# Source code for networkx.algorithms.similarity

""" Functions measuring similarity using graph edit distance.

The graph edit distance is the number of edge/node changes needed
to make two graphs isomorphic.

The default algorithm/implementation is sub-optimal for some graphs.
The problem of finding the exact Graph Edit Distance (GED) is NP-hard
so it is often slow. If the simple interface graph_edit_distance
takes too long for your graph, try optimize_graph_edit_distance
and/or optimize_edit_paths.

At the same time, I encourage capable people to investigate
alternative GED algorithms, in order to improve the choices available.
"""

import math
import time
import warnings
from dataclasses import dataclass
from itertools import product

import networkx as nx

__all__ = [
"graph_edit_distance",
"optimal_edit_paths",
"optimize_graph_edit_distance",
"optimize_edit_paths",
"simrank_similarity",
"panther_similarity",
"generate_random_paths",
]

def debug_print(*args, **kwargs):
print(*args, **kwargs)

[docs]
@nx._dispatch(
graphs={"G1": 0, "G2": 1}, preserve_edge_attrs=True, preserve_node_attrs=True
)
def graph_edit_distance(
G1,
G2,
node_match=None,
edge_match=None,
node_subst_cost=None,
node_del_cost=None,
node_ins_cost=None,
edge_subst_cost=None,
edge_del_cost=None,
edge_ins_cost=None,
roots=None,
upper_bound=None,
timeout=None,
):
"""Returns GED (graph edit distance) between graphs G1 and G2.

Graph edit distance is a graph similarity measure analogous to
Levenshtein distance for strings.  It is defined as minimum cost
of edit path (sequence of node and edge edit operations)
transforming graph G1 to graph isomorphic to G2.

Parameters
----------
G1, G2: graphs
The two graphs G1 and G2 must be of the same type.

node_match : callable
A function that returns True if node n1 in G1 and n2 in G2
should be considered equal during matching.

The function will be called like

node_match(G1.nodes[n1], G2.nodes[n2]).

That is, the function will receive the node attribute
dictionaries for n1 and n2 as inputs.

Ignored if node_subst_cost is specified.  If neither
node_match nor node_subst_cost are specified then node
attributes are not considered.

edge_match : callable
A function that returns True if the edge attribute dictionaries
for the pair of nodes (u1, v1) in G1 and (u2, v2) in G2 should
be considered equal during matching.

The function will be called like

edge_match(G1[u1][v1], G2[u2][v2]).

That is, the function will receive the edge attribute
dictionaries of the edges under consideration.

Ignored if edge_subst_cost is specified.  If neither
edge_match nor edge_subst_cost are specified then edge
attributes are not considered.

node_subst_cost, node_del_cost, node_ins_cost : callable
Functions that return the costs of node substitution, node
deletion, and node insertion, respectively.

The functions will be called like

node_subst_cost(G1.nodes[n1], G2.nodes[n2]),
node_del_cost(G1.nodes[n1]),
node_ins_cost(G2.nodes[n2]).

That is, the functions will receive the node attribute
dictionaries as inputs.  The functions are expected to return
positive numeric values.

Function node_subst_cost overrides node_match if specified.
If neither node_match nor node_subst_cost are specified then
default node substitution cost of 0 is used (node attributes
are not considered during matching).

If node_del_cost is not specified then default node deletion
cost of 1 is used.  If node_ins_cost is not specified then
default node insertion cost of 1 is used.

edge_subst_cost, edge_del_cost, edge_ins_cost : callable
Functions that return the costs of edge substitution, edge
deletion, and edge insertion, respectively.

The functions will be called like

edge_subst_cost(G1[u1][v1], G2[u2][v2]),
edge_del_cost(G1[u1][v1]),
edge_ins_cost(G2[u2][v2]).

That is, the functions will receive the edge attribute
dictionaries as inputs.  The functions are expected to return
positive numeric values.

Function edge_subst_cost overrides edge_match if specified.
If neither edge_match nor edge_subst_cost are specified then
default edge substitution cost of 0 is used (edge attributes
are not considered during matching).

If edge_del_cost is not specified then default edge deletion
cost of 1 is used.  If edge_ins_cost is not specified then
default edge insertion cost of 1 is used.

roots : 2-tuple
Tuple where first element is a node in G1 and the second
is a node in G2.
These nodes are forced to be matched in the comparison to
allow comparison between rooted graphs.

upper_bound : numeric
Maximum edit distance to consider.  Return None if no edit
distance under or equal to upper_bound exists.

timeout : numeric
Maximum number of seconds to execute.
After timeout is met, the current best GED is returned.

Examples
--------
>>> G1 = nx.cycle_graph(6)
>>> G2 = nx.wheel_graph(7)
>>> nx.graph_edit_distance(G1, G2)
7.0

>>> G1 = nx.star_graph(5)
>>> G2 = nx.star_graph(5)
>>> nx.graph_edit_distance(G1, G2, roots=(0, 0))
0.0
>>> nx.graph_edit_distance(G1, G2, roots=(1, 0))
8.0

--------
optimal_edit_paths, optimize_graph_edit_distance,

is_isomorphic: test for graph edit distance of 0

References
----------
..  Zeina Abu-Aisheh, Romain Raveaux, Jean-Yves Ramel, Patrick
Martineau. An Exact Graph Edit Distance Algorithm for Solving
Pattern Recognition Problems. 4th International Conference on
Pattern Recognition Applications and Methods 2015, Jan 2015,
Lisbon, Portugal. 2015,
<10.5220/0005209202710278>. <hal-01168816>
https://hal.archives-ouvertes.fr/hal-01168816

"""
bestcost = None
for _, _, cost in optimize_edit_paths(
G1,
G2,
node_match,
edge_match,
node_subst_cost,
node_del_cost,
node_ins_cost,
edge_subst_cost,
edge_del_cost,
edge_ins_cost,
upper_bound,
True,
roots,
timeout,
):
# assert bestcost is None or cost < bestcost
bestcost = cost
return bestcost

[docs]
@nx._dispatch(graphs={"G1": 0, "G2": 1})
def optimal_edit_paths(
G1,
G2,
node_match=None,
edge_match=None,
node_subst_cost=None,
node_del_cost=None,
node_ins_cost=None,
edge_subst_cost=None,
edge_del_cost=None,
edge_ins_cost=None,
upper_bound=None,
):
"""Returns all minimum-cost edit paths transforming G1 to G2.

