photometry of solar system targets that can suitably be performed only from spacecraft, like the Kepler in the K2 mission, and the currently running TESS mission. © 2020 The Authors ; The research leading to these results has received funding from the European Union 's Horizon 2020 Research and Innovation Programme, under Grant Agreement No. 687378 ; from the K-125015 , PD-116175 , PD-128360 , and GINOP-2.3.2-15-2016-00003 grants of the National Research Development and Innovation Office (NKFIH, Hungary); and from the LP2012-31 and LP2018-7/2019 Lendület grants of the Hungarian Academy of Sciences . L. M. was supported by the Premium Postdoctoral Research Program of the Hungarian Academy of Sciences . The research leading to these results have been supported by the ÚNKP-19-2 New National Excellence Program of the Ministry of Human Capacities , Hungary. Funding for the Kepler and K2 missions are provided by the NASA Science Mission Directorate. The data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555 . Support for MAST for non-HST data is provided by the NASA Office of Space Science via grant NNX09AF08G and by other grants and contracts. The authors thank the hospitality of the Veszprém Regional Centre of the Hungarian Academy of Sciences (MTA VEAB), where part of this project was carried out. We also indebted to S. Benecchi and an anonymous reviewer for their comments which have helped to improve the paper. ; Peer reviewed

Context. Thanks to the Gaia mission, it will be possible to determine the masses of approximately hundreds of large main belt asteroids with very good precision. We currently have diameter estimates for all of them that can be used to compute their volume and hence their density. However, some of those diameters are still based on simple thermal models, which can occasionally lead to volume uncertainties as high as 20–30%. Aims. The aim of this paper is to determine the 3D shape models and compute the volumes for 13 main belt asteroids that were selected from those targets for which Gaia will provide the mass with an accuracy of better than 10%. Methods. We used the genetic Shaping Asteroids with Genetic Evolution (SAGE) algorithm to fit disk-integrated, dense photometric lightcurves and obtain detailed asteroid shape models. These models were scaled by fitting them to available stellar occultation and/or thermal infrared observations. Results. We determine the spin and shape models for 13 main belt asteroids using the SAGE algorithm. Occultation fitting enables us to confirm main shape features and the spin state, while thermophysical modeling leads to more precise diameters as well as estimates of thermal inertia values. Conclusions. We calculated the volume of our sample of main-belt asteroids for which the Gaia satellite will provide precise mass determinations. From our volumes, it will then be possible to more accurately compute the bulk density, which is a fundamental physical property needed to understand the formation and evolution processes of small Solar System bodies. ; The research leading to these results has received funding from the European Union's Horizon 2020 Research and Innovation Programme, under Grant Agreement no 687378 (SBNAF). Funding for the Kepler and K2 missions is provided by the NASA Science Mission directorate. L.M. was supported by the Premium Postdoctoral Research Program of the Hungarian Academy of Sciences. The research leading to these results has received funding from the LP2012-31 and LP2018-7 Lendület grants of the Hungarian Academy of Sciences. This project has been supported by the Lendület grant LP2012-31 of the Hungarian Academy of Sciences and by the GINOP-2.3.2-15-2016-00003 grant of the Hungarian National Research, Development and Innovation Office (NKFIH). TRAPPIST-South is a project funded by the Belgian Fonds de la Recherche Scientifique (F.R.S.-FNRS) under grant FRFC 2.5.594.09.F. TRAPPIST-North is a project funded by the University of Liège, and performed in collaboration with Cadi Ayyad University of Marrakesh. E.J. is a FNRS Senior Research Associate. "The Joan Oró Telescope (TJO) of the Montsec Astronomical Observatory (OAdM) is owned by the Catalan Government and operated by the Institute for Space Studies of Catalonia (IEEC)." "This article is based on observations made with the SARA telescopes (Southeastern Association for Research in Astronomy), whose node is located at the Kitt Peak National Observatory, AZ under the auspices of the National Optical Astronomy Observatory (NOAO)." "This project uses data from the SuperWASP archive. The WASP project is currently funded and operated by Warwick University and Keele University, and was originally set up by Queen's University Belfast, the Universities of Keele, St. Andrews, and Leicester, the Open University, the Isaac Newton Group, the Instituto de Astrofisica de Canarias, the South African Astronomical Observatory, and by STFC." "This publication makes use of data products from theWide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration." The work of TSR was carried out through grant APOSTD/2019/046 by Generalitat Valenciana (Spain).

Haumea—one of the four known trans-Neptunian dwarf planets—is a very elongated and rapidly rotating body1, 2, 3. In contrast to other dwarf planets4, 5, 6, its size, shape, albedo and density are not well constrained. The Centaur Chariklo was the first body other than a giant planet known to have a ring system7, and the Centaur Chiron was later found to possess something similar to Chariklo's rings8, 9. Here we report observations from multiple Earth-based observatories of Haumea passing in front of a distant star (a multi-chord stellar occultation). Secondary events observed around the main body of Haumea are consistent with the presence of a ring with an opacity of 0.5, width of 70 kilometres and radius of about 2,287 kilometres. The ring is coplanar with both Haumea's equator and the orbit of its satellite Hi'iaka. The radius of the ring places it close to the 3:1 mean-motion resonance with Haumea's spin period—that is, Haumea rotates three times on its axis in the time that a ring particle completes one revolution. The occultation by the main body provides an instantaneous elliptical projected shape with axes of about 1,704 kilometres and 1,138 kilometres. Combined with rotational light curves, the occultation constrains the three-dimensional orientation of Haumea and its triaxial shape, which is inconsistent with a homogeneous body in hydrostatic equilibrium. Haumea's largest axis is at least 2,322 kilometres, larger than previously thought, implying an upper limit for its density of 1,885 kilograms per cubic metre and a geometric albedo of 0.51, both smaller than previous estimates1, 10, 11. In addition, this estimate of the density of Haumea is closer to that of Pluto than are previous estimates, in line with expectations. No global nitrogen- or methane-dominated atmosphere was detected. ; J.L.O. acknowledges funding from Spanish and Andalusian grants MINECO AYA-2014-56637-C2-1-P and J. A. 2012-FQM1776 as well as FEDER funds. Part of the research leading to these results received funding from the European Union's Horizon 2020 Research and Innovation Programme, under grant agreement no. 687378. B.S. acknowledges support from the French grants 'Beyond Neptune' ANR-08-BLAN-0177 and 'Beyond Neptune II' ANR-11-IS56-0002. Part of the research leading to these results has received funding from the European Research Council under the European Community's H2020 (2014-2020/ERC grant agreement no. 669416 'Lucky Star'). A.P. and R.S. have been supported by the grant LP2012-31 of the Hungarian Academy of Sciences. All of the Hungarian contributors acknowledge the partial support from K-125015 grant of the National Research, Development and Innovation Office (NKFIH). G.B.-R., F.B.-R., F.L.R., R.V.-M., J.I.B.C., M.A., A.R.G.-J. and B.E.M. acknowledge support from CAPES, CNPq and FAPERJ. J.C.G. acknowledges funding from AYA2015-63939-C2-2-P and from the Generalitat Valenciana PROMETEOII/2014/057. K.H. and P.P. were supported by the project RVO:67985815. The Astronomical Observatory of the Autonomous Region of the Aosta Valley acknowledges a Shoemaker NEO Grant 2013 from The Planetary Society. We acknowledge funds from a 2016 'Research and Education' grant from Fondazione CRT. We also acknowledge the Slovakian project ITMS no. 26220120029.