Abstract
The present thesis is devoted to an application of the ideas of complex networks theory for analysing, modelling, and, finally, optimising different processes that occur in transportation networks. In particular we will be primarily concerned with the public transport networks, understanding them as the assembly of all means of public transport in a given city. Although studies of such networks have a long standing tradition and are the subject of different applied mathematical and technological disciplines, the novelty of our approach is that for the first time a complex network theory is used for such analysis. A network is a set of items which we will call nodes with connections between them, which we will call links. Network is a central notion of our time and the explosion of interest to networks is a social and cultural phenomenon which arrived at the end of last century [3, 38, 89, 39, 113, 21, 10, 114]. In mathematical literature the term ’graph’ is used and the graph theory constitutes a part of discrete mathematics [17]. A typical example of graph is given in Fig. 1.Systems, which have the form of network are numerous: these are the internet, www, neural networks, metabolic networks, transportation networks, wood webs, distribution networks (such as blood vessels or postal delivery routes), social networks of communication between people, networks of citation between papers and many, many others. Physicists started the empirical and theoretical analysis of networks only very recently, seminal papers are dated by late 1990th. From an analysis of single small graphs and properties of individual vertices or edges (see Fig. 1.1) the task of the research shifted to consideration of statistical properties
of graphs (networks). This change in the task caused also a change in the way of analysis. The breakthrough and the ”birth of network science” occurred due to new technologies, both are due to computers: www allowed for comparatively easy access to databases on different networks whereas computer power allowed
for their detailed statistical analysis (which would have been simply impossible without computers for majority of networks of interest). To denote such objects and the type of question one is interested in the term complex networks often is used.
It appeared [10, 11, 12, 13], that the most important natural and manmade networks have a special structure, which is characterized by a fattail distribution of the number of node links and drastically differs from the classical random graphs,
studied before by mathematicians [14, 15]. As a rule, these networks are not static, but they evolve and one cannot understand their structure without understanding the principles of their evolution.
The main motivation of our research was an expectation that applying concepts and ideas of complex network science to the public transport networks will result in a better understanding of their structure, their functioning, in particular their robustness to targeted attacks and random failures. In turn this will allow for
an effective modelling of this networks and their optimisations. A certain novelty of our studies is that for the first time we have analysed a public transportation system of a city as a whole (previously only certain subnetworks of public transport were considered [16, 17, 18]); another particular feature of an empirical part of our analysis is considerably large database (before much smaller cities were considered [19, 20]; last but not least one should mention the specific features of public transport networks that were for the first time analysed in our studies (in particular, we were interested in their vulnerability and resilience under attacks,
in their specific features such as generalized assortativities, centralities, harness  see below for more details and definitions). Such a comprehensive analysis allowed us to offer public transport networks models: such networks were never modeled before our study. Moreover, the majority of current models of complex
networks consider network growth in terms of nodes, the novelty of one of our models consists in its growth in terms of chains. The main results of the thesis are published in: [21, 22, 23, 24, 25, 26, 27, 28, 29]. They were reported at the following meetings: COST ACTION P10 Physics of Risk (Vilnius, Lithuania, 1316 May 2006); MECO32 Conference of the Middle European Cooperation in
Statistical Physics, (Ladek Zdroj, Poland, 1618 April 2007); ANet07 Conference on Applications of Networks (Krakow, Poland, 15 Nov 2007); MECO33 Conference of the Middle European Cooperation in Statistical Physics( Wels, Austria, 1317 Apr 2008); AGSOE, DY, DPG Meeting (Dresden, Germany, 2327 Mar 2009); Statistical Physics and Low Dimensional Systems (Nancy, France, 1315 May 2009); Statistical Physics: Modern Trends and Applications (Lviv, Ukraine,
2325 June 2009); MECO35 Conference of the Middle European Cooperation in Statistical Physics(Pont´Mousson, France, 1519 Apr 2010); AGSOE, DY, DPG Meeting (Regensburg, Germany, 21  26 Mar 2010); MECO36 Conference of the Middle European Cooperation in Statistical Physics (Lviv, Ukraine, 57 Apr 2011)
The setup of the thesis is the following. In the chapter 1 we give a sketch of the evolution of the networks science, introduce some complex network models, review some of the previous studies about network vulnerability, about public transport networks and their optimisation. Chapter 2 is devoted to an empirical analysis of public transport networks. There, we will introduce different representations for the networks and define different observables in terms of which an analysis will be performed. Different public transport network models will be introduced and analysed in Chapter 3. Some of them will allow for analytic solutions, the other will be considered by numerical simulations. We will show that such models correctly describe essential features of the public transport networks. In Chapter 4 we present results about public transport network vulnerability and resilience. In particular, this will allow to elaborate criteria to determine network stability prior to an attack as well as will allow us to find certain correlation between the theoretical predictions for idealized networks with data for the
realworld networks. Chapter 5 deals with network optimisation, conclusions and outlook are collected in the last chapter.
Date of Award  2011 

Original language  English 
Awarding Institution 

Supervisor  Bertrand Berche (Supervisor) & Christian von Ferber (Supervisor) 