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betweenness.c
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betweenness.c
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/*
** Computes the Brandes betweenness centrality for nodes and links,
** and reports min/max/avg node/edge betweenness (normalized).
**
** See Ulrik Brandes, "A faster algorithm for betweenness centrality,"
** Journal of Mathematical Sociology, v25, n2, pp. 163-177, 2001
** for the fundamental algorithm for computing node betweenness.
**
** See Brandes, "On variants of shortest-path betweenness
** centrality and their generic computation," Social Networks, Nov 2007
** for the slight changes needed to compute edge betweenness.
**
** ---------------------------------------------------------------------
** Copyright (C) 2010 The Regents of the University of California.
**
** This program is free software: you can redistribute it and/or modify
** it under the terms of the GNU General Public License as published by
** the Free Software Foundation, either version 3 of the License, or
** (at your option) any later version.
**
** This program is distributed in the hope that it will be useful,
** but WITHOUT ANY WARRANTY; without even the implied warranty of
** MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
** GNU General Public License for more details.
**
** You should have received a copy of the GNU General Public License
** along with this program. If not, see <http://www.gnu.org/licenses/>.
*/
#include <unistd.h>
#include <stdio.h>
#include <math.h>
#include <Judy.h>
/*
** Whether to normalize betweenness in the old way, by dividing by n(n-1)
** instead of (n-1)(n-2)/2. The factor n(n-1) was used in "Lessons from
** Three Views of the Internet Topology" and is based on the upper bound
** given in "Exploring networks with traceroute-like probes: theory and
** simulation". However, the exact upper bound on betweenness is
** (n-1)(n-2)/2, as shown in "A Set of Measures of Centrality Based on
** Betweenness", and this is the normalization factor used by Boost and R.
** Therefore, to produce widely comparable values, we should normalize by
** (n-1)(n-2)/2 by default and only normalize by n(n-1) when validating
** against the Three Views paper.
*/
int loose_normalization = 0;
Pvoid_t nodes = (Pvoid_t)NULL;
Pvoid_t links = (Pvoid_t)NULL;
Pvoid_t queue = (Pvoid_t)NULL;
Pvoid_t visited = (Pvoid_t)NULL; /* Judy1 */
Pvoid_t distances = (Pvoid_t)NULL;
/*
** Because delta and centrality are floating values, we can't use Judy
** arrays to store them. Therefore, we use {node_index} to map the index
** of nodes in {nodes} to indexes in {delta} and {centrality}, since the
** indexes in {nodes} may be sparse.
*/
Pvoid_t node_index = (Pvoid_t)NULL;
double *delta = NULL; /* delta[t]: dependency of s on t */
double *node_centrality = NULL; /* C_B[v]: betweenness centrality for node v */
/*
** C_B[(v,w)]: betweenness centrality for edge (v, w)
**
** Warning: We represent undirected links with a pair of symmetric directed
** links and the link betweenness is calculated per *directed*
** link. For summary statistics like mean, min, and max, this
** doesn't matter, but it might matter for some things, so you
** (the programmer who is thinking of implementing some new
** analysis) should keep this in mind. Specifically, it may
** mean you need to limit yourself to the link betweenness value
** for just one of the two symmetric directed links (e.g., only
** use the value for the direction going from a lower-numbered
** node to a higher-numbered node).
**
** Unlike with nodes, there is no need to map a sparse link ID to an array
** index because link IDs aren't present in the input graph and therefore
** aren't subject to the whims of the user.
**
** To index into {edge_centrality}, we need a link ID. Link IDs are only
** implicit in our graph representation; namely, each directed link is
** stored in {links}, and the index of a link in {links} is its ID for the
** purposes of indexing into {edge_centrality}. The {links} array also
** contains some metadata (specifically, the degree of each node), so our
** use of this implicit link ID will mean {edge_centrality} needs to be
** larger than it absolutely needs to be. However, this overhead will be
** less than using a separate link ID map. In the worst case of every node
** having a degree of 1, {edge_centrality} will be twice as large as it
** needs to be, but the overhead decreases quickly as node degree increases
** (for example, if all nodes have degree 2, then {edge_centrality} will
** only be 1.3x as large as it needs to be). So in practice, the overhead
** is not an issue.
*/
double *edge_centrality = NULL;
unsigned long num_nodes, num_links;