Graph edit path is a sequence of node and edge edit operations
transforming graph G1 to graph isomorphic to G2.  Edit operations
include substitutions, deletions, and insertions.

Parameters
----------
G1, G2: graphs
The two graphs G1 and G2 must be of the same type.

node_match : callable
A function that returns True if node n1 in G1 and n2 in G2
should be considered equal during matching.

The function will be called like

node_match(G1.nodes[n1], G2.nodes[n2]).

That is, the function will receive the node attribute
dictionaries for n1 and n2 as inputs.

Ignored if node_subst_cost is specified.  If neither
node_match nor node_subst_cost are specified then node
attributes are not considered.

edge_match : callable
A function that returns True if the edge attribute dictionaries
for the pair of nodes (u1, v1) in G1 and (u2, v2) in G2 should
be considered equal during matching.

The function will be called like

edge_match(G1[u1][v1], G2[u2][v2]).

That is, the function will receive the edge attribute
dictionaries of the edges under consideration.

Ignored if edge_subst_cost is specified.  If neither
edge_match nor edge_subst_cost are specified then edge
attributes are not considered.

node_subst_cost, node_del_cost, node_ins_cost : callable
Functions that return the costs of node substitution, node
deletion, and node insertion, respectively.

The functions will be called like

node_subst_cost(G1.nodes[n1], G2.nodes[n2]),
node_del_cost(G1.nodes[n1]),
node_ins_cost(G2.nodes[n2]).

That is, the functions will receive the node attribute
dictionaries as inputs.  The functions are expected to return
positive numeric values.

Function node_subst_cost overrides node_match if specified.
If neither node_match nor node_subst_cost are specified then
default node substitution cost of 0 is used (node attributes
are not considered during matching).

If node_del_cost is not specified then default node deletion
cost of 1 is used.  If node_ins_cost is not specified then
default node insertion cost of 1 is used.

edge_subst_cost, edge_del_cost, edge_ins_cost : callable
Functions that return the costs of edge substitution, edge
deletion, and edge insertion, respectively.

The functions will be called like

edge_subst_cost(G1[u1][v1], G2[u2][v2]),
edge_del_cost(G1[u1][v1]),
edge_ins_cost(G2[u2][v2]).

That is, the functions will receive the edge attribute
dictionaries as inputs.  The functions are expected to return
positive numeric values.

Function edge_subst_cost overrides edge_match if specified.
If neither edge_match nor edge_subst_cost are specified then
default edge substitution cost of 0 is used (edge attributes
are not considered during matching).

If edge_del_cost is not specified then default edge deletion
cost of 1 is used.  If edge_ins_cost is not specified then
default edge insertion cost of 1 is used.

upper_bound : numeric
Maximum edit distance to consider.

Returns
-------
edit_paths : list of tuples (node_edit_path, edge_edit_path)
node_edit_path : list of tuples (u, v)
edge_edit_path : list of tuples ((u1, v1), (u2, v2))

cost : numeric
Optimal edit path cost (graph edit distance).

Examples
--------
>>> G1 = nx.cycle_graph(4)
>>> G2 = nx.wheel_graph(5)
>>> paths, cost = nx.optimal_edit_paths(G1, G2)
>>> len(paths)
40
>>> cost
5.0

--------
graph_edit_distance, optimize_edit_paths

References
----------
..  Zeina Abu-Aisheh, Romain Raveaux, Jean-Yves Ramel, Patrick
Martineau. An Exact Graph Edit Distance Algorithm for Solving
Pattern Recognition Problems. 4th International Conference on
Pattern Recognition Applications and Methods 2015, Jan 2015,
Lisbon, Portugal. 2015,
<10.5220/0005209202710278>. <hal-01168816>
https://hal.archives-ouvertes.fr/hal-01168816

"""
paths = []
bestcost = None
for vertex_path, edge_path, cost in optimize_edit_paths(
G1,
G2,
node_match,
edge_match,
node_subst_cost,
node_del_cost,
node_ins_cost,
edge_subst_cost,
edge_del_cost,
edge_ins_cost,
upper_bound,
False,
):
# assert bestcost is None or cost <= bestcost
if bestcost is not None and cost < bestcost:
paths = []
paths.append((vertex_path, edge_path))
bestcost = cost
return paths, bestcost

[docs]
@nx._dispatch(graphs={"G1": 0, "G2": 1})
def optimize_graph_edit_distance(
G1,
G2,
node_match=None,
edge_match=None,
node_subst_cost=None,
node_del_cost=None,
node_ins_cost=None,
edge_subst_cost=None,
edge_del_cost=None,
edge_ins_cost=None,
upper_bound=None,
):
"""Returns consecutive approximations of GED (graph edit distance)
between graphs G1 and G2.

Graph edit distance is a graph similarity measure analogous to
Levenshtein distance for strings.  It is defined as minimum cost
of edit path (sequence of node and edge edit operations)
transforming graph G1 to graph isomorphic to G2.

Parameters
----------
G1, G2: graphs
The two graphs G1 and G2 must be of the same type.

node_match : callable
A function that returns True if node n1 in G1 and n2 in G2
should be considered equal during matching.

The function will be called like

node_match(G1.nodes[n1], G2.nodes[n2]).

That is, the function will receive the node attribute
dictionaries for n1 and n2 as inputs.

Ignored if node_subst_cost is specified.  If neither
node_match nor node_subst_cost are specified then node
attributes are not considered.

edge_match : callable
A function that returns True if the edge attribute dictionaries
for the pair of nodes (u1, v1) in G1 and (u2, v2) in G2 should
be considered equal during matching.

The function will be called like

edge_match(G1[u1][v1], G2[u2][v2]).

That is, the function will receive the edge attribute
dictionaries of the edges under consideration.

Ignored if edge_subst_cost is specified.  If neither
edge_match nor edge_subst_cost are specified then edge
attributes are not considered.

node_subst_cost, node_del_cost, node_ins_cost : callable
Functions that return the costs of node substitution, node
deletion, and node insertion, respectively.

The functions will be called like

node_subst_cost(G1.nodes[n1], G2.nodes[n2]),
node_del_cost(G1.nodes[n1]),
node_ins_cost(G2.nodes[n2]).

That is, the functions will receive the node attribute
dictionaries as inputs.  The functions are expected to return
positive numeric values.

Function node_subst_cost overrides node_match if specified.
If neither node_match nor node_subst_cost are specified then
default node substitution cost of 0 is used (node attributes
are not considered during matching).

