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LON Module

LON dataclass

Local Optima Network (LON) representation.

A LON is a directed graph where nodes represent local optima and edges represent transitions between them discovered during basin-hopping search.

Attributes:

Name Type Description
graph Graph

The underlying igraph Graph object.
best_fitness float | None

The best (minimum) fitness value found.
Source code in src/lonpy/lon.py 
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@dataclass
class LON:
    """
    Local Optima Network (LON) representation.

    A LON is a directed graph where nodes represent local optima and edges
    represent transitions between them discovered during basin-hopping search.

    Attributes:
        graph: The underlying igraph Graph object.
        best_fitness: The best (minimum) fitness value found.
    """

    graph: ig.Graph = field(default_factory=lambda: ig.Graph(directed=True))
    best_fitness: float | None = None

    @classmethod
    def from_trace_data(
        cls,
        trace: pd.DataFrame,
    ) -> "LON":
        """
        Create a LON from trace data.

        Args:
            trace: DataFrame with columns [run, fit1, node1, fit2, node2] where:
                - run: integer run number
                - fit1: integer fitness of source node (scaled)
                - node1: string hash of source node
                - fit2: integer fitness of target node (scaled)
                - node2: string hash of target node

        Returns:
            LON instance with constructed graph.
        """
        trace = trace.copy()
        trace.columns = pd.Index(["run", "fit1", "node1", "fit2", "node2"])

        # Combine nodes from both columns
        lnodes1 = trace[["node1", "fit1"]].rename(columns={"node1": "Node", "fit1": "Fitness"})
        lnodes2 = trace[["node2", "fit2"]].rename(columns={"node2": "Node", "fit2": "Fitness"})
        lnodes = pd.concat([lnodes1, lnodes2], ignore_index=True)

        ledges = trace[["node1", "node2"]].copy()

        # Count node and edge occurrences
        nodes = lnodes.groupby(["Node", "Fitness"], as_index=False).size()
        nodes.columns = pd.Index(["Node", "Fitness", "Count"])

        edges = ledges.groupby(["node1", "node2"], as_index=False).size()
        edges.columns = pd.Index(["Start", "End", "Count"])

        graph = ig.Graph(directed=True)

        for _, row in nodes.iterrows():
            graph.add_vertex(name=str(row["Node"]), Fitness=row["Fitness"], Count=row["Count"])

        for _, row in edges.iterrows():
            with contextlib.suppress(ValueError):
                graph.add_edge(str(row["Start"]), str(row["End"]), Count=row["Count"])

        # Remove self-loops
        graph = graph.simplify(multiple=False, loops=True)

        best = nodes["Fitness"].min()

        return cls(graph=graph, best_fitness=best)

    @property
    def n_vertices(self) -> int:
        """Number of vertices (local optima) in the LON."""
        return int(self.graph.vcount())

    @property
    def n_edges(self) -> int:
        """Number of edges in the LON."""
        return int(self.graph.ecount())

    @property
    def vertex_names(self) -> list[str]:
        """List of vertex names (node hashes)."""
        return list(self.graph.vs["name"])

    @property
    def vertex_fitness(self) -> list[float]:
        """List of vertex fitness values."""
        return list(self.graph.vs["Fitness"])

    @property
    def vertex_count(self) -> list[int]:
        """List of vertex counts (times visited)."""
        return list(self.graph.vs["Count"])

    def get_sinks(self) -> list[int]:
        """Get indices of sink nodes (nodes with no outgoing edges)."""
        out_degrees = self.graph.degree(mode="out")
        return [i for i, d in enumerate(out_degrees) if d == 0]

    def get_global_optima_indices(self) -> list[int]:
        """Get indices of global optima nodes (nodes at best fitness)."""
        return [i for i, f in enumerate(self.vertex_fitness) if f == self.best_fitness]

    def compute_metrics(self, known_best: float | None = None) -> dict[str, Any]:
        """
        Compute LON metrics.

        Args:
            known_best: Known global optimum value. If None, uses the best
                fitness found in the network.