unsigned long long num_pairs; /* C(n, 2) pairs of nodes */
/*
** The number of elements in the {links} array.
**
** This is a fairly low-level bit of information and closely tied to the
** exact details of the graph representation in {nodes} and {links}, but we
** need this information to determine what size {edge_centrality} should
** have and for iterating over {edge_centrality}.
*/
unsigned long links_array_size;
/* ====================================================================== */
void load_graph(void);
void dump_graph(void);
void dump_links(Word_t i, Word_t li);
void compute_brandes_betweenness_centrality(void);
void compute_node_brandes_betweenness_centrality(Word_t s);
void compute_node_dependency(Word_t s, Pvoid_t L, Word_t ltail, Pvoid_t P,
Pvoid_t sigma);
void normalize_centrality(void);
void compute_centrality_statistics(double *centrality, unsigned long len,
const char *name);
void free_predecessor_array(Pvoid_t P);
void fill_array(double *x, unsigned long len, double value);
/* ====================================================================== */
void
load_graph(void)
{
Word_t pi, i, v, l0, li, d, *pv;
int rc;
num_nodes = num_links = 0;
pi = 0; /* previous node ID; 0 == no previous node */
l0 = 0; /* starting index for the links of the current node */
li = 1; /* index of current link */
d = 0; /* node degree */
while ((rc = scanf("%lu %lu", &i, &v)) == 2) {
if (i == 0 || v == 0) {
fputs("ERROR: node IDs must be > 0\n", stderr);
exit(1);
}
if (i != pi) {
if (pi != 0) {
JLI(pv, links, l0); *pv = d; /* save the degree of the previous node */
l0 = li++;
d = 0;
}
JLI(pv, node_index, i); *pv = num_nodes;
++num_nodes;
pi = i;
JLI(pv, nodes, i); *pv = l0;
}
++num_links;
JLI(pv, links, li); *pv = v;
++li;
++d;
}
if (rc != EOF && rc != 2) {
fputs("ERROR: malformed line in input graph\n", stderr);
exit(1);
}
if (num_nodes > 0) {
JLI(pv, links, l0); *pv = d; /* save the degree of the last node */
}
links_array_size = (unsigned long)li;
num_pairs = (unsigned long long)num_nodes * (num_nodes - 1) / 2;
num_links /= 2; /* count undirected links */
printf("loaded %lu nodes, %lu undirected links, %llu pairs\n",
num_nodes, num_links, num_pairs);
}
/* ====================================================================== */
void
dump_graph(void)
{
Word_t i, *pv;
i = 0;
JLF(pv, nodes, i);
while (pv != NULL) {
dump_links(i, *pv);
JLN(pv, nodes, i);
}
}
/* ====================================================================== */
void
dump_links(Word_t i, Word_t li)
{
Word_t d, *pv;
JLG(pv, links, li);
d = *pv;
while (d > 0) {
--d;
++li;
JLG(pv, links, li);
printf("%lu %lu\n", i, *pv);
}
}
/* ====================================================================== */
void
compute_brandes_betweenness_centrality(void)
{
Word_t s, *pv;
s = 0;
JLF(pv, nodes, s);
while (pv != NULL) {
#ifdef DEBUG
printf("\n* node %lu:\n", s);
#endif
compute_node_brandes_betweenness_centrality(s);
JLN(pv, nodes, s);
}
normalize_centrality();
compute_centrality_statistics(node_centrality, num_nodes, "node");
compute_centrality_statistics(edge_centrality, links_array_size, "edge");