If node_del_cost is not specified then default node deletion
cost of 1 is used.  If node_ins_cost is not specified then
default node insertion cost of 1 is used.

edge_subst_cost, edge_del_cost, edge_ins_cost : callable
Functions that return the costs of edge substitution, edge
deletion, and edge insertion, respectively.

The functions will be called like

edge_subst_cost(G1[u1][v1], G2[u2][v2]),
edge_del_cost(G1[u1][v1]),
edge_ins_cost(G2[u2][v2]).

That is, the functions will receive the edge attribute
dictionaries as inputs.  The functions are expected to return
positive numeric values.

Function edge_subst_cost overrides edge_match if specified.
If neither edge_match nor edge_subst_cost are specified then
default edge substitution cost of 0 is used (edge attributes
are not considered during matching).

If edge_del_cost is not specified then default edge deletion
cost of 1 is used.  If edge_ins_cost is not specified then
default edge insertion cost of 1 is used.

upper_bound : numeric
Maximum edit distance to consider.

Returns
-------
Generator of consecutive approximations of graph edit distance.

Examples
--------
>>> G1 = nx.cycle_graph(6)
>>> G2 = nx.wheel_graph(7)
>>> for v in nx.optimize_graph_edit_distance(G1, G2):
...     minv = v
>>> minv
7.0

--------
graph_edit_distance, optimize_edit_paths

References
----------
..  Zeina Abu-Aisheh, Romain Raveaux, Jean-Yves Ramel, Patrick
Martineau. An Exact Graph Edit Distance Algorithm for Solving
Pattern Recognition Problems. 4th International Conference on
Pattern Recognition Applications and Methods 2015, Jan 2015,
Lisbon, Portugal. 2015,
<10.5220/0005209202710278>. <hal-01168816>
https://hal.archives-ouvertes.fr/hal-01168816
"""
for _, _, cost in optimize_edit_paths(
G1,
G2,
node_match,
edge_match,
node_subst_cost,
node_del_cost,
node_ins_cost,
edge_subst_cost,
edge_del_cost,
edge_ins_cost,
upper_bound,
True,
):
yield cost

[docs]
@nx._dispatch(
graphs={"G1": 0, "G2": 1}, preserve_edge_attrs=True, preserve_node_attrs=True
)
def optimize_edit_paths(
G1,
G2,
node_match=None,
edge_match=None,
node_subst_cost=None,
node_del_cost=None,
node_ins_cost=None,
edge_subst_cost=None,
edge_del_cost=None,
edge_ins_cost=None,
upper_bound=None,
strictly_decreasing=True,
roots=None,
timeout=None,
):
"""GED (graph edit distance) calculation: advanced interface.

Graph edit path is a sequence of node and edge edit operations
transforming graph G1 to graph isomorphic to G2.  Edit operations
include substitutions, deletions, and insertions.

Graph edit distance is defined as minimum cost of edit path.

Parameters
----------
G1, G2: graphs
The two graphs G1 and G2 must be of the same type.

node_match : callable
A function that returns True if node n1 in G1 and n2 in G2
should be considered equal during matching.

The function will be called like

node_match(G1.nodes[n1], G2.nodes[n2]).

That is, the function will receive the node attribute
dictionaries for n1 and n2 as inputs.

Ignored if node_subst_cost is specified.  If neither
node_match nor node_subst_cost are specified then node
attributes are not considered.

edge_match : callable
A function that returns True if the edge attribute dictionaries
for the pair of nodes (u1, v1) in G1 and (u2, v2) in G2 should
be considered equal during matching.

The function will be called like

edge_match(G1[u1][v1], G2[u2][v2]).

That is, the function will receive the edge attribute
dictionaries of the edges under consideration.

Ignored if edge_subst_cost is specified.  If neither
edge_match nor edge_subst_cost are specified then edge
attributes are not considered.

node_subst_cost, node_del_cost, node_ins_cost : callable
Functions that return the costs of node substitution, node
deletion, and node insertion, respectively.

The functions will be called like

node_subst_cost(G1.nodes[n1], G2.nodes[n2]),
node_del_cost(G1.nodes[n1]),
node_ins_cost(G2.nodes[n2]).

That is, the functions will receive the node attribute
dictionaries as inputs.  The functions are expected to return
positive numeric values.

Function node_subst_cost overrides node_match if specified.
If neither node_match nor node_subst_cost are specified then
default node substitution cost of 0 is used (node attributes
are not considered during matching).

If node_del_cost is not specified then default node deletion
cost of 1 is used.  If node_ins_cost is not specified then
default node insertion cost of 1 is used.

edge_subst_cost, edge_del_cost, edge_ins_cost : callable
Functions that return the costs of edge substitution, edge
deletion, and edge insertion, respectively.

The functions will be called like

edge_subst_cost(G1[u1][v1], G2[u2][v2]),
edge_del_cost(G1[u1][v1]),
edge_ins_cost(G2[u2][v2]).

That is, the functions will receive the edge attribute
dictionaries as inputs.  The functions are expected to return
positive numeric values.

Function edge_subst_cost overrides edge_match if specified.
If neither edge_match nor edge_subst_cost are specified then
default edge substitution cost of 0 is used (edge attributes
are not considered during matching).

If edge_del_cost is not specified then default edge deletion
cost of 1 is used.  If edge_ins_cost is not specified then
default edge insertion cost of 1 is used.

upper_bound : numeric
Maximum edit distance to consider.

strictly_decreasing : bool
If True, return consecutive approximations of strictly
decreasing cost.  Otherwise, return all edit paths of cost
less than or equal to the previous minimum cost.

roots : 2-tuple
Tuple where first element is a node in G1 and the second
is a node in G2.
These nodes are forced to be matched in the comparison to
allow comparison between rooted graphs.

timeout : numeric
Maximum number of seconds to execute.
After timeout is met, the current best GED is returned.

