András Czirok
János Török
Contact hours
2x2 hours/week
4 credits

Short Description of the Course: 
Complex systems consist of many interacting units and are characterized by nonlinearity, positive and negative feedback, and emergent cooperative phenomena. Examples are the brain, the cell, the internet, the economy etc. Recent developments in network science provide efficient tools to handle such systems. In the first part of the course we will present the basics of complex network theory, including small world and scale-free properties, important measures, spreading and breakdown phenomena, and weighted and directed graphs. In the second part examples will be analyzed from information technology, economy, sociology and biology. The course also contains a laboratory part, where network analytical skills can be acquired during hands-on interactive classes focusing on python scripting but also covering graphical network analysis tools. We also offer optional participation in network-related research projects. 

Aim of the Course: 
The aim of the course is to enable the enrolling students to acquire up-to-date knowledge of the basic principles, analytical and visualization tools of network topology and dynamics. The practical (laboratory) part of the course will enable students to construct networks from raw data-sets, to determine their properties (degree distribution, small worldness, clustering, etc.) and compare real networks to random network models, as well as utilize networks in simulations, like infection spreading and opinion dynamics. Both interactive tools and short python scripts are used to run experiments and investigate data during the lab sessions. The course will also enable students to apply their acquired network-related skills to a variety of real-world complex networks such as the Internet, social networks, networks of the economy and biological systems. 

Basic concepts of calculus and probability theory are required for this course.
Programming aptitude is required as the lab sessions involve writing python code, but knowledge of python is not, as all features that are used are covered during the labs.

Detailed Program and Class Schedule: 

  1. Introduction
    • Complex systems and complex networks
    • Major graph types, bipartite networks
    • Adjacency matrix, sparse and dense networks
  2. Basic network characteristics
    • Node degree, clustering coefficient, distance and paths
    • The small word property
    • Centralities, components
  3. Advanced network characteristics
    • Degree distribution
    • Scale-free networks and classical random graphs
    • Scale-free degree distribution and divergence
  4. Degree correlations
    • Assortative and disassortative networks
    • Full statistical description of degree correlations, Average Nearest Neighbors Degree, Pearson-correlation
  5. Network models I.
    • The Erdős-Rényi model
    • The Watts Strogatz small world network model
    • The Barabási-Albert model
  6. Network models II.
    • Variations of the Barabási-Albert model: Holme-Kim model, fitness models
    • The configuration model
    • Network randomization
  7. Robustness and spreading
    • Percolation transition in networks
    • Resilience of networks against random break down and intentional attack
    • Epidemic spreading on networks, SIS and SIR models, epidemic threshold
  8. Motifs and communities
    • Network motifs, Feed Forward Loop, motif significance profile and z-score
    • Communities, hierarchical clustering, the Girvan-Newman algorithm
    • Modularity, modularity optimization, resolution limit
    • Overlapping communities, the Clique Percolation Method
  9. Biological aspects of network analysis
    • Network hubs and modules in cellular changes, network cores, skeletons, backbones
    • Differences between changes in evolution and engineering
  10. Dynamics of biological networks
    • Signal and noise in biological networks; topological phase transitions of biological and other networks
    • Changes of modular and core/periphery network topologies in stress and in environmental changes
    • Rigid and flexible network topologies and their transitions
  11. Cellular networks
    • Protein structures: hot-spots, dynamics and evolution
    • Protein-protein interaction networks: hub and module dynamics in cellular changes
    • Signaling and gene regulatory networks
  12. Networks of neurons and ecosystems
    • Network construction via reverse engineering and from correlations
    • Changes in brain default networks during aging and disease
    • Ecological networks: keystone species and prediction of ecosystem damage
  13. Networks and drug design
    • Human disease and drug-target networks
    • Determination of drug binding sites in amino acid networks
    • Prediction of novel drug targets by network topology and dynamics


Method of instruction: 
Lectures, recitations, practical sessions, and project-based computer assignments

Smaller homework assignments are given weekly on the practical sessions.
Download the homework assignments for 2012

A 70 minutes written test is taken at midterm.
Download the test questions for 2012

Term projects:
A longer term project has to be fulfilled during the semester. The students can choose between simple projects (usually corresponding to numerical studies of one of the problems discussed in the course) and research projects (aimed at involving the student in ongoing network research). A short presentation summarizing the achieved results is expected on the last lecture of the course.
Download the project descriptions for 2012


Optional Textbooks:

A. Barrat, M. Barthélemy, and A. Vespignani: Dynamical Processes on Complex Networks,(Cambridge UP, 2008)

Boccaletti, et al.: Complex networks: structure and dynamics, Phys. Rep. 424, 175-308;

M. E. J. Newman: Networks: An Introduction (Oxford UP, 2010)

P. Csermely: Weak Links: The Universal Key to the Stability of Networks and Complex Systems, Springer (2009).

Instructors' bio:

Andras Czirok (born 1973) is an associate professor of the Department of Biological Phsyics, Eötvös University. He obtained his Ph.D. in physics in 2000 at the same institution, where he studied biophysics and collective behaviour. He was a tenured associate professor of Cell Biology and Physiology at the University of Kansas Medical Center until his retirement in 2024. He published 100+ refereed journal articles. In 2022 Dr Czirok with his colleagues founded BioPhys-Concepts LLC, a startup commercializing various in vitro cell technologies.

János Török (born 1972) is an associate professor of the Department of Theoretical Physics, Budapest University of Technology and Economics. He received his Ph.D. in physics in 2000 jointly at the same institution and Université Paris-Sud. His major research topics are granular materials, rock fracture, opinion models, and social networks. He published more than 60 papers in refereed journals with an average of 30 citations per paper. He is a regular visitor of Aalto University, Finland and was twice awarded the Teaching Excellence prize. 

Students' Review About This Course

"I thoroughly enjoyed the combination of foundational concepts in graph theory and applications of complex networks across various domains that was taught. Additionally, I conducted research with one of the professors, which allowed me to merge my prior research experience in computational neuroscience with the material taught in class. Throughout the semester, I helped build a network model to simulate global activity states of the C. elegans nervous system to search for stable attractor states and map them to distinct behavioral patterns."

Ionatan Kuperwajs

Ionatan Kuperwajs

Macalester College

"Professor Abel is extremely organized in his lectures. He has planned out every step and it is apparent. He is also readily available for as long as I need him after class and over email."

"The networks course I took at AIT was perhaps the most influential of my entire undergraduate experience. Being exposed to multiple experts in the field at such an early stage in my career as a network scientist was an enormous boon, giving me a broad introduction to a variety of topics in an important emerging field that most do not have an opportunity to learn until graduate school. The breadth of the course was an excellent introduction, covering the theory from biology to statistical mechanics, with plenty of hands-on coding “lab” experience to truly get a feel for how the science is done in practice, covering all parts of the usual project pipeline from data collection/cleaning, to analysis and simulation, to data visualization. I continue to refer back to slides and bits of code from the course even today. (St. Mary's College of Maryland)" 

Samuel Rosenblatt

Samuel Rosenblatt