diff --git a/esda/moran_local_mv.py b/esda/moran_local_mv.py
index 087bae12..fda9a744 100644
--- a/esda/moran_local_mv.py
+++ b/esda/moran_local_mv.py
@@ -1,6 +1,7 @@
 import numpy as np
 import esda
 from libpysal.weights import lag_spatial
+from libpysal.graph import Graph
 
 try:
     from tqdm import tqdm
@@ -48,7 +49,7 @@ def __init__(
 
         Attributes
         ----------
-        W      : The weights matrix inputted, but row standardized
+        connectivity : The weights matrix inputted, but row standardized
         D      : The "design" matrix used in computation. If X is
                       not None, this will be [1 y X]
         R      : the "response" matrix used in computation. Will
@@ -79,9 +80,10 @@ def __init__(
         """
         self._mvquads = mvquads
         y = np.asarray(y).reshape(-1, 1)
-        if hasattr(W, "to_W"):
-            W = W.to_W()
-        W.transform = "r"
+        if isinstance(W, Graph):
+            W = W.transform("R")
+        else:
+            W.transform = "r" # TODO: as a function for graph
         y = y - y.mean()
         if unit_scale:
             y /= y.std()
@@ -97,17 +99,17 @@ def __init__(
         self.DtDi = np.linalg.inv(
             self.D.T @ self.D
         )  # this is only PxP, so not too bad...
-        self._left_component_ = (self.D @ self.DtDi) * (W.n - 1)
+        self._left_component_ = (self.D @ self.DtDi) * (self.N - 1)
         self._lmos_ = self._left_component_ * self.R
-        self.W = W
+        self.connectivity = W
         self.permutations = permutations
         if permutations is not None:  # NOQA necessary to avoid None > 0
             if permutations > 0:
                 self._crand(y, X, W)
 
 
-                self._rlmos_ *= W.n - 1
-                self._p_sim_ = np.zeros((W.n, self.P + 1))
+                self._rlmos_ *= self.N - 1
+                self._p_sim_ = np.zeros((self.N, self.P + 1))
                 # TODO: this should be changed to the general p-value framework
                 for permutation in range(self.permutations):
                     self._p_sim_ += (
@@ -167,12 +169,18 @@ def _crand(self, y, X, W):
         N = W.n
         N_permutations = self.permutations
         prange = range(N_permutations)
-        max_neighbs = W.max_neighbors + 1
+        if isinstance(W, Graph):
+            max_neighbs = W.cardinalities.max() + 1
+        else:
+            max_neighbs = W.max_neighbors + 1
         pre_permutations = np.array(
             [np.random.permutation(N - 1)[0:max_neighbs] for i in prange]
         )
         straight_ids = np.arange(N)
-        id_order = W.id_order
+        if isinstance(W, Graph): # NOQA
+            id_order = W.unique_ids
+        else:
+            id_order = W.id_order
         DtDi = self.DtDi
         ordered_weights = [W.weights[id_order[i]] for i in straight_ids]
         ordered_cardinalities = [W.cardinalities[id_order[i]] for i in straight_ids]
@@ -309,12 +317,14 @@ def __init__(
             X /= X.std(axis=0)
         self.y = y
         self.X = X
-        if hasattr(W, "to_W"):
-            W = W.to_W()
-        W.transform = "r"
-        self.W = W
         y_filtered_ = self.y_filtered_ = self._part_regress_transform(y, X)
-        Wyf = lag_spatial(self.W, y_filtered_)
+        if isinstance(W, Graph):
+            W = W.transform("R")
+            Wyf = W.lag(y_filtered_)
+        else:
+            W.transform = "r"
+            Wyf = lag_spatial(W, y_filtered_) # TODO: graph
+        self.connectivity = W
         self.partials_ = np.column_stack((y_filtered_, Wyf))
         self.permutations = permutations
         y_out = self.y_filtered_
@@ -355,18 +365,21 @@ def _crand(self):
         neighbors to i in each randomization.
         """
         _, z = self.partials_.T
-        lisas = np.zeros((self.W.n, self.permutations))
-        n_1 = self.W.n - 1
+        lisas = np.zeros((self.connectivity.n, self.permutations))
+        n_1 = self.connectivity.n - 1
         prange = list(range(self.permutations))
-        k = self.W.max_neighbors + 1
-        nn = self.W.n - 1
+        k = self.connectivity.cardinalities.max() + 1
+        nn = self.connectivity.n - 1
         rids = np.array([np.random.permutation(nn)[0:k] for i in prange])
-        ids = np.arange(self.W.n)
-        ido = self.W.id_order
-        w = [self.W.weights[ido[i]] for i in ids]
-        wc = [self.W.cardinalities[ido[i]] for i in ids]
+        ids = np.arange(self.connectivity.n)
+        if hasattr(self.connectivity, "id_order"):
+            ido = self.connectivity.id_order
+        else:
+            ido = self.connectivity.unique_ids.values
+        w = [self.connectivity.weights[ido[i]] for i in ids]
+        wc = [self.connectivity.cardinalities[ido[i]] for i in ids]
 
-        for i in tqdm(range(self.W.n), desc="Simulating by site"):
+        for i in tqdm(range(self.connectivity.n), desc="Simulating by site"):
             idsi = ids[ids != i]
             np.random.shuffle(idsi)
             tmp = z[idsi[rids[:, 0 : wc[i]]]]