diff --git a/CHANGELOG.md b/CHANGELOG.md
index 38dd3094..3ba8af51 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -41,7 +41,8 @@ and this project adheres to [Semantic Versioning](https://semver.org/spec/v2.0.0
 * The term `edges` in `metrics` and `adata` is now replaced with `molecules`.
 * Renaming of variables in per-stage JSON reports.
 * Changed name of TCRb to TCRVb5 antibody in human-immunology-panel file and bumped to version 0.5.0.
-* Renaming of component metrics in adata and
+* Renaming of component metrics in adata
+* Use MPX graph compatible permutation strategy when calculating Moran's I.
 
 ### Fixed
 
diff --git a/poetry.lock b/poetry.lock
index ae5985a9..515b1719 100644
--- a/poetry.lock
+++ b/poetry.lock
@@ -103,25 +103,6 @@ files = [
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 [[package]]
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@@ -418,40 +399,6 @@ files = [
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@@ -1137,24 +1013,6 @@ docs = ["POT", "Pyro4", "Pyro4 (>=4.27)", "annoy", "matplotlib", "memory-profile
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@@ -1623,34 +1481,6 @@ files = [
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-    {file = "shapely-2.0.3-cp39-cp39-win32.whl", hash = "sha256:58afbba12c42c6ed44c4270bc0e22f3dadff5656d711b0ad335c315e02d04707"},
-    {file = "shapely-2.0.3-cp39-cp39-win_amd64.whl", hash = "sha256:5026b30433a70911979d390009261b8c4021ff87c7c3cbd825e62bb2ffa181bc"},
-    {file = "shapely-2.0.3.tar.gz", hash = "sha256:4d65d0aa7910af71efa72fd6447e02a8e5dd44da81a983de9d736d6e6ccbe674"},
-]
-
-[package.dependencies]
-numpy = ">=1.14,<2"
-
-[package.extras]
-docs = ["matplotlib", "numpydoc (==1.1.*)", "sphinx", "sphinx-book-theme", "sphinx-remove-toctrees"]
-test = ["pytest", "pytest-cov"]
-
 [[package]]
 name = "six"
 version = "1.16.0"
@@ -4563,4 +4297,4 @@ files = [
 [metadata]
 lock-version = "2.0"
 python-versions = ">=3.10,<3.12"
-content-hash = "75ca943bb7fe811d8e0cff717d81c81f34189097ff7dc771562d9420235c7e74"
+content-hash = "58a04a367ecd21c757e71bc26fe247571e0314f6419877b9eb3502fe300eb8b5"
diff --git a/pyproject.toml b/pyproject.toml
index 6aa6caec..704b8115 100644
--- a/pyproject.toml
+++ b/pyproject.toml
@@ -36,8 +36,6 @@ yappi = "*"
 pandas = "^2.0.0"
 numpy = "<1.24.0"
 annoy = "<=1.17.0"
-esda = "*"
-libpysal = ">=4.9.2"
 xopen = "*"
 anndata = ">=0.8.0"
 numba = ">=0.56.4"
diff --git a/src/pixelator/analysis/__init__.py b/src/pixelator/analysis/__init__.py
index bd62f901..ad8fc9a8 100644
--- a/src/pixelator/analysis/__init__.py
+++ b/src/pixelator/analysis/__init__.py
@@ -104,7 +104,7 @@ def analyse_pixels(
             edgelist=edgelist,
             use_full_bipartite=use_full_bipartite,
             normalization=polarization_normalization,
-            permutations=polarization_n_permutations,
+            n_permutations=polarization_n_permutations,
         )
         dataset.polarization = scores
 
@@ -137,7 +137,9 @@ def analyse_pixels(
         )
     }
     # save dataset
-    dataset.save(Path(output) / f"{output_prefix}.analysis.dataset.pxl")
+    dataset.save(
+        Path(output) / f"{output_prefix}.analysis.dataset.pxl", force_overwrite=True
+    )
 