Returns
-------
Generator of tuples (node_edit_path, edge_edit_path, cost)
node_edit_path : list of tuples (u, v)
edge_edit_path : list of tuples ((u1, v1), (u2, v2))
cost : numeric

--------
graph_edit_distance, optimize_graph_edit_distance, optimal_edit_paths

References
----------
..  Zeina Abu-Aisheh, Romain Raveaux, Jean-Yves Ramel, Patrick
Martineau. An Exact Graph Edit Distance Algorithm for Solving
Pattern Recognition Problems. 4th International Conference on
Pattern Recognition Applications and Methods 2015, Jan 2015,
Lisbon, Portugal. 2015,
<10.5220/0005209202710278>. <hal-01168816>
https://hal.archives-ouvertes.fr/hal-01168816

"""
# TODO: support DiGraph

import numpy as np
import scipy as sp

@dataclass
class CostMatrix:
C: ...
lsa_row_ind: ...
lsa_col_ind: ...
ls: ...

def make_CostMatrix(C, m, n):
# assert(C.shape == (m + n, m + n))
lsa_row_ind, lsa_col_ind = sp.optimize.linear_sum_assignment(C)

# Fixup dummy assignments:
# each substitution i<->j should have dummy assignment m+j<->n+i
# NOTE: fast reduce of Cv relies on it
# assert len(lsa_row_ind) == len(lsa_col_ind)
indexes = zip(range(len(lsa_row_ind)), lsa_row_ind, lsa_col_ind)
subst_ind = [k for k, i, j in indexes if i < m and j < n]
indexes = zip(range(len(lsa_row_ind)), lsa_row_ind, lsa_col_ind)
dummy_ind = [k for k, i, j in indexes if i >= m and j >= n]
# assert len(subst_ind) == len(dummy_ind)
lsa_row_ind[dummy_ind] = lsa_col_ind[subst_ind] + m
lsa_col_ind[dummy_ind] = lsa_row_ind[subst_ind] + n

return CostMatrix(
C, lsa_row_ind, lsa_col_ind, C[lsa_row_ind, lsa_col_ind].sum()
)

def extract_C(C, i, j, m, n):
# assert(C.shape == (m + n, m + n))
row_ind = [k in i or k - m in j for k in range(m + n)]
col_ind = [k in j or k - n in i for k in range(m + n)]
return C[row_ind, :][:, col_ind]

def reduce_C(C, i, j, m, n):
# assert(C.shape == (m + n, m + n))
row_ind = [k not in i and k - m not in j for k in range(m + n)]
col_ind = [k not in j and k - n not in i for k in range(m + n)]
return C[row_ind, :][:, col_ind]

def reduce_ind(ind, i):
# assert set(ind) == set(range(len(ind)))
rind = ind[[k not in i for k in ind]]
for k in set(i):
rind[rind >= k] -= 1
return rind

def match_edges(u, v, pending_g, pending_h, Ce, matched_uv=None):
"""
Parameters:
u, v: matched vertices, u=None or v=None for
deletion/insertion
pending_g, pending_h: lists of edges not yet mapped
Ce: CostMatrix of pending edge mappings
matched_uv: partial vertex edit path
list of tuples (u, v) of previously matched vertex
mappings u<->v, u=None or v=None for
deletion/insertion

Returns:
list of (i, j): indices of edge mappings g<->h
localCe: local CostMatrix of edge mappings
(basically submatrix of Ce at cross of rows i, cols j)
"""
M = len(pending_g)
N = len(pending_h)
# assert Ce.C.shape == (M + N, M + N)

# only attempt to match edges after one node match has been made
# this will stop self-edges on the first node being automatically deleted
# even when a substitution is the better option
if matched_uv is None or len(matched_uv) == 0:
g_ind = []
h_ind = []
else:
g_ind = [
i
for i in range(M)
if pending_g[i][:2] == (u, u)
or any(
pending_g[i][:2] in ((p, u), (u, p), (p, p)) for p, q in matched_uv
)
]
h_ind = [
j
for j in range(N)
if pending_h[j][:2] == (v, v)
or any(
pending_h[j][:2] in ((q, v), (v, q), (q, q)) for p, q in matched_uv
)
]

m = len(g_ind)
n = len(h_ind)

if m or n:
C = extract_C(Ce.C, g_ind, h_ind, M, N)
# assert C.shape == (m + n, m + n)

# Forbid structurally invalid matches
# NOTE: inf remembered from Ce construction
for k, i in enumerate(g_ind):
g = pending_g[i][:2]
for l, j in enumerate(h_ind):
h = pending_h[j][:2]
if nx.is_directed(G1) or nx.is_directed(G2):
if any(
g == (p, u) and h == (q, v) or g == (u, p) and h == (v, q)
for p, q in matched_uv
):
continue
else:
if any(
g in ((p, u), (u, p)) and h in ((q, v), (v, q))
for p, q in matched_uv
):
continue
if g == (u, u) or any(g == (p, p) for p, q in matched_uv):
continue
if h == (v, v) or any(h == (q, q) for p, q in matched_uv):
continue
C[k, l] = inf

localCe = make_CostMatrix(C, m, n)
ij = [
(
g_ind[k] if k < m else M + h_ind[l],
h_ind[l] if l < n else N + g_ind[k],
)
for k, l in zip(localCe.lsa_row_ind, localCe.lsa_col_ind)
if k < m or l < n
]

else:
ij = []
localCe = CostMatrix(np.empty((0, 0)), [], [], 0)

return ij, localCe

def reduce_Ce(Ce, ij, m, n):
if len(ij):
i, j = zip(*ij)
m_i = m - sum(1 for t in i if t < m)
n_j = n - sum(1 for t in j if t < n)
return make_CostMatrix(reduce_C(Ce.C, i, j, m, n), m_i, n_j)
return Ce

def get_edit_ops(
matched_uv, pending_u, pending_v, Cv, pending_g, pending_h, Ce, matched_cost
):
"""
Parameters:
matched_uv: partial vertex edit path
list of tuples (u, v) of vertex mappings u<->v,
u=None or v=None for deletion/insertion
pending_u, pending_v: lists of vertices not yet mapped
Cv: CostMatrix of pending vertex mappings
pending_g, pending_h: lists of edges not yet mapped
Ce: CostMatrix of pending edge mappings
matched_cost: cost of partial edit path

Returns:
sequence of
(i, j): indices of vertex mapping u<->v
Cv_ij: reduced CostMatrix of pending vertex mappings
(basically Cv with row i, col j removed)
list of (x, y): indices of edge mappings g<->h
Ce_xy: reduced CostMatrix of pending edge mappings
(basically Ce with rows x, cols y removed)
cost: total cost of edit operation
NOTE: most promising ops first
"""
m = len(pending_u)
n = len(pending_v)
# assert Cv.C.shape == (m + n, m + n)

