Last Friday we had the third, and unfortunately last, online lecture of the masterclass. This masterclass on the mathematics of networks was held in English and hence suitable for international schools as well.

The masterclass this year was held online via Zoom in the form of three independent online classes of two hours each. We enjoyed it a lot and we would like to thank all students and teachers who participated in the masterclass! We are looking forward to seeing you next year again the next masterclass "NETWORKS goes to school". Although the masterclass is normally organised on location, the advantages of online lectures were evident. It was much easier for students and teachers from different regions to participate, we will take these advantages under consideration for the masterclass of next year.

Below you can find an overview of the lectures with the material we discussed.

**Youri Raaijmakers – Queueing theory – ****Preventing queues growing large**

In this talk you will see how a mathematical model of a queue can be constructed and analysed. Some typical everyday examples of queues can be found in supermarkets, industrial production systems and hospitals. In a supermarket customers arrive to the counters, they may have to wait in the queue until their turn comes, they are served and then leave the supermarket. But an important question is, can we change the policy serving the customers to serve them faster?

**Material used**: Section 1.2 (up to Section 1.2.6) and Section 2.1 from this booklet. In Sections 3.1 and 3.2 you can find exercises, in the end of the booklet we have added the solutions.

**Rens Kamphuis – Graph theory – Finding the optimal route in a road traffic network**

In this talk you will see how graphs can be used to represent networks and how algorithms can be constructed to solve questions about networks. For instance, what is the shortest route from one location in a network to another? This is a question that we ask our favourite route planner on a daily basis.

**Material used**: Sections 1.2 (mainly 1.2.6), 1.3 and Section 2.2 from this booklet. In Section 3.3 you can find exercises, in the end of the booklet we have added the solutions.

**Janusz Meylahn – Complex networks – Learning how synchronisation works**

In this talk you will have a glimpse in how synchronisation works. Synchronisation is the coordination of events to operate a system in unison. Simply put, multiple parts work together to produce one whole. Think of phenomena like fireflies flashing in synchrony, neurons ringing in the brain, the gravitational synchronisation of meteors or an audience clapping after a concert.

**Material used**: Section 1.3 and Section 3.4 from this booklet. In Section 3.4 you can also find exercises, in the end of the booklet we have added the solutions.

You can read more about the previous editions of the masterclass and also find all the material presented in the previous editions here. In case you have any questions regarding the material or you are interested to use the material feel free to contact us via editor@networkpages.nl.

The organisers,

Jan-Pieter Dorsman and Nicos Starreveld.