     polarization_report = PolarizationReport()
     colocalization_report = ColocalizationReport()
diff --git a/src/pixelator/analysis/colocalization/__init__.py b/src/pixelator/analysis/colocalization/__init__.py
index 893b0ec1..5c39127e 100644
--- a/src/pixelator/analysis/colocalization/__init__.py
+++ b/src/pixelator/analysis/colocalization/__init__.py
@@ -16,7 +16,6 @@
     estimate_observation_statistics,
     permutation_analysis_results,
 )
-from pixelator.analysis.colocalization.permute import permutations
 from pixelator.analysis.colocalization.prepare import (
     filter_by_region_counts,
     filter_by_unique_values,
@@ -31,6 +30,7 @@
     MarkerColocalizationResults,
     TransformationTypes,
 )
+from pixelator.analysis.permute import permutations
 from pixelator.graph.utils import Graph
 from pixelator.statistics import correct_pvalues, log1p_transformation
 
diff --git a/src/pixelator/analysis/colocalization/estimate.py b/src/pixelator/analysis/colocalization/estimate.py
index 38a62e92..3776f512 100644
--- a/src/pixelator/analysis/colocalization/estimate.py
+++ b/src/pixelator/analysis/colocalization/estimate.py
@@ -12,8 +12,8 @@
 
 from pixelator.analysis.colocalization.types import (
     CoLocalizationFunction,
-    RegionByCountsDataFrame,
 )
+from pixelator.analysis.types import RegionByCountsDataFrame
 
 logger = logging.getLogger(__name__)
 
diff --git a/src/pixelator/analysis/colocalization/prepare.py b/src/pixelator/analysis/colocalization/prepare.py
index 4c0e77a7..8936bc35 100644
--- a/src/pixelator/analysis/colocalization/prepare.py
+++ b/src/pixelator/analysis/colocalization/prepare.py
@@ -7,7 +7,7 @@
 
 import pandas as pd
 
-from pixelator.analysis.colocalization.types import RegionByCountsDataFrame
+from pixelator.analysis.types import RegionByCountsDataFrame
 from pixelator.graph.utils import Graph, create_node_markers_counts
 
 
diff --git a/src/pixelator/analysis/colocalization/statistics.py b/src/pixelator/analysis/colocalization/statistics.py
index 6f92eafa..a7d714e0 100644
--- a/src/pixelator/analysis/colocalization/statistics.py
+++ b/src/pixelator/analysis/colocalization/statistics.py
@@ -12,8 +12,8 @@
 from pixelator.analysis.colocalization.types import (
     CoLocalizationFunction,
     MarkerColocalizationResults,
-    RegionByCountsDataFrame,
 )
+from pixelator.analysis.types import RegionByCountsDataFrame
 
 
 def _wide_correlation_matrix_to_long_correlation_results(
diff --git a/src/pixelator/analysis/colocalization/types.py b/src/pixelator/analysis/colocalization/types.py
index 9bf24237..db64cf7a 100644
--- a/src/pixelator/analysis/colocalization/types.py
+++ b/src/pixelator/analysis/colocalization/types.py
@@ -8,7 +8,8 @@
 
 import pandas as pd
 
-RegionByCountsDataFrame = pd.DataFrame
+from pixelator.analysis.types import RegionByCountsDataFrame
+
 MarkerColocalizationResults = pd.DataFrame
 
 TransformationTypes = Literal["raw", "log1p"]
diff --git a/src/pixelator/analysis/colocalization/permute.py b/src/pixelator/analysis/permute.py
similarity index 98%
rename from src/pixelator/analysis/colocalization/permute.py
rename to src/pixelator/analysis/permute.py
index 770f1a38..5f7d98b0 100644
--- a/src/pixelator/analysis/colocalization/permute.py
+++ b/src/pixelator/analysis/permute.py
@@ -10,7 +10,7 @@
 from numpy.random import Generator as RandomNumberGenerator
 from numpy.random import default_rng
 
-from pixelator.analysis.colocalization.types import (
+from pixelator.analysis.types import (
     RegionByCountsDataFrame,
 )
 
diff --git a/src/pixelator/analysis/polarization/__init__.py b/src/pixelator/analysis/polarization/__init__.py
index 2c69858e..025f2b58 100644
--- a/src/pixelator/analysis/polarization/__init__.py
+++ b/src/pixelator/analysis/polarization/__init__.py
@@ -1,93 +1,97 @@
 """Copyright © 2023 Pixelgen Technologies AB."""
 
 import logging
-import warnings
-from concurrent import futures
-from typing import get_args
+import time
+from functools import partial
+from typing import Iterable, get_args
 
-import esda.moran
 import numpy as np
 import pandas as pd
+from scipy.sparse import csr_matrix
+from scipy.stats import norm
 
