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r""" 

Centrality 

  

This module is meant for all functions related to centrality in networks. 

  

.. csv-table:: 

:class: contentstable 

:widths: 30, 70 

:delim: | 

  

:func:`centrality_betweenness` | Return the centrality betweenness of `G` 

:func:`centrality_closeness_top_k` | Return the k most closeness central vertices of `G` 

  

Functions 

--------- 

""" 

from __future__ import print_function, absolute_import 

  

from libc.string cimport memset 

from libc.stdint cimport uint32_t 

from cysignals.memory cimport check_allocarray, sig_free 

from cysignals.signals cimport sig_check 

  

include "sage/data_structures/bitset.pxi" 

from sage.graphs.base.static_sparse_graph cimport * 

from sage.libs.gmp.mpq cimport * 

from sage.rings.rational cimport Rational 

from sage.ext.memory_allocator cimport MemoryAllocator 

  

ctypedef fused numerical_type: 

mpq_t 

double 

  

cimport cython 

  

def centrality_betweenness(G, exact=False, normalize=True): 

r""" 

Return the centrality betweenness of `G` 

  

The centrality betweenness of a vertex `v\in G` is defined by: 

  

.. MATH:: 

  

c(v) = \sum_{s\neq v \neq t} \frac{\#\{\text{shortest } st-\text{paths containing v}\}} 

{\#\{\text{shortest } st-\text{paths}\}} 

  

For more information, see the :wikipedia:`Betweenness_centrality`. 

  

INPUT: 

  

- ``G`` -- a (di)graph 

  

- ``exact`` (boolean, default: ``False``) -- whether to compute over 

rationals or on ``double`` C variables. 

  

- ``normalize`` (boolean; default: ``True``) -- whether to renormalize the 

values by dividing them by `\binom {n-1} 2` (for graphs) or `2\binom {n-1} 

2` (for digraphs). 

  

ALGORITHM: 

  

To compute `c(v)`, we fix `s` and define `c_s(v)` as the centrality of `v` 

*due to* `s`, obtained from the formula above by running the sum over `t` 

only. We obtain `c(v)=\sum_{s\neq v} c_s(v)`. 

  

For every vertex `s`, we compute the value of `c_s(v)` for all `v`, using 

the following remark (see [Brandes01]_): 

  

Let `v_1,...,v_k` be the out-neighbors of `v` such that 

`dist(s,v_i)=dist(s,v)+1`. Then 

  

.. MATH:: 

  

c_s(v) = \sum_{1\leq i \leq k} c_s(v_i) 

\frac{\#\{\text{shortest } sv_i-\text{paths}\}} 

{\#\{\text{shortest } sv -\text{paths}\}} 

  

The number of shortest paths between `s` and every other vertex can be 

computed with a slightly modified BFS. While running this BFS we can also 

store the list of the vertices `v_1,...,v_k` associated with each `v`. 

  

EXAMPLES:: 

  

sage: from sage.graphs.centrality import centrality_betweenness 

sage: centrality_betweenness(digraphs.Circuit(6)) # abs tol 1e-10 

{0: 0.5, 1: 0.5, 2: 0.5, 3: 0.5, 4: 0.5, 5: 0.5} 

sage: centrality_betweenness(graphs.CycleGraph(6)) # abs tol 1e-10 

{0: 0.2, 1: 0.2, 2: 0.2, 3: 0.2, 4: 0.2, 5: 0.2} 

  

Exact computations:: 

  

sage: graphs.PetersenGraph().centrality_betweenness(exact=True) 

{0: 1/12, 1: 1/12, 2: 1/12, 3: 1/12, 4: 1/12, 5: 1/12, 6: 1/12, 7: 1/12, 8: 1/12, 9: 1/12} 

  

TESTS: 

  

Compare with NetworkX:: 

  

sage: import networkx 

sage: g = graphs.RandomGNP(100,.2) 

sage: nw = networkx.betweenness_centrality(g.networkx_graph(copy=False)) 

sage: sg = centrality_betweenness(g) 

sage: max(abs(nw[x]-sg[x]) for x in g) # abs tol 1e-10 

0 

  

