The Bethe Hessian, also called the deformed Laplacian, of a finite simple graph with adjacency matrix 
and degree matrix 
is the real symmetric matrix
 |
where 
is the identity matrix and 
is a real number with 
(Saade et al. 2014).
For suitable choices of 
,
the eigenvectors belonging to the negative eigenvalues
of 
encode a vertex partition correlated
with the graph's communities. Here, communities are vertex subsets with comparatively
many internal edges and fewer edges to other subsets in the usual assortative setting;
see community detection. These eigenvectors
carry the same partition information as the informative real eigenvalues
of the nonbacktracking matrix. The Bethe
Hessian therefore provides a symmetric 
alternative to a generally nonsymmetric 
matrix. Applying cluster
analysis to these eigenvectors gives a form of spectral
graph partitioning; the number of negative eigenvalues
can also suggest the number of communities under a stochastic
block model.
Li et al. (2026) extend the construction to a nonuniform hypergraph. Let 
be its set of hyperedge sizes,
let 
be the diagonal matrix of vertex degrees contributed
by hyperedges of size 
, and let ![[A^((k))]_(ij)](/images/equations/BetheHessian/Inline13.svg)
for 
count the hyperedges of size 
containing both 
and 
, with zero diagonal. Their order-resolved
Bethe Hessian is
 |
where 
is a regularization parameter and 
. The separate matrices for each hyperedge size allow
different orders to make different contributions to the inferred set
partition.