+from pixelator.analysis.permute import permutations
 from pixelator.analysis.polarization.types import PolarizationNormalizationTypes
-from pixelator.graph.utils import Graph, create_node_markers_counts
-from pixelator.pixeldataset import (
-    MIN_VERTICES_REQUIRED,
-)
-from pixelator.statistics import (
-    clr_transformation,
-    correct_pvalues,
-)
+from pixelator.graph.utils import Graph
+from pixelator.pixeldataset import MIN_VERTICES_REQUIRED
+from pixelator.statistics import clr_transformation, correct_pvalues
+from pixelator.utils import get_pool_executor
 
-with warnings.catch_warnings():
-    warnings.filterwarnings("ignore", module="libpysal")
-    from libpysal.weights import WSP
+logger = logging.getLogger(__name__)
 
 
-logger = logging.getLogger(__name__)
+def _compute_for_marker(
+    marker, marker_count_matrix, weight_matrix, normalization_factor
+):
+    try:
+        x_centered = np.array(
+            marker_count_matrix[marker] - marker_count_matrix[marker].mean()
+        )
+        x_centered_squared_sum = (x_centered * x_centered).sum()
+        element_wise_weighted_sum = (x_centered @ weight_matrix @ x_centered).sum()
+        r = normalization_factor * (element_wise_weighted_sum / x_centered_squared_sum)
+        return r
+    except ValueError as e:
+        logger.warning("Error computing Moran's I for marker %s: %s", marker, e)
+        return np.nan
 
 
-def morans_autocorr(
-    w: np.ndarray, y: np.ndarray, permutations: int
-) -> esda.moran.Moran:
-    """Calculate Moran's I statistics.
-
-    Computes the Moran's I autocorrelation statistics for the given spatial
-    weights (`w`) and target variable (`y`). The target variable can be the counts
-    of a specific antibody so the Moran's I value could indicate if the
-    antibody has a localized spatial pattern. The function returns an object
-    with the Moran's statistics: I, p_rand and z_rand (as well as p_sim and z_sim
-    if `permutations` > 0).
-    :param w: the weights matrix (connectivity matrix)
-    :param y: the counts vector
-    :param permutations: the number of permutations for simulated Z-score (z_sim)
-                         estimation (if permutations > 0)
-    :returns: the Moran's statistics
-    :rtype: esda.moran.Moran
-    """
-    # Default Moran's transformation is row-standardized "r".
-    # https://github.com/pysal/esda/blob/main/esda/moran.py
-    return esda.moran.Moran(y, w, permutations=permutations)
+def _compute_morans_i(
+    marker: str,
+    marker_count_matrix: pd.DataFrame,
+    marker_count_matrix_permuted: Iterable[pd.DataFrame],
+    weight_matrix: csr_matrix,
+    normalization_factor: float,
+):
+    r = _compute_for_marker(
+        marker, marker_count_matrix, weight_matrix, normalization_factor
+    )
+    perm_rs = np.fromiter(
+        (
+            _compute_for_marker(marker, perm, weight_matrix, normalization_factor)
+            for perm in marker_count_matrix_permuted
+        ),
+        dtype=np.float64,
+    )
+    perm_rs_mean = np.nanmean(perm_rs)
+    perm_rs_std_dev = np.nanstd(perm_rs)
+    perm_rs_z_score = (r - perm_rs_mean) / perm_rs_std_dev
+    p_value = norm.sf(np.abs(perm_rs_z_score))
+    return marker, r, perm_rs_z_score, perm_rs_mean, perm_rs_std_dev, p_value
 
 
-def polarization_scores_component(
+def polarization_scores_component_graph(
     graph: Graph,
     component_id: str,
     normalization: PolarizationNormalizationTypes = "clr",
-    permutations: int = 0,
+    n_permutations: int = 50,
+    random_seed: int | None = None,
 ) -> pd.DataFrame:
-    """Calculate Moran's I statistics for a component.
+    """Calculate Moran's I statistics for a component graph.
 
