On Compact Routing for the Internet

While there exist compact routing schemes designed for grids, trees, and Internet-like topologies that offer routing tables of sizes that scale logarithmically with the network size, we demonstrate in this paper that in view of recent results in compact routing research, such logarithmic scaling on Internet-like topologies is fundamentally impossible in the presence of topology dynamics or topology-independent (flat) addressing. We use analytic arguments to show that the number of routing control messages per topology change cannot scale better than linearly on Internet-like topologies. We also employ simulations to confirm that logarithmic routing table size scaling gets broken by topology-independent addressing, a cornerstone of popular locator-identifier split proposals aiming at improving routing scaling in the presence of network topology dynamics or host mobility. These pessimistic findings lead us to the conclusion that a fundamental re-examination of assumptions behind routing models and abstractions is needed in order to find a routing architecture that would be able to scale ``indefinitely.''Comment: This is a significantly revised, journal version of cs/050802

Chimera states in networks of Van der Pol oscillators with hierarchical connectivities

This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in Chaos 26, 094825 (2016) and may be found at https://doi.org/10.1063/1.4962913.Chimera states are complex spatio-temporal patterns that consist of coexisting domains of coherent and incoherent dynamics. We analyse chimera states in ring networks of Van der Pol oscillators with hierarchical coupling topology. We investigate the stepwise transition from a nonlocal to a hierarchical topology and propose the network clustering coefficient as a measure to establish a link between the existence of chimera states and the compactness of the initial base pattern of a hierarchical topology; we show that a large clustering coefficient promotes the occurrence of chimeras. Depending on the level of hierarchy and base pattern, we obtain chimera states with different numbers of incoherent domains. We investigate the chimera regimes as a function of coupling strength and nonlinearity parameter of the individual oscillators. The analysis of a network with larger base pattern resulting in larger clustering coefficient reveals two different types of chimera states and highlights the increasing role of amplitude dynamics. Chimera states are an example of intriguing partial synchronization patterns appearing in networks of identical oscillators. They exhibit a hybrid structure combining coexisting spatial domains of coherent (synchronized) and incoherent (desynchronized) dynamics.1,2 Recent studies have demonstrated the emergence of chimera states in a variety of topologies and for different types of individual dynamics. In this paper, we analyze chimera states in networks with complex coupling topologies arising in neuroscience. We provide a systematic analysis of the transition from nonlocal to hierarchical (quasi-fractal) connectivities in ring networks of identical Van der Pol oscillators and use the clustering coefficient and the symmetry properties to classify different topologies with respect to the occurrence of chimera states. We show that symmetric connectivities with large clustering coefficients promote the emergence of chimera states, while they are suppressed by slight topological asymmetries or small clustering coefficient.DFG, 163436311, SFB 910: Kontrolle selbstorganisierender nichtlinearer Systeme: Theoretische Methoden und Anwendungskonzept

Graph Summarization

The continuous and rapid growth of highly interconnected datasets, which are both voluminous and complex, calls for the development of adequate processing and analytical techniques. One method for condensing and simplifying such datasets is graph summarization. It denotes a series of application-specific algorithms designed to transform graphs into more compact representations while preserving structural patterns, query answers, or specific property distributions. As this problem is common to several areas studying graph topologies, different approaches, such as clustering, compression, sampling, or influence detection, have been proposed, primarily based on statistical and optimization methods. The focus of our chapter is to pinpoint the main graph summarization methods, but especially to focus on the most recent approaches and novel research trends on this topic, not yet covered by previous surveys.Comment: To appear in the Encyclopedia of Big Data Technologie