Magnitude function determines generic finite metric spaces

Magnitude function determines generic finite metric spaces, Discrete Analysis 2025:13, 19 pp.

The magnitude function of a metric space is a concept introduced by Tom Leinster in 2010 that associates with a metric space a real valued magnitude function that in some sense describes the size of the space at different distance scales.

For finite metric spaces, the formal definition is as follows. If \(X\) has elements \(x_1,\dots,x_n\) and \(t\) is a positive real number, let \(Z_X(t)\) be the matrix with \(ij\)th entry equal to \(e^{-td(x_i,x_j)}\). The magnitude of \(X\) at scale \(t\) is then defined to be the sum of the entries of \(Z_X(t)^{-1}\) (assuming that \(Z_X(t)\) is invertible, but it is also possible to formulate a sensible definition when it is not invertible). Note that if \(t\) is very large, then the off-diagonal entries of \(Z_X(t)\) are very close to zero, so \(Z_X(t)\), and therefore \(Z_X(t)^{-1}\) as well, is approximately the identity. Thus, the magnitude at scale \(t\) tends to \(|X|\) as \(t\to\infty\).

Now consider an example where \(X\) consists of \(m\) clusters of points \(X_1,\dots,X_m\), with the diameter of each cluster being at most some very small positive \(\epsilon\) and the distances between the clusters being at least 1. If we choose \(t\) in such a way that \(t\) is large but \(\epsilon t\) is small, then \(Z_X(t)_{ij}\) is close to 1 when \(x_i\) and \(x_j\) belong to the same cluster and close to 0 otherwise. If the \(i\)th cluster \(X_i\) contains \(m_i\) points, then let \(f\) be the vector such that \(f(x_j)=1/m_i\) when \(x_j\in X_i\). We then have for \(x_r\in X_i\) that
\[Z_X(t)f(x_r)=\sum_sZ_X(t)_{rs}f(x_s)\approx m_i^{-1}m_i=1.\]
If \(Z_X(t)\) is invertible, this shows that the magnitude of \(X\) at scale \(t\) is approximately equal to the sum of the values of \(f\), which is equal to \(m\), the number of clusters. Thus, the magnitude function is expressing the fact that at distance scale \(t^{-1}\) the metric space looks like a collection of \(m\) points. If we then increase \(t\) so that \(t^{-1}\) is much smaller than all the distances between distinct points of \(X\), then it is as though we are looking through a sufficiently powerful microscope for the internal structure of the clusters to become detectable.

Magnitude can also be defined for compact metric spaces, either by replacing sums by integrals or by taking limits of discretizations. There turn out to be many parallels, drawn by Leinster and others, between magnitude and the more classical invariants of integral geometry; and authors including Barcelo, Carbery, Gimperlein, Goffeng and Meckes have demonstrated that the large-scale asymptotics of the magnitude function carry a remarkable amount of information about the intrinsic volumes and other size-related features of a compact metric space.

This paper augments that literature, focusing on finite metric spaces. The author proves that almost every finite metric space can be reconstructed completely from the asymptotics of its magnitude function. In particular, this is the case if the distances are rationally independent. This is an interesting and unexpected result: it is not so surprising that the set of distances can be recovered, but much less obvious that one can determine the entire space. The techniques of the proof are also novel in that they make use of the small-scale asymptotics of the magnitude function (and its derivatives) as well as the large-scale asymptotics, which may well stimulate further work in this direction.

The paper also establishes the same conclusion under different hypotheses for small spaces: if \(|X|=3\) then no hypotheses are needed at all, and if \(|X|=4\) then it is sufficient if the largest distance between two points is less than twice the smallest distance between two (distinct) points. Finally, for all sizes of metric space, this last condition on distances is sufficient if in addition no two different sums of at most five non-zero distances are equal.