  The airline industry was severely hit by the COVID-19 crisis with an average
demand decrease of about $64\%$ (IATA, April 2020) which triggered already
several bankruptcies of airline companies all over the world. While the
robustness of the world airline network (WAN) was mostly studied as an
homogeneous network, we introduce a new tool for analyzing the impact of a
company failure: the `airline company network' where two airlines are connected
if they share at least one route segment. Using this tool, we observe that the
failure of companies well connected with others has the largest impact on the
connectivity of the WAN. We then explore how the global demand reduction
affects airlines differently, and provide an analysis of different scenarios if
its stays low and does not come back to its pre-crisis level. Using traffic
data from the Official Aviation Guide (OAG) and simple assumptions about
customer's airline choice strategies, we find that the local effective demand
can be much lower than the average one, especially for companies that are not
monopolistic and share their segments with larger companies. Even if the
average demand comes back to $60\%$ of the total capacity, we find that between
$46\%$ and $59\%$ of the companies could experience a reduction of more than
$50\%$ of their traffic, depending on the type of competitive advantage that
drives customer's airline choice. These results highlight how the complex
competitive structure of the WAN weakens its robustness when facing such a
large crisis.
