Spatial networks with wireless applications

Many networks have nodes located in physical space, with links more common between closely spaced pairs of nodes. For example, the nodes could be wireless devices and links communication channels in a wireless mesh network. We describe recent work involving such networks, considering effects due to the geometry (convex,non-convex, and fractal), node distribution, distance-dependent link probability, mobility, directivity and interference.Comment: Review article- an amended version with a new title from the origina

Renormalization group theory for percolation in time-varying networks

Motivated by multi-hop communication in unreliable wireless networks, we present a percolation theory for time-varying networks. We develop a renormalization group theory for a prototypical network on a regular grid, where individual links switch stochastically between active and inactive states. The question whether a given source node can communicate with a destination node along paths of active links is equivalent to a percolation problem. Our theory maps the temporal existence of multi-hop paths on an effective two-state Markov process. We show analytically how this Markov process converges towards a memory-less Bernoulli process as the hop distance between source and destination node increases. Our work extends classical percolation theory to the dynamic case and elucidates temporal correlations of message losses. Quantification of temporal correlations has implications for the design of wireless communication and control protocols, e.g. in cyber-physical systems such as self-organized swarms of drones or smart traffic networks.Comment: 8 pages, 3 figure

Organic Design of Massively Distributed Systems: A Complex Networks Perspective

The vision of Organic Computing addresses challenges that arise in the design of future information systems that are comprised of numerous, heterogeneous, resource-constrained and error-prone components or devices. Here, the notion organic particularly highlights the idea that, in order to be manageable, such systems should exhibit self-organization, self-adaptation and self-healing characteristics similar to those of biological systems. In recent years, the principles underlying many of the interesting characteristics of natural systems have been investigated from the perspective of complex systems science, particularly using the conceptual framework of statistical physics and statistical mechanics. In this article, we review some of the interesting relations between statistical physics and networked systems and discuss applications in the engineering of organic networked computing systems with predictable, quantifiable and controllable self-* properties.Comment: 17 pages, 14 figures, preprint of submission to Informatik-Spektrum published by Springe

Continuous Time Monte Carlo and Spatial Ordering in Driven Lattice Gases: Application to Driven Vortices in Periodic Superconducting Networks

We consider the two dimensional (2D) classical lattice Coulomb gas as a model for magnetic field induced vortices in 2D superconducting networks. Two different dynamical rules are introduced to investigate driven diffusive steady states far from equilibrium as a function of temperature and driving force. The resulting steady states differ dramatically depending on which dynamical rule is used. We show that the commonly used driven diffusive Metropolis Monte Carlo dynamics contains unphysical intrinsic randomness that destroys the spatial ordering present in equilibrium (the vortex lattice) over most of the driven phase diagram. A continuous time Monte Carlo (CTMC) is then developed, which results in spatially ordered driven states at low temperature in finite sized systems. We show that CTMC is the natural discretization of continuum Langevin dynamics, and argue that it gives the correct physical behavior when the discrete grid represents the minima of a periodic potential. We use detailed finite size scaling methods to analyze the spatial structure of the steady states. We find that finite size effects can be subtle and that very long simulation times can be needed to arrive at the correct steady state. For particles moving on a triangular grid, we find that the ordered moving state is a transversely pinned smectic that becomes unstable to an anisotropic liquid on sufficiently large length scales. For particles moving on a square grid, the moving state is a similar smectic at large drives, but we find evidence for a possible moving solid at lower drives. We find that the driven liquid on the square grid has long range hexatic order, and we explain this as a specifically non-equilibrium effect. We show that, in the liquid, fluctuations are diffusive in both the transverse and longitudinal directions.Comment: 29 pages, 35 figure

The Dynamics of Coalition Formation on Complex Networks

Acknowledgements This work was carried out within the framework of PIK’s COPAN project. We gratefully acknowledge funding by the German Federal Ministry of Education and Research (BMBF) via the CoNDyNet project, grant no. 03SF0472A, and of the Government of the Russian Federation (Agreement No. 14.Z50.31.0033 with Institute of Applied Physics RAS). We thank Jonathan Donges for helpful discussions on this manuscript.Peer reviewedPublisher PD

Network Geometry

Networks are finite metric spaces, with distances defined by the shortest paths between nodes. However, this is not the only form of network geometry: two others are the geometry of latent spaces underlying many networks and the effective geometry induced by dynamical processes in networks. These three approaches to network geometry are intimately related, and all three of them have been found to be exceptionally efficient in discovering fractality, scale invariance, self-similarity and other forms of fundamental symmetries in networks. Network geometry is also of great use in a variety of practical applications, from understanding how the brain works to routing in the Internet. We review the most important theoretical and practical developments dealing with these approaches to network geometry and offer perspectives on future research directions and challenges in this frontier in the study of complexity

The Kuramoto model in complex networks

181 pages, 48 figures. In Press, Accepted Manuscript, Physics Reports 2015 Acknowledgments We are indebted with B. Sonnenschein, E. R. dos Santos, P. Schultz, C. Grabow, M. Ha and C. Choi for insightful and helpful discussions. T.P. acknowledges FAPESP (No. 2012/22160-7 and No. 2015/02486-3) and IRTG 1740. P.J. thanks founding from the China Scholarship Council (CSC). F.A.R. acknowledges CNPq (Grant No. 305940/2010-4) and FAPESP (Grants No. 2011/50761-2 and No. 2013/26416-9) for financial support. J.K. would like to acknowledge IRTG 1740 (DFG and FAPESP).Peer reviewedPreprin