Applied Maths Honours, 1st semester 2018, Lecturer: Eduardo G. Altmann
this www: http://www.maths.usyd.edu.au/u/ega/networks2018
Codes and Notebooks: https://github.sydney.edu.au/egol5583/networks2018/
Classes: Tuesdays and Thursdays 910AM, AGR room
Consultation: Wednesday 910AM, in my office (Carslaw 615)
Assessment: Exam (50%), Assignment (25%), and Project (25%). Attendance to class is essential.
Abstract: The representation of the relationship (links) between objects (nodes) in form of a network is increasingly popular in social, economical, biological, and technological sciences. Realworld examples of networks coming from these various fields show striking differences to the randomgraph models proposed by mathematicians (in the mid XX century). For instance, real networks show (i) a large number of triangles, short loops, and other small subgraphs which are absent in simple randomgraph models; and (ii) the distribution of the number of links that nodes receive deviates from a Binomial and shows large skewness (fattailed distribution). The goals of this course are to characterize these disagreements between complex networks and random graphs, to show why they matter, and to present more advanced mathematical methods to model complex networks. The course will start with a general introduction to the field of complex networks, discussing different examples of networks and how to characterize them (e.g., clustering and centrality measures). We will then introduce and apply computational methods to empiricallymeasured networks. The main part of the course will be the definition and characterization of different randomgraph ensembles. Some significant deviations between random graphs and real networks will be explored, with focus on the smallworld effect, the fattailed degree distribution, and the consequences to network robustness. In the final part of the course we will discuss how more advanced randomgraph ensembles (e.g., exponential randomgraph models) can be used to characterize realworld networks. This will be interpreted in terms of – and serve as an introduction to – more general ideas and methods, such as the Principle of Maximum Entropy and Importance Sampling Monte Carlo methods (e.g., the Metropolis Algorithm). This course involves computer simulations and data analysis. Familiarity with a programming language (e.g., Python or Matlab) and basic statistical concepts are desired.
Objectives and learning outcome: develop analytical, numerical, and modeling skills that help to connect abstract mathematical ideas to realworld systems represented as networks.
References:
Software for computation:
Network data:
Week  Date  Topic  Computation / Homeworks  
1  68/3  I  Introduction  Motivation, definitions, notation  Visualize Networks 
2  1315/3  II Network Characterization:  Local and global measures  Compute measures 
3  2022/3 
 Centrality Measures  Compute most central nodes 
4  2729/3  III  Network Models
 Poisson Random Graphs  Generate networks 
Break        Assignment due April 6 
5  1012/4 
 Small World Networks  Simulate WS model 
6  1719/4 
 Scalefree networks 

7  2426/4 
 Random graphs, Maximum Entropy Principle, ERGMs  Shuffling links 
8  13/5 
 Importance Sampling, Stochastic Block Models  Choice of Projects 
9  810/5  IV – Network Resilience  Percolation and random failures  Project 
10  1517/5 
 Targeted attacks  Project 
11  2024/5  V – Dynamics on Networks  Cascades, Epidemics, etc.  Project 
12 
2931/5 

Project Presentation 

13  57/6  Project Presentation(?) 
 
14 or 15  ?  Exam  Exam 

Week 1:
Week 2: