def test_Degree_Reach(self):
"""
Test reach at radius 1 - the reach values are no the degrees of the nodes
"""
compute_centrality(self.graph, self.nodes, True, False, False, False,
False, 1, True, 1, [], [])
assert eq_tol(getattr(self.graph["A"], REACH), 2)
assert eq_tol(getattr(self.graph["B"], REACH), 2)
assert eq_tol(getattr(self.graph["C"], REACH), 3)
assert eq_tol(getattr(self.graph["D"], REACH), 1)
def test_Betweenness(self):
"""
Test betweenness at infinite radius
"""
compute_centrality(self.graph, self.nodes, False, False, True, False,
False, INFINITE_RADIUS, True, 1, [], [])
assert eq_tol(getattr(self.graph["A"], BETWEENNESS), 0)
assert eq_tol(getattr(self.graph["B"], BETWEENNESS), 0)
assert eq_tol(getattr(self.graph["C"], BETWEENNESS), 6)
assert eq_tol(getattr(self.graph["D"], BETWEENNESS), 0)
def test_Infinite_Reach(self):
"""
Test reach at infinite radius
"""
compute_centrality(self.graph, self.nodes, True, False, False, False,
False, INFINITE_RADIUS, True, 1, [], [])
assert eq_tol(getattr(self.graph["A"], REACH), 3)
assert eq_tol(getattr(self.graph["B"], REACH), 3)
assert eq_tol(getattr(self.graph["C"], REACH), 3)
assert eq_tol(getattr(self.graph["D"], REACH), 3)
def test_Closeness(self):
"""
Test closeness at infinite radius
"""
compute_centrality(self.graph, self.nodes, False, False, False, True,
False, INFINITE_RADIUS, True, 1, [], [])
assert eq_tol(getattr(self.graph["A"], CLOSENESS), 1.0 / 7)
assert eq_tol(getattr(self.graph["B"], CLOSENESS), 1.0 / 7)
assert eq_tol(getattr(self.graph["C"], CLOSENESS), 1.0 / 5)
assert eq_tol(getattr(self.graph["D"], CLOSENESS), 1.0 / 11)
def test_Gravity(self):
"""
Test gravity at infinite radius and a beta value of 1
"""
compute_centrality(self.graph, self.nodes, False, True, False, False, False,
INFINITE_RADIUS, 1, [], [])
assert eq_tol(getattr(self.graph["A"], GRAVITY), 1)
assert eq_tol(getattr(self.graph["B"], GRAVITY), 1)
assert eq_tol(getattr(self.graph["C"], GRAVITY), 4.0 / 3)
assert eq_tol(getattr(self.graph["D"], GRAVITY), 2.0 / 3)
def test_Gravity(self):
"""
Test gravity at infinite radius and a beta value of 1
"""
compute_centrality(self.graph, self.nodes, False, True, False, False,
False, INFINITE_RADIUS, True, 1, [], [])
assert eq_tol(getattr(self.graph["A"], GRAVITY), 1)
assert eq_tol(getattr(self.graph["B"], GRAVITY), 1)
assert eq_tol(getattr(self.graph["C"], GRAVITY), 4.0 / 3)
assert eq_tol(getattr(self.graph["D"], GRAVITY), 2.0 / 3)
def test_Degree_Reach(self):
"""
Test reach at radius 1 - the reach values are no the degrees of the nodes
"""
compute_centrality(self.graph, self.nodes, True, False, False, False, False,
1, 1, [], [])
assert eq_tol(getattr(self.graph["A"], REACH), 2)
assert eq_tol(getattr(self.graph["B"], REACH), 2)
assert eq_tol(getattr(self.graph["C"], REACH), 3)
assert eq_tol(getattr(self.graph["D"], REACH), 1)
def test_Infinite_Reach(self):