}
/* ====================================================================== */
/*
** Variable names are taken from Brandes' Algorithm 1 where possible:
**
** s -- (lower case s) The starting node for traversals.
**
** L -- Brandes calls this S (capital S), but we use L to avoid confusion
** with s (lower case s).
**
** L is a list of nodes reachable from the starting node s in
** non-decreasing order of distance from s. Brandes uses a stack,
** but we simply use a list, since iterating over the final L in
** reverse order achieves the desired effect of a stack without the
** complexity.
**
** P[w] -- A list of the predecessors of node w in the shortest
** paths from s to w. Each P[w] is a Judy array.
**
** We need to know the predecessor nodes for computing the _node_
** betweenness, and we need to know the predecessor links for
** computing the _edge_ betweenness. For this reason, the P[w]
** subarray contains pairs of values: (link ID, node ID). For
** example, if w has the incoming shortest-path links (x, w), (y,
** w), and (z, w), with link IDs 1, 2, and 3, then P[w] has the
** values [1, x, 2, y, 3, z]. We could just store [1, 2, 3] and
** obtain the values x, y, z by indexing into {links}, but storing
** the node IDs directly in P[w] improves the locality of reference.
**
** sigma[t] -- The number of shortest paths from s to t.
** sigma[s] is 1 by convention.
**
** distances[t] -- The shortest path distance from s to t. This
** corresponds to Brandes' d[t]. We use a global variable to avoid
** repeatedly re-allocating this array.
**
** delta[t] -- The dependency of s on t (delta_s_dot(t) in the paper).
*/
void
compute_node_brandes_betweenness_centrality(Word_t s)
{
Pvoid_t L = (Pvoid_t)NULL;
Pvoid_t P = (Pvoid_t)NULL;
Pvoid_t P_entry = (Pvoid_t)NULL;
Pvoid_t sigma = (Pvoid_t)NULL;
Word_t *pv, Rc_word, ltail, qhead, qtail, v, w, li, deg;
Word_t dv, dw, sigma_v, sigma_w;
int Rc_int;
ltail = 0; /* index of next available entry in L */
JLI(pv, sigma, s); *pv = 1; /* sigma[s] = 1 by convention */
qhead = qtail = 0;
JLI(pv, queue, qtail); *pv = s; ++qtail;
J1FA(Rc_word, visited);
J1S(Rc_int, visited, s);
JLI(pv, distances, s); *pv = 0;
while (qhead != qtail) {
JLG(pv, queue, qhead); v = *pv;
JLD(Rc_int, queue, qhead);
++qhead;
JLI(pv, L, ltail); *pv = v; ++ltail;
JLG(pv, distances, v); dv = *pv;
JLG(pv, nodes, v); li = *pv;
JLG(pv, links, li); deg = *pv;
JLG(pv, sigma, v); sigma_v = *pv;
while (deg > 0) {
--deg;
++li;
JLG(pv, links, li); w = *pv;
#ifdef DEBUG
printf(" ? %lu %lu\n", v, w);
#endif
J1T(Rc_int, visited, w);
if (Rc_int) { /* already visited */
JLG(pv, distances, w); dw = *pv;
}
else {
dw = dv + 1;
#ifdef DEBUG
printf(" q %lu %lu = %lu\n", v, w, dw);
#endif
J1S(Rc_int, visited, w);
JLI(pv, distances, w); *pv = dw;
JLI(pv, queue, qtail); *pv = w; ++qtail;
}
if (dw == dv + 1) { /* the shortest path from s to w is via v? */
#ifdef DEBUG
printf(" P[%lu] << %lu\n", w, v);
#endif
JLG(pv, sigma, w); sigma_w = (pv ? *pv : 0);
JLI(pv, sigma, w); *pv = sigma_w + sigma_v;
/* Append v to P[w]. We actually append the pair (link ID, v);
see the comments above this function for details. */
JLG(pv, P, w); P_entry = (pv ? (Pvoid_t)*pv : (Pvoid_t)NULL);
JLC(Rc_word, P_entry, 0, -1); /* Rc_word == 0 if P_entry == NULL */
JLI(pv, P_entry, Rc_word); *pv = li; /* array is created if needed */
JLI(pv, P_entry, Rc_word + 1); *pv = v;
JLI(pv, P, w); *pv = (Word_t)P_entry;
}
}
}
compute_node_dependency(s, L, ltail, P, sigma);
JLFA(Rc_word, L);
free_predecessor_array(P);
JLFA(Rc_word, sigma);
}
/* ====================================================================== */
/*
** Computes delta[t], the dependency of s on t (delta_s_dot(t) in the
** paper), and partially computes C_B[w] for all nodes reachable from a
** single source node s.
*/
void
compute_node_dependency(Word_t s, Pvoid_t L, Word_t ltail, Pvoid_t P,
Pvoid_t sigma)
{
Pvoid_t P_entry = (Pvoid_t)NULL;
Word_t *pv, j, li, v, w, vi, wi, sigma_v, sigma_w;
double factor;
fill_array(delta, num_nodes, 0.0);
while (ltail-- > 0) {
JLG(pv, L, ltail); w = *pv;
JLG(pv, sigma, w); sigma_w = *pv;
JLG(pv, node_index, w); wi = *pv;
#ifdef DEBUG
printf(" L: w = %lu; sigma_w = %lu; wi = %lu\n", w, sigma_w, wi);
#endif
JLG(pv, P, w);
if (pv) {
P_entry = (Pvoid_t)*pv;
for (j = 0; ; j += 2) {
JLG(pv, P_entry, j);
if (!pv) break; /* end of P_entry array */
li = *pv; /* link ID for incoming link */
JLG(pv, P_entry, j + 1);
v = *pv; /* predecessor node */
JLG(pv, sigma, v); sigma_v = *pv;
JLG(pv, node_index, v); vi = *pv;
factor = (sigma_v / (double)sigma_w) * (1.0 + delta[wi]);
delta[vi] += factor;
if (edge_centrality[li] < 0.0) {
edge_centrality[li] = factor;
}
else {
edge_centrality[li] += factor;
}
#ifdef DEBUG
printf(" P[w]: v = %lu, sigma_v = %lu; vi = %lu; d[vi] = %.5f\n",
v, sigma_v, vi, delta[vi]);
#endif
}
}
if (w != s) {
node_centrality[wi] += delta[wi];
#ifdef DEBUG
printf(" C_B[w] = %.5f\n", node_centrality[wi]);
