Skip to content

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.
final_run_values Series | None

Dictionary mapping run number to final fitness value.
Source code in src/lonpy/lon.py 
 37
 38
 39
 40
 41
 42
 43
 44
 45
 46
 47
 48
 49
 50
 51
 52
 53
 54
 55
 56
 57
 58
 59
 60
 61
 62
 63
 64
 65
 66
 67
 68
 69
 70
 71
 72
 73
 74
 75
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
@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.
        final_run_values: Dictionary mapping run number to final fitness value.
    """

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

    @classmethod
    def from_trace_data(
        cls,
        trace: pd.DataFrame,
        config: LONConfig | None = None,
    ) -> "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
            config: Optional configuration for LON construction. If None, uses default
                configuration with minimum fitness aggregation.

        Returns:
            LON instance with constructed graph.

        Raises:
            ValueError: If fitness_aggregation is "strict" and duplicates are detected,
                or if max_fitness_deviation threshold is exceeded.
        """
        config = config or LONConfig()
        trace = trace.copy()
        trace.columns = pd.Index(["run", "fit1", "node1", "fit2", "node2"])

        # Extract final fitness value from each run as a Series
        final_run_values = trace.groupby("run").tail(1).set_index("run")["fit2"]

        lnodes = pd.concat(
            [
                trace[["node1", "fit1"]].rename(columns={"node1": "Node", "fit1": "Fitness"}),
                trace[["node2", "fit2"]].rename(columns={"node2": "Node", "fit2": "Fitness"}),
            ],
            ignore_index=True,
        )

        # Node deduplication by grouping and aggregation
        node_agg = (
            lnodes.groupby("Node").agg({"Fitness": ["min", "max", "mean", "nunique"]}).reset_index()
        )
        node_agg.columns = pd.Index(
            [
                "Node",
                "Fitness_min",
                "Fitness_max",
                "Fitness_mean",
                "Fitness_nunique",
            ]
        )

        visit_counts = lnodes.groupby("Node").size().reset_index(name="Count")

        duplicates = node_agg[node_agg["Fitness_nunique"] > 1]

        # Check for duplication issues, raise errors or warning according to config
        if not duplicates.empty:
            _validate_duplicate_nodes(node_agg, duplicates, config)

        match config.fitness_aggregation:
            case "first":
                fitness_values = lnodes.groupby("Node", as_index=False).first()[["Node", "Fitness"]]
            case "max":
                fitness_values = node_agg[["Node", "Fitness_max"]].rename(
                    columns={"Fitness_max": "Fitness"}
                )
            case "mean":
                fitness_values = node_agg[["Node", "Fitness_mean"]].rename(
                    columns={"Fitness_mean": "Fitness"}
                )
            case _:  # "min" or default
                fitness_values = node_agg[["Node", "Fitness_min"]].rename(
                    columns={"Fitness_min": "Fitness"}
                )

        # Merge fitness values with visit counts
        nodes = pd.merge(fitness_values, visit_counts, on="Node")

        edges = trace.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,
            final_run_values=final_run_values,
            eq_atol=config.eq_atol,
        )

    @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 _allclose(self, f1: float | None, f2: float | None) -> bool:
        """Check if two fitness values are equal within tolerance."""
        if f1 is None or f2 is None:
            return f1 == f2
        atol = self.eq_atol if self.eq_atol is not None else DEFAULT_ATOL
        return np.allclose(f1, f2, atol=atol, rtol=0.0)

    def _isclose_series(self, f1: pd.Series, f2: float):
        """Check element-wise if a Series of fitness values are equal to a scalar within tolerance."""
        atol = self.eq_atol if self.eq_atol is not None else DEFAULT_ATOL
        return np.isclose(f1, f2, atol=atol, rtol=0.0)

    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 compute_network_metrics(self, known_best: float | None = None) -> dict[str, Any]:
        """
        Compute LON network 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
                - global_strength: Proportion of global optima incoming strength to total incoming strength of all nodes
                - sink_strength: Proportion of global sinks incoming strength to incoming strength of all sink nodes
        """
        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 self._allclose(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 self._allclose(fits[src], fits[tgt]):
                neutral_edge_indices.append(i)

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

        # Strength (global): incoming strength to global optima / total incoming strength
        igs = [i for i, f in enumerate(self.vertex_fitness) if self._allclose(f, best)]
        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))
            global_strength = round(stren_igs / stren_all, 4) if stren_all > 0 else 0.0
        else:
            global_strength = 0.0

        # Strength (sinks only): incoming strength to global sinks / incoming strength to all sinks
        global_sinks = [s for s in sinks_id if self._allclose(self.vertex_fitness[s], best)]
        local_sinks = [s for s in sinks_id if not self._allclose(self.vertex_fitness[s], best)]
        if self.n_edges > 0 and global_sinks:
            edge_weights = self.graph.es["Count"]
            sing = sum(
                self.graph.strength(global_sinks, mode="in", loops=False, weights=edge_weights)
            )
            sinl = (
                sum(self.graph.strength(local_sinks, mode="in", loops=False, weights=edge_weights))
                if local_sinks
                else 0
            )
            sink_strength = round(sing / (sing + sinl), 4) if (sing + sinl) > 0 else 0.0
        else:
            sink_strength = 0.0

        return {
            "n_optima": n_optima,
            "n_funnels": n_funnels,
            "n_global_funnels": n_global_funnels,
            "neutral": neutral,
            "global_strength": global_strength,
            "sink_strength": sink_strength,
        }

    def compute_performance_metrics(self, known_best: float | None = None) -> dict[str, Any]:
        """
        Compute performance metrics based on sampling runs.