"""
Test reach at infinite radius
"""
compute_centrality(self.graph, self.nodes, True, False, False, False, False,
INFINITE_RADIUS, 1, [], [])
assert eq_tol(getattr(self.graph["A"], REACH), 3)
assert eq_tol(getattr(self.graph["B"], REACH), 3)
assert eq_tol(getattr(self.graph["C"], REACH), 3)
assert eq_tol(getattr(self.graph["D"], REACH), 3)
def test_Closeness(self):
"""
Test closeness at infinite radius
"""
compute_centrality(self.graph, self.nodes, False, False, False, True, False,
INFINITE_RADIUS, 1, [], [])
assert eq_tol(getattr(self.graph["A"], CLOSENESS), 1.0 / 7)
assert eq_tol(getattr(self.graph["B"], CLOSENESS), 1.0 / 7)
assert eq_tol(getattr(self.graph["C"], CLOSENESS), 1.0 / 5)
assert eq_tol(getattr(self.graph["D"], CLOSENESS), 1.0 / 11)
def test_Betweenness(self):
"""
Test betweenness at infinite radius
"""
compute_centrality(self.graph, self.nodes, False, False, True, False, False,
INFINITE_RADIUS, 1, [], [])
assert eq_tol(getattr(self.graph["A"], BETWEENNESS), 0)
assert eq_tol(getattr(self.graph["B"], BETWEENNESS), 0)
assert eq_tol(getattr(self.graph["C"], BETWEENNESS), 6)
assert eq_tol(getattr(self.graph["D"], BETWEENNESS), 0)
def test_Straightness(self):
"""
Test straightness at infinite radius
"""
compute_centrality(self.graph, self.nodes, False, False, False, False,
True, INFINITE_RADIUS, True, 1, [], [])
assert eq_tol(getattr(self.graph["A"], STRAIGHTNESS),
2 + sqrt(5) / (1 + sqrt(2)))
assert eq_tol(getattr(self.graph["B"], STRAIGHTNESS),
2 + sqrt(5) / (1 + sqrt(2)))
assert eq_tol(getattr(self.graph["C"], STRAIGHTNESS), 3)
assert eq_tol(getattr(self.graph["D"], STRAIGHTNESS),
1 + 2 * sqrt(5) / (1 + sqrt(2)))
def test_Straightness(self):
"""
Test straightness at infinite radius
"""
compute_centrality(self.graph, self.nodes, False, False, False, False, True,
INFINITE_RADIUS, 1, [], [])
assert eq_tol(getattr(self.graph["A"], STRAIGHTNESS),
2 + sqrt(5) / (1 + sqrt(2)))
assert eq_tol(getattr(self.graph["B"], STRAIGHTNESS),
2 + sqrt(5) / (1 + sqrt(2)))
assert eq_tol(getattr(self.graph["C"], STRAIGHTNESS), 3)
assert eq_tol(getattr(self.graph["D"], STRAIGHTNESS),
1 + 2 * sqrt(5) / (1 + sqrt(2)))
def compute_centrality(nodes, origins, compute_r, compute_g, compute_b,
compute_c, compute_s, radius, network_radius, beta, measures_to_normalize,
accumulator_fields):
"""
Computes reach, gravity, betweenness, closeness, and straightness on a graph.
|nodes|: graph representation; dictionary mapping node id's to |Node| objects
|origins|: subset of nodes that will be used as sources of shortest path trees
|compute_r|: compute reach?
|compute_g|: compute gravity type index?
|compute_b|: compute betweenness?
|compute_c|: compute closeness?
|compute_s|: compute straightness?
|radius|: for each node, only consider other nodes that can be reached within
this distance
|network_radius|: use network radius or birds-eye radius?
