# The Effects of Edge Weights on Correlating Dynamical Networks - Comparing Unweighted and Weighted Brain Graphs of nervus opticus Patients

### Christian Moewes, Rudolf Kruse

#### Abstract

We are interested in the regression analysis of dynamical networks. Our goal is to predict real-valued function values from a given observation which is manifested as series of graphs. Every observation is described by a set of dependent variables that we want to predict using the dynamical graphs. These graphs change their edges over time, while the set of nodes is assumed to be constant. Such settings can be found in many real-world applications, e.g., communication networks, brain connectivity, microblogging. We apply several measures to every graph in the series to globally describe its evolution. The resulting multivariate time series is used to learn vector autoregressive (VAR) models. The parameters of these models can be used to correlate them with the dependent variables. The graph measures typically depend on the type of edges, i.e., weighted or unweighted. So do the VAR models and thus the regression results. In this paper we argue that it is beneficial to keep edge weights in this setting. To support this claim, we analyze electroencephalographic (EEG) networks from patients suffering from visual field defects. The edge weights are in the unit interval and might be thresholded. We show that dynamical network models for weighted edges lead to similar regression performances compared to those of unweighted graphs.

#### References

- Bailey, I. L. and Lovie, J. E. (1976). New design principles for visual acuity letter charts. Am J Optom Physiol Opt, 53(11):740-745.
- Bunke, H. (1997). On a relation between graph edit distance and maximum common subgraph. Pattern Recognit Lett, 18(8):689-694.
- Csárdi, G. and Nepusz, T. (2006). The igraph software package for complex network research. InterJournal Complex Systems, 1695.
- Delorme, A. and Makeig, S. (2004). EEGLAB: an open source toolbox for analysis of single-trial EEG dynamics including independent component analysis. J Neurosci Methods, 134(1):9-21.
- Edwards, E. (2007). Electrocortical Activation and Human Brain Mapping. PhD thesis, University of California, Berkeley, CA, USA.
- Faloutsos, M., Faloutsos, P., and Faloutsos, C. (1999). On power-law relationships of the internet topology. In Proceedings of the conference on Applications, technologies, architectures, and protocols for computer communication, SIGCOMM 7899, pages 251- 262, New York, NY, USA. ACM.
- Fischhoff, I. R., Sundaresan, S. R., Cordingley, J., Larkin, H. M., Sellier, M., and Rubenstein, D. I. (2007). Social relationships and reproductive state influence leadership roles in movements of plains zebra, equus burchellii. Anim Behav, 73(5):825-831.
- Held, P., Moewes, C., Braune, C., Kruse, R., and Sabel, B. A. (2012). Advanced analysis of dynamic graphs in social and neural networks. In Borgelt, C., Gil, M. Í ., Sousa, J. M. C., and Verleysen, M., editors, Towards Advanced Data Analysis by Combining Soft Computing and Statistics, pages 205-222. Springer-Verlag.
- Kasten, E., Wst, S., Behrens-Baumann, W., and Sabel, B. A. (1998). Computer-based training for the treatment of partial blindness. Nat Med, 4(9):1083-1087.
- Kleinberg, J. M., Kumar, R., Raghavan, P., Rajagopalan, S., and Tomkins, A. S. (1999). The web as a graph: measurements, models, and methods. In Proceedings of the 5th annual international conference on Computing and combinatorics, COCOON'99, pages 1-17, Berlin, Heidelberg. Springer-Verlag.
- L ütkepohl, H. (2005). New Introduction to Multiple Time Series Analysis. Econometrics / Statistics. SpringerVerlag, Berlin / Heidelberg, Germany.
- Makeig, S., Bell, A. J., Jung, T., and Sejnowski, T. J. (1996). Independent component analysis of electroencephalographic data. In Touretzky, D. S., Mozer, M. C., and Hasselmo, M. E., editors, Advances in Neural Information Processing Systems, volume 8, pages 145- 151, Cambridge, MA, USA. MIT Press.
- Moewes, C., Kruse, R., and Sabel, B. A. (2013). Analysis of dynamic brain networks using var models. In Kruse, R., Berthold, M. R., Moewes, C., Gil, M. Í ., Grzegorzewski, P., and Hryniewicz, O., editors, Synergies of Soft Computing and Statistics for Intelligent Data Analysis, pages 525-532. Proc. of the 6th Int'l Conf. on Soft Methods in Probability and Statistics (SMPS2012), 4-6 Oct. 2012, Konstanz, Germany.
- Montez, T., Linkenkaer-Hansen, K., van Dijk, B. W., and Stam, C. J. (2006). Synchronization likelihood with explicit time-frequency priors. Neuroimage, 33(4):1117-1125.
- Pearl, J. (2009). Causal inference in statistics: An overview. Stat Surv, 3:96-146.
- Pedregosa, F., Varoquaux, G., Gramfort, and et al. (2011). Scikit-learn: Machine learning in python. JMLR, 12:2825-2830.
- Pereira-Leal, J. B., Enright, A. J., and Ouzounis, C. A. (2004). Detection of functional modules from protein interaction networks. Proteins: Struct Funct Bioinf, 54(1):49-57.
- Sabel, B. A., Fedorov, A. B., Naue, N., Borrmann, A., Herrmann, C., and Gall, C. (2011). Non-invasive alternating current stimulation improves vision in optic neuropathy. Restor Neurol Neurosci, 29(6):493-505.
- Seabold, S. and Perktold, J. (2010). Statsmodels: Econometric and statistical modeling with python. In van der Walt, S. and Millman, J., editors, Proceedings of the 9th Python in Science Conference (SciPy 2010), pages 57-61.
- Sporns, O. (2010). Networks of the Brain. MIT Press, Cambridge, MA, USA.
- Stam, C., Jones, B., Nolte, G., Breakspear, M., and Scheltens, P. (2007). Small-World networks and functional connectivity in alzheimer's disease. Cerebral Cortex, 17(1):92-99.
- Stam, C. J. and van Dijk, B. W. (2002). Synchronization likelihood: an unbiased measure of generalized synchronization in multivariate data sets. J Phys D: Nonlinear Phenom, 163(3-4):236-251.
- Varela, F., Lachaux, J., Rodriguez, E., and Martinerie, J. (2001). The brainweb: phase synchronization and large-scale integration. Nat Rev Neurosci, 2(4):229- 239.
- Wassermann, S. and Faust, K. (1994). Social Network Analysis: Methods and Applications, volume 8 of Stuctural Analysis in the Social Sciences. Cambridge University Press, Cambridge, UK.
- Wendling, F., Ansari-Asl, K., Bartolomei, F., and Senhadji, L. (2009). From EEG signals to brain connectivity: A model-based evaluation of interdependence measures. J Neurosci Methods, 183(1):9-18.
- Wüst, S., Kasten, E., and Sabel, B. A. (2002). Blindsight after optic nerve injury indicates functionality of spared fibers. J Cogn Neurosci, 14(2):243-253.

