Current and future projects

A comprehensive Wiki describing ongoing and possible future research concerning network theory and its various applications within Jürgen Kurths' group, RD IV, PIK.

Remarks

Here we present some ideas concerning future research on climate and recurrence networks as well as on more general issues in complex network theory. We see this as a pool of projects for our own joint work as well as a collection of useful suggestions for diploma students, research interns and the like.

Your are all invited to contribute to this collection and to extend the contributions of others.

Please indicate a date of your update in brackets, e.g., "(3.10.09)".

General network theory

Achievements and manuscripts in preparation

  • In preparation:
    • Paper on "Significance tests for spatially embedded complex networks" (Jonathan, Reik).

Jobst's node splitting invariance

  • To do list:
  • Extend and apply Jobst's concept of node splitting invariance.
  • Think about notion of continuous spatial networks.
  • Relationships to the theory of estimation.

Influence of spatial embedding on network measures

  • Not much literature on this.
  • Probably a lot of theoretical work can be done in this field.
  • Important applications in our main research domains of climate and recurrence networks. Both are spatially embedded and strongly spatially constrained
  • Application also in power grids!
  • Evaluate the effect of spatial constraints on network measures.
  • Derive theoretical bounds, estimates etc. If possible...
  • Manuscript in preparation by Jonathan, Reik et al., "Significance tests for spatially embedded complex networks" (3.10.09).

Networks in time series analysis (general)

Achievements and manuscripts in preparation

  • In preparation?
    • Scaling in degree distribution of recurrence networks

Ambiguities in the visibility graph approach to time series analysis (Reik)

  • Particularly related to the most recent papers/preprints.
  • What are potentials of this approach?
  • What is the actual (dynamically relevant) meaning of network-theoretic measures computed on VGs?

Recurrence Networks

Achievements and manuscripts in preparation

  • Already done:
    • Physics Letters A paper published.
    • NJP paper published.
    • PRE paper published.
  • In review:

  • Paper investigating Shrimps using recurrence network measures.

Comparison between RQA and network measures (Yong)

  • We've made the following statement several times without proof: RQA measures are much local while network measures (e.g., betweenness) incorporate global information of the adjacency matrix.
  • We need to find a model (probably with "well-fragmented" phase space) to show the network measures. I'll have a look at the literature.
  • According to my preliminary results, shrimp structures could be a good candidate in showing the difference between RQA and network measures (see work in preparation above).

Apply RQA to quantify general complex networks (Jonathan)

  • Problem: General complex networks do not possess a unique labeling of nodes. Therefore there is no time ordering given and hence, RQA measures are not uniquely defined for a general complex network (graph).
  • Possibly solution: Find an "optimal" labeling, that extremizes some energy-like scalar quantity, such as maximum diagonal line length, determinism or the like.
  • Construct an algorithm to find an "optimal" labeling. Extremalization strategy could be simulated annealing, a genetic (evolutionary) algorithm, ...
  • Test the algorithm by randomly relabeling recurrence networks and then trying to recover the original labeling using the algorithm. Since this is probably impossible, at least the RQA measures related to the original labeling should be recovered by the new algorithmically found labeling.
  • If the algorithm is found to work reasonably well, apply it systematically to study and classify artificial network models and natural networks (internet, gene/protein interaction, ...).

Rényi-entropies and recurrence network motifs (Jonathan)

  • Possibly there is a formal correspondence between the definition of Rényi entropies for a dynamical system and the frequency of occurrence of certain types of motifs in the corresponding recurrence networks.
  • It would be interesting to further investigate this correspondence.
  • The issue of counting boxes or states seems to be crucial here.

Robustness of recurrence network statistics

  • Investigate systematically the influence of initial conditions, noise, embedding, ... on the statistics (network measures) of recurrence networks.

Recurrence network analysis of real-world time series and significance tests

  • Apply recurrence network analysis to real-world time series from paleoclimatology, economics, biology, medicine, ... .
  • Develop appropriate significance tests for these applications.
  • In preparation:
    • Synchronous changes in African climate during the Plio-Pleistocene and implications for hominin evolution, Nature Geosciences, together with M. Trauth.
    • Technical twin paper for NPG.

Climate networks

Achievements and manuscripts in preparation

  • Already done:
    • EPJ ST, "Complex networks in climate dynamics".
    • EPL, "The backbone of the climate network".