     Computes polarization statistics for all antibodies in the `graph` given
     as input (a single connected component). The statistics are computed using
     Moran's I autocorrelation to measure how clustered/localized the spatial
-    patterns of the antibody is in the graph. Spatial weights (`w`) are derived
-    directly from the graph. The statistics contain the I value, the p-value,
-    the adjusted p-value and the z-score under the randomization assumption.
+    patterns of the antibody is in the graph.
+
+    The statistics contain the I value, as well as a p-value, the adjusted p-value
+    and the z-score.The later are based on permuting the input graph to estimate a
+    null-hypothesis for the Moran's I statistic.
+
     The function returns a pd.DataFrame with the following columns:
-      morans_i, morans_p_value, morans_z, marker, component (morans_p_value_sim
-      and morans_z_sim if `permutations` > 0)
+      morans_i, morans_p_value, morans_z, marker, component
+
     :param graph: a graph (it must be a single connected component)
     :param component_id: the id of the component
     :param normalization: the normalization method to use (raw or clr)
-    :param permutations: the number of permutations for simulated Z-score (z_sim)
-                         estimation (if permutations>0)
+    :param n_permutations: the number of permutations to use to estimate the
+                           null-hypothesis for the Moran's I statistic
+    :param random_seed: the random seed to use to ensure that the permutations
+                        are reproducible across runs
     :returns: a pd.DataFrame with the polarization statistics for each antibody
     :rtype: pd.DataFrame
     :raises: AssertionError when the input is not valid
     """
-    if len(graph.connected_components()) > 1:
-        raise AssertionError("The graph given as input is not a single component")
-
-    if normalization not in get_args(PolarizationNormalizationTypes):
-        raise AssertionError(f"incorrect value for normalization {normalization}")
-
-    logger.debug(
-        "Computing polarization scores for component %s with %i nodes",
-        component_id,
-        graph.vcount(),
-    )
-
     if graph.vcount() < MIN_VERTICES_REQUIRED:
         logger.debug(
             (
@@ -99,60 +103,113 @@ def polarization_scores_component(
         )
         return pd.DataFrame()
 
-    # create antibody node counts
-    counts_df = create_node_markers_counts(graph=graph, k=0)
+    def _compute_morans_on_current_graph():
+        N = float(graph.vcount())
 
-    # remove markers with zero variance
-    counts_df = counts_df.loc[
-        :, (counts_df != 0).any(axis=0) & (counts_df.nunique() > 1)
-    ]
+        W = graph.get_adjacency_sparse()
+        # Normalize the weights by the degree of the node
+        W = W / W.sum(axis=0)
+        X = graph.node_marker_counts
 
-    # clr transformation
-    if normalization == "clr":
-        counts_df = clr_transformation(df=counts_df, non_negative=True, axis=0)
+        # remove markers with zero variance
+        X = X.loc[:, (X != 0).any(axis=0) & (X.nunique() > 1)]
 
-    # compute the spatial weights matrix (w) from the graph
-    w = WSP(graph.get_adjacency_sparse()).to_W(silence_warnings=True)
+        # Calculate normalization factor
+        C = N / W.sum()
 
-    # compute polarization statistics for each marker using the spatial weights (w)
-    # and the markers counts distribution (y) (Morans I autocorrelation)
-    def data():
-        for m in counts_df.columns:
-            yield morans_autocorr(w, counts_df[m], permutations)
+        X_perm = [
+            clr_transformation(x) if normalization == "clr" else x
+            for x in permutations(X, n=n_permutations, random_seed=random_seed)
+        ]
+
+        if normalization == "clr":
+            X = clr_transformation(X, non_negative=True, axis=0)
 
-    if permutations > 0:
-        df = pd.DataFrame(
-            ((m.I, m.p_rand, m.z_rand, m.p_sim, m.z_sim) for m in data()),
+        _compute_morans_i_per_marker = partial(
+            _compute_morans_i,
+            marker_count_matrix=X,
+            marker_count_matrix_permuted=X_perm,
+            weight_matrix=W,
+            normalization_factor=C,
+        )
+        markers = X.columns
+
+        results = pd.DataFrame(
+            map(_compute_morans_i_per_marker, markers),
             columns=[
+                "marker",
                 "morans_i",
-                "morans_p_value",
                 "morans_z",
-                "morans_p_value_sim",
-                "morans_z_sim",
-            ],
-        ).fillna(0)
-    else:
-        df = pd.DataFrame(
-            ((m.I, m.p_rand, m.z_rand) for m in data()),
-            columns=[
-                "morans_i",
+                "perm_mean",
+                "perm_std",
                 "morans_p_value",
-                "morans_z",
             ],
-        ).fillna(0)
+        )
+        return results
+
+    start_time = time.perf_counter()
+    logger.info("Computing Moran's I for component: %s", component_id)
 