Stupid cases:: 

  

sage: centrality_betweenness(Graph()) 

{} 

sage: centrality_betweenness(Graph(2)) 

{0: 0.0, 1: 0.0} 

sage: centrality_betweenness(Graph(2),exact=1) 

{0: 0, 1: 0} 

  

REFERENCES: 

  

.. [Brandes01] Ulrik Brandes, 

A faster algorithm for betweenness centrality, 

Journal of Mathematical Sociology 25.2 (2001): 163-177, 

http://www.inf.uni-konstanz.de/algo/publications/b-fabc-01.pdf 

""" 

if exact: 

return centrality_betweenness_C(G,<mpq_t> 0,normalize=normalize) 

else: 

return centrality_betweenness_C(G,<double>0,normalize=normalize) 

  

@cython.cdivision(True) 

cdef dict centrality_betweenness_C(G, numerical_type _, normalize=True): 

r""" 

Return the centrality betweenness of G (C implementation) 

  

INPUT: 

  

- ``G`` -- a graph 

  

- ``_`` -- this variable is ignored, only its type matters. If it is of type 

`mpq_t` then computations are made on `Q`, if it is ``double`` the 

computations are made on ``double``. 

  

- ``normalize`` (boolean; default: ``True``) -- whether to renormalize the 

values by dividing them by `\binom {n-1} 2` (for graphs) or `2\binom {n-1} 

2` (for digraphs). 

  

For more information, see the documentation of ``centrality_betweenness``. 

""" 

# Trivial case 

if G.order() <= 2: 

zero = 0. if numerical_type is double else Rational(0) 

return {v:zero for v in G} 

  

# A copy of G, for faster neighbor enumeration 

cdef short_digraph g 

  

# A second copy, to remember the edges used during the BFS (see doc) 

cdef short_digraph bfs_dag 

  

cdef int n = G.order() 

  

cdef bitset_t seen # Vertices whose neighbors have been explored 

cdef bitset_t next_layer # Unexplored neighbors of vertices in 'seen' 

  

cdef uint32_t * queue # BFS queue 

cdef uint32_t * degrees # degree[v] = nb of vertices which discovered v 

  

cdef numerical_type * n_paths_from_source 

cdef numerical_type * betweenness_source 

cdef numerical_type * betweenness 

cdef numerical_type mpq_tmp 

  

cdef int layer_current_beginning 

cdef int layer_current_end 

cdef int layer_next_end 

  

cdef int source,i,j,u,v 

cdef uint32_t * p_tmp 

  

if numerical_type is mpq_t: 

mpq_init(mpq_tmp) 

  

try: 

init_short_digraph(g, G, edge_labelled = False) 

init_reverse(bfs_dag, g) 

  

queue = <uint32_t *> check_allocarray(n, sizeof(uint32_t)) 

degrees = <uint32_t *> check_allocarray(n, sizeof(uint32_t)) 

n_paths_from_source = <numerical_type *> check_allocarray(n, sizeof(numerical_type)) 

betweenness_source = <numerical_type *> check_allocarray(n, sizeof(numerical_type)) 

betweenness = <numerical_type *> check_allocarray(n, sizeof(numerical_type)) 

  

bitset_init(seen,n) 

bitset_init(next_layer,n) 

  

if numerical_type is double: 

memset(betweenness,0,n*sizeof(double)) 

else: 

for i in range(n): 

mpq_init(betweenness[i]) 

mpq_set_ui(betweenness[i],0,1) 

mpq_init(betweenness_source[i]) 

mpq_init(n_paths_from_source[i]) 

  

for source in range(n): 

