# Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

__Mittagsseminar Talk Information__ | |

**Date and Time**: Thursday, December 21, 2023, 12:15 pm

**Duration**: 30 minutes

**Location**: CAB G51

**Speaker**: Kalina Petrova

## On Connectivity in Random Graph Models with Limited Dependencies

For any positive edge density $p$, a random graph in the Erdös-Rényi $G(n,p)$ model is connected with non-zero probability, since all edges are mutually independent. We consider random graph models in which edges that do not share endpoints are independent while incident edges may be dependent and ask: what is the minimum probability $\rho(n)$, such that for any distribution $D$ (in this model) on graphs with $n$ vertices in which each potential edge has a marginal probability of being present at least $\rho(n)$, a graph drawn from $D$ is connected with non-zero probability? As it turns out, the condition "edges that do not share endpoints are independent" needs to be clarified and the answer to the question above is sensitive to the specification. In fact, we formalize this intuitive description into a strict hierarchy of five independence conditions, which we show to have at least three different behaviors for the threshold $\rho(n)$. For each condition, we provide upper and lower bounds for $\rho(n)$. In the strongest condition, the *coloring model* (which includes, e.g., random geometric graphs), we prove that $\rho(n) → 2 - \phi ≈ 0.38$ for $n \rightarrow \infty$. This separates it from the weaker independence conditions we consider, as there we prove that $\rho(n) > 0.5 - o(n)$. In stark contrast to the coloring model, for our weakest independence condition — pairwise independence of non-adjacent edges — we show that $\rho(n)$ lies within $O(1/n^2)$ of the threshold $1-2/n$ for completely arbitrary distributions. This is joint work with Johannes Lengler, Anders Martinsson, Patrick Schnider, Raphael Steiner, Simon Weber, and Emo Welzl.

Upcoming talks | All previous talks | Talks by speaker | Upcoming talks in iCal format (beta version!)

Previous talks by year: 2024 2023 2022 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996

Information for students and suggested topics for student talks

Automatic MiSe System Software Version 1.4803M | admin login