-    df["marker"] = counts_df.columns.tolist()
-    df["component"] = component_id
+    results = _compute_morans_on_current_graph()
+    results["component"] = component_id
 
-    logger.debug("Polarization scores for components %s computed", component_id)
-    return df
+    run_time = time.perf_counter() - start_time
+    logger.info(
+        "Finished computing Moran's I for component: %s in %.2fs",
+        component_id,
+        run_time,
+    )
+    results = results.drop(columns=["perm_mean", "perm_std"])
+    return results
+
+
+def polarization_scores_component_df(
+    component_id: str,
+    component_df: pd.DataFrame,
+    use_full_bipartite: bool,
+    normalization: PolarizationNormalizationTypes = "clr",
+    n_permutations: int = 50,
+    random_seed: int | None = None,
+):
+    """Calculate Moran's I statistics for a component.
+
+    See `polarization_scores_component_graph` for details.
+
+    :param component_id: the id of the component
+    :param component_df: A data frame with an edgelist for a single connected component
+    :param use_full_bipartite: use the bipartite graph instead of the projection (UPIA)
+    :param normalization: the normalization method to use (raw or clr)
+    :param n_permutations: the number of permutations to use to estimate the
+                           null-hypothesis for the Moran's I statistic
+    :param random_seed: the random seed to use to ensure that the permutations
+                        are reproducible across runs
+    :returns: a pd.DataFrame with the polarization statistics for each antibody
+    :rtype: pd.DataFrame
+    :raises: AssertionError when the input is not valid
+    """
+    graph = Graph.from_edgelist(
+        edgelist=component_df,
+        add_marker_counts=True,
+        simplify=True,
+        use_full_bipartite=use_full_bipartite,
+    )
+
+    component_result = polarization_scores_component_graph(
+        graph=graph,
+        component_id=component_id,
+        normalization=normalization,
+        n_permutations=n_permutations,
+        random_seed=random_seed,
+    )
+    return component_result
 
 
 def polarization_scores(
     edgelist: pd.DataFrame,
     use_full_bipartite: bool = False,
     normalization: PolarizationNormalizationTypes = "clr",
-    permutations: int = 0,
+    n_permutations: int = 0,
+    random_seed: int | None = None,
 ) -> pd.DataFrame:
     """Calculate Moran's I statistics for an edgelist.
 
@@ -168,50 +225,52 @@ def polarization_scores(
     :param edgelist: an edge list (pd.DataFrame) with a component column
     :param use_full_bipartite: use the bipartite graph instead of the projection (UPIA)
     :param normalization: the normalization method to use (raw or clr)
-    :param permutations: the number of permutations for simulated Z-score (z_sim)
-                         estimation (if permutations>0)
+    :param n_permutations: the number of permutations for simulated Z-score (z_sim)
+                           estimation (if n_permutations>0)
+    :param random_seed: the random seed to use for reproducibility
     :returns: a pd.DataFrames with all the polarization scores
     :rtype: pd.DataFrame
     :raises: AssertionError when the input is not valid
     """
+    if normalization not in get_args(PolarizationNormalizationTypes):
+        raise AssertionError(f"incorrect value for normalization {normalization}")
+
     if "component" not in edgelist.columns:
         raise AssertionError("Edge list is missing the component column")
 
+    if n_permutations < 0:
+        logger.warning(
+            "Setting `n_permutations < 0` will mean no z-scores and p-values will be calculated."
+        )
+
     logger.debug(
         "Computing polarization for edge list with %i elements",
         edgelist.shape[0],
     )
 
-    # we make the computation in parallel (for each component)
-    with futures.ThreadPoolExecutor() as executor:
-        tasks = []
-        for component_id, component_df in edgelist.groupby("component", observed=True):
-            # build the graph from the component
-            graph = Graph.from_edgelist(
-                edgelist=component_df,
-                add_marker_counts=True,
-                simplify=False,
+    def data():
+        with get_pool_executor() as pool:
+            polarization_function = partial(
+                polarization_scores_component_df,
                 use_full_bipartite=use_full_bipartite,
+                normalization=normalization,
+                n_permutations=n_permutations,
+                random_seed=random_seed,
             )
-            tasks.append(
-                executor.submit(
-                    polarization_scores_component,
-                    graph=graph,
-                    component_id=component_id,
-                    normalization=normalization,
-                    permutations=permutations,
-                )
+            yield from pool.starmap(
+                polarization_function,
+                (
+                    (component_id, component_df)
+                    for component_id, component_df in edgelist.groupby(
+                        "component", observed=True
+                    )
+                ),
+                chunksize=10,
             )
-        results = futures.wait(tasks)
-        data = []
-        for result in results.done:
-            if result.exception() is not None:
-                raise result.exception()  # type: ignore
-            scores = result.result()
-            data.append(scores)
 