#### Paper Citation

#### in Harvard Style

Moewes C. and Kruse R. (2013). **The Effects of Edge Weights on Correlating Dynamical Networks - Comparing Unweighted and Weighted Brain Graphs of nervus opticus Patients** . In *Proceedings of the 5th International Joint Conference on Computational Intelligence - Volume 1: FCTA, (IJCCI 2013)* ISBN 978-989-8565-77-8, pages 279-284. DOI: 10.5220/0004641402790284

#### in Bibtex Style

@conference{fcta13,

author={Christian Moewes and Rudolf Kruse},

title={The Effects of Edge Weights on Correlating Dynamical Networks - Comparing Unweighted and Weighted Brain Graphs of nervus opticus Patients},

booktitle={Proceedings of the 5th International Joint Conference on Computational Intelligence - Volume 1: FCTA, (IJCCI 2013)},

year={2013},

pages={279-284},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0004641402790284},

isbn={978-989-8565-77-8},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 5th International Joint Conference on Computational Intelligence - Volume 1: FCTA, (IJCCI 2013)

TI - The Effects of Edge Weights on Correlating Dynamical Networks - Comparing Unweighted and Weighted Brain Graphs of nervus opticus Patients

SN - 978-989-8565-77-8

AU - Moewes C.

AU - Kruse R.

PY - 2013

SP - 279

EP - 284

DO - 10.5220/0004641402790284