You can "hear" the shape of a simple network, but as soon as you tell the data which way to flow, the shape becomes invisible.
April 3, 2026
Original Paper
One can almost never hear the shape of a digraph
arXiv · 2604.02165
The Takeaway
A long-standing theory was that a network's mathematical 'fingerprint' reveals its structure, but this study proves that's almost never true for directed networks. It reveals that symmetry is the secret ingredient that makes complex structures understandable to us.
From the abstract
Kac raised the question `Can one hear the shape of a drum?' The answer is negative. On the other hand, one can hear the shape of a drum in the generic case. Haemers conjectured that almost all graphs are determined by their spectra, which can be regarded as a discrete version of Kac's question. Vu conjectured a similar phenomenon for symmetric $\pm 1$ matrices. The first main result of this paper shows that one can almost never hear the shape of a digraph by proving that almost all digraphs are