def EigenValueGap(t):
"""Returns the eigenvalue gap in the second-order transition matrix of a
temporal network, as well as in the corresponding null model. Returns
the tuple (lambda(T2), lambda(T2_null))
@param t: The temporalnetwork instance to work on
"""
#NOTE to myself: most of the time goes for construction of the 2nd order
#NOTE null graph, then for the 2nd order null transition matrix
g2 = t.igraphSecondOrder().components(mode="STRONG").giant()
g2n = t.igraphSecondOrderNull().components(mode="STRONG").giant()
Log.add('Calculating eigenvalue gap ... ', Severity.INFO)
# Build transition matrices
T2 = Utilities.RWTransitionMatrix(g2)
T2n = Utilities.RWTransitionMatrix(g2n)
# Compute eigenvector sequences
# NOTE: ncv=13 sets additional auxiliary eigenvectors that are computed
# NOTE: in order to be more confident to find the one with the largest
# NOTE: magnitude, see
# NOTE: https://github.com/scipy/scipy/issues/4987
w2 = sla.eigs(T2, which="LM", k=2, ncv=13, return_eigenvectors=False)
evals2_sorted = np.sort(-np.absolute(w2))
w2n = sla.eigs(T2n, which="LM", k=2, ncv=13, return_eigenvectors=False)
evals2n_sorted = np.sort(-np.absolute(w2n))
Log.add('finished.', Severity.INFO)
return (np.abs(evals2_sorted[1]), np.abs(evals2n_sorted[1]))
def SlowDownFactor(t):
"""Returns a factor S that indicates how much slower (S>1) or faster (S<1)
a diffusion process in the temporal network evolves on a second-order model
compared to a first-order model. This value captures the effect of order
correlations on a diffusion process in the temporal network.
@param t: The temporalnetwork instance to work on
"""
#NOTE to myself: most of the time goes for construction of the 2nd order
#NOTE null graph, then for the 2nd order null transition matrix
g2 = t.igraphSecondOrder().components(mode="STRONG").giant()
g2n = t.igraphSecondOrderNull().components(mode="STRONG").giant()
Log.add('Calculating slow down factor ... ', Severity.INFO)
# Build transition matrices
T2 = Utilities.RWTransitionMatrix(g2)
T2n = Utilities.RWTransitionMatrix(g2n)
# Compute eigenvector sequences
# NOTE: ncv=13 sets additional auxiliary eigenvectors that are computed
# NOTE: in order to be more confident to find the one with the largest
# NOTE: magnitude, see
# NOTE: https://github.com/scipy/scipy/issues/4987
w2 = sla.eigs(T2, which="LM", k=2, ncv=13, return_eigenvectors=False)
evals2_sorted = np.sort(-np.absolute(w2))
w2n = sla.eigs(T2n, which="LM", k=2, ncv=13, return_eigenvectors=False)
evals2n_sorted = np.sort(-np.absolute(w2n))
Log.add('finished.', Severity.INFO)
return np.log(np.abs(evals2n_sorted[1])) / np.log(np.abs(evals2_sorted[1]))
def RWDiffusion(g, samples=5, epsilon=0.01, max_iterations=100000):
"""Computes the average number of steps requires by a random walk process
to fall below a total variation distance below epsilon (TVD computed between the momentary
visitation probabilities \pi^t and the stationary distribution \pi = \pi^{\infty}. This time can be
used to measure diffusion speed in a given (weighted and directed) network."""