        Returns:
            Dictionary containing:
                - n_optima: Number of local optima (vertices)
                - n_funnels: Number of funnels (sinks)
                - n_global_funnels: Number of funnels at global optimum
                - neutral: Proportion of nodes with equal-fitness connections
                - strength: Proportion of incoming strength to global optima
        """
        best = known_best if known_best is not None else self.best_fitness

        n_optima = self.n_vertices

        sinks_id = self.get_sinks()
        n_funnels = len(sinks_id)

        sinks_fit = [self.vertex_fitness[i] for i in sinks_id]
        n_global_funnels = sum(1 for f in sinks_fit if f == best)

        # Neutral: proportion of nodes with equal-fitness connections
        el = self.graph.get_edgelist()
        fits = self.vertex_fitness
        neutral_edge_indices = []
        for i, (src, tgt) in enumerate(el):
            if fits[src] == fits[tgt]:
                neutral_edge_indices.append(i)

        if neutral_edge_indices:
            gnn = self.graph.subgraph_edges(neutral_edge_indices, delete_vertices=True)
            gnn = gnn.simplify(multiple=False, loops=True)
            neutral = round(gnn.vcount() / n_optima, 4)
        else:
            neutral = 0.0

        # Strength: incoming strength to global optima
        igs = self.get_global_optima_indices()
        if self.n_edges > 0 and igs:
            edge_weights = self.graph.es["Count"]
            stren_igs = sum(self.graph.strength(igs, mode="in", loops=False, weights=edge_weights))
            stren_all = sum(self.graph.strength(mode="in", loops=False, weights=edge_weights))
            strength = round(stren_igs / stren_all, 4) if stren_all > 0 else 0.0
        else:
            strength = 0.0

        return {
            "n_optima": n_optima,
            "n_funnels": n_funnels,
            "n_global_funnels": n_global_funnels,
            "neutral": neutral,
            "strength": strength,
        }

    def to_cmlon(self) -> "CMLON":
        """
        Convert LON to Compressed Monotonic LON (CMLON).

        Returns:
            CMLON instance with contracted neutral nodes.
        """
        return CMLON.from_lon(self)

n_vertices: int property

Number of vertices (local optima) in the LON.

n_edges: int property

Number of edges in the LON.

vertex_names: list[str] property

List of vertex names (node hashes).

vertex_fitness: list[float] property

List of vertex fitness values.

vertex_count: list[int] property

List of vertex counts (times visited).

from_trace_data(trace: pd.DataFrame) -> LON classmethod

Create a LON from trace data.

Parameters:

Name Type Description Default
trace DataFrame

DataFrame with columns [run, fit1, node1, fit2, node2] where: - run: integer run number - fit1: integer fitness of source node (scaled) - node1: string hash of source node - fit2: integer fitness of target node (scaled) - node2: string hash of target node

 required

Returns:

Type Description
LON

LON instance with constructed graph.
Source code in src/lonpy/lon.py 
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@classmethod
def from_trace_data(
    cls,
    trace: pd.DataFrame,
) -> "LON":
    """
    Create a LON from trace data.

    Args:
        trace: DataFrame with columns [run, fit1, node1, fit2, node2] where:
            - run: integer run number
            - fit1: integer fitness of source node (scaled)
            - node1: string hash of source node
            - fit2: integer fitness of target node (scaled)
            - node2: string hash of target node

    Returns:
        LON instance with constructed graph.
    """
    trace = trace.copy()
    trace.columns = pd.Index(["run", "fit1", "node1", "fit2", "node2"])

    # Combine nodes from both columns
    lnodes1 = trace[["node1", "fit1"]].rename(columns={"node1": "Node", "fit1": "Fitness"})
    lnodes2 = trace[["node2", "fit2"]].rename(columns={"node2": "Node", "fit2": "Fitness"})
    lnodes = pd.concat([lnodes1, lnodes2], ignore_index=True)

    ledges = trace[["node1", "node2"]].copy()

    # Count node and edge occurrences
    nodes = lnodes.groupby(["Node", "Fitness"], as_index=False).size()
    nodes.columns = pd.Index(["Node", "Fitness", "Count"])

    edges = ledges.groupby(["node1", "node2"], as_index=False).size()
    edges.columns = pd.Index(["Start", "End", "Count"])

    graph = ig.Graph(directed=True)

    for _, row in nodes.iterrows():
        graph.add_vertex(name=str(row["Node"]), Fitness=row["Fitness"], Count=row["Count"])

    for _, row in edges.iterrows():
        with contextlib.suppress(ValueError):
            graph.add_edge(str(row["Start"]), str(row["End"]), Count=row["Count"])

    # Remove self-loops
    graph = graph.simplify(multiple=False, loops=True)

    best = nodes["Fitness"].min()

    return cls(graph=graph, best_fitness=best)

get_sinks() -> list[int]

Get indices of sink nodes (nodes with no outgoing edges).