Time dependent networks and related significance tests

  • Reproduction of Yamasaki's results would be the most important. Yamasaki's results were reproduced to some degree of accuracy (Jakob Tzscheischler, Jakob Runge, Alex Radebach). Contact with Avi Gozolchiani established. Some conceptual problems remain (3.10.09).
  • Construct time dependent climate networks from data and study the evolution of network measures.
  • Focus on "true" network measures, i.e., clustering coefficient, average path length, assortativity ..., NOT only link density, as done by Yamasaki et al.
  • Develop meaningful significance tests for the time dependent network measures.
  • Think about the evolution of maximum degree, maximum betweenness etc. over time (Gorka's recent papers). Can we learn something new here?
  • We can locate these "extremal" nodes on a map and visualize the evolution over time. I expect this would be interesting for interpretation.

    Tipping points (Norbert)

    • Scott
    • Can tipping points be identified by the network (perhaps hubs?), identify important tipping points of the climate system in the climate network.
    • Extraction of the structural network features of these tipping points.
    • Comparison of hubs corresponding to tipping points with hubs of non-tipping points.
    • Are tipping points connected.
    • Dynamics of the tipping points due to changes in the climate system.

    Regional scale networks

    • Effects of extreme events, El Nino etc.
    • It is crucial to look at regional subnetworks embedded in a global network! For example, one could look at the effect of removing such a regional subnetwork on the global network (Scott, relation to tipping points).
    • Africa: we have now new satellite based data (TRMM) from Bodo Bookhagen
    • Monsoon and Amazonia: as far as I remember, Jakob should focus on this?

    Clustering and community structure in climate networks

    • Tsonis has a manuscript out there (rejected by PRL), but we can possibly do better. The manuscript can be found here.
    • Can we speculate on the reasons why Tsonis' paper was rejected by PRL? Is it because of the community detection method or because of the underlying climatological phenomenon? If this becomes clearer, we are sure we gonna do better.
    • Apply and compare different clustering methods, e.g., "classical" methods by Newman, fancy method by Jobst, new method by Thilo Gross (submitted to PNAS recently).
    • Clustering - Thilo Gross is applying his newly developed method to one of our correlation matrices (for global HadCM3 SAT data set, as used in the EPL and EPJ ST papers). He will contact us once he obtains interesting results (3.10.09).

    Extensions of layered climate network approach (Hanna)

    • Study more advanced statistics and network measures.
    • We can only learn something really new if we study more that just degree centrality / area weighted connectivity.
    • Think about a better network construction method here. As the average correlation decays quickly with increasing vertical distance, when one uses a fixed link density the upper layers of the atmosphere receive too many connections. This leads to artificial structures.
    • Extend 3D layered network approach (Hanna) to layered networks of different observables.
    • Use this approach to look at the jet stream, particularly on the impact of its possible relocation due to climate change (Norbert). (3.10.09)

    "Pacemaker networks" (Reik)

    • For certain oscillatory modes in the climate system (ENSO band, 7.8 years band, 11 years cycle) based on a combination of
      • wavelet methods (What are locations with significant variability? What are the phase relationships? Causality???)
      • and network theory.

    Systematic intercomparison of direct observations, reanalysis, and different climate models (Reik)

    • Also different IPCC scenarios?!

    • Excellent project for some student...

     How far is the network approach from climate research? (Yong)

    • We can basically apply all methods and ideas from network theory (beyond those measures we've used) to climate networks. However, we should go one step further, dealing with one particular problem, like the regional scale and
    • to develop some particular algorithms/measures for the climate networks.

    Definition of Links in Complex Networks extracted from data

    Significance based links (decays of correlation) (Yong)

    • Concerning the quick decays of the correlation with vertical distances, I suppose that applying a significance test before claiming a network link would be helpful. We can use a significance level (instead of a constant link density) to construct the layered network. It seems that I myself should test this idea very soon before making suggestion.
    • May be problematic because the link density could be arbitrary high.

    Links based on joint recurrences (Yong)

    • Computationally extensive.
    • As far as I remember, one of the diploma students is working on this problem? Jakob R. put this aside for now (3.10.09).

    Visualization

    • Coarse graining
    • PCA
    • Joint work with Thomas Nocke et al., a software for selecting and visualizing edges will be available soon (3.10.09).