avg_speed = 0
T = Utilities.RWTransitionMatrix(g)
pi = Utilities.StationaryDistribution(T)
n = len(g.vs())
for s in range(samples):
x = np.zeros(n)
seed = np.random.randint(n)
x[seed] = 1
while Utilities.TVD(x, pi) > epsilon:
avg_speed += 1
# NOTE x * T = (T^T * x^T)^T
# NOTE T is already transposed to get the left EV
x = (T.dot(x.transpose())).transpose()
if avg_speed > max_iterations:
Log.add("x[0:10] = " + str(x[0:10]))
Log.add("pi[0:10] = " + str(pi[0:10]))
raise RuntimeError(
"Failed to converge within maximal number of iterations. Start of current x and pi are printed above"
)
return avg_speed / samples
def exportDiffusionMovieFrames(g,
file_prefix='diffusion',
visual_style=None,
steps=100,
initial_index=-1):
"""Exports an animation showing the evolution of a diffusion
process on the network"""
T = Utilities.RWTransitionMatrix(g)
if visual_style == None:
visual_style = {}
visual_style["vertex_color"] = "lightblue"
visual_style["vertex_label"] = g.vs["name"]
visual_style["edge_curved"] = .5
visual_style["vertex_size"] = 30
# lambda expression for the coloring of nodes according to some quantity p \in [0,1]
# p = 1 ==> color red
# p = 0 ==> color white
color_p = lambda p: "rgb(255," + str(int(
(1 - p) * 255)) + "," + str(int((1 - p) * 255)) + ")"
# Initial state of random walker
if initial_index < 0:
initial_index = np.random.randint(0, len(g.vs()))
x = np.zeros(len(g.vs()))
x[initial_index] = 1
# compute stationary state
pi = Utilities.StationaryDistribution(T)
scale = np.mean(np.abs(x - pi))
# Plot network (useful as poster frame of video)
igraph.plot(g, file_prefix + "_network.pdf", **visual_style)
# Create frames
for i in range(0, steps):
visual_style["vertex_color"] = [color_p(p**0.1) for p in x]
igraph.plot(g, file_prefix + "_frame_" + str(i).zfill(5) + ".png",
**visual_style)
if i % 10 == 0:
Log.add('Step' + str(i) + '\tTVD = ' + str(Utilities.TVD(x, pi)),
Severity.INFO)
# NOTE x * T = (T^T * x^T)^T
# NOTE T is already transposed to get the left EV
x = (T.dot(x.transpose())).transpose()
def Laplacian(temporalnet, model="SECOND"):
"""Returns the transposed Laplacian matrix corresponding to the the second-order (model=SECOND) or
the second-order null (model=NULL) model for a temporal network.
@param temporalnet: The temporalnetwork instance to work on
@param model: either C{"SECOND"} or C{"NULL"}, where C{"SECOND"} is the
the default value.
"""
if (model is "SECOND" or "NULL") == False:
raise ValueError("model must be one of \"SECOND\" or \"NULL\"")
if model == "SECOND":
network = temporalnet.igraphSecondOrder().components(
mode="STRONG").giant()
elif model == "NULL":
network = temporalnet.igraphSecondOrderNull().components(
mode="STRONG").giant()
T2 = Utilities.RWTransitionMatrix(network)
I = sparse.identity(len(network.vs()))
return I - T2
def exportDiffusionMovieFramesFirstOrder(t,
file_prefix='diffusion',
visual_style=None,
steps=100,
initial_index=-1,
model='SECOND',
dynamic=False,
NWframesPerRWStep=5):
"""Exports an animation showing the evolution of a diffusion
process on the first-order aggregate network, where random walk dynamics
either follows a first-order (mode='NULL') or second-order (model='SECOND') Markov
model"""
assert model == 'SECOND' or model == 'NULL'
g1 = t.igraphFirstOrder()
if model == 'SECOND':
g2 = t.igraphSecondOrder()
temporal = tn.TemporalNetwork.ShuffleTwoPaths(t, l=steps)
elif model == 'NULL':
g2 = t.igraphSecondOrderNull()
temporal = tn.TemporalNetwork.ShuffleEdges(t, l=steps)
T = Utilities.RWTransitionMatrix(g2)
# visual style is for *first-order* aggregate network
if visual_style == None:
visual_style = {}
visual_style["vertex_color"] = "lightblue"
visual_style["vertex_label"] = g1.vs["name"]
visual_style["edge_curved"] = .5
visual_style["vertex_size"] = 30
visual_style["edge_color"] = ["darkgrey"] * g1.ecount()
visual_style["edge_width"] = [.5] * g1.ecount()
# Initial state of random walker
if initial_index < 0:
initial_index = np.random.randint(0, len(g2.vs()))
exp = 1.0 / .75
x = np.zeros(len(g2.vs()))
x[initial_index] = 1
# This index allows to quickly map node names to indices in the first-order network
map_name_to_id = {}
for i in range(len(g1.vs())):
map_name_to_id[g1.vs()['name'][i]] = i
# Index to quickly map second-order node indices to first-order node indices
map_2_to_1 = {}
for j in range(len(g2.vs())):
# j is index of node in *second-order* network
# we first get the name of the *target* of the underlying edge
node = g2.vs()["name"][j].split(t.separator)[1]
# we map the target of second-order node j to the index of the *first-order* node
map_2_to_1[j] = map_name_to_id[node]