     # create dataframe with all the scores
-    scores = pd.concat(data, axis=0)
+    logger.debug("Concatenating the polarization dataframes")
+    scores = pd.concat(data(), axis=0)
     if scores.empty:
         logger.warning("Polarization results were empty")
         return scores
diff --git a/src/pixelator/analysis/types.py b/src/pixelator/analysis/types.py
new file mode 100644
index 00000000..8e543475
--- /dev/null
+++ b/src/pixelator/analysis/types.py
@@ -0,0 +1,8 @@
+"""Module with types related to analysis.
+
+Copyright © 2024 Pixelgen Technologies AB.
+"""
+
+import pandas as pd
+
+RegionByCountsDataFrame = pd.DataFrame
diff --git a/src/pixelator/annotate/__init__.py b/src/pixelator/annotate/__init__.py
index 0a5809ad..5612ccdd 100644
--- a/src/pixelator/annotate/__init__.py
+++ b/src/pixelator/annotate/__init__.py
@@ -237,7 +237,9 @@ def annotate_components(
     dataset = PixelDataset.from_data(
         edgelist=filtered_edgelist, adata=adata, metadata=metadata
     )
-    dataset.save(Path(output) / f"{output_prefix}.annotate.dataset.pxl")
+    dataset.save(
+        Path(output) / f"{output_prefix}.annotate.dataset.pxl", force_overwrite=True
+    )
 
     edgelist_metrics_dict = edgelist_metrics(filtered_edgelist)
     adata_metrics_dict = anndata_metrics(adata)
diff --git a/src/pixelator/cli/analysis.py b/src/pixelator/cli/analysis.py
index c4783950..f4388db1 100644
--- a/src/pixelator/cli/analysis.py
+++ b/src/pixelator/cli/analysis.py
@@ -68,14 +68,13 @@
 )
 @click.option(
     "--polarization-n-permutations",
-    default=0,
+    default=50,
     required=False,
-    type=click.IntRange(min=0),
+    type=click.IntRange(min=5),
     show_default=True,
     help=(
         "Set the number of permutations use to compute the empirical"
-        " z-score and p-value for the polarization score. If not set,"
-        " only the analytical z-score estimation will be performed"
+        " z-score and p-value for the polarization score."
     ),
 )
 @click.option(
diff --git a/src/pixelator/pixeldataset/__init__.py b/src/pixelator/pixeldataset/__init__.py
index 519ead49..56db69bf 100644
--- a/src/pixelator/pixeldataset/__init__.py
+++ b/src/pixelator/pixeldataset/__init__.py
@@ -37,10 +37,6 @@
 )
 from pixelator.types import PathType
 
-with warnings.catch_warnings():
-    warnings.filterwarnings("ignore", module="libpysal")
-
-
 logger = logging.getLogger(__name__)
 
 SIZE_DEFINITION = "molecules"
diff --git a/tests/analysis/colocalization/test_estimate.py b/tests/analysis/colocalization/test_estimate.py
index 2e547bc9..506ff609 100644
--- a/tests/analysis/colocalization/test_estimate.py
+++ b/tests/analysis/colocalization/test_estimate.py
@@ -12,12 +12,12 @@
     estimate_observation_statistics,
     permutation_analysis_results,
 )
-from pixelator.analysis.colocalization.permute import permutations
 from pixelator.analysis.colocalization.statistics import (
     Jaccard,
     Pearson,
     apply_multiple_stats,
 )
+from pixelator.analysis.permute import permutations
 