Source code in src/lonpy/lon.py 
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def get_sinks(self) -> list[int]:
    """Get indices of sink nodes (nodes with no outgoing edges)."""
    out_degrees = self.graph.degree(mode="out")
    return [i for i, d in enumerate(out_degrees) if d == 0]

get_global_optima_indices() -> list[int]

Get indices of global optima nodes (nodes at best fitness).

Source code in src/lonpy/lon.py 
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def get_global_optima_indices(self) -> list[int]:
    """Get indices of global optima nodes (nodes at best fitness)."""
    return [i for i, f in enumerate(self.vertex_fitness) if f == self.best_fitness]

compute_metrics(known_best: float | None = None) -> dict[str, Any]

Compute LON metrics.

Parameters:

Name Type Description Default
known_best float | None

Known global optimum value. If None, uses the best fitness found in the network.

 None

Returns:

Type Description
dict[str, Any]

Dictionary containing: - n_optima: Number of local optima (vertices) - n_funnels: Number of funnels (sinks) - n_global_funnels: Number of funnels at global optimum - neutral: Proportion of nodes with equal-fitness connections - strength: Proportion of incoming strength to global optima
Source code in src/lonpy/lon.py 
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def compute_metrics(self, known_best: float | None = None) -> dict[str, Any]:
    """
    Compute LON metrics.

    Args:
        known_best: Known global optimum value. If None, uses the best
            fitness found in the network.

    Returns:
        Dictionary containing:
            - n_optima: Number of local optima (vertices)
            - n_funnels: Number of funnels (sinks)
            - n_global_funnels: Number of funnels at global optimum
            - neutral: Proportion of nodes with equal-fitness connections
            - strength: Proportion of incoming strength to global optima
    """
    best = known_best if known_best is not None else self.best_fitness

    n_optima = self.n_vertices

    sinks_id = self.get_sinks()
    n_funnels = len(sinks_id)

    sinks_fit = [self.vertex_fitness[i] for i in sinks_id]
    n_global_funnels = sum(1 for f in sinks_fit if f == best)

    # Neutral: proportion of nodes with equal-fitness connections
    el = self.graph.get_edgelist()
    fits = self.vertex_fitness
    neutral_edge_indices = []
    for i, (src, tgt) in enumerate(el):
        if fits[src] == fits[tgt]:
            neutral_edge_indices.append(i)

    if neutral_edge_indices:
        gnn = self.graph.subgraph_edges(neutral_edge_indices, delete_vertices=True)
        gnn = gnn.simplify(multiple=False, loops=True)
        neutral = round(gnn.vcount() / n_optima, 4)
    else:
        neutral = 0.0

    # Strength: incoming strength to global optima
    igs = self.get_global_optima_indices()
    if self.n_edges > 0 and igs:
        edge_weights = self.graph.es["Count"]
        stren_igs = sum(self.graph.strength(igs, mode="in", loops=False, weights=edge_weights))
        stren_all = sum(self.graph.strength(mode="in", loops=False, weights=edge_weights))
        strength = round(stren_igs / stren_all, 4) if stren_all > 0 else 0.0
    else:
        strength = 0.0

    return {
        "n_optima": n_optima,
        "n_funnels": n_funnels,
        "n_global_funnels": n_global_funnels,
        "neutral": neutral,
        "strength": strength,
    }

to_cmlon() -> CMLON

Convert LON to Compressed Monotonic LON (CMLON).

Returns:

Type Description
CMLON

CMLON instance with contracted neutral nodes.
Source code in src/lonpy/lon.py 
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def to_cmlon(self) -> "CMLON":
    """
    Convert LON to Compressed Monotonic LON (CMLON).

    Returns:
        CMLON instance with contracted neutral nodes.
    """
    return CMLON.from_lon(self)

CMLON dataclass

Compressed Monotonic Local Optima Network (CMLON).

CMLON contracts nodes with equal fitness that are connected, creating a compressed representation of the fitness landscape.

Attributes:

Name Type Description
graph Graph

The underlying igraph Graph object.
best_fitness float | None

The best (minimum) fitness value.
source_lon LON | None

Reference to the original LON (optional).
Source code in src/lonpy/lon.py 
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@dataclass
class CMLON:
    """
    Compressed Monotonic Local Optima Network (CMLON).

    CMLON contracts nodes with equal fitness that are connected,
    creating a compressed representation of the fitness landscape.