  

if numerical_type is double: 

memset(betweenness_source ,0,n*sizeof(double)) 

memset(n_paths_from_source,0,n*sizeof(double)) 

n_paths_from_source[source]=1 

else: 

for i in range(n): 

mpq_set_ui(betweenness_source[i] ,0,1) 

mpq_set_ui(n_paths_from_source[i],0,1) 

mpq_set_ui(n_paths_from_source[source],1,1) 

  

# initialize data 

bitset_set_first_n(seen, 0) 

bitset_add(seen,source) 

bitset_set_first_n(next_layer, 0) 

  

memset(degrees,0,n*sizeof(uint32_t)) 

  

queue[0] = source 

layer_current_beginning = 0 

layer_current_end = 1 

layer_next_end = 1 

  

# The number of shortest paths from 'source' to every other vertex. 

# 

# It is a BFS. The graph is explored layer by layer. 

while layer_current_beginning<layer_current_end: 

  

# Looking for all non-discovered neighbors of some vertex of the 

# current layer. 

for j in range(layer_current_beginning,layer_current_end): 

u = queue[j] 

  

# List the neighbors of u 

p_tmp = g.neighbors[u] 

while p_tmp<g.neighbors[u+1]: 

v = p_tmp[0] 

p_tmp += 1 

  

# Is it a new vertex ? 

if bitset_in(seen,v): 

continue 

  

# Is it the first time we see it ? 

elif not bitset_in(next_layer,v): 

bitset_add(next_layer,v) 

queue[layer_next_end] = v 

layer_next_end += 1 

  

# update the count of paths and the BFS dag. 

bfs_dag.neighbors[v][degrees[v]] = u 

degrees[v] += 1 

if numerical_type is double: 

n_paths_from_source[v] += n_paths_from_source[u] 

else: 

mpq_add(n_paths_from_source[v],n_paths_from_source[v],n_paths_from_source[u]) 

  

# 'next_layer' becomes 'current_layer' 

for j in range(layer_current_end, layer_next_end): 

bitset_add(seen,queue[j]) 

  

layer_current_beginning = layer_current_end 

layer_current_end = layer_next_end 

  

# Compute the betweenness from the number of paths 

# 

# We enumerate vertices in reverse order of discovery. 

for i in range(layer_current_end-1,-1,-1): 

u = queue[i] 

for j in range(degrees[u]): 

v = bfs_dag.neighbors[u][j] 

if v != source: # better to not 'if' but set it to 0 afterwards? 

if numerical_type is double: 

betweenness_source[v] += (betweenness_source[u]+1)*(n_paths_from_source[v]/n_paths_from_source[u]) 

else: 

mpq_set_ui(mpq_tmp,1,1) 

mpq_add(mpq_tmp,betweenness_source[u],mpq_tmp) 

mpq_mul(mpq_tmp,mpq_tmp,n_paths_from_source[v]) 

mpq_div(mpq_tmp,mpq_tmp,n_paths_from_source[u]) 

mpq_add(betweenness_source[v],betweenness_source[v],mpq_tmp) 

  

# update betweenness from betweenness_source 

for i in range(n): 

if numerical_type is double: 

betweenness[i] += betweenness_source[i] 

else: 

mpq_add(betweenness[i],betweenness[i],betweenness_source[i]) 

  

sig_check() # check for KeyboardInterrupt 

  

if numerical_type is double: 

betweenness_list = [betweenness[i] for i in range(n)] 

else: 

betweenness_list = [Rational(None) for x in range(n)] 

  

for i in range(n): 

(<Rational> (betweenness_list[i])).set_from_mpq(betweenness[i]) 

for i in range(n): 

mpq_clear(betweenness_source[i]) 

mpq_clear(betweenness[i]) 

mpq_clear(n_paths_from_source[i]) 

mpq_clear(mpq_tmp) 

  

finally: 

free_short_digraph(g) 

free_short_digraph(bfs_dag) 

bitset_free(seen) 

bitset_free(next_layer) 

sig_free(queue) 

sig_free(n_paths_from_source) 

sig_free(degrees) 

sig_free(betweenness_source) 

sig_free(betweenness) 

