The floodlight problem

Given three angles summing to 2, given n points in the plane and a tripartition k1 + k2 + k3 = n, we can tripartition the plane into three wedges of the given angles so that the i-th wedge contains ki of the points. This new result on dissecting point sets is used to prove that lights of specied angles not exceeding can be placed at n xed points in the plane to illuminate the entire plane if and only if the angles sum to at least 2. We give O(n log n) algorithms for both these problems

Dynamic reconfiguration of human brain networks during learning

Human learning is a complex phenomenon requiring flexibility to adapt existing brain function and precision in selecting new neurophysiological activities to drive desired behavior. These two attributes -- flexibility and selection -- must operate over multiple temporal scales as performance of a skill changes from being slow and challenging to being fast and automatic. Such selective adaptability is naturally provided by modular structure, which plays a critical role in evolution, development, and optimal network function. Using functional connectivity measurements of brain activity acquired from initial training through mastery of a simple motor skill, we explore the role of modularity in human learning by identifying dynamic changes of modular organization spanning multiple temporal scales. Our results indicate that flexibility, which we measure by the allegiance of nodes to modules, in one experimental session predicts the relative amount of learning in a future session. We also develop a general statistical framework for the identification of modular architectures in evolving systems, which is broadly applicable to disciplines where network adaptability is crucial to the understanding of system performance.Comment: Main Text: 19 pages, 4 figures Supplementary Materials: 34 pages, 4 figures, 3 table

Detection of Core-Periphery Structure in Networks Using Spectral Methods and Geodesic Paths

We introduce several novel and computationally efficient methods for detecting "core--periphery structure" in networks. Core--periphery structure is a type of mesoscale structure that includes densely-connected core vertices and sparsely-connected peripheral vertices. Core vertices tend to be well-connected both among themselves and to peripheral vertices, which tend not to be well-connected to other vertices. Our first method, which is based on transportation in networks, aggregates information from many geodesic paths in a network and yields a score for each vertex that reflects the likelihood that a vertex is a core vertex. Our second method is based on a low-rank approximation of a network's adjacency matrix, which can often be expressed as a tensor-product matrix. Our third approach uses the bottom eigenvector of the random-walk Laplacian to infer a coreness score and a classification into core and peripheral vertices. We also design an objective function to (1) help classify vertices into core or peripheral vertices and (2) provide a goodness-of-fit criterion for classifications into core versus peripheral vertices. To examine the performance of our methods, we apply our algorithms to both synthetically-generated networks and a variety of networks constructed from real-world data sets.Comment: This article is part of EJAM's December 2016 special issue on "Network Analysis and Modelling" (available at https://www.cambridge.org/core/journals/european-journal-of-applied-mathematics/issue/journal-ejm-volume-27-issue-6/D245C89CABF55DBF573BB412F7651ADB

Computation in Complex Networks

Complex networks are one of the most challenging research focuses of disciplines, including physics, mathematics, biology, medicine, engineering, and computer science, among others. The interest in complex networks is increasingly growing, due to their ability to model several daily life systems, such as technology networks, the Internet, and communication, chemical, neural, social, political and financial networks. The Special Issue “Computation in Complex Networks" of Entropy offers a multidisciplinary view on how some complex systems behave, providing a collection of original and high-quality papers within the research fields of: • Community detection • Complex network modelling • Complex network analysis • Node classification • Information spreading and control • Network robustness • Social networks • Network medicin

