ACM40970 Maths of Complex Networks

Academic Year 2022/2023

*** Not available in the academic year indicated above ***

1. Recap on graph theory: representation of graphs, adjacency matrix, directed and weighted graphs, trees, paths, bipartite and multilayer networks, etc. (2-3 weeks)

2. Measures and metrics: definition of centralities (degree, eigenvector, pagerank, closeness, betweenness), clustering coefficient, assortativity, etc. (2-3 weeks)

3. Models of networks: random graphs (Erdos-Renyi), small-world graphs (Watts-Newman-Strogatz), scale-free networks (Barabasi-Albert), etc. (2-3 weeks)

4. Community structure: modularity, Fiedler eigenvector, spectral and algorithmic methods for community detection, the Stochastic Block Model, etc (1-2 weeks)

5. Dynamical systems on networks: systems with one or more variables, Master Stability Function, Laplacian, random walks, synchronization, pattern formation, epidemics spreading models etc. (2-3 weeks)

Reading list
M. E. J. Newman (2010). Networks: An Introduction. Oxford: Oxford University Press.
M. A. Porter & J. P. Gleeson (2016) Dynamical systems on networks. Springer
E. Estrada (2011). The Structure of Complex Networks: Theory and Applications, Oxford: Oxford University Press.

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Curricular information is subject to change

Learning Outcomes:

1. Represent different types of networks with a corresponding appropriate mathematical (matrix) definition

2. Using different network centralities to classify and rank the nodes of a complex network

3. Identify the most suitable algorithm for community detection

4. Discuss the physical interpretation of the respective network generation model

5. Set-up the correct mathematical model for a given dynamical process on a complex network

Indicative Module Content:

This module aims to introduce the student to the central concepts related to the subject of complex networks. In particular, the student will learn about the structural and dynamical properties of complex systems and their network representation. The module starts with a short recap on the graph-theoretical concepts (adjacency/incidence matrix, bipartite networks, multilayer networks, Laplacian matrix, random walks, etc.). In the second stage, the student will learn about centrality measures (degree, closeness, betweenness, PageRank, assortativity, etc.) and algorithms associated with them. The statistical structural properties such as the degree distribution or the clustering coefficient will the topics related to the several generation algorithms that will be introduced for describing random network models such as Scale-Free or Small-World ones. Detecting communities in complex networks is another important topic that the student will face in a later stage. For the last part instead, the focus will be on the dynamical systems based on complex networks. The Master Stability Function will be the crucial tool for students to illustrate dynamical models such as synchronization, spreading, or pattern formation.

Student Effort Hours: 
Student Effort Type Hours
Lectures

36
Autonomous Student Learning

40
Total

76
Approaches to Teaching and Learning:
The teaching will be delivered in the Active Learning modality, where students will be first presented to pre-recorded video lectures and participate in classroom workshops. During the workshops, the Lecturer will discuss theoretical concepts and will illustrate them through examples and exercises.
Aiming to formative assessment, the students might choose to present the problem sheets (max. three non-consecutive) as an alternative to the Class Tests. 
Requirements, Exclusions and Recommendations

Not applicable to this module.


Module Requisites and Incompatibles
Not applicable to this module.
 
Assessment Strategy  
Description Timing Open Book Exam Component Scale Must Pass Component % of Final Grade
Continuous Assessment: The assessment process will be organized in 4 Class Tests throughout the Trimester. Each assessment will consist in a 45 mins written exam each 3 weeks and will have 25% of the final grade. Throughout the Trimester n/a Standard conversion grade scale 40% Yes

100

Carry forward of passed components
No
 
Resit In Terminal Exam
Spring Yes - 2 Hour
Please see Student Jargon Buster for more information about remediation types and timing. 
Feedback Strategy/Strategies

• Feedback individually to students, post-assessment
• Group/class feedback, post-assessment

How will my Feedback be Delivered?

Students will receive individual feedback post-assessment via Brightspace in the Assessment section.