  

if not G.is_directed(): 

betweenness_list = [x/2 for x in betweenness_list] 

  

if normalize: 

if G.is_directed(): 

betweenness_list = [ x/((n-1)*(n-2)) for x in betweenness_list] 

else: 

betweenness_list = [2*x/((n-1)*(n-2)) for x in betweenness_list] 

  

return {vv:betweenness_list[i] for i,vv in enumerate(G.vertices())} 

  

cdef void _estimate_reachable_vertices_dir(short_digraph g, int* reachL, int* reachU): 

r""" 

For each vertex ``v``, bounds the number of vertices reachable from ``v``. 

  

The lower bound is stored in reachL[v], while the upper bound is stored 

in reachU[v]. These two arrays must be pre-allocated and they must 

have size at least `n`, where `n` is the number of nodes of `g`. 

  

The estimate works as follows: first, we compute the graph of strongly 

connected components `\mathcal{G=(V,E)}`, then, for each SCC C, we set: 

  

.. MATH:: 

  

L(C)=|C|+\max_{(C,C') \in \mathcal{E}}L(C') \\ 

U(C)=|C|+\max_{(C,C') \in \mathcal{E}}L(C') 

  

By analyzing strongly connected components in reverse topological order, 

we are sure that, as soon as we process component `C`, all components 

`C'` appearing on the right hand side have already been processed. 

A further improvement on these bounds is obtained by exactly computing 

the number of vertices reachable from the biggest strongly connected 

component, and handle this component separately. 

  

Then, for each vertex ``v``, we set ``reachL[v]=L(C)``, where ``C`` is 

the strongly connected component containing ``v``. 

  

INPUT: 

  

``g`` (short_digraph): the input graph; 

  

OUTPUT: 

  

``reachL``, ``reachU``: two arrays that should be allocated outside 

this function and that should have size at least ``g.n``. At the end, 

``reachL[v]`` (resp., ``reachU[v]``) will contain the lower (resp., upper) 

bound on the number of reachable vertices from ``v``. 

""" 

cdef MemoryAllocator mem = MemoryAllocator() 

cdef int n = g.n 

cdef int* scc = <int *> mem.malloc(n * sizeof(int)) 

cdef int i, v, w, maxscc = 0 

cdef int nscc = tarjan_strongly_connected_components_C(g, scc) 

cdef short_digraph sccgraph 

strongly_connected_components_digraph_C(g, nscc, scc, sccgraph) 

  

cdef int* scc_sizes = <int *> mem.calloc(nscc, sizeof(int)) 

cdef int nreach_maxscc = 0 

cdef short* reach_max_scc = <short *> mem.calloc(nscc, sizeof(short)) 

cdef int* reachL_scc = <int *> mem.calloc(nscc, sizeof(int)) 

cdef uint64_t* reachU_scc = <uint64_t *> mem.calloc(nscc, sizeof(uint64_t)) 

cdef uint64_t* reachU_without_maxscc = <uint64_t *> mem.calloc(nscc, sizeof(uint64_t)) 

# We need uint64_t because these values may become much bigger than g.n, 

# up to g.n^2, during the computation. Only at the end, we set reachL and 

# reachU as the maximum between g.n and the computed value (so that they 

# can be converted to int without overflow). 

  

# Variables used in BFS from the largest strongly connected component 

cdef uint32_t startq, endq 

cdef int* q = <int *> mem.malloc(nscc * sizeof(int)) 

cdef short* reached = <short *> mem.calloc(nscc, sizeof(short)) 

cdef uint32_t *neigh_start 

cdef uint32_t *neigh_end 

  

# Compute scc_sizes 

for i in range(g.n): 

scc_sizes[scc[i]] += 1 

  

# Compute maxscc 

for i in range(nscc): 

if scc_sizes[maxscc] < scc_sizes[i]: 

maxscc = i 

reach_max_scc[maxscc] = 1 

  