 
 def test_estimate_observation_statistics():
diff --git a/tests/analysis/colocalization/test_permute.py b/tests/analysis/colocalization/test_permute.py
index 3a3851f4..5426b1c2 100644
--- a/tests/analysis/colocalization/test_permute.py
+++ b/tests/analysis/colocalization/test_permute.py
@@ -7,7 +7,7 @@
 import pandas as pd
 from numpy.random import default_rng
 from pandas.testing import assert_series_equal
-from pixelator.analysis.colocalization.permute import (
+from pixelator.analysis.permute import (
     permutations,
     permute,
 )
diff --git a/tests/analysis/polarization/test_polarization.py b/tests/analysis/polarization/test_polarization.py
index f45a5ee3..c41ae3cd 100644
--- a/tests/analysis/polarization/test_polarization.py
+++ b/tests/analysis/polarization/test_polarization.py
@@ -12,7 +12,8 @@
 from pandas.testing import assert_frame_equal
 from pixelator.analysis.polarization import (
     polarization_scores,
-    polarization_scores_component,
+    polarization_scores_component_df,
+    polarization_scores_component_graph,
 )
 
 from tests.graph.networkx.test_tools import create_randomly_connected_bipartite_graph
@@ -23,45 +24,68 @@ def test_polarization(enable_backend, full_graph_edgelist: pd.DataFrame):
     # TODO we should test more scenarios here (sparse and clustered patterns)
 
     scores = polarization_scores(
-        edgelist=full_graph_edgelist, permutations=0, use_full_bipartite=True
+        edgelist=full_graph_edgelist,
+        n_permutations=10,
+        use_full_bipartite=True,
+        random_seed=1,
     )
 
-    expected = pd.DataFrame(
-        data={
-            "morans_i": [-0.058418204066991254, -0.05841820406699129],
-            "morans_p_value": [0.0006289931974147407, 0.0006289931974147351],
-            "morans_p_adjusted": [0.0006289931974147407, 0.0006289931974147407],
-            "morans_z": [-3.418794027777849, -3.4187940277778512],
-            "marker": ["A", "B"],
-            "component": ["PXLCMP0000000", "PXLCMP0000000"],
+    expected = pd.DataFrame.from_dict(
+        {
+            0: {
+                "marker": "A",
+                "morans_i": -0.058418204066991254,
+                "morans_z": -7.844216486178604,
+                "morans_p_value": 2.1783248596911517e-15,
+                "morans_p_adjusted": 4.356649719382303e-15,
+                "component": "PXLCMP0000000",
+            },
+            1: {
+                "marker": "B",
+                "morans_i": -0.05841820406699129,
+                "morans_z": -7.747319374160002,
+                "morans_p_value": 4.692634773999269e-15,
+                "morans_p_adjusted": 4.692634773999269e-15,
+                "component": "PXLCMP0000000",
+            },
         },
-        index=pd.Index([0, 1]),
+        orient="index",
     )
+
     # test polarization scores
-    assert_frame_equal(scores, expected)
+    assert_frame_equal(scores.sort_index(axis=1), expected.sort_index(axis=1))
 
 
 @pytest.mark.parametrize("enable_backend", ["networkx"], indirect=True)
 def test_permuted_polarization(enable_backend, full_graph_edgelist: pd.DataFrame):
     # TODO we should test more scenarios here (sparse and clustered patterns)
-    np.random.seed(1)
-
     scores = polarization_scores(
-        edgelist=full_graph_edgelist, permutations=999, use_full_bipartite=True
+        edgelist=full_graph_edgelist,
+        n_permutations=10,
+        use_full_bipartite=True,
+        random_seed=1,
     )
 
-    expected = pd.DataFrame(
-        data={
-            "morans_i": [-0.058418204066991254, -0.05841820406699129],
-            "morans_p_value": [0.0006289931974147407, 0.0006289931974147351],
-            "morans_p_adjusted": [0.0006289931974147407, 0.0006289931974147407],
-            "morans_z": [-3.418794027777849, -3.4187940277778512],
-            "morans_p_value_sim": {0: 0.02, 1: 0.01},
-            "morans_z_sim": {0: -3.2672829000275176, 1: -3.9734152894187327},
-            "marker": ["A", "B"],
-            "component": ["PXLCMP0000000", "PXLCMP0000000"],
+    expected = pd.DataFrame.from_dict(
+        {
+            0: {
+                "marker": "A",
+                "morans_i": -0.058418204066991254,
+                "morans_z": -7.844216486178604,
+                "morans_p_value": 2.1783248596911517e-15,
+                "morans_p_adjusted": 4.356649719382303e-15,
+                "component": "PXLCMP0000000",
+            },
+            1: {
+                "marker": "B",
+                "morans_i": -0.05841820406699129,
+                "morans_z": -7.747319374160002,
+                "morans_p_value": 4.692634773999269e-15,
+                "morans_p_adjusted": 4.692634773999269e-15,
+                "component": "PXLCMP0000000",
+            },
         },
-        index=pd.Index([0, 1]),
+        orient="index",
     )
     # test polarization scores
     assert_frame_equal(scores, expected)
@@ -85,18 +109,30 @@ def test_polarization_with_differentially_polarized_markers():
     random_vertex = graph.vs.get_vertex(random.randint(0, graph.vcount()))
     random_vertex["markers"]["C"] = 10
 
