diff --git a/src/torch_ppr/utils.py b/src/torch_ppr/utils.py
index 398d5cf..ca8cd75 100644
--- a/src/torch_ppr/utils.py
+++ b/src/torch_ppr/utils.py
@@ -95,7 +95,7 @@ def edge_index_to_sparse_matrix(
     )
 
 
-def validate_adjacency(adj: torch.Tensor, n: Optional[int] = None):
+def validate_adjacency(adj: torch.Tensor, n: Optional[int] = None, rtol: float = 1.0e-04):
     """
     Validate the page-rank adjacency matrix.
 
@@ -108,6 +108,8 @@ def validate_adjacency(adj: torch.Tensor, n: Optional[int] = None):
         the adjacency matrix
     :param n:
         the number of nodes
+    :param rtol:
+        the tolerance for checking the sum is close to 1.0
 
     :raises ValueError:
         if the adjacency matrix is invalid
@@ -143,8 +145,16 @@ def validate_adjacency(adj: torch.Tensor, n: Optional[int] = None):
     else:
         # hotfix until torch.sparse.sum is implemented
         adj_sum = adj.t() @ torch.ones(adj.shape[0])
-    if not torch.allclose(adj_sum, torch.ones_like(adj_sum), rtol=1.0e-04):
-        raise ValueError(f"Invalid column sum: {adj_sum}. expected 1.0")
+    exp_sum = torch.ones_like(adj_sum)
+    mask = adj_sum == 0
+    if mask.any():
+        logger.warning(f"Adjacency contains {mask.sum().item()} isolated nodes.")
+        exp_sum[mask] = 0.0
+    if not torch.allclose(adj_sum, exp_sum, rtol=rtol):
+        raise ValueError(
+            f"Invalid column sum: {adj_sum} (min: {adj_sum.min().item()}, max: {adj_sum.max().item()}). "
+            f"Expected 1.0 with a relative tolerance of {rtol}.",
+        )
 
 
 def prepare_page_rank_adjacency(
@@ -187,8 +197,11 @@ def prepare_page_rank_adjacency(
     adj = edge_index_to_sparse_matrix(edge_index=edge_index, num_nodes=num_nodes)
     # symmetrize
     adj = adj + adj.t()
+    # TODO: should we add an identity matrix here?
     # adjacency normalization: normalize to col-sum = 1
-    degree_inv = torch.reciprocal(torch.sparse.sum(adj, dim=0).to_dense())
+    degree_inv = torch.reciprocal(
+        torch.sparse.sum(adj, dim=0).to_dense().clamp_min(min=torch.finfo(adj.dtype).eps)
+    )
     degree_inv = torch.sparse_coo_tensor(
         indices=torch.arange(degree_inv.shape[0], device=adj.device).unsqueeze(dim=0).repeat(2, 1),
         values=degree_inv,