|beta|: parameter for gravity type index
|measures_to_normalize|: a list of measures to normalize
|accumulator_fields|: a list of cost attributes to accumulate
"""
# Number of nodes in the graph
N = len(nodes)
O = len(origins)
if O > N:
raise Invalid_Parameters_Exception("size of origins exceeds size of nodes")
elif O == 0:
return
# Preprocessing
have_accumulations = len(accumulator_fields) > 0
if have_accumulations:
empty_accumulations = lambda: dict((field, 0.0) for field in
accumulator_fields)
have_locations = hasattr(nodes.values()[0], LOCATION)
if compute_s and not have_locations:
# We cannot compute straightness without node locations
compute_s = False
if compute_b:
# Initialize betweenness values
for id in nodes:
setattr(nodes[id], BETWEENNESS, 0.0)
# Initialize the sum of all node weights (normalization)
sum_weights = 0.0
# Computation
progress = Progress_Bar(O, 1, STEP_4)
for s in origins:
if s not in nodes:
continue
weight_s = getattr(nodes[s], WEIGHT)
if have_locations: location_s = getattr(nodes[s], LOCATION)
sum_weights += weight_s
# Initialize reach (weighted and unweighted) computation for |s|
# (normalization)
reach_s = -1
weighted_reach_s = -weight_s
# Initialize measures
if compute_g: gravity_s = 0.0
if compute_b:
P = {s: []} # Predecessors
S = [] # Stack containing nodes in the order they are extended
sigma = {s: 1.0} # Number of shortest paths from |s| to other nodes
delta = {} # Dependency of |s| on other nodes
if compute_c: d_sum_s = 0.0
if compute_s: straightness_s = 0.0
if have_accumulations:
accumulations_s = {s: empty_accumulations()}
d = {s: 0.0} # Shortest distance from |s| to other nodes
# Queue for Dijkstra
Q = [(0.0, s)] if network_radius else [(0.0, s, 0.0)]
# If we use euclidean radius, make a list of all reachable nodes
if not network_radius:
reachable_s = set()
for t in nodes:
location_t = getattr(nodes[t], LOCATION)
if dist(location_s, location_t) <= radius:
reachable_s.add(t)
# Dijkstra
while Q and (True if network_radius else reachable_s):
# Pop the closest node to |s| from |Q|
if network_radius:
d_sv, v = heappop(Q)
else:
d_sv, v, dist_sv = heappop(Q)
if v in reachable_s:
reachable_s.remove(v)
weight_v = getattr(nodes[v], WEIGHT)
if have_locations: location_v = getattr(nodes[v], LOCATION)
compute = network_radius or dist_sv <= radius
if compute:
reach_s += 1
weighted_reach_s += weight_v
if d_sv > 0:
if compute_g: gravity_s += weight_v * exp(-d_sv * beta)
if compute_c: d_sum_s += weight_v * d_sv
if compute_s: straightness_s += (weight_v *
dist(location_s, location_v) / d_sv)
if compute_b: S.append(v)
for w, d_vw, accumulations_vw in getattr(nodes[v], NEIGHBORS):
# s ~ ... ~ v ~ w
d_sw = d_sv + d_vw
if not network_radius:
# Use Euclidean distance
location_w = getattr(nodes[w], LOCATION)
dist_sw = dist(location_s, location_w)
if compute_b: b_refresh = False
add_w_to_Q = False
if not w in d: # Found a path from |s| to |w| for the first time
if d_sw <= radius or not network_radius:
add_w_to_Q = True
d[w] = d_sw
if compute_b: b_refresh = True
elif lt_tol(d_sw, d[w]): # Found a better path from |s| to |w|
if d_sw <= radius or not network_radius:
if d[w] <= radius or not network_radius:
longer_path_node = (d[w], w) if network_radius else (d[w], w,
dist_sw)
Q.remove(longer_path_node)
heapify(Q)
add_w_to_Q = True
d[w] = d_sw