-    scores = polarization_scores_component(graph, component_id="PXLCMP0000000")
+    scores = polarization_scores_component_graph(
+        graph, component_id="PXLCMP0000000", n_permutations=10, random_seed=1
+    )
 
     # We don't expect to get a value for B, since it has only one value in it.
     # Hence it is filtered out.
     expected = pd.DataFrame.from_dict(
         {
-            "morans_i": {0: 0.37307837882553935, 1: -0.0037658463832960496},
-            "morans_p_value": {0: 1.317601591958555e-17, 1: 0.5910333410276116},
-            "morans_z": {0: 8.54213868598478, 1: 0.537339187874427},
-            "marker": {0: "A", 1: "C"},
-            "component": {0: "PXLCMP0000000", 1: "PXLCMP0000000"},
-        }
+            0: {
+                "marker": "A",
+                "morans_i": 0.37307837882553935,
+                "morans_z": 4.095594525493692,
+                "morans_p_value": 2.1054318495343033e-05,
+                "component": "PXLCMP0000000",
+            },
+            1: {
+                "marker": "C",
+                "morans_i": -0.0037658463832960483,
+                "morans_z": -2.1799059127756326,
+                "morans_p_value": 0.014632218279985308,
+                "component": "PXLCMP0000000",
+            },
+        },
+        orient="index",
     )
     # test polarization scores
     assert_frame_equal(scores, expected, check_exact=False, atol=1e-3)
@@ -121,22 +157,30 @@ def test_permuted_polarization_with_differentially_polarized_markers():
     random_vertex["markers"]["C"] = 10
 
     np.random.seed(1)
-    scores = polarization_scores_component(
-        graph, component_id="PXLCMP0000000", permutations=999
+    scores = polarization_scores_component_graph(
+        graph, component_id="PXLCMP0000000", n_permutations=10, random_seed=1
     )
 
     # We don't expect to get a value for B, since it has only one value in it.
     # Hence it is filtered out.
     expected = pd.DataFrame.from_dict(
         {
-            "morans_i": {0: 0.37307837882553935, 1: -0.0037658463832960496},
-            "morans_p_value": {0: 1.317601591958555e-17, 1: 0.5910333410276116},
-            "morans_z": {0: 8.54213868598478, 1: 0.537339187874427},
-            "morans_p_value_sim": {0: 0.001, 1: 0.411},
-            "morans_z_sim": {0: 8.91274668296225, 1: 0.5422508893213437},
-            "marker": {0: "A", 1: "C"},
-            "component": {0: "PXLCMP0000000", 1: "PXLCMP0000000"},
-        }
+            0: {
+                "marker": "A",
+                "morans_i": 0.37307837882553935,
+                "morans_z": 4.095594525493692,
+                "morans_p_value": 2.1054318495343033e-05,
+                "component": "PXLCMP0000000",
+            },
+            1: {
+                "marker": "C",
+                "morans_i": -0.0037658463832960483,
+                "morans_z": -2.1799059127756326,
+                "morans_p_value": 0.014632218279985308,
+                "component": "PXLCMP0000000",
+            },
+        },
+        orient="index",
     )
     # test polarization scores
     assert_frame_equal(scores, expected, check_exact=False, atol=1e-3)
diff --git a/tests/integration/test_small_D21.yaml b/tests/integration/test_small_D21.yaml
index c5456800..8940ee1c 100644
--- a/tests/integration/test_small_D21.yaml
+++ b/tests/integration/test_small_D21.yaml
@@ -27,7 +27,7 @@ small-D21:
           "--compute-colocalization",
           "--use-full-bipartite",
           "--polarization-n-permutations",
-          "0",
+          "5",
           "--colocalization-min-region-count",
           "0",
           "--colocalization-n-permutations",