# 1) a vertex mapping from optimal linear sum assignment
i, j = min(
(k, l) for k, l in zip(Cv.lsa_row_ind, Cv.lsa_col_ind) if k < m or l < n
)
xy, localCe = match_edges(
pending_u[i] if i < m else None,
pending_v[j] if j < n else None,
pending_g,
pending_h,
Ce,
matched_uv,
)
Ce_xy = reduce_Ce(Ce, xy, len(pending_g), len(pending_h))
# assert Ce.ls <= localCe.ls + Ce_xy.ls
if prune(matched_cost + Cv.ls + localCe.ls + Ce_xy.ls):
pass
else:
# get reduced Cv efficiently
Cv_ij = CostMatrix(
reduce_C(Cv.C, (i,), (j,), m, n),
reduce_ind(Cv.lsa_row_ind, (i, m + j)),
reduce_ind(Cv.lsa_col_ind, (j, n + i)),
Cv.ls - Cv.C[i, j],
)
yield (i, j), Cv_ij, xy, Ce_xy, Cv.C[i, j] + localCe.ls

# 2) other candidates, sorted by lower-bound cost estimate
other = []
fixed_i, fixed_j = i, j
if m <= n:
candidates = (
(t, fixed_j)
for t in range(m + n)
if t != fixed_i and (t < m or t == m + fixed_j)
)
else:
candidates = (
(fixed_i, t)
for t in range(m + n)
if t != fixed_j and (t < n or t == n + fixed_i)
)
for i, j in candidates:
if prune(matched_cost + Cv.C[i, j] + Ce.ls):
continue
Cv_ij = make_CostMatrix(
reduce_C(Cv.C, (i,), (j,), m, n),
m - 1 if i < m else m,
n - 1 if j < n else n,
)
# assert Cv.ls <= Cv.C[i, j] + Cv_ij.ls
if prune(matched_cost + Cv.C[i, j] + Cv_ij.ls + Ce.ls):
continue
xy, localCe = match_edges(
pending_u[i] if i < m else None,
pending_v[j] if j < n else None,
pending_g,
pending_h,
Ce,
matched_uv,
)
if prune(matched_cost + Cv.C[i, j] + Cv_ij.ls + localCe.ls):
continue
Ce_xy = reduce_Ce(Ce, xy, len(pending_g), len(pending_h))
# assert Ce.ls <= localCe.ls + Ce_xy.ls
if prune(matched_cost + Cv.C[i, j] + Cv_ij.ls + localCe.ls + Ce_xy.ls):
continue
other.append(((i, j), Cv_ij, xy, Ce_xy, Cv.C[i, j] + localCe.ls))

yield from sorted(other, key=lambda t: t + t.ls + t.ls)

def get_edit_paths(
matched_uv,
pending_u,
pending_v,
Cv,
matched_gh,
pending_g,
pending_h,
Ce,
matched_cost,
):
"""
Parameters:
matched_uv: partial vertex edit path
list of tuples (u, v) of vertex mappings u<->v,
u=None or v=None for deletion/insertion
pending_u, pending_v: lists of vertices not yet mapped
Cv: CostMatrix of pending vertex mappings
matched_gh: partial edge edit path
list of tuples (g, h) of edge mappings g<->h,
g=None or h=None for deletion/insertion
pending_g, pending_h: lists of edges not yet mapped
Ce: CostMatrix of pending edge mappings
matched_cost: cost of partial edit path

Returns:
sequence of (vertex_path, edge_path, cost)
vertex_path: complete vertex edit path
list of tuples (u, v) of vertex mappings u<->v,
u=None or v=None for deletion/insertion
edge_path: complete edge edit path
list of tuples (g, h) of edge mappings g<->h,
g=None or h=None for deletion/insertion
cost: total cost of edit path
NOTE: path costs are non-increasing
"""
# debug_print('matched-uv:', matched_uv)
# debug_print('matched-gh:', matched_gh)
# debug_print('matched-cost:', matched_cost)
# debug_print('pending-u:', pending_u)
# debug_print('pending-v:', pending_v)
# debug_print(Cv.C)
# assert list(sorted(G1.nodes)) == list(sorted(list(u for u, v in matched_uv if u is not None) + pending_u))
# assert list(sorted(G2.nodes)) == list(sorted(list(v for u, v in matched_uv if v is not None) + pending_v))
# debug_print('pending-g:', pending_g)
# debug_print('pending-h:', pending_h)
# debug_print(Ce.C)
# assert list(sorted(G1.edges)) == list(sorted(list(g for g, h in matched_gh if g is not None) + pending_g))
# assert list(sorted(G2.edges)) == list(sorted(list(h for g, h in matched_gh if h is not None) + pending_h))
# debug_print()

if prune(matched_cost + Cv.ls + Ce.ls):
return

if not max(len(pending_u), len(pending_v)):
# assert not len(pending_g)
# assert not len(pending_h)
# path completed!
# assert matched_cost <= maxcost_value
nonlocal maxcost_value
maxcost_value = min(maxcost_value, matched_cost)
yield matched_uv, matched_gh, matched_cost

else:
edit_ops = get_edit_ops(
matched_uv,
pending_u,
pending_v,
Cv,
pending_g,
pending_h,
Ce,
matched_cost,
)
for ij, Cv_ij, xy, Ce_xy, edit_cost in edit_ops:
i, j = ij
# assert Cv.C[i, j] + sum(Ce.C[t] for t in xy) == edit_cost
if prune(matched_cost + edit_cost + Cv_ij.ls + Ce_xy.ls):
continue

# dive deeper
u = pending_u.pop(i) if i < len(pending_u) else None
v = pending_v.pop(j) if j < len(pending_v) else None
matched_uv.append((u, v))
for x, y in xy:
len_g = len(pending_g)
len_h = len(pending_h)
matched_gh.append(
(
pending_g[x] if x < len_g else None,
pending_h[y] if y < len_h else None,
)
)
sortedx = sorted(x for x, y in xy)
sortedy = sorted(y for x, y in xy)
G = [
(pending_g.pop(x) if x < len(pending_g) else None)
for x in reversed(sortedx)
]
H = [
(pending_h.pop(y) if y < len(pending_h) else None)
for y in reversed(sortedy)
]

yield from get_edit_paths(
matched_uv,
pending_u,
pending_v,
Cv_ij,
matched_gh,
pending_g,
pending_h,
Ce_xy,
matched_cost + edit_cost,
)

# backtrack
if u is not None:
pending_u.insert(i, u)
if v is not None:
pending_v.insert(j, v)
matched_uv.pop()
for x, g in zip(sortedx, reversed(G)):
if g is not None:
pending_g.insert(x, g)
for y, h in zip(sortedy, reversed(H)):
if h is not None:
pending_h.insert(y, h)
for _ in xy:
matched_gh.pop()

# Initialization

pending_u = list(G1.nodes)
pending_v = list(G2.nodes)

initial_cost = 0
if roots:
root_u, root_v = roots
if root_u not in pending_u or root_v not in pending_v:
raise nx.NodeNotFound("Root node not in graph.")