    Attributes:
        graph: The underlying igraph Graph object.
        best_fitness: The best (minimum) fitness value.
        source_lon: Reference to the original LON (optional).
    """

    graph: ig.Graph = field(default_factory=lambda: ig.Graph(directed=True))
    best_fitness: float | None = None
    source_lon: LON | None = None

    @classmethod
    def from_lon(cls, lon: LON) -> "CMLON":
        """
        Create CMLON from LON by contracting neutral nodes.

        The compression process:
        1. Mark edges as "improving" (f2 < f1) or "equal" (f2 == f1)
        2. Create subgraph of equal-fitness edges
        3. Find weakly connected components
        4. Contract vertices using component membership
        5. Combine parallel edge weights

        Args:
            lon: Source LON instance.

        Returns:
            CMLON with contracted neutral components.
        """
        if lon.n_edges == 0:
            cmlon_graph = lon.graph.copy()
            return cls(graph=cmlon_graph, best_fitness=lon.best_fitness, source_lon=lon)

        # Create a working copy
        mlon = lon.graph.copy()
        mlon.vs["Count"] = [1] * mlon.vcount()

        el = mlon.get_edgelist()
        fits = mlon.vs["Fitness"]

        f1 = [fits[src] for src, _ in el]
        f2 = [fits[tgt] for _, tgt in el]

        # Mark edge types and find equal-fitness edges
        edge_types = []
        equal_edge_indices = []
        for i, (fit1, fit2) in enumerate(zip(f1, f2)):
            if fit2 < fit1:
                edge_types.append("improving")
            elif fit2 == fit1:
                edge_types.append("equal")
                equal_edge_indices.append(i)
            else:
                edge_types.append("worsening")
        mlon.es["type"] = edge_types

        # Create subgraph of equal-fitness edges (keep all vertices)
        if equal_edge_indices:
            gnn = mlon.subgraph_edges(equal_edge_indices, delete_vertices=False)
        else:
            gnn = ig.Graph(n=mlon.vcount(), directed=True)

        # Find weakly connected components
        nn_memb = gnn.components(mode="weak").membership

        # Contract vertices using component membership
        cmlon_graph = _contract_vertices(
            mlon,
            nn_memb,
            vertex_attr_comb={"Fitness": "first", "Count": "sum", "name": "first"},
        )

        # Combine parallel edges by summing Count
        cmlon_graph = _simplify_with_edge_sum(cmlon_graph)

        return cls(graph=cmlon_graph, best_fitness=lon.best_fitness, source_lon=lon)

    @property
    def n_vertices(self) -> int:
        """Number of vertices in CMLON."""
        return int(self.graph.vcount())

    @property
    def n_edges(self) -> int:
        """Number of edges in CMLON."""
        return int(self.graph.ecount())

    @property
    def vertex_fitness(self) -> list[float]:
        """List of vertex fitness values."""
        return list(self.graph.vs["Fitness"])

    @property
    def vertex_count(self) -> list[int]:
        """List of vertex counts (contracted nodes)."""
        return list(self.graph.vs["Count"])

    def get_sinks(self) -> list[int]:
        """Get indices of sink nodes (nodes with no outgoing edges)."""
        out_degrees = self.graph.degree(mode="out")
        return [i for i, d in enumerate(out_degrees) if d == 0]

    def get_global_sinks(self) -> list[int]:
        """Get indices of global sinks (sinks at best fitness)."""
        sinks = self.get_sinks()
        fits = self.vertex_fitness
        return [s for s in sinks if fits[s] == self.best_fitness]

    def get_local_sinks(self) -> list[int]:
        """Get indices of local sinks (sinks not at best fitness)."""
        sinks = self.get_sinks()
        fits = self.vertex_fitness
        if self.best_fitness is None:
            return []
        return [s for s in sinks if fits[s] > self.best_fitness]

    def compute_metrics(self, known_best: float | None = None) -> dict[str, Any]:
        """
        Compute CMLON metrics.

        Args:
            known_best: Known global optimum value. If None, uses the best
                fitness found in the network.

