# 15th NETWORKS Day

**On Tuesday 29 November the 15th NETWORKS Day will take place from 9.30 - 17.30 in Domusdela in Eindhoven. **

9:30 – 10:00 | Coffee, welcome and opening |

10:00 – 11:00 | Sandjai Bhulai (VU) |

11:00 –11:30 | Coffee break |

11:30 – 12:30 | Pim van der Hoorn (TU/e) |

12:30 - 14:00 | Lunch |

14:00 - 15:00 | Ross Kang (UvA) |

15:00 - 15:30 | Coffee break |

15:30 - 16:30 | Aida Abiad Monge (TU/e) |

16:30 - 17.30 | Drinks |

## Pim van der Hoorn (TU/e)

**Estimating tails of degree distributions in complex networks, a tale of extremes. **

A common first-order property of networks that is often studied is the distribution of the degrees of its nodes. When researchers started looking at these distributions, they made an interesting discovery: For many networks, coming from a variety of different domains, the degree distribution showed strikingly similar behaviour. The tail of the distribution seemed to decrease as a power-law. This gave rise to a large body of research, studying the impact of power-law distributions on other network properties and designing models for networks that have such power-law degree distributions. It also led to many claims about the universality of power-laws in real-world networks and the underlying causes for this, fuelling generations of network scientists.

All of this came to an abrupt halt, when a paper in 2019 claimed, based on statistical analysis of a large body of networks, that power-laws are actually very rare. This then led to a heated debate among scientists about the existence of power-laws in networks.

In this talk I will address the problem of inferring the existence of power-laws in networks. I will explain that part of the problem is a lack of a proper definition of what these power-laws are. The other is that when researchers do use a definition, this is often too restrictive and hence leads to the conclusion that power-laws are rare. We will see how this problem can be addressed using Extreme-Value theory and what tools exists to properly study the existence of power-laws in networks. The conclusion is that power laws are not rare or everywhere. The true answer is somewhere in the middle.

## Aida Abiad Monge (TU/e)

**Are almost all graphs determined by their spectrum?**

We look at the spectrum (eigenvalues) of the adjacency matrix of a graph, and ask whether the eigenvalues determine the graph. This is a difficult, but important problem which plays a special role in the famous graph isomorphism problem. It has been conjectured by van Dam and Haemers that almost every graph is determined by its spectrum. The mentioned problem has been solved for several families of graphs; sometimes by proving that the spectrum determines the graph, and sometimes by constructing nonisomorphic graphs with the same spectrum. In recent years this problem has attracted much interest. In this talk we will report on several results concerning this conjecture.