# BFS to compute number of reachable vertices for the biggest SCC. 

q[0] = maxscc 

nreach_maxscc = scc_sizes[maxscc] 

reached[maxscc] = 1 

startq = 0 

endq = 1 

while startq < endq: 

v = q[startq] 

startq += 1 

neigh_start = sccgraph.neighbors[v] 

neigh_end = sccgraph.neighbors[v+1] 

  

while (neigh_start < neigh_end): 

w = neigh_start[0] 

if not reached[w]: 

reached[w] = 1 

nreach_maxscc += scc_sizes[w] 

q[endq] = w 

endq += 1 

neigh_start += 1 

  

reachL_scc[maxscc] = nreach_maxscc 

reachU_scc[maxscc] = nreach_maxscc 

reachU_without_maxscc[maxscc] = 0 

# Dynamic programming to estimate number of reachable vertices for other 

# SCCs 

for i in range(nscc): 

if i == maxscc: 

continue 

  

neigh_start = sccgraph.neighbors[i] 

neigh_end = sccgraph.neighbors[i+1] 

  

while (neigh_start < neigh_end): 

w = neigh_start[0] 

neigh_start += 1 

  

reachL_scc[i] = max(reachL_scc[i], reachL_scc[w]) 

reachU_scc[i] += reachU_scc[w] 

# Note that this might become much bigger than g.n, up to g.n*g.n. 

# Hence we used uint64_t, and only at the end we take the minimum 

# between this value and g.n (since g.n is an upper bound on 

# the number of reachable vertices). 

if not reached[w]: 

reachU_without_maxscc[i] += reachU_without_maxscc[w] 

reach_max_scc[i] = reach_max_scc[i] or reach_max_scc[w] 

  

if reach_max_scc[i]: 

reachU_scc[i] = reachU_without_maxscc[i] + nreach_maxscc 

  

reachL_scc[i] += scc_sizes[i] 

reachU_scc[i] += scc_sizes[i] 

if not reached[i]: 

reachU_without_maxscc[i] += scc_sizes[i] 

  

for i in range(n): 

reachL[i] = reachL_scc[scc[i]] 

reachU[i] = <int> min(reachU_scc[scc[i]], g.n) 

  

cdef void _compute_reachable_vertices_undir(short_digraph g, int* reachable): 

r""" 

For each vertex ``v``, computes the number of vertices reachable from ``v``. 

  

The number of vertices reachable from ``v`` (which is the size of the 

connected component containing ``v``) is stored in variable 

``reachable[v]``. The array ``reachable`` is assumed to be allocated 

outside this function, and it is assumed to have size at least ``g.n``. 

""" 

cdef MemoryAllocator mem = MemoryAllocator() 

cdef int i 

cdef int n = g.n 

cdef int* q = <int *> mem.malloc(n*sizeof(int)) 

cdef short* reached = <short *> mem.calloc(n, sizeof(short)) 

  

cdef int v, w 

cdef uint32_t *neigh_start 

cdef uint32_t *neigh_end 

cdef uint32_t startq, endq 

cdef list currentcc 

  

memset(reachable, 0, n * sizeof(int)) 

  

for i in range(n): 

# BFS from i 

if reachable[i] != 0: 

continue 

  

reached[i] = 1 

currentcc = [i] 

  

q[0] = i 

startq = 0 

endq = 1 

while startq < endq: 

v = q[startq] 

startq += 1 

neigh_start = g.neighbors[v] 

neigh_end = g.neighbors[v+1] 

  

while (neigh_start < neigh_end): 

w = neigh_start[0] 

if not reached[w]: 

reached[w] = 1 

currentcc.append(w) 

q[endq] = w 

endq += 1 

neigh_start += 1 

  

for v in currentcc: 

reachable[v] = len(currentcc) 

  

cdef void _sort_vertices_degree(short_digraph g, int *sorted_verts): 

r""" 

Sorts vertices in decreasing order of degree. 

  

Uses counting sort, since degrees are between `0` and `n-1`: the running 

time is then `O(n)`. 