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

        Returns:
            Dictionary containing:
                - success: Proportion of runs that reached the global optimum
                - deviation: Mean absolute deviation from the global optimum
        """
        best = known_best if known_best is not None else self.best_fitness
        # Success: proportion of runs that reached the global optimum
        success = (
            self._isclose_series(self.final_run_values, best).sum() / len(self.final_run_values)
            if self.final_run_values is not None and best is not None
            else 0.0
        )

        # Deviation: mean deviation from the global optimum value
        deviation = (
            (self.final_run_values - best).abs().mean()
            if self.final_run_values is not None and best is not None
            else 0.0
        )

        return {"success": success, "deviation": deviation}

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

        This is a convenience method that combines both network metrics
        (topology-based) and performance metrics (run-based).

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

        Returns:
            Dictionary containing all network and performance metrics:
                Network metrics: n_optima, n_funnels, n_global_funnels, neutral, global_strength, sink_strength
                Performance metrics: success, deviation
        """
        network_metrics = self.compute_network_metrics(known_best)
        performance_metrics = self.compute_performance_metrics(known_best)
        return {**network_metrics, **performance_metrics}

    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, config: LONConfig | None = None) -> 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
config LONConfig | None

Optional configuration for LON construction. If None, uses default configuration with minimum fitness aggregation.

 None

Returns:

Type Description
LON

LON instance with constructed graph.

Raises:

Type Description
ValueError

If fitness_aggregation is "strict" and duplicates are detected, or if max_fitness_deviation threshold is exceeded.
Source code in src/lonpy/lon.py 
 56
 57
 58
 59
 60
 61
 62
 63
 64
 65
 66
 67
 68
 69
 70
 71
 72
 73
 74
 75
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
@classmethod
def from_trace_data(
    cls,
    trace: pd.DataFrame,
    config: LONConfig | None = None,
) -> "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
        config: Optional configuration for LON construction. If None, uses default
            configuration with minimum fitness aggregation.

    Returns:
        LON instance with constructed graph.

    Raises:
        ValueError: If fitness_aggregation is "strict" and duplicates are detected,
            or if max_fitness_deviation threshold is exceeded.
    """
    config = config or LONConfig()
    trace = trace.copy()
    trace.columns = pd.Index(["run", "fit1", "node1", "fit2", "node2"])

    # Extract final fitness value from each run as a Series
    final_run_values = trace.groupby("run").tail(1).set_index("run")["fit2"]

    lnodes = pd.concat(
        [
            trace[["node1", "fit1"]].rename(columns={"node1": "Node", "fit1": "Fitness"}),
            trace[["node2", "fit2"]].rename(columns={"node2": "Node", "fit2": "Fitness"}),
        ],
        ignore_index=True,
    )

    # Node deduplication by grouping and aggregation
    node_agg = (
        lnodes.groupby("Node").agg({"Fitness": ["min", "max", "mean", "nunique"]}).reset_index()
    )
    node_agg.columns = pd.Index(
        [
            "Node",
            "Fitness_min",
            "Fitness_max",
            "Fitness_mean",
            "Fitness_nunique",
        ]
    )

    visit_counts = lnodes.groupby("Node").size().reset_index(name="Count")

    duplicates = node_agg[node_agg["Fitness_nunique"] > 1]

    # Check for duplication issues, raise errors or warning according to config
    if not duplicates.empty:
        _validate_duplicate_nodes(node_agg, duplicates, config)

    match config.fitness_aggregation:
        case "first":
            fitness_values = lnodes.groupby("Node", as_index=False).first()[["Node", "Fitness"]]
        case "max":
            fitness_values = node_agg[["Node", "Fitness_max"]].rename(
                columns={"Fitness_max": "Fitness"}
            )
        case "mean":
            fitness_values = node_agg[["Node", "Fitness_mean"]].rename(
                columns={"Fitness_mean": "Fitness"}
            )
        case _:  # "min" or default
            fitness_values = node_agg[["Node", "Fitness_min"]].rename(
                columns={"Fitness_min": "Fitness"}
            )

    # Merge fitness values with visit counts
    nodes = pd.merge(fitness_values, visit_counts, on="Node")

    edges = trace.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,
        final_run_values=final_run_values,
        eq_atol=config.eq_atol,
    )

get_sinks() -> list[int]

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

Source code in src/lonpy/lon.py 
199
200
201
202
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]

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

Compute all LON metrics (network topology + performance).

This is a convenience method that combines both network metrics (topology-based) and performance metrics (run-based).