# remove roots from pending
pending_u.remove(root_u)
pending_v.remove(root_v)

# cost matrix of vertex mappings
m = len(pending_u)
n = len(pending_v)
C = np.zeros((m + n, m + n))
if node_subst_cost:
C[0:m, 0:n] = np.array(
[
node_subst_cost(G1.nodes[u], G2.nodes[v])
for u in pending_u
for v in pending_v
]
).reshape(m, n)
if roots:
initial_cost = node_subst_cost(G1.nodes[root_u], G2.nodes[root_v])
elif node_match:
C[0:m, 0:n] = np.array(
[
1 - int(node_match(G1.nodes[u], G2.nodes[v]))
for u in pending_u
for v in pending_v
]
).reshape(m, n)
if roots:
initial_cost = 1 - node_match(G1.nodes[root_u], G2.nodes[root_v])
else:
# all zeroes
pass
# assert not min(m, n) or C[0:m, 0:n].min() >= 0
if node_del_cost:
del_costs = [node_del_cost(G1.nodes[u]) for u in pending_u]
else:
del_costs =  * len(pending_u)
# assert not m or min(del_costs) >= 0
if node_ins_cost:
ins_costs = [node_ins_cost(G2.nodes[v]) for v in pending_v]
else:
ins_costs =  * len(pending_v)
# assert not n or min(ins_costs) >= 0
inf = C[0:m, 0:n].sum() + sum(del_costs) + sum(ins_costs) + 1
C[0:m, n : n + m] = np.array(
[del_costs[i] if i == j else inf for i in range(m) for j in range(m)]
).reshape(m, m)
C[m : m + n, 0:n] = np.array(
[ins_costs[i] if i == j else inf for i in range(n) for j in range(n)]
).reshape(n, n)
Cv = make_CostMatrix(C, m, n)
# debug_print(f"Cv: {m} x {n}")
# debug_print(Cv.C)

pending_g = list(G1.edges)
pending_h = list(G2.edges)

# cost matrix of edge mappings
m = len(pending_g)
n = len(pending_h)
C = np.zeros((m + n, m + n))
if edge_subst_cost:
C[0:m, 0:n] = np.array(
[
edge_subst_cost(G1.edges[g], G2.edges[h])
for g in pending_g
for h in pending_h
]
).reshape(m, n)
elif edge_match:
C[0:m, 0:n] = np.array(
[
1 - int(edge_match(G1.edges[g], G2.edges[h]))
for g in pending_g
for h in pending_h
]
).reshape(m, n)
else:
# all zeroes
pass
# assert not min(m, n) or C[0:m, 0:n].min() >= 0
if edge_del_cost:
del_costs = [edge_del_cost(G1.edges[g]) for g in pending_g]
else:
del_costs =  * len(pending_g)
# assert not m or min(del_costs) >= 0
if edge_ins_cost:
ins_costs = [edge_ins_cost(G2.edges[h]) for h in pending_h]
else:
ins_costs =  * len(pending_h)
# assert not n or min(ins_costs) >= 0
inf = C[0:m, 0:n].sum() + sum(del_costs) + sum(ins_costs) + 1
C[0:m, n : n + m] = np.array(
[del_costs[i] if i == j else inf for i in range(m) for j in range(m)]
).reshape(m, m)
C[m : m + n, 0:n] = np.array(
[ins_costs[i] if i == j else inf for i in range(n) for j in range(n)]
).reshape(n, n)
Ce = make_CostMatrix(C, m, n)
# debug_print(f'Ce: {m} x {n}')
# debug_print(Ce.C)
# debug_print()

maxcost_value = Cv.C.sum() + Ce.C.sum() + 1

if timeout is not None:
if timeout <= 0:
raise nx.NetworkXError("Timeout value must be greater than 0")
start = time.perf_counter()

def prune(cost):
if timeout is not None:
if time.perf_counter() - start > timeout:
return True
if upper_bound is not None:
if cost > upper_bound:
return True
if cost > maxcost_value:
return True
if strictly_decreasing and cost >= maxcost_value:
return True
return False

# Now go!

done_uv = [] if roots is None else [roots]

for vertex_path, edge_path, cost in get_edit_paths(
done_uv, pending_u, pending_v, Cv, [], pending_g, pending_h, Ce, initial_cost
):
# assert sorted(G1.nodes) == sorted(u for u, v in vertex_path if u is not None)
# assert sorted(G2.nodes) == sorted(v for u, v in vertex_path if v is not None)
# assert sorted(G1.edges) == sorted(g for g, h in edge_path if g is not None)
# assert sorted(G2.edges) == sorted(h for g, h in edge_path if h is not None)
# print(vertex_path, edge_path, cost, file = sys.stderr)
# assert cost == maxcost_value
yield list(vertex_path), list(edge_path), cost

[docs]
@nx._dispatch
def simrank_similarity(
G,
source=None,
target=None,
importance_factor=0.9,
max_iterations=1000,
tolerance=1e-4,
):
"""Returns the SimRank similarity of nodes in the graph G.

SimRank is a similarity metric that says "two objects are considered
to be similar if they are referenced by similar objects." _.

The pseudo-code definition from the paper is::

def simrank(G, u, v):
in_neighbors_u = G.predecessors(u)
in_neighbors_v = G.predecessors(v)
scale = C / (len(in_neighbors_u) * len(in_neighbors_v))
return scale * sum(simrank(G, w, x)
for w, x in product(in_neighbors_u,
in_neighbors_v))

where G is the graph, u is the source, v is the target,
and C is a float decay or importance factor between 0 and 1.

The SimRank algorithm for determining node similarity is defined in
_.

