How to calculate the barycenter of a weighted graph

Discrete structures like graphs make it possible to naturally and exibly model complex phenomena. Since graphs that represent various types of information are increasingly available today, their analysis has become a popular subject of research. The graphs studied in the field of data science at this time generally have a large number of nodes that are not fairly weighted and connected to each other, translating a structural specification of the data. Yet, even an algorithm for locating the average position in graphs is lacking although this knowledge would be of primary interest for statistical or representation problems. In this work, we develop a stochastic algorithm for finding the Fréchet mean of weighted undirected metric graphs. This method relies on a noisy simulated annealing algorithm dealt with using homogenization. We then illustrate our algorithm with two examples (subgraphs of a social network and of a collaboration and citation network)

How to calculate the barycenter of a weighted graph

Discrete structures like graphs make it possible to naturally and exibly model complex phenomena. Since graphs that represent various types of information are increasingly available today, their analysis has become a popular subject of research. The graphs studied in the field of data science at this time generally have a large number of nodes that are not fairly weighted and connected to each other, translating a structural specification of the data. Yet, even an algorithm for locating the average position in graphs is lacking although this knowledge would be of primary interest for statistical or representation problems. In this work, we develop a stochastic algorithm for finding the Fréchet mean of weighted undirected metric graphs. This method relies on a noisy simulated annealing algorithm dealt with using homogenization. We then illustrate our algorithm with two examples (subgraphs of a social network and of a collaboration and citation network)

Convex Hull Realizations of the Multiplihedra

We present a simple algorithm for determining the extremal points in Euclidean space whose convex hull is the nth polytope in the sequence known as the multiplihedra. This answers the open question of whether the multiplihedra could be realized as convex polytopes. We use this realization to unite the approach to A_n-maps of Iwase and Mimura to that of Boardman and Vogt. We include a review of the appearance of the nth multiplihedron for various n in the studies of higher homotopy commutativity, (weak) n-categories, A_infinity-categories, deformation theory, and moduli spaces. We also include suggestions for the use of our realizations in some of these areas as well as in related studies, including enriched category theory and the graph associahedra.Comment: typos fixed, introduction revise

Evolution of circumbinary planets around eccentric binaries: The case of Kepler-34

The existence of planets orbiting a central binary star system immediately raises questions regarding their formation and dynamical evolution. Recent discoveries of circumbinary planets by the Kepler space telescope has shown that some of these planets reside close to the dynamical stability limit where it is very difficult to form planets in situ. For binary systems with nearly circular orbits, such as Kepler-38, the observed proximity of planetary orbits to the stability limit can be understood by an evolutionary process in which planets form farther out in the disk and migrate inward to their observed position. The Kepler-34 system has a high orbital eccentricity of 0.52. Here, we analyse evolutionary scenarios for the planet observed around this system using two-dimensional hydrodynamical simulations. The highly eccentric binary opens a wide inner hole in the disk which is also eccentric, and displays a slow prograde precession. As a result of the large, eccentric inner gap, an embedded planet settles in a final equilibrium position that lies beyond the observed location of Kepler-34 b, but has the correct eccentricity. In this configuration the planetary orbit is aligned with the disk in a state of apsidal corotation.To account for the closer orbit of Kepler-34 b to the central binary, we considered a two-planet scenario and examined the evolution of the system through joint inward migration and capture into mean-motion resonances. When the inner planet orbits inside the gap of the disk, planet-planet scattering ensues. While often one object is thrown into a large, highly eccentric orbit, at times the system is left with a planet close to the observed orbit, suggesting that Kepler 34 might have had two circumbinary planets where one might have been scattered out of the system or into an orbit where it did not transit the central binary during the operation of Kepler.Comment: 11 pages, 13 figures, accepted by Astronomy & Astrophysics. arXiv admin note: text overlap with arXiv:1401.764

Towards Swarm Calculus: Urn Models of Collective Decisions and Universal Properties of Swarm Performance

Methods of general applicability are searched for in swarm intelligence with the aim of gaining new insights about natural swarms and to develop design methodologies for artificial swarms. An ideal solution could be a `swarm calculus' that allows to calculate key features of swarms such as expected swarm performance and robustness based on only a few parameters. To work towards this ideal, one needs to find methods and models with high degrees of generality. In this paper, we report two models that might be examples of exceptional generality. First, an abstract model is presented that describes swarm performance depending on swarm density based on the dichotomy between cooperation and interference. Typical swarm experiments are given as examples to show how the model fits to several different results. Second, we give an abstract model of collective decision making that is inspired by urn models. The effects of positive feedback probability, that is increasing over time in a decision making system, are understood by the help of a parameter that controls the feedback based on the swarm's current consensus. Several applicable methods, such as the description as Markov process, calculation of splitting probabilities, mean first passage times, and measurements of positive feedback, are discussed and applications to artificial and natural swarms are reported