#### For citation:

Pitsik E. N., Goremyko M. V., Makarov V. V., Hramov A. E. Mathematical modelling of the network of professional interactions. *Izvestiya VUZ. Applied Nonlinear Dynamics*, 2018, vol. 26, iss. 1, pp. 21-32. DOI: 10.18500/0869-6632-2018-26-1-21-32

# Mathematical modelling of the network of professional interactions

Description of real-world systems of interacting units by the means of network model is an effective method of research both in macro- and microscale. In addition, using the simple onelayer networks with one type of connections between the nodes when describing real-world networks is inefficiently because of their complex structural and dynamical nature. Besides, presence of similar features in real networks that are fundamentally different by their nature provided a wide spread of proposed model in many fields of science for the acquisition of new fundamental knowledge about functioning of the real network structures. For this reason the object of this article is modelling of multiplex network build on the basis of real data about professional interactions in world-wide musical community. The changes in characteristics in in proposed model reflects structural and dynamical features of real network, such as scale-free connection structure and clusters formation. Results obtained for multiplex network shows that after uniting the isolated systems their topologies undergo noticeable changes. In particular, significant changes in centrality values and in cluster formation inside the network were obtained. Besides, the correlations between the characteristics and dynamics of these correlations in process of uniting the isolated systems in general network. Obtained results confirm the effectiveness of multiplex network model for studying structural and dynamical processes of many real systems.