if compute_b: b_refresh = True
if add_w_to_Q:
new_node = (d_sw, w) if network_radius else (d_sw, w, dist_sw)
heappush(Q, new_node)
if have_accumulations:
accumulations_s[w] = merge_maps(accumulations_s[v],
dict(accumulations_vw), add)
if compute_b:
if b_refresh:
sigma[w] = 0.0
P[w] = []
if eq_tol(d_sw, d[w]): # Count all shortest paths from |s| to |w|
sigma[w] += sigma[v] # Update the number of shortest paths
P[w].append(v) # |v| is a predecessor of |w|
delta[v] = 0.0 # Recognize |v| as a predecessor
if compute_r: setattr(nodes[s], REACH, weighted_reach_s)
if compute_g: setattr(nodes[s], GRAVITY, gravity_s)
if compute_b:
while S: # Revisit nodes in reverse order of distance from |s|
w = S.pop()
delta_w = delta[w] if w in delta else 0.0 # Dependency of |s| on |w|
for v in P[w]:
weight_w = getattr(nodes[w], WEIGHT)
delta[v] += sigma[v] / sigma[w] * (weight_w + delta_w)
if w != s:
between_w = getattr(nodes[w], BETWEENNESS)
setattr(nodes[w], BETWEENNESS, between_w + delta_w)
if compute_c: setattr(nodes[s], CLOSENESS, (1.0 / d_sum_s if d_sum_s > 0
else 0.0))
if compute_s: setattr(nodes[s], STRAIGHTNESS, straightness_s)
nodes[s].reach = reach_s
nodes[s].weighted_reach = weighted_reach_s
if have_accumulations:
total_accumulations_s = empty_accumulations()
for v in accumulations_s:
total_accumulations_s = merge_maps(total_accumulations_s,
accumulations_s[v], add)
for field in accumulator_fields:
setattr(nodes[s], field, total_accumulations_s[field])
progress.step()
# Normalization
if BETWEENNESS in measures_to_normalize and O < N:
measures_to_normalize.remove(BETWEENNESS)
AddWarning(WARNING_NO_BETWEENNESS_NORMALIZATION)
if measures_to_normalize:
norm_progress = Progress_Bar(O, 1, PROGRESS_NORMALIZATION)
for s in origins:
if s not in nodes:
continue
reach_s = nodes[s].reach
weighted_reach_s = nodes[s].weighted_reach
# Normalize reach
if compute_r and REACH in measures_to_normalize:
weight_s = getattr(nodes[s], WEIGHT)
try: setattr(nodes[s], NORM_REACH, reach_s / (sum_weights - weight_s))
except: setattr(nodes[s], NORM_REACH, 0.0)
# Normalize gravity
if compute_g and GRAVITY in measures_to_normalize:
gravity_s = getattr(nodes[s], GRAVITY)
try: setattr(nodes[s], NORM_GRAVITY, (exp(beta) * gravity_s /
weighted_reach_s))
except: setattr(nodes[s], NORM_GRAVITY, 0.0)
# Normalize betweenness
if compute_b and BETWEENNESS in measures_to_normalize:
betweenness_s = getattr(nodes[s], BETWEENNESS)
try: setattr(nodes[s], NORM_BETWEENNESS, (betweenness_s /
(weighted_reach_s * (reach_s - 1))))
except: setattr(nodes[s], NORM_BETWEENNESS, 0.0)
# Normalize closeness
if compute_c and CLOSENESS in measures_to_normalize:
closeness_s = getattr(nodes[s], CLOSENESS)
try: setattr(nodes[s], NORM_CLOSENESS, closeness_s * weighted_reach_s)
except: setattr(nodes[s], NORM_CLOSENESS, 0.0)
# Normalize straightness
if compute_s and STRAIGHTNESS in measures_to_normalize:
straightness_s = getattr(nodes[s], STRAIGHTNESS)
try: setattr(nodes[s], NORM_STRAIGHTNESS, (straightness_s /
weighted_reach_s))
except: setattr(nodes[s], NORM_STRAIGHTNESS, 0.0)
norm_progress.step()
def compute_centrality(nodes, origins, compute_r, compute_g, compute_b,
compute_c, compute_s, radius, beta, measures_to_normalize,
accumulator_fields):
"""
Computes reach, gravity, betweenness, closeness, and straightness on a graph.