# compute stationary state of random walk process
pi = Utilities.StationaryDistribution(T)
scale = np.mean(np.abs(x - pi))
# lambda expression for the coloring of nodes according to some quantity p \in [0,1]
# p = 1 ==> color red
# p = 0 ==> color white
color_p = lambda p: "rgb(255," + str(int(
(1 - p) * 255)) + "," + str(int((1 - p) * 255)) + ")"
visual_style["edge_color"] = ["darkgrey"] * g1.ecount()
visual_style["edge_width"] = [.5] * g1.ecount()
# Create video frames
for i in range(0, steps):
# based on visitation probabilities in *second-order* aggregate network,
# we need to compute visitation probabilities of nodes in the *first-order*
# aggregate network
x_firstorder = np.zeros(len(g1.vs()))
# j is the index of nodes in the *second-order* network, which we need to map
# to nodes in the *first-order* network
for j in range(len(x)):
if x[j] > 0:
x_firstorder[
map_2_to_1[j]] = x_firstorder[map_2_to_1[j]] + x[j]
# Perform some reasonable color scaling
visual_style["vertex_color"] = [
color_p(np.power((p - min(x)) / (max(x) - min(x)), exp))
for p in x_firstorder
]
# Visualize illustrative network dynamics
if dynamic == True:
#slice = igraph.Graph(n=len(g1.vs()))
#slice.vs["name"] = g1.vs["name"]
L = len(temporal.ordered_times)
for e in temporal.time[temporal.ordered_times[i % L]]:
e_id = g1.get_eid(e[0], e[1])
visual_style["edge_width"][e_id] = 5
visual_style["edge_color"][e_id] = "black"
#slice.add_edge(e[0], e[1])
igraph.plot(g1, file_prefix + "_frame_" + str(i).zfill(5) + ".png",
**visual_style)
else:
igraph.plot(g1, file_prefix + "_frame_" + str(i).zfill(5) + ".png",
**visual_style)
# Plot first-order aggregate network after 30 RW steps (particularly useful as poster frame of video)
if i == 30:
visual_style["edge_color"] = "black"
visual_style["edge_width"] = 1
igraph.plot(g1, file_prefix + "_network.pdf", **visual_style)
# Reset edge color and width
visual_style["edge_color"] = ["darkgrey"] * g1.ecount()
visual_style["edge_width"] = [.5] * g1.ecount()
if i % 50 == 0:
Log.add('Frame ' + str(i) + '\tTVD = ' + str(Utilities.TVD(x, pi)))
# every NWframesPerRWStep frames, perform one random walk step
if i % NWframesPerRWStep == 0:
# NOTE x * T = (T^T * x^T)^T
# NOTE T is already transposed to get the left EV
x = (T.dot(x.transpose())).transpose()
def EntropyGrowthRateRatio(t, mode='FIRSTORDER', method='MLE'):
"""Computes the ratio between the entropy growth rate ratio between
the second-order and first-order model of a temporal network t. Ratios smaller
than one indicate that the temporal network exhibits non-Markovian characteristics"""
# NOTE to myself: most of the time here goes into computation of the
# NOTE EV of the transition matrix for the bigger of the
# NOTE two graphs below (either 2nd-order or 2nd-order null)
assert method == 'MLE' or method == 'Miller'
# Generate strongly connected component of second-order network
g2 = t.igraphSecondOrder().components(mode="STRONG").giant()
Log.add('Calculating entropy growth rate ratio ... ', Severity.INFO)
# Compute entropy growth rate of observed transition matrix
T2 = Utilities.RWTransitionMatrix(g2)
T2_pi = Utilities.StationaryDistribution(T2)
T2.data *= np.log2(T2.data)
# For T2, we can apply a Miller correction to the entropy estimation
if method == 'Miller':
# K is the number of possible two-paths that can exist based on the
# time-stamped edge sequence (or observed two paths)
if len(t.tedges) > 0:
edges = set((e[0], e[1]) for e in t.tedges)
else:
edges = [(tp[0], tp[1]) for tp in t.twopaths]
for tp in t.twopaths:
edges.append((tp[1], tp[2]))
edges = set(edges)
K = len(Utilities.getPossibleTwoPaths(edges))
#print('K = ', K)
# N is the number of observations used to estimate the transition probabilities
# in the second-order network. This corresponds to the total link weight in the
# second-order network (accounting for the fractional couting of two-paths)
N = np.sum(g2.es["weight"])
#print('N = ', N)
H2 = np.sum(T2 * T2_pi) + (K - 1) / (2 * N)
else:
# simple MLE estimation
H2 = -np.sum(T2 * T2_pi)
H2 = np.absolute(H2)
# Compute entropy rate of null model
if mode == 'FIRSTORDER':
g2n = t.igraphFirstOrder().components(mode="STRONG").giant()
else:
g2n = t.igraphSecondOrderNull().components(mode="STRONG").giant()
# For the entropy rate of the null model, no Miller correction is needed
# since we assume that transitions correspond to the true probabilities
T2n = Utilities.RWTransitionMatrix(g2n)
T2n_pi = Utilities.StationaryDistribution(T2n)
T2n.data *= np.log2(T2n.data)
H2n = -np.sum(T2n * T2n_pi)
H2n = np.absolute(H2n)
Log.add('finished.', Severity.INFO)
# Return ratio
return H2 / H2n