""" 

cdef MemoryAllocator mem = MemoryAllocator() 

cdef uint32_t *verts_of_degree = <uint32_t*> mem.calloc(g.n, sizeof(uint32_t)) 

cdef uint32_t *next_vert_of_degree = <uint32_t*> mem.malloc(g.n * sizeof(uint32_t)) 

cdef int d, v 

  

# Otherwise, segmentation fault 

if g.n == 0: 

return 

  

for v in range(g.n): 

verts_of_degree[out_degree(g, v)] += 1 

  

next_vert_of_degree[g.n - 1] = 0 

for i in range(g.n-2,-1,-1): 

next_vert_of_degree[i] = next_vert_of_degree[i+1] + verts_of_degree[i+1] 

  

for v in range(g.n): 

d = out_degree(g, v) 

sorted_verts[next_vert_of_degree[d]] = v 

next_vert_of_degree[d] += 1 

  

  

def centrality_closeness_top_k(G, int k=1, int verbose=0): 

r""" 

Computes the k vertices with largest closeness centrality. 

  

The algorithm is based on performing a breadth-first-search (BFS) from each 

vertex, and to use bounds in order to cut these BFSes as soon as possible. 

If k is small, it is much more efficient than computing all centralities 

with :meth:`~sage.graphs.generic_graph.GenericGraph.centrality_closeness`. 

Conversely, if k is close to the number of nodes, the running-time is 

approximately the same (it might even be a bit longer, because more 

computations are needed). 

For more information, see [BCM15]_. The algorithm does not work on 

weighted graphs. 

  

INPUT: 

  

- ``G`` a Sage Graph or DiGraph; 

  

- ``k`` (integer, default: 1): the algorithm will return the ``k`` 

vertices with largest closeness centrality. This value should be between 

1 and the number of vertices with positive (out)degree, because the 

closeness centrality is not defined for vertices with (out)degree 0. If 

``k`` is bigger than this value, the output will contain all vertices 

of positive (out)degree. 

  

- ``verbose`` (integer, default: 0): an integer defining 

how "verbose" the algorithm should be. If 

0, nothing is printed, if 1, we print only the performance ratio at 

the end of the algorithm, if 2, we print partial results every 1000 

visits, if 3, we print partial results after every visit. 

  

OUTPUT: 

  

An ordered list of ``k`` pairs ``(closv, v)``, where ``v`` is one of the 

``k`` most central vertices, and ``closv`` is its closeness centrality. 

If ``k`` is bigger than the number of vertices with positive (out)degree, 

the list might be smaller. 

  

REFERENCES: 

  

.. [BCM15] Michele Borassi, Pierluigi Crescenzi, and Andrea Marino, 

*Fast and Simple Computation of Top-k Closeness Centralities*. 

:arxiv:`1507.01490` 

  

EXAMPLES:: 

  

sage: from sage.graphs.centrality import centrality_closeness_top_k 

sage: g = graphs.PathGraph(10) 

sage: centrality_closeness_top_k(g, 4, 1) 

Final performance ratio: 0.711111111111 

[(0.36, 5), 

(0.36, 4), 

(0.3333333333333333, 6), 

(0.3333333333333333, 3)] 

sage: g = digraphs.Path(10) 

sage: centrality_closeness_top_k(g, 5, 1) 

Final performance ratio: 0.422222222222 

[(0.2, 0), 

(0.19753086419753085, 1), 

(0.19444444444444442, 2), 

(0.19047619047619047, 3), 

(0.18518518518518517, 4)] 

  

TESTS: 

  

If ``k`` or ``verbose`` is not an integer:: 

  

sage: from sage.graphs.centrality import centrality_closeness_top_k 

sage: g = digraphs.Path(10) 

sage: centrality_closeness_top_k(g, 'abc', 1) 

Traceback (most recent call last): 

... 

TypeError: an integer is required 

sage: centrality_closeness_top_k(g, 1, 'abc') 

Traceback (most recent call last): 

... 