Parameters
----------
G : NetworkX graph
A NetworkX graph

source : node
If this is specified, the returned dictionary maps each node
v in the graph to the similarity between source and
v.

target : node
If both source and target are specified, the similarity
value between source and target is returned. If
target is specified but source is not, this argument is
ignored.

importance_factor : float
The relative importance of indirect neighbors with respect to
direct neighbors.

max_iterations : integer
Maximum number of iterations.

tolerance : float
Error tolerance used to check convergence. When an iteration of
the algorithm finds that no similarity value changes more than
this amount, the algorithm halts.

Returns
-------
similarity : dictionary or float
If source and target are both None, this returns a
dictionary of dictionaries, where keys are node pairs and value
are similarity of the pair of nodes.

If source is not None but target is, this returns a
dictionary mapping node to the similarity of source and that
node.

If neither source nor target is None, this returns
the similarity value for the given pair of nodes.

Examples
--------
>>> G = nx.cycle_graph(2)
>>> nx.simrank_similarity(G)
{0: {0: 1.0, 1: 0.0}, 1: {0: 0.0, 1: 1.0}}
>>> nx.simrank_similarity(G, source=0)
{0: 1.0, 1: 0.0}
>>> nx.simrank_similarity(G, source=0, target=0)
1.0

The result of this function can be converted to a numpy array
representing the SimRank matrix by using the node order of the
graph to determine which row and column represent each node.
Other ordering of nodes is also possible.

>>> import numpy as np
>>> sim = nx.simrank_similarity(G)
>>> np.array([[sim[u][v] for v in G] for u in G])
array([[1., 0.],
[0., 1.]])
>>> sim_1d = nx.simrank_similarity(G, source=0)
>>> np.array([sim[v] for v in G])
array([1., 0.])

References
----------
..  https://en.wikipedia.org/wiki/SimRank
..  G. Jeh and J. Widom.
"SimRank: a measure of structural-context similarity",
In KDD'02: Proceedings of the Eighth ACM SIGKDD
International Conference on Knowledge Discovery and Data Mining,
pp. 538--543. ACM Press, 2002.
"""
import numpy as np

nodelist = list(G)
s_indx = None if source is None else nodelist.index(source)
t_indx = None if target is None else nodelist.index(target)

x = _simrank_similarity_numpy(
G, s_indx, t_indx, importance_factor, max_iterations, tolerance
)

if isinstance(x, np.ndarray):
if x.ndim == 1:
return dict(zip(G, x))
# else x.ndim == 2
return {u: dict(zip(G, row)) for u, row in zip(G, x)}
return x

def _simrank_similarity_python(
G,
source=None,
target=None,
importance_factor=0.9,
max_iterations=1000,
tolerance=1e-4,
):
"""Returns the SimRank similarity of nodes in the graph G.

This pure Python version is provided for pedagogical purposes.

Examples
--------
>>> G = nx.cycle_graph(2)
>>> nx.similarity._simrank_similarity_python(G)
{0: {0: 1, 1: 0.0}, 1: {0: 0.0, 1: 1}}
>>> nx.similarity._simrank_similarity_python(G, source=0)
{0: 1, 1: 0.0}
>>> nx.similarity._simrank_similarity_python(G, source=0, target=0)
1
"""
# build up our similarity adjacency dictionary output
newsim = {u: {v: 1 if u == v else 0 for v in G} for u in G}

# These functions compute the update to the similarity value of the nodes
# u and v with respect to the previous similarity values.
def avg_sim(s):
return sum(newsim[w][x] for (w, x) in s) / len(s) if s else 0.0

def sim(u, v):

for its in range(max_iterations):
oldsim = newsim
newsim = {u: {v: sim(u, v) if u is not v else 1 for v in G} for u in G}
is_close = all(
all(
abs(newsim[u][v] - old) <= tolerance * (1 + abs(old))
for v, old in nbrs.items()
)
for u, nbrs in oldsim.items()
)
if is_close:
break

if its + 1 == max_iterations:
raise nx.ExceededMaxIterations(
f"simrank did not converge after {max_iterations} iterations."
)

if source is not None and target is not None:
return newsim[source][target]
if source is not None:
return newsim[source]
return newsim

def _simrank_similarity_numpy(
G,
source=None,
target=None,
importance_factor=0.9,
max_iterations=1000,
tolerance=1e-4,
):
"""Calculate SimRank of nodes in G using matrices with numpy.

The SimRank algorithm for determining node similarity is defined in
_.

Parameters
----------
G : NetworkX graph
A NetworkX graph

source : node
If this is specified, the returned dictionary maps each node
v in the graph to the similarity between source and
v.

target : node
If both source and target are specified, the similarity
value between source and target is returned. If
target is specified but source is not, this argument is
ignored.

importance_factor : float
The relative importance of indirect neighbors with respect to
direct neighbors.

max_iterations : integer
Maximum number of iterations.

tolerance : float
Error tolerance used to check convergence. When an iteration of
the algorithm finds that no similarity value changes more than
this amount, the algorithm halts.

Returns
-------
similarity : numpy array or float
If source and target are both None, this returns a
2D array containing SimRank scores of the nodes.

If source is not None but target is, this returns an
1D array containing SimRank scores of source and that
node.

If neither source nor target is None, this returns
the similarity value for the given pair of nodes.

Examples
--------
>>> G = nx.cycle_graph(2)
>>> nx.similarity._simrank_similarity_numpy(G)
array([[1., 0.],
[0., 1.]])
>>> nx.similarity._simrank_similarity_numpy(G, source=0)
array([1., 0.])
>>> nx.similarity._simrank_similarity_numpy(G, source=0, target=0)
1.0

References
----------
..  G. Jeh and J. Widom.
"SimRank: a measure of structural-context similarity",
In KDD'02: Proceedings of the Eighth ACM SIGKDD
International Conference on Knowledge Discovery and Data Mining,
pp. 538--543. ACM Press, 2002.
"""
# This algorithm follows roughly
#
#     S = max{C * (A.T * S * A), I}
#
# where C is the importance factor, A is the column normalized
# adjacency matrix, and I is the identity matrix.
import numpy as np

# column-normalize the adjacency_matrix
s[s == 0] = 1

newsim = np.eye(len(G), dtype=np.float64)
for its in range(max_iterations):
prevsim = newsim.copy()
np.fill_diagonal(newsim, 1.0)

if np.allclose(prevsim, newsim, atol=tolerance):
break

if its + 1 == max_iterations:
raise nx.ExceededMaxIterations(
f"simrank did not converge after {max_iterations} iterations."
)

if source is not None and target is not None:
return newsim[source, target]
if source is not None:
return newsim[source]
return newsim

[docs]
@nx._dispatch(edge_attrs="weight")
def panther_similarity(
G, source, k=5, path_length=5, c=0.5, delta=0.1, eps=None, weight="weight"
):
r"""Returns the Panther similarity of nodes in the graph G to node v.