        Returns:
            Dictionary containing:
                - n_optima: Number of optima in CMLON
                - n_funnels: Number of funnels (sinks)
                - n_global_funnels: Number of funnels at global optimum
                - neutral: Proportion of contracted nodes
                - strength: Ratio of incoming strength to global vs local sinks
                - global_funnel_proportion: Proportion of nodes that can reach
                  a global optimum
        """
        best = known_best if known_best is not None else self.best_fitness

        n_optima = self.n_vertices

        sinks_id = self.get_sinks()
        n_funnels = len(sinks_id)

        sinks_fit = [self.vertex_fitness[i] for i in sinks_id]
        n_global_funnels = sum(1 for f in sinks_fit if f == best)

        # Neutral: proportion of contracted nodes
        if self.source_lon is not None:
            neutral = round(1.0 - self.n_vertices / self.source_lon.n_vertices, 4)
        else:
            neutral = 0.0

        # Strength: ratio of incoming strength to global vs local sinks
        igs = [s for s, f in zip(sinks_id, sinks_fit) if f == best]
        ils = [s for s, f in zip(sinks_id, sinks_fit) if best is not None and f > best]

        if self.n_edges > 0:
            edge_weights = self.graph.es["Count"] if "Count" in self.graph.es.attributes() else None
            sing = (
                sum(self.graph.strength(igs, mode="in", loops=False, weights=edge_weights))
                if igs
                else 0
            )
            sinl = (
                sum(self.graph.strength(ils, mode="in", loops=False, weights=edge_weights))
                if ils
                else 0
            )
            total = sing + sinl
            strength = round(sing / total, 4) if total > 0 else 0.0
        else:
            strength = 0.0

        gfunnel = self._compute_global_funnel_proportion()

        return {
            "n_optima": n_optima,
            "n_funnels": n_funnels,
            "n_global_funnels": n_global_funnels,
            "neutral": neutral,
            "strength": strength,
            "global_funnel_proportion": gfunnel,
        }

    def _compute_global_funnel_proportion(self) -> float:
        """Compute proportion of nodes that can reach a global optimum."""
        igs = self.get_global_sinks()
        if not igs:
            return 0.0

        # Get all nodes that can reach any global sink
        reachable = set()
        for sink in igs:
            component = self.graph.subcomponent(sink, mode="in")
            reachable.update(component)

        return len(reachable) / self.n_vertices if self.n_vertices > 0 else 0.0

n_vertices: int property

Number of vertices in CMLON.

n_edges: int property

Number of edges in CMLON.

vertex_fitness: list[float] property

List of vertex fitness values.

vertex_count: list[int] property

List of vertex counts (contracted nodes).

from_lon(lon: LON) -> CMLON classmethod

Create CMLON from LON by contracting neutral nodes.

The compression process: 1. Mark edges as "improving" (f2 < f1) or "equal" (f2 == f1) 2. Create subgraph of equal-fitness edges 3. Find weakly connected components 4. Contract vertices using component membership 5. Combine parallel edge weights

Parameters:

Name Type Description Default
lon LON

Source LON instance.

 required

Returns:

Type Description
CMLON

CMLON with contracted neutral components.
Source code in src/lonpy/lon.py 
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@classmethod
def from_lon(cls, lon: LON) -> "CMLON":
    """
    Create CMLON from LON by contracting neutral nodes.

    The compression process:
    1. Mark edges as "improving" (f2 < f1) or "equal" (f2 == f1)
    2. Create subgraph of equal-fitness edges
    3. Find weakly connected components
    4. Contract vertices using component membership
    5. Combine parallel edge weights

    Args:
        lon: Source LON instance.

    Returns:
        CMLON with contracted neutral components.
    """
    if lon.n_edges == 0:
        cmlon_graph = lon.graph.copy()
        return cls(graph=cmlon_graph, best_fitness=lon.best_fitness, source_lon=lon)

    # Create a working copy
    mlon = lon.graph.copy()
    mlon.vs["Count"] = [1] * mlon.vcount()

    el = mlon.get_edgelist()
    fits = mlon.vs["Fitness"]

    f1 = [fits[src] for src, _ in el]
    f2 = [fits[tgt] for _, tgt in el]

    # Mark edge types and find equal-fitness edges
    edge_types = []
    equal_edge_indices = []
    for i, (fit1, fit2) in enumerate(zip(f1, f2)):
        if fit2 < fit1:
            edge_types.append("improving")
        elif fit2 == fit1:
            edge_types.append("equal")
            equal_edge_indices.append(i)
        else:
            edge_types.append("worsening")
    mlon.es["type"] = edge_types

    # Create subgraph of equal-fitness edges (keep all vertices)
    if equal_edge_indices:
        gnn = mlon.subgraph_edges(equal_edge_indices, delete_vertices=False)
    else:
        gnn = ig.Graph(n=mlon.vcount(), directed=True)

    # Find weakly connected components
    nn_memb = gnn.components(mode="weak").membership

    # Contract vertices using component membership
    cmlon_graph = _contract_vertices(
        mlon,
        nn_memb,
        vertex_attr_comb={"Fitness": "first", "Count": "sum", "name": "first"},
    )

    # Combine parallel edges by summing Count
    cmlon_graph = _simplify_with_edge_sum(cmlon_graph)

    return cls(graph=cmlon_graph, best_fitness=lon.best_fitness, source_lon=lon)

get_sinks() -> list[int]

Get indices of sink nodes (nodes with no outgoing edges).