TypeError: an integer is required 

  

If ``k`` is bigger than the number of nodes:: 

  

sage: from sage.graphs.centrality import centrality_closeness_top_k 

sage: g = graphs.PathGraph(5) 

sage: centrality_closeness_top_k(g, 10, 0) 

[(0.6666666666666666, 2), 

(0.5714285714285714, 3), 

(0.5714285714285714, 1), 

(0.4, 4), 

(0.4, 0)] 

  

Empty graph:: 

  

sage: from sage.graphs.centrality import centrality_closeness_top_k 

sage: g = Graph() 

sage: centrality_closeness_top_k(g, 10, 0) 

[] 

sage: g = Graph(10) 

sage: centrality_closeness_top_k(g, 10, 0) 

[] 

  

The result is correct:: 

  

sage: from sage.graphs.centrality import centrality_closeness_top_k 

sage: import random 

sage: n = 20 

sage: m = random.randint(1, n*(n-1) / 2) 

sage: k = random.randint(1, n) 

sage: g = graphs.RandomGNM(n,m) 

sage: topk = centrality_closeness_top_k(g, k) 

sage: centr = g.centrality_closeness(algorithm='BFS') 

sage: sorted_centr = sorted(centr.values(), reverse=True) 

sage: assert(len(topk)==min(k, len(sorted_centr))) 

sage: for i in range(len(topk)): 

....: assert(abs(topk[i][0] - sorted_centr[i]) < 1e-12) 

  

Directed case:: 

  

sage: from sage.graphs.centrality import centrality_closeness_top_k 

sage: import random 

sage: n = 20 

sage: m = random.randint(1, n*(n-1)) 

sage: k = random.randint(1, n) 

sage: g = digraphs.RandomDirectedGNM(n,m) 

sage: topk = centrality_closeness_top_k(g, k) 

sage: centr = g.centrality_closeness(algorithm='BFS') 

sage: sorted_centr = sorted(centr.values(), reverse=True) 

sage: assert(len(topk)==min(k, len(sorted_centr))) 

sage: for i in range(len(topk)): 

....: assert(abs(topk[i][0] - sorted_centr[i]) < 1e-12) 

""" 

  

if k >= G.num_verts(): 

closeness_dict = G.centrality_closeness(by_weight=False,algorithm='BFS') 

return sorted([(closz, z) for z,closz in closeness_dict.iteritems()], reverse=True) 

if G.num_verts()==0 or G.num_verts()==1: 

return [] 

  

cdef MemoryAllocator mem = MemoryAllocator() 

cdef short_digraph sd 

# Copying the whole graph to obtain the list of neighbors quicker than by 

# calling out_neighbors. This data structure is well documented in the 

# module sage.graphs.base.static_sparse_graph 

init_short_digraph(sd, G) 

cdef int n = sd.n 

cdef int *reachL = <int *> mem.malloc(n * sizeof(int)) 

cdef int *reachU 

cdef int *pred = <int *> mem.calloc(n, sizeof(int)) 

cdef double *farness = <double *> mem.malloc(n * sizeof(double)) 

cdef int d, nd, x, v, w 

cdef long f, gamma 

cdef int *queue = <int*> mem.malloc(n * sizeof(int)) 

cdef double tildefL, tildefU 

cdef bint stopped 

cdef uint32_t * p_tmp 

cdef int layer_current_beginning, layer_current_end, layer_next_end=0 

cdef long visited = 0 

cdef int nvis = 0 

cdef short *seen = <short *> mem.calloc(n, sizeof(short)) 

cdef bint directed = G.is_directed() 

  

cdef int *topk = <int*> mem.malloc(k * sizeof(int)) 

for i in range(k): 

topk[i] = -1 

for i in range(n): 

pred[i] = -1 

  

cdef double kth = n 

cdef int *sorted_vert = <int *> mem.malloc(n * sizeof(int)) 

if directed: 

reachU = <int *> mem.malloc(n * sizeof(int)) 

