Spectral plot properties: Towards a qualitative classification of networks
Anirban Banerjee Jürgen Jost
Networks & Heterogeneous Media 2008, 3(2): 395-411 doi: 10.3934/nhm.2008.3.395
We introduce a tentative classification scheme for empirical networks based on global qualitative properties detected through the spectrum of the Laplacian of the graph underlying the network. Our method identifies several distinct types of networks across different domains of applications, indicates hidden regularity properties and provides evidence for processes like node duplication behind the evolution or construction of a given class of networks.
keywords: graph Laplacian Network classification graph evolution graph spectrum spectral plots
Dirk Helbing Jürgen Jost Alessandro Vespignani
Networks & Heterogeneous Media 2008, 3(2): i-ii doi: 10.3934/nhm.2008.3.2i
This issue of Networks and Heterogeneous Media (NHM) collects selected papers of the European Conference on Complex Systems ’07, which took place in Dresden, Germany, from October 1 to 6, 2007.
Studying complex systems has enormously changed our view of the world. The discovery that actions and reactions are often disproportionate and that small per- turbations can cause tremendous responses, has led to new scientific disciplines such as catastrophe theory, chaos theory, and the theory of phase transitions.

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keywords: XXXX
A new nonlocal variational setting for image processing
Yan Jin Jürgen Jost Guofang Wang
Inverse Problems & Imaging 2015, 9(2): 415-430 doi: 10.3934/ipi.2015.9.415
We introduce a new nonlocal variational scheme for image denoising. This scheme is motivated by, but different from the nonlocal means filter of Buades et al [9] and the nonlocal TV model proposed by Gilboa-Osher by using nonlocal operators. Our approach is based on general geometric considerations. Experiments show that the corresponding TV model yields denoising results that can compare favorably with those obtained by other methods.
keywords: Image denoising denoising weight. nonlocal TV nonlocal means
Consensus and synchronization in discrete-time networks of multi-agents with stochastically switching topologies and time delays
Wenlian Lu Fatihcan M. Atay Jürgen Jost
Networks & Heterogeneous Media 2011, 6(2): 329-349 doi: 10.3934/nhm.2011.6.329
We analyze stability of consensus algorithms in networks of multi-agents with time-varying topologies and delays. The topology and delays are modeled as induced by an adapted process and are rather general, including i.i.d. topology processes, asynchronous consensus algorithms, and Markovian jumping switching. In case the self-links are instantaneous, we prove that the network reaches consensus for all bounded delays if the graph corresponding to the conditional expectation of the coupling matrix sum across a finite time interval has a spanning tree almost surely. Moreover, when self-links are also delayed and when the delays satisfy certain integer patterns, we observe and prove that the algorithm may not reach consensus but instead synchronize at a periodic trajectory, whose period depends on the delay pattern. We also give a brief discussion on the dynamics in the absence of self-links.
keywords: Consensus network of multi-agents synchronization adapted process switching topology. delay

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