- Sporns O., Chialvo D.R., Hilgetag C.C. Organization, development and function of complex brain networks. Trends Cogn. Sci., 2004, vol. 8, pp. 418–425.
- Maksimenko V.A., Luttjohann A., Makarov V.V., Goremyko M.V., Koronovskii A.A., Nedaivozov V., Runnova A.E., van Luijtelaar G., Hramov A.E., Boccaletti S. Macroscopic and microscopic spectral properties of brain networks during local and global synchronization. Phys. Rev. E., 2017, vol. 96, no. 1.
- Kirsanov D.V., Nedaivozov V.O., Makarov V.V., Goremyko M.V., Hramov A.E. Study of pattern formation in multilayer adaptive network of phase oscillators in application to brain dynamics analysis. Proc. SPIE, 2017, vol. 10337, pp. 103370Z1–7.
- Kai S., Gon¸calves J.P., Larminie C., Przuli N. Predicting disease associations via ˇ biological network analysis. BMC Bioinformatics, 2014, vol. 15.
- Sharan R. and Ideker T. Modeling cellular machinery through biological network comparison. Nat. Biotechnol., 2006, vol. 24, pp. 427–433.
- Ma X., Yu H., Wang Y., Wang Y. Large-Scale transportation network congestion evolution prediction using deep learning theory. PLoS ONE, 2015, vol. 10, no. 3.
- He X., Liu X.H. Modeling the day-to-day traffic evolution process after an unexpected network disruption. Transp. Res. Part B Methodol., 2012, vol. 46, no. 1, pp. 50–71.
- Wang L., Kuo G.S. Modeling for network selection in heterogeneous wireless networks. A tutorial. IEEE Commun. Surveys Tuts., 2013, vol. 15, no. 1, pp. 271–292.
- Yang S., Yang X., Zhang C., Spyrou E. Using social network theory for modeling human mobility. IEEE Netw., 2010, vol. 24, no. 5, pp. 6–13.
- Dawson S., Gasevic D., Siemens G., Joksimovic S. Current State and Future Trends: A Citation Network Analysis of the Learning Analytics Field, ACM, 2014, pp. 231– 240.
- Erdi P., Makovi K., Somogyvari Z., Strandburg K., Volf P., Zalanyi L. Prediction of emerging technologies based on analysis of the US patent citation network. Scientometrics, 2013, vol. 95, pp. 225.
- Ta-Shun C., Hsin-Yu S. Using social network theory for modeling human mobility. IEEE Netw., 2010, vol. 24, no. 5, pp. 6–13.
- Shakkottai S., Srikant R. Network optimization and control. Found. Trends. Network., 2008, vol. 2, no. 3, pp. 271–379.
- Battiston F., Nicosia V., Latora V. Structural measures for multiplex networks. Phys. Rev. E, 2014, vol. 89, no. 3.
- Makarov V.V., Koronovskii A.A., Maksimenko V.A., Hramov A.E., Moskalenko O.I., Buldu J.M., Boccaletti S. Emergence of a multilayer structure in adaptive networks of phase oscillators. Chaos, Solitons, Fractals, 2016, vol. 84, pp. 23–30.
- Turrigiano G.G., Nelson S.B. Homeostatic plasticity in the developing nervous system. Nat. Rev. Neurosci., 2004, Vol. 5, pp. 97–107.
- Arnaboldi V., Conti M., Passarella A., Pezzoni F. Ego networks in Twitter: An experimental analysis. IEEE CCW, 2013, pp. 229–234.
- Gon¸calves B., Perra N., Vespignani A. Modeling users’ activity on twitter networks: Validation of Dunbar’s number. PLoS ONE, 2011, vol. 6, no. 8, pp. 1–5.
- de Ruiter J., Weston G., Lyon S.M. Dunbar’s Number: Group Size and Brain Physiology in Humans Reexamined. Am Anthropol., 2011, vol. 113, no. 4, pp. 557–568.
- Menichetti G., Remondini D., Panzarasa P., Mondragon R.J., Bianconi G. Weighted ´ multiplex networks. PLoS ONE, 2014, vol. 9, no. 6, pp. 1–8.
- Ru-Ya T., Xue-Fu Z., Yi-Jun L. SSIC model: A multi-layer model for intervention of online rumors spreading. Physica A, 2015, vol. 427, pp. 181–191.
- Zhao L., Wang J., Chen Y., Wang Q., Cheng J., Cui H. SIHR rumor spreading model in social networks. Physica A: Statistical Menics and its Applications, 2012, vol. 391, no. 7, pp. 2444–2453.
- Hu Q., Gao Y., Ma P., Yin Y., Zhang Y., Xing C. A New Approach to Identify Influential Spreaders in Complex Networks / Web-Age Information Management: 14th International Conference. Springer Berlin Heidelberg, 2013, pp. 99–104.
- Roshani F., Naimi Y. Effects of degree-biased transmission rate and nonlinear infectivity on rumor spreading in complex social networks. Phys. Rev. E, 2012, vol. 85, no. 3.
- Buono C., Alvarez-Zuzek L.G., Macri P.A., Braunstein L.A. Epidemics in partially overlapped multiplex networks. PLoS ONE, 2014, vol. 9, no. 3, pp. 1–5.
- Allmusic.com http://www.allmusic.com/
- Callaway D.S., Newman M.E.J., Strogatz S.H., Watts D.J. Network robustness and fragility: Percolation on random graphs. Phys. Rev. Lett., 2000, vol. 85, no. 25, pp. 5468–5471.
- Brandes U. A faster algorithm for betweenness centrality. J. Math. Sociol., 2001, vol. 25, no. 3, pp. 163–177.
- Papadopoulos S., Kompatsiaris Y., Vakali A., Spyridonos P. Community detection in Social Media. Data Min Knowl Discov., 2012, vol. 24, no. 3, pp. 515–554.
- Aggarwal C.C. An Introduction to Social Network Data Analytics. Social Network Data Analytics. Springer US, 2011, pp. 1–15.
- Leskovec J., Lang Kevin J., Mahoney M. Empirical comparison of algorithms for network community detection. Proceedings of the 19th International Conference on World Wide Web. ACM, 2010, pp. 631–640.
- Newman M.E.J. Equivalence between modularity optimization and maximum likelihood methods for community detection. Phys. Rev. E, 2016, vol. 94, no. 5.
- Newman M.E.J. Mathematics of Networks. The New Palgrave Dictionary of Economics. Palgrave Macmillan, 2008.

- 1415 reads