Panther is a similarity metric that says "two objects are considered
to be similar if they frequently appear on the same paths." _.

Parameters
----------
G : NetworkX graph
A NetworkX graph
source : node
Source node for which to find the top k similar other nodes
k : int (default = 5)
The number of most similar nodes to return
path_length : int (default = 5)
How long the randomly generated paths should be (T in _)
c : float (default = 0.5)
A universal positive constant used to scale the number
of sample random paths to generate.
delta : float (default = 0.1)
The probability that the similarity $S$ is not an epsilon-approximation to (R, phi),
where $R$ is the number of random paths and $\phi$ is the probability
that an element sampled from a set $A \subseteq D$, where $D$ is the domain.
eps : float or None (default = None)
The error bound. Per _, a good value is sqrt(1/|E|). Therefore,
if no value is provided, the recommended computed value will be used.
weight : string or None, optional (default="weight")
The name of an edge attribute that holds the numerical value
used as a weight. If None then each edge has weight 1.

Returns
-------
similarity : dictionary
Dictionary of nodes to similarity scores (as floats). Note:
the self-similarity (i.e., v) will not be included in
the returned dictionary.

Examples
--------
>>> G = nx.star_graph(10)
>>> sim = nx.panther_similarity(G, 0)

References
----------
..  Zhang, J., Tang, J., Ma, C., Tong, H., Jing, Y., & Li, J.
Panther: Fast top-k similarity search on large networks.
In Proceedings of the ACM SIGKDD International Conference
on Knowledge Discovery and Data Mining (Vol. 2015-August, pp. 1445–1454).
Association for Computing Machinery. https://doi.org/10.1145/2783258.2783267.
"""
import numpy as np

num_nodes = G.number_of_nodes()
if num_nodes < k:
warnings.warn(
f"Number of nodes is {num_nodes}, but requested k is {k}. "
"Setting k to number of nodes."
)
k = num_nodes
# According to , they empirically determined
# a good value for eps to be sqrt( 1 / |E| )
if eps is None:
eps = np.sqrt(1.0 / G.number_of_edges())

inv_node_map = {name: index for index, name in enumerate(G.nodes)}
node_map = np.array(G)

# Calculate the sample size R for how many paths
# to randomly generate
t_choose_2 = math.comb(path_length, 2)
sample_size = int((c / eps**2) * (np.log2(t_choose_2) + 1 + np.log(1 / delta)))
index_map = {}
_ = list(
generate_random_paths(
G, sample_size, path_length=path_length, index_map=index_map, weight=weight
)
)
S = np.zeros(num_nodes)

inv_sample_size = 1 / sample_size

source_paths = set(index_map[source])

# Calculate the path similarities
# between source (v) and node (v_j)
# using our inverted index mapping of
# vertices to paths
for node, paths in index_map.items():
# Only consider paths where both
# node and source are present
common_paths = source_paths.intersection(paths)
S[inv_node_map[node]] = len(common_paths) * inv_sample_size

# Retrieve top k similar
# Note: the below performed anywhere from 4-10x faster
# (depending on input sizes) vs the equivalent np.argsort(S)[::-1]
top_k_unsorted = np.argpartition(S, -k)[-k:]
top_k_sorted = top_k_unsorted[np.argsort(S[top_k_unsorted])][::-1]

# Add back the similarity scores
top_k_sorted_names = (node_map[n] for n in top_k_sorted)
top_k_with_val = dict(zip(top_k_sorted_names, S[top_k_sorted]))

# Remove the self-similarity
top_k_with_val.pop(source, None)

[docs]
@nx._dispatch(edge_attrs="weight")
def generate_random_paths(
G, sample_size, path_length=5, index_map=None, weight="weight"
):
"""Randomly generate sample_size paths of length path_length.

Parameters
----------
G : NetworkX graph
A NetworkX graph
sample_size : integer
The number of paths to generate. This is R in _.
path_length : integer (default = 5)
The maximum size of the path to randomly generate.
This is T in _. According to the paper, T >= 5 is
recommended.
index_map : dictionary, optional
If provided, this will be populated with the inverted
index of nodes mapped to the set of generated random path
indices within paths.
weight : string or None, optional (default="weight")
The name of an edge attribute that holds the numerical value
used as a weight. If None then each edge has weight 1.

Returns
-------
paths : generator of lists
Generator of sample_size paths each with length path_length.

Examples
--------
Note that the return value is the list of paths:

>>> G = nx.star_graph(3)
>>> random_path = nx.generate_random_paths(G, 2)

By passing a dictionary into index_map, it will build an
inverted index mapping of nodes to the paths in which that node is present:

>>> G = nx.star_graph(3)
>>> index_map = {}
>>> random_path = nx.generate_random_paths(G, 3, index_map=index_map)
>>> paths_containing_node_0 = [random_path[path_idx] for path_idx in index_map.get(0, [])]

References
----------
..  Zhang, J., Tang, J., Ma, C., Tong, H., Jing, Y., & Li, J.
Panther: Fast top-k similarity search on large networks.
In Proceedings of the ACM SIGKDD International Conference
on Knowledge Discovery and Data Mining (Vol. 2015-August, pp. 1445–1454).
Association for Computing Machinery. https://doi.org/10.1145/2783258.2783267.
"""
import numpy as np

# Calculate transition probabilities between
# every pair of vertices according to Eq. (3)

node_map = np.array(G)
num_nodes = G.number_of_nodes()

for path_index in range(sample_size):
# Sample current vertex v = v_i uniformly at random
node_index = np.random.randint(0, high=num_nodes)
node = node_map[node_index]

# Add v into p_r and add p_r into the path set
# of v, i.e., P_v
path = [node]

# Build the inverted index (P_v) of vertices to paths
if index_map is not None:
if node in index_map:
else:
index_map[node] = {path_index}

starting_index = node_index
for _ in range(path_length):
# Randomly sample a neighbor (v_j) according
# to transition probabilities from node (v) to its neighbors
neighbor_index = np.random.choice(
num_nodes, p=transition_probabilities[starting_index]
)

# Set current vertex (v = v_j)
starting_index = neighbor_index