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 all network and performance metrics: Network metrics: n_optima, n_funnels, n_global_funnels, neutral, global_strength, sink_strength Performance metrics: success, deviation
Source code in src/lonpy/lon.py 
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
def compute_metrics(self, known_best: float | None = None) -> dict[str, Any]:
    """
    Compute all LON metrics (network topology + performance).

    This is a convenience method that combines both network metrics
    (topology-based) and performance metrics (run-based).

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

    Returns:
        Dictionary containing all network and performance metrics:
            Network metrics: n_optima, n_funnels, n_global_funnels, neutral, global_strength, sink_strength
            Performance metrics: success, deviation
    """
    network_metrics = self.compute_network_metrics(known_best)
    performance_metrics = self.compute_performance_metrics(known_best)
    return {**network_metrics, **performance_metrics}

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 
331
332
333
334
335
336
337
338
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 
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
@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
    eq_atol: float | 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,
                eq_atol=lon.eq_atol,
            )

        # 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 lon._allclose(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"},
        )

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

    def _allclose(self, f1: float | None, f2: float | None) -> bool:
        """Check if two fitness values are equal within tolerance."""
        if f1 is None or f2 is None:
            return f1 == f2
        atol = self.eq_atol if self.eq_atol is not None else DEFAULT_ATOL
        return np.allclose(f1, f2, atol=atol, rtol=0.0)

    @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 self._allclose(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_network_metrics(self, known_best: float | None = None) -> dict[str, Any]:
        """
        Compute CMLON network 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
                - global_strength: Proportion of global sinks incoming strength to total incoming strength of all nodes
                - sink_strength: Proportion of global sinks incoming strength to incoming strength of all sink nodes
                - 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 self._allclose(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 (global): incoming strength to global sinks / total incoming strength
        igs = [s for s, f in zip(sinks_id, sinks_fit) if self._allclose(f, best)]
        ils = [s for s, f in zip(sinks_id, sinks_fit) if not self._allclose(f, best)]

        if self.n_edges > 0:
            edge_weights = self.graph.es["Count"]
            sing = (
                sum(self.graph.strength(igs, mode="in", loops=False, weights=edge_weights))
                if igs
                else 0
            )
            total = sum(self.graph.strength(mode="in", loops=False, weights=edge_weights))
            global_strength = round(sing / total, 4) if total > 0 else 0.0
        else:
            global_strength = 0.0

        # Strength (sinks only): incoming strength to global sinks / incoming strength to all sinks
        if self.n_edges > 0 and igs:
            edge_weights = self.graph.es["Count"]
            sinl = (
                sum(self.graph.strength(ils, mode="in", loops=False, weights=edge_weights))
                if ils
                else 0
            )
            sink_strength = round(sing / (sing + sinl), 4) if (sing + sinl) > 0 else 0.0
        else:
            sink_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,
            "global_strength": global_strength,
            "sink_strength": sink_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

    def compute_performance_metrics(self, known_best: float | None = None) -> dict[str, Any]:
        """
        Compute performance metrics from the source LON.

        CMLON delegates to its source LON for performance metrics since
        it doesn't have its own sampling run data.

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

        Returns:
            Dictionary containing performance metrics from source LON, or
            empty dict if no source LON is available.
        """
        return (
            self.source_lon.compute_performance_metrics(known_best)
            if self.source_lon is not None
            else {}
        )

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

        This is a convenience method that combines both CMLON-specific network
        metrics and performance metrics from the source LON.

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

        Returns:
            Dictionary containing all network and performance metrics:
                Network metrics: n_optima, n_funnels, n_global_funnels, neutral,
                    global_strength, sink_strength, global_funnel_proportion
                Performance metrics: success, deviation (from source LON)
        """
        network_metrics = self.compute_network_metrics(known_best)
        performance_metrics = self.compute_performance_metrics(known_best)
        return {**network_metrics, **performance_metrics}

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 
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
@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,
            eq_atol=lon.eq_atol,
        )

    # 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 lon._allclose(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"},
    )

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

get_sinks() -> list[int]

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

Source code in src/lonpy/lon.py 
460
461
462
463
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 
465
466
467
468
469
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 self._allclose(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 
471
472
473
474
475
476
477
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 all CMLON metrics (network topology + performance).

This is a convenience method that combines both CMLON-specific network metrics and performance metrics from the source LON.

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 all network and performance metrics: Network metrics: n_optima, n_funnels, n_global_funnels, neutral, global_strength, sink_strength, global_funnel_proportion Performance metrics: success, deviation (from source LON)
Source code in src/lonpy/lon.py 
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
def compute_metrics(self, known_best: float | None = None) -> dict[str, Any]:
    """
    Compute all CMLON metrics (network topology + performance).

    This is a convenience method that combines both CMLON-specific network
    metrics and performance metrics from the source LON.

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

    Returns:
        Dictionary containing all network and performance metrics:
            Network metrics: n_optima, n_funnels, n_global_funnels, neutral,
                global_strength, sink_strength, global_funnel_proportion
            Performance metrics: success, deviation (from source LON)
    """
    network_metrics = self.compute_network_metrics(known_best)
    performance_metrics = self.compute_performance_metrics(known_best)
    return {**network_metrics, **performance_metrics}