|nodes|: graph representation; dictionary mapping node id's to |Node| objects
|origins|: subset of nodes that will be used as sources of shortest path trees
|compute_r|: compute reach?
|compute_g|: compute gravity type index?
|compute_b|: compute betweenness?
|compute_c|: compute closeness?
|compute_s|: compute straightness?
|radius|: for each node, only consider other nodes that can be reached within
this distance
|beta|: parameter for gravity type index
|measures_to_normalize|: a list of measures to normalize
|accumulator_fields|: a list of cost attributes to accumulate
"""
# Number of nodes in the graph
N = len(nodes)
O = len(origins)
if N == 0 or O == 0:
return
# Preprocessing
have_accumulations = len(accumulator_fields) > 0
have_locations = hasattr(nodes.values()[0], LOCATION)
if compute_s and not have_locations:
# We cannot compute straightness without node locations
compute_s = False
if compute_b:
# Initialize betweenness values
for id in nodes:
setattr(nodes[id], BETWEENNESS, 0.0)
# Initialize the sum of all node weights (normalization)
sum_weights = 0.0
# Computation
progress = Progress_Bar(O, 1, STEP_4)
for s in origins:
weight_s = getattr(nodes[s], WEIGHT)
if have_locations: location_s = getattr(nodes[s], LOCATION)
sum_weights += weight_s
# Initialize reach (weighted and unweighted) computation for |s|
# (normalization)
reach_s = -1
weighted_reach_s = -weight_s
# Initialize measures
if compute_g: gravity_s = 0.0
if compute_b:
P = {s: []} # Predecessors
S = [] # Stack containing nodes in the order they are extended
sigma = {s: 1.0} # Number of shortest paths from |s| to other nodes
delta = {} # Dependency of |s| on other nodes
if compute_c: d_sum_s = 0.0
if compute_s: straightness_s = 0.0
if have_accumulations:
empty_accumulations = lambda: dict((field, 0.0) for field in
accumulator_fields)
accumulations_s = {s: empty_accumulations()}
d = {s: 0.0} # Shortest distance from |s| to other nodes
Q = [(0.0, s)] # Queue for Dijkstra
# Dijkstra
while Q:
# Pop the closest node to |s| from |Q|
d_sv, v = heappop(Q)
weight_v = getattr(nodes[v], WEIGHT)
if have_locations: location_v = getattr(nodes[v], LOCATION)
reach_s += 1
weighted_reach_s += weight_v
if d_sv > 0:
if compute_g: gravity_s += weight_v * exp(-d_sv * beta)
if compute_c: d_sum_s += weight_v * d_sv
if compute_s: straightness_s += (weight_v * dist(location_s, location_v)
/ d_sv)
if compute_b: S.append(v)
for w, d_vw, accumulations_vw in getattr(nodes[v], NEIGHBORS):
# s ~ ... ~ v ~ w
d_sw = d_sv + d_vw
if compute_b: b_refresh = False
if not w in d: # Found a path from |s| to |w| for the first time
if d_sw <= radius:
heappush(Q, (d_sw, w)) # Add |w| to |Q|
if have_accumulations:
accumulations_s[w] = merge_maps(accumulations_s[v],
dict(accumulations_vw), add)
d[w] = d_sw
if compute_b: b_refresh = True
elif lt_tol(d_sw, d[w]): # Found a better path from |s| to |w|
if d_sw <= radius:
if d[w] <= radius:
Q.remove((d[w], w))
heapify(Q)
heappush(Q, (d_sw, w)) # Add |w| to |Q|