#endif
}
}
}
/* ====================================================================== */
/*
** Normalizes the absolute node and edge centrality values.
**
** Note: Because we represent each undirected link with a pair of symmetric
** directed links, we need to divide the computed absolute node
** centrality values by 2 in addition to normalizing by n(n - 1) or
** (n - 1)(n - 2)/2.
**
** In contrast, we shouldn't divide edge centrality by 2.
*/
void normalize_centrality(void)
{
double factor = (loose_normalization ? num_nodes * (num_nodes - 1)
: (num_nodes - 1) * (num_nodes - 2) / 2.0);
double *x = node_centrality;
double *end = x + num_nodes;
double d = 2.0 * factor;
while (x < end) {
*x++ /= d;
}
x = edge_centrality;
end = x + links_array_size;
d = factor;
while (x < end) {
if (*x >= 0.0) { /* edge centrality has -1.0 at non-link entries */
*x /= d;
}
++x;
}
}
/* ====================================================================== */
/*
** Computes the average, std dev, min, and max of the node and edge
** betweenness centrality.
**
** The sample standard deviation code is efficient and numerically stable.
** However, we compute the average betweenness separately rather than
** simply re-using the mean calculated by the incremental standard
** deviation code because the latter seems to produce a mean that is
** slightly less accurate due to round-off errors.
*/
void
compute_centrality_statistics(double *centrality, unsigned long len,
const char *name)
{
unsigned long i, num_values=0;
double x, sum, min_betweenness, max_betweenness;
#ifdef DEBUG
printf("\n* %s centrality distribution:\n", name);
#endif
sum = 0.0;
min_betweenness = -1.0;
max_betweenness = 0.0;
for (i = 0; i < len; i++) {
x = centrality[i];
if (x < 0.0) continue; /* edge centrality has -1.0 at non-link entries */
num_values += 1;
sum += x;
#ifdef DEBUG
printf(" %.5f\n", x);
#endif
if (x > max_betweenness) {
max_betweenness = x;
}
if (x < min_betweenness || min_betweenness < 0) {
min_betweenness = x;
}
}
printf("min %s betweenness = %.4e\n", name, min_betweenness);
printf("average %s betweenness = %.4e\n", name, sum / num_values);
printf("max %s betweenness = %.4e\n", name, max_betweenness);
}
/* ====================================================================== */
void
free_predecessor_array(Pvoid_t P)
{
Pvoid_t P_entry = (Pvoid_t)NULL;
Word_t Rc_word, i, *pv;
i = 0;
JLF(pv, P, i);
while (pv != NULL) {
P_entry = (Pvoid_t)*pv;
JLFA(Rc_word, P_entry);
JLN(pv, P, i);
}
JLFA(Rc_word, P);
}
/* ====================================================================== */
void fill_array(double *x, unsigned long len, double value)
{
double *end = x + len;
while (x < end) {
*x++ = value;
}
}
/* ====================================================================== */
int
main(int argc, char *argv[])
{
int c;
while ((c = getopt(argc, argv, "z")) != -1) {
switch (c) {
case 'z':
loose_normalization = 1;
break;
case '?':
default:
fprintf(stderr, "Usage: betweenness [-z]\n"
"Options:\n"
" -z to normalize betweenness with n(n-1) [NOT recommended]\n");
exit(1);
}
}
argc -= optind;
argv += optind;
load_graph();
#ifdef DEBUG
dump_graph();
#endif
/* Allocate {delta} once globally to avoid allocating for each source node. */
delta = (double *)malloc(num_nodes * sizeof(double));
node_centrality = (double *)malloc(num_nodes * sizeof(double));
fill_array(node_centrality, num_nodes, 0.0);
edge_centrality = (double *)malloc(links_array_size * sizeof(double));
fill_array(edge_centrality, links_array_size, -1);
compute_brandes_betweenness_centrality();
return 0;
}