2nd Workshop "Complex network concept in problems of quantum physics and cosmology"

# 2nd Workshop "Complex network concept in problems of quantum physics and cosmology"

### Astronomical observatory of the Ivan Franko National University of Lviv (8 Kyrylo and Mefodij Str., Lviv), June 22, 2018. Starting at 15.00.

The task of the Workshop is to present common research as well as the principal research fields of the Laboratory for Statistical Physics of Complex Systems of the ICMP NAS of Ukraine, Astronomical Observatory and Department of Theoretical Physics of the Ivan Franko National University of Lviv. The Workshop is organized in the frames and under support of the Project DFFD №76/105-2017 "* Complex network concept in problems of quantum physics and cosmology*" (0117U003869). Within the project we plan to use the methods of complex network theory for the explanation and quantitative description of behavior of matter on different space and time scales.

## Program:

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*On Distance between graphs*

*M. Krasnytska*^{1,2}

^{1} Institute for Condensed Matter Physics, National Acad. Sci. of Ukraine, 79011 Lviv, Ukraine

^{2} L ^{4} Collaboration & Doctoral College for the Statistical Physics of Complex Systems, Leipzig- Lorraine-Lviv-Coventry, Europe

We use in our everyday life Euclidean distance as a measure of distance between two objects in d-dimensional space space. But how to measure distance between objects in the case when the very notion of dimensionality is ill-defined, as it is in the case of graphs and networks? In this talk I will review different definitions of distance which are used with a purpose of modeling and tackling practical problems of object matching and comparison [1]. Due to the ever rising importance of the network paradigm across several areas of science, comparing and classifying graphs represent essential steps in the networks analysis of complex systems [2]. Building then the HIM kernel function derived from the Hamming-Ipsen-Mikhailov (HIM) distance it is possible to move from network comparison to network classification via the Support Vector Machine (SVM) algorithm. Applications of HIM distance and HIM kernel in computational biology and social networks science demonstrate the effectiveness of the proposed functions as a general purpose solution [2].

[1] F. Memoli. Gromov-Wasserstein Distances and the Metric Approach to Object Matching. Found. Comput. Math (2011) 11, 417-487.

[2] Giuseppe Jurman, Roberto Visintainer, Michele Filosi, Samantha Riccadonna, Cesare Furlanello The HIM glocal metric and kernel for network comparison and classification 2015 IEEE International Conference on Data Science and Advanced Analytics (DSAA) arXiv:1201.2931

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*Observation of Lee-Yang zeros of an arbitrary spin bath*

*A. R. Kuzmak *

*Department for Theoretical Physics, Ivan Franko National University of Lviv, 12 Drahomanov St., Lviv, UA-79005, Ukraine*

Recently in paper [Peng et al., Phys. Rev. Lett. 114, 010601 (2015)] the experimental observation of Lee-Yang zeros of an Ising-type spin-1/2 bath, by mesuring coherece of a probe spin, was reported. We generalize this problem on the case of an arbitrary spin bath. Namely, we consider the evolution of arbitrary probe spin s ′ which interacts with bath composed by the spins of an arbitrary value s. As a result, a connection between the observed values of probe spin, such as magnetization and susceptibility, and Lee-Yang zeros is found.

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*Combined analysis of topological and spatial aspects of a real-world complex network: a case study of public transportation in UK*

*R. de Regt ^{ 1,2}, C. von Ferber^{ 1,2}, Yu. Holovatch^{ 3,2}, M. Lebovka^{4,5}*

^{1}Applied Mathematics Research Centre, Coventry University, Coventry, CV1 5FB, UK

^{2}L^{4}Collaboration & Doctoral College for the Statistical Physics of Complex Systems, Leipzig- Lorraine-Lviv-Coventry, Europe

^{3}Institute for Condensed Matter Physics, National Acad. Sci. of Ukraine, 79011 Lviv, Ukraine

^{4}F.D. Ovcharenko Institute of Biocolloidal Chemistry, National Acad. Sci. of Ukraine, 03142 Kyiv, Ukraine

^{5}Sorbonne Universités, Université de Technologie de Compiègne, EA 4297, Centre de Recherches de Royallieu, BP 20529-60205 Compiègne Cedex, FranceWe investigate topological and spatial features of public transport networks (PTN) within the UK on both a national and local scale. The networks studied include those of Greater London, Greater Manchester, West Midlands, Bristol, the national rail and coach networks of mainland UK for the period of 2011. Using methods developed in complex network theory [1,2] we analyse and compare the statistics of previous studies on other PTN [2] with the results obtained in the present study [3]. In particular our analysis allows to discriminate PTN in respect to their stability to targeted attacks and random failures. Such behaviour has a close analogy to the physical process known as percolation. Based on other analogies in behaviour of interacting many-particle systems and systems of many interacting agents of non-physical nature, we use framework of statistical physics to further quantify public transport networks. In particular, we suggest to quantify PTN shape by familiar shape characteristics of particle aggregates. In turn, analysis of their scaling properties leads to fractal measures of public transportation networks that enable one to gain useful insights into the serviceable area of stations. Moreover, we investigate universal load dynamics of these systems.

We hope that methods of combined analysis of topological and spatial aspects of network-like structures exploited in this case study may be useful for further exploitation in problems of quantum mechanics and astrophysics.

[1]. Yu. Holovatch, R. Kenna, S. Thurner. Complex systems: physics beyond physics. Eur. Journ. Phys. 38 (2017) 023002

[2]. B. Berche, C. von Ferber, T. Holovatch, Yu. Holovatch. Transportation network stability: a case study of city transit. Advances in Complex Systems, 15 (2012) 1250063

[3]. R. de Regt, C. von Ferber, Yu. Holovatch, M. Lebovka. Topological and spatial aspects of public transportation in UK viewed as a complex network. Transportmetrica (submitted) arXiv:1705.07266

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*Geometry in quantum mechanics: degree of mixing of quantum states*

*V. M. Tkachuk *

*Department for Theoretical Physics, Ivan Franko National University of Lviv, 12 Drahomanov St., Lviv, UA-79005, Ukraine*

Pure and mixed states are key concept in quantum mechanics and in quantum information theory. Therefore there is important question about the degree of mixing of a quantum state. Geometric ideas play important role in quantum mechanics and in quantum information theory. We use Hilbert Schmidt distance in order to measure the degree of mixing of quantum state. Namely, we define the measure of mixing of quantum state as minimal Hilbert Schmidt distance to the set of pure states.

###
*Evolution of Cosmic web as complex network*

*M. Tsizh*

*Astronomical Observatory of Ivan Franko National University of Lviv, Kyryla and Methodij str. 8, Lviv 79005, Ukraine*

In this work I use catalog based on MultiDark cosmological simulation to track how the complex network, build on halos as vertexes, changes with large scale evolution of simulated Universe. Starting with snapshots at z~5 and up to z=0 I compute different averaged metrics (degree, clustering coefficient, different kind of centralities, assortativity and others) for each snapshot. Also few types of distance between graphs for network at different z (graph edit distance) are computed to test which of them in the best way displays the evolution of Cosmic web. The 100x100x100 h^{-1} Mpc cube of total 1 Gb^{3} simulated volume of Multidark simulation were used for computation, several values of linking length for network were tested in range from 1.2 to 2 Mpc.

**Organizers:**

Institute for Condensed Matter Physics NAS of Ukraine

Department of Theoretical Physics Ivan Franko National University of Lviv

Astronomical Observatory of Ivan Franko National University of Lviv