_estimate_reachable_vertices_dir(sd, reachL, reachU) 

else: 

_compute_reachable_vertices_undir(sd, reachL) 

reachU = reachL 

_sort_vertices_degree(sd, sorted_vert) 

  

for x in sorted_vert[:n]: 

sig_check() 

  

if out_degree(sd, x) == 0: 

break 

# We start a BFSCut from x: 

  

# We reset variable seen: 

for v in queue[:layer_next_end]: 

seen[v] = 0 

pred[v] = -1 

  

layer_current_beginning = 0 

layer_current_end = 1 

layer_next_end = 1 

d = 0 

f = 0 

# We are at level 0, and gamma is the number of arcs exiting level 0 

# (hence, deg(x)). 

gamma = out_degree(sd, x) 

nd = 1 

queue[0] = x 

stopped = False 

seen[x] = 1 

nvis += 1 

  

# The graph is explored layer by layer. 

while layer_current_beginning<layer_current_end and not stopped: 

sig_check() 

  

# We update our estimate of the farness of v. 

# The estimate sets distance d+1 to gamma vertices (which is an 

# upper bound on the number of vertices at distance d+1 from v), 

# and distance d+2 to all other vertices reachable from x. 

tildefL = ((f - gamma + (d+2) * (<double>(reachL[x]-nd))) * (n-1)) / ((<double>(reachL[x]-1))*(reachL[x]-1)) 

tildefU = ((f - gamma + (d+2) * (<double>(reachU[x]-nd))) * (n-1)) / ((<double>(reachU[x]-1))*(reachU[x]-1)) 

d += 1 

gamma = 0 

  

if tildefL >= kth and tildefU >= kth: 

farness[x] = n 

stopped = True 

break 

# Looking for all non-discovered neighbors of some vertex of the 

# current layer. 

for j in range(layer_current_beginning,layer_current_end): 

sig_check() 

  

u = queue[j] 

  

# List the neighbors of u 

p_tmp = sd.neighbors[u] 

while p_tmp<sd.neighbors[u+1] and not stopped: 

sig_check() 

  

visited += 1 

v = p_tmp[0] 

p_tmp += 1 

# Is it a new vertex ? 

if not seen[v]: 

seen[v] = 1 

queue[layer_next_end] = v 

layer_next_end += 1 

f = f + d 

gamma += out_degree(sd, v) if directed else out_degree(sd, v) - 1 

nd = nd + 1 

pred[v] = u 

elif directed or pred[u] != v: 

tildefL += (n-1) / (<double>(reachL[x]-1)*(reachL[x]-1)) 

tildefU += (n-1) / (<double>(reachU[x]-1)*(reachU[x]-1)) 

if tildefL >= kth and tildefU >= kth: 

farness[x] = n 

stopped = True 

if stopped: 

break 

# 'next_layer' becomes 'current_layer' 

layer_current_beginning = layer_current_end 

layer_current_end = layer_next_end 

  

if not stopped: 

farness[x] = ((<double> f) * (n-1)) / (<double>(nd-1) * (nd-1)) 

  

if farness[x] < kth: 

for i in range(k): 

if topk[i] == -1 or farness[topk[i]] == kth: 

topk[i] = x 

break 

kth = 0 

for i in range(k): 

if topk[i] == -1: 

kth = n 

break 

kth = max(kth, farness[topk[i]]) 

if verbose >= 3 or (verbose == 2 and nvis % 1000 == 0): 

print("Visit {} from {}:".format(nvis, x)) 

print(" Lower bound: {}".format(1 / kth)) 

print(" Perf. ratio: {}".format(visited / (nvis * <double> (sd.neighbors[sd.n]-sd.edges)))) 

  

if verbose > 0: 

print("Final performance ratio: {}".format(visited / (n * <double> (sd.neighbors[sd.n]-sd.edges)))) 

  

cdef list V = G.vertices() 

return sorted([(1.0/farness[v], V[v]) for v in topk[:k] if v != -1], reverse=True)