Statistical physics approaches to large-scale socio-economic networks

Die statistische Physik erforschte im letzten Jahrzehnt eine Fülle von wissenschaftlichen Gebieten, was zu einem besseren quantitativen Verständnis von verschiedenen, aus vielen Elementen bestehenden Systemen, z.B. von sozialen Systemen, geführt hat. Eine empirische Quantifizierung von menschlichem Verhalten auf gesellschaftlichem Niveau hat sich allerdings bisher als sehr schwierig erwiesen, wegen Problemen bei der Gewinnung und Qualität von Daten. In dieser Doktorarbeit erstellen wir zum ersten mal einen umfangreichen über fünf Jahre gesammelten Datensatz, der praktisch alle Aktionen und Eigenschaften der 350.000 Teilnehmer einer gesamten menschlichen Gesellschaft aus einem selbstentwickelten Massive Multiplayer Online Game enthält. Wir beschreiben dieses aus stark wechselwirkenden Spielern bestehende soziale System in drei Ebenen. In einem ersten Schritt analysieren wir die Individuen und deren Verhalten im Verlauf der Zeit. Eine Skalen- und Fluktuationsanalyse von Aktions-Reaktions-Zeitreihen enthüllt Persistenz der möglichen Aktionen und qualitative Unterschiede zwischen "guten" und "schlechten" Spielern. Wir untersuchen danach den Diffusionsprozess der im Spieluniversum stattfindenden Bewegungen der Individuen. Wir finden Subdiffusivität und eine durch ein Potenzgesetz verteilte Präferenz zu kürzlich besuchten Orten zurückzukehren. Zweitens, auf der nächsthöheren Ebene, verwenden wir Netzwerktheorie um die topologische Struktur der Interaktionen zwischen Individuen zu quantifizieren. Wir konzentrieren uns auf sechs durch direkte Interaktionen definierte Netzwerke, drei davon positiv (Handel, Freundschaft, Kommunikation), drei negativ (Feindschaft, Attacke, Bestrafung). Diese Netzwerke weisen nichttriviale statistische Eigenschaften auf, z.B. skaleninvariante Topologie, und entwickeln sich in der Zeit, was uns erlaubt eine Reihe von Hypothesen über sozialdynamische Phänomene zu testen. Wir finden qualitative Unterschiede zwischen positiven und negativen Netzwerken in Evolution und Struktur. Schließlich untersuchen wir das Multiplex-Netzwerk der Spielergesellschaft, das sich aus den einzelnen Netzwerk-Schichten zusammensetzt. Wir quantifizieren Interaktionen zwischen verschiedenen Netzwerken und zeigen die nichttrivialen Organisationsprinzipien auf die auch in echten menschlichen Gesellschaften beobachtet wurden. Unsere Erkenntnisse liefern Belege für die Hypothese der strukturellen Balance, die eine Vermeidung von gewissen frustrierten Zuständen auf mikroskopischem Niveau postuliert. Mit diesem Aufbau demonstrieren wir die Möglichkeit der Gewinnung neuartiger wissenschaftlicher Erkenntnisse über die Natur von kollektivem menschlichen Verhalten in großangelegten sozialen Systemen.In the past decade a variety of fields has been explored by statistical physicists, leading to an increase of our quantitative understanding of various systems composed of many interacting elements, such as social systems. However, an empirical quantification of human behavior on a societal level has so far proved to be tremendously difficult due to problems in data availability, quality and ways of acquisition. In this doctoral thesis we compile for the first time a large-scale data set consisting of practically all actions and properties of 350,000 odd participants of an entire human society interacting in a self-developed Massive Multiplayer Online Game, over a period of five years. We describe this social system composed of strongly interacting players in the game in three consecutive levels. In a first step, we examine the individuals and their behavioral properties over time. A scaling and fluctuation analysis of action-reaction time-series reveals persistence of the possible actions and qualitative differences between "good" and "bad" players. We then study and model the diffusion process of human mobility occurring within the "game universe". We find subdiffusion and a power-law distributed preference to return to more recently visited locations. Second, on a higher level, we use network theory to quantify the topological structure of interactions between the individuals. We focus on six network types defined by direct interactions, three of them with a positive connotation (trade, friendship, communication), three with a negative one (enmity, attack, punishment). These networks exhibit non-trivial statistical properties, e.g. scale-free topology, and evolve over time, allowing to test a series of long-standing social-dynamics hypotheses. We find qualitative differences in evolution and topological structure between positive and negative tie networks. Finally, on a yet higher level, we consider the multiplex network of the player society, constituted by the coupling of the single network layers. We quantify interactions between different networks and detect the non-trivial organizational principles which lead to the observed structure of the system and which have been observed in real human societies as well. Our findings with the multiplex framework provide evidence for the half-century old hypothesis of structural balance, where certain frustrated states on a microscopic level tend to be avoided. Within this setup we demonstrate the feasibility for generating novel scientific insights on the nature of collective human behavior in large-scale social systems