Source code in src/lonpy/lon.py 
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def get_sinks(self) -> list[int]:
    """Get indices of sink nodes (nodes with no outgoing edges)."""
    out_degrees = self.graph.degree(mode="out")
    return [i for i, d in enumerate(out_degrees) if d == 0]

get_global_sinks() -> list[int]

Get indices of global sinks (sinks at best fitness).

Source code in src/lonpy/lon.py 
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def get_global_sinks(self) -> list[int]:
    """Get indices of global sinks (sinks at best fitness)."""
    sinks = self.get_sinks()
    fits = self.vertex_fitness
    return [s for s in sinks if fits[s] == self.best_fitness]

get_local_sinks() -> list[int]

Get indices of local sinks (sinks not at best fitness).

Source code in src/lonpy/lon.py 
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def get_local_sinks(self) -> list[int]:
    """Get indices of local sinks (sinks not at best fitness)."""
    sinks = self.get_sinks()
    fits = self.vertex_fitness
    if self.best_fitness is None:
        return []
    return [s for s in sinks if fits[s] > self.best_fitness]

compute_metrics(known_best: float | None = None) -> dict[str, Any]

Compute CMLON metrics.

Parameters:

Name Type Description Default
known_best float | None

Known global optimum value. If None, uses the best fitness found in the network.

 None

Returns:

Type Description
dict[str, Any]

Dictionary containing: - n_optima: Number of optima in CMLON - n_funnels: Number of funnels (sinks) - n_global_funnels: Number of funnels at global optimum - neutral: Proportion of contracted nodes - strength: Ratio of incoming strength to global vs local sinks - global_funnel_proportion: Proportion of nodes that can reach a global optimum
Source code in src/lonpy/lon.py 
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def compute_metrics(self, known_best: float | None = None) -> dict[str, Any]:
    """
    Compute CMLON metrics.

    Args:
        known_best: Known global optimum value. If None, uses the best
            fitness found in the network.

    Returns:
        Dictionary containing:
            - n_optima: Number of optima in CMLON
            - n_funnels: Number of funnels (sinks)
            - n_global_funnels: Number of funnels at global optimum
            - neutral: Proportion of contracted nodes
            - strength: Ratio of incoming strength to global vs local sinks
            - global_funnel_proportion: Proportion of nodes that can reach
              a global optimum
    """
    best = known_best if known_best is not None else self.best_fitness

    n_optima = self.n_vertices

    sinks_id = self.get_sinks()
    n_funnels = len(sinks_id)

    sinks_fit = [self.vertex_fitness[i] for i in sinks_id]
    n_global_funnels = sum(1 for f in sinks_fit if f == best)

    # Neutral: proportion of contracted nodes
    if self.source_lon is not None:
        neutral = round(1.0 - self.n_vertices / self.source_lon.n_vertices, 4)
    else:
        neutral = 0.0

    # Strength: ratio of incoming strength to global vs local sinks
    igs = [s for s, f in zip(sinks_id, sinks_fit) if f == best]
    ils = [s for s, f in zip(sinks_id, sinks_fit) if best is not None and f > best]

    if self.n_edges > 0:
        edge_weights = self.graph.es["Count"] if "Count" in self.graph.es.attributes() else None
        sing = (
            sum(self.graph.strength(igs, mode="in", loops=False, weights=edge_weights))
            if igs
            else 0
        )
        sinl = (
            sum(self.graph.strength(ils, mode="in", loops=False, weights=edge_weights))
            if ils
            else 0
        )
        total = sing + sinl
        strength = round(sing / total, 4) if total > 0 else 0.0
    else:
        strength = 0.0

    gfunnel = self._compute_global_funnel_proportion()

    return {
        "n_optima": n_optima,
        "n_funnels": n_funnels,
        "n_global_funnels": n_global_funnels,
        "neutral": neutral,
        "strength": strength,
        "global_funnel_proportion": gfunnel,
    }