if have_accumulations:
accumulations_s[w] = merge_maps(accumulations_s[v],
dict(accumulations_vw), add)
d[w] = d_sw
if compute_b: b_refresh = True
if compute_b:
if b_refresh:
sigma[w] = 0.0
P[w] = []
if eq_tol(d_sw, d[w]): # Count all shortest paths from |s| to |w|
sigma[w] += sigma[v] # Update the number of shortest paths
P[w].append(v) # |v| is a predecessor of |w|
delta[v] = 0.0 # Recognize |v| as a predecessor
if compute_r: setattr(nodes[s], REACH, weighted_reach_s)
if compute_g: setattr(nodes[s], GRAVITY, gravity_s)
if compute_b:
while S: # Revisit nodes in reverse order of distance from |s|
w = S.pop()
delta_w = delta[w] if w in delta else 0.0 # Dependency of |s| on |w|
for v in P[w]:
weight_w = getattr(nodes[w], WEIGHT)
delta[v] += sigma[v] / sigma[w] * (weight_w + delta_w)
if w != s:
between_w = getattr(nodes[w], BETWEENNESS)
setattr(nodes[w], BETWEENNESS, between_w + delta_w)
if compute_c: setattr(nodes[s], CLOSENESS, (1.0 / d_sum_s if d_sum_s > 0
else 0.0))
if compute_s: setattr(nodes[s], STRAIGHTNESS, straightness_s)
nodes[s].reach = reach_s
nodes[s].weighted_reach = weighted_reach_s
if have_accumulations:
total_accumulations_s = empty_accumulations()
for v in accumulations_s:
total_accumulations_s = merge_maps(total_accumulations_s,
accumulations_s[v], add)
for field in accumulator_fields:
setattr(nodes[s], field, total_accumulations_s[field])
progress.step()
# Normalization
if BETWEENNESS in measures_to_normalize and O < N:
measures_to_normalize.remove(BETWEENNESS)
AddWarning(WARNING_NO_BETWEENNESS_NORMALIZATION)
if measures_to_normalize:
norm_progress = Progress_Bar(O, 1, PROGRESS_NORMALIZATION)
for s in origins:
reach_s = nodes[s].reach
weighted_reach_s = nodes[s].weighted_reach
# Normalize reach
if compute_r and REACH in measures_to_normalize:
weight_s = getattr(nodes[s], WEIGHT)
try: setattr(nodes[s], NORM_REACH, reach_s / (sum_weights - weight_s))
except: setattr(nodes[s], NORM_REACH, 0.0)
# Normalize gravity
if compute_g and GRAVITY in measures_to_normalize:
gravity_s = getattr(nodes[s], GRAVITY)
try: setattr(nodes[s], NORM_GRAVITY, (exp(beta) * gravity_s /
weighted_reach_s))
except: setattr(nodes[s], NORM_GRAVITY, 0.0)
# Normalize betweenness
if compute_b and BETWEENNESS in measures_to_normalize:
betweenness_s = getattr(nodes[s], BETWEENNESS)
try: setattr(nodes[s], NORM_BETWEENNESS, (betweenness_s /
(weighted_reach_s * (reach_s - 1))))
except: setattr(nodes[s], NORM_BETWEENNESS, 0.0)
# Normalize closeness
if compute_c and CLOSENESS in measures_to_normalize:
closeness_s = getattr(nodes[s], CLOSENESS)
try: setattr(nodes[s], NORM_CLOSENESS, closeness_s * weighted_reach_s)
except: setattr(nodes[s], NORM_CLOSENESS, 0.0)
# Normalize straightness
if compute_s and STRAIGHTNESS in measures_to_normalize:
straightness_s = getattr(nodes[s], STRAIGHTNESS)
try: setattr(nodes[s], NORM_STRAIGHTNESS, (straightness_s /
weighted_reach_s))
except: setattr(nodes[s], NORM_STRAIGHTNESS, 0.0)
norm_progress.step()
