### Documentation improvements and fixes

parent be6f450d
 ... ... @@ -161,10 +161,10 @@ figure. .. note:: We emphasize that no constraints are imposed on what kind of modular structure is allowed, as the matrix of edge counts :math:e is unconstrained. Hence, we can detect the putatively typical pattern of "community structure" With the SBM no constraints are imposed on what kind of modular structure is allowed, as the matrix of edge counts :math:e is unconstrained. Hence, we can detect the putatively typical pattern of "community structure" _, i.e. when nodes are connected mostly to other nodes of the same group, if it happens to be the most likely network description, but we can also ... ...
 ... ... @@ -13,6 +13,7 @@ the observed data, the network can be reconstructed according to the posterior distribution, .. math:: :label: posterior-reconstruction P(\boldsymbol A, \boldsymbol b | \boldsymbol{\mathcal{D}}) = \frac{P(\boldsymbol{\mathcal{D}} | \boldsymbol A)P(\boldsymbol A, \boldsymbol b)}{P(\boldsymbol{\mathcal{D}})} ... ... @@ -111,14 +112,14 @@ In this situation the priors :math:P(p|\alpha=1,\beta=1) and .. note:: It is important to emphasize that since this approach makes use of the *correlations* between edges to inform the reconstruction, as described by the inferred SBM, this means it can also be used when only single measurements have been performed, :math:n_{ij}=1, and the error magnitudes :math:p and :math:q are unknown. Since every arbitrary adjacency matrix can be cast in this setting, this method can be used to reconstruct networks for which no error assessments of any kind have been provided. Since this approach also makes use of the *correlations* between edges to inform the reconstruction, as described by the inferred SBM, this means it can also be used when only single measurements have been performed, :math:n_{ij}=1, and the error magnitudes :math:p and :math:q are unknown. Since every arbitrary adjacency matrix can be cast in this setting, this method can be used to reconstruct networks for which no error assessments of any kind have been provided. Below, we illustrate how the reconstruction can be performed with a simple example, using ... ... @@ -173,7 +174,8 @@ simple example, using global pv, u, cs u = s.collect_marginal(u) bstate = s.get_block_state() pv = bstate.levels.collect_vertex_marginals(pv) b = gt.perfect_prop_hash([bstate.levels.b]) pv = bstate.levels.collect_vertex_marginals(pv, b=b) cs.append(gt.local_clustering(s.get_graph()).fa.mean()) gt.mcmc_equilibrate(state, force_niter=10000, mcmc_args=dict(niter=10), ... ... @@ -310,9 +312,9 @@ Which yields: .. testoutput:: measured Posterior probability of edge (11, 36): 0.515651... Posterior probability of non-edge (15, 73): 0.009000... Estimated average local clustering: 0.571673 ± 0.003228... Posterior probability of edge (11, 36): 0.967896... Posterior probability of non-edge (15, 73): 0.038703... Estimated average local clustering: 0.572129 ± 0.005409... The results are very similar to the ones obtained with the uniform model in this case, but can be quite different in situations where a large ... ... @@ -423,7 +425,8 @@ inference: global pv, u, cs u = s.collect_marginal(u) bstate = s.get_block_state() pv = bstate.levels.collect_vertex_marginals(pv) b = gt.perfect_prop_hash([bstate.levels.b]) pv = bstate.levels.collect_vertex_marginals(pv, b=b) cs.append(gt.local_clustering(s.get_graph()).fa.mean()) gt.mcmc_equilibrate(state, force_niter=10000, mcmc_args=dict(niter=10), ... ...
 ... ... @@ -3,7 +3,76 @@ Reconstruction from dynamics ++++++++++++++++++++++++++++ In some cases direct measurements of the edges of a network are either impossible to be done, or can be done only at significant experimental cost. In such situations, we are required to infer the network of interactions from the observed functional behavior [peixoto-network-2019]_. In graph-tool this can be done for epidemic spreading (via :class:~graph_tool.inference.uncertain_blockmodel.EpidemicsBlockState), for the kinetic Ising model (via :class:~graph_tool.inference.uncertain_blockmodel.IsingGlauberBlockState and :class:~graph_tool.inference.uncertain_blockmodel.CIsingGlauberBlockState), and for the equilibrium Ising model (via :class:~graph_tool.inference.uncertain_blockmodel.PseudoIsingBlockState and :class:~graph_tool.inference.uncertain_blockmodel.PseudoCIsingBlockState). We consider the general reconstruction framework outlined above, where the observed data :math:\mathcal{D} in Eq. :eq:posterior-reconstruction are a time-series generated by some of the supported processes. Just like before, the posterior distribution includes not only the adjacency matrix, but also the parameters of the dynamical model and of the SBM that is used as a prior. For example, in the case of a SIS epidemics, where :math:\sigma_i(t)=1 means node :math:i is infected at time :math:t, or :math:0 otherwise, the likelihood for a time-series :math:\boldsymbol\sigma is .. math:: P(\boldsymbol\sigma|\boldsymbol A,\boldsymbol\beta,\gamma)=\prod_t\prod_iP(\sigma_i(t)|\boldsymbol\sigma(t-1)), where .. math:: P(\sigma_i(t)|\boldsymbol\sigma(t-1)) = f(e^{m_i(t-1)}, \sigma_i(t))^{1-\sigma_i(t-1)} \times f(\gamma,\sigma_i(t))^{\sigma_i(t-1)} is the transition probability for node :math:i at time :math:t, with :math:f(p,\sigma) = (1-p)^{\sigma}p^{1-\sigma}, and where .. math:: m_i(t) = \sum_jA_{ij}\ln(1-\beta_{ij})\sigma_j(t) is the contribution from all neighbors of node :math:i to its infection probability at time :math:t. In the equations above the value :math:\beta_{ij} is the probability of an infection via an existing edge :math:(i,j), and :math:\gamma is the :math:1\to 0 recovery probability. With these additional parameters, the full posterior distribution for the reconstruction becomes .. math:: P(\boldsymbol A,\boldsymbol b,\boldsymbol\beta|\boldsymbol\sigma) = \frac{P(\boldsymbol\sigma|\boldsymbol A,\boldsymbol b,\gamma)P(\boldsymbol A|\boldsymbol b)P(\boldsymbol b)P(\boldsymbol\beta)}{P(\boldsymbol\sigma|\gamma)}. Since :math:\beta_{ij}\in[0,1] we use the uniform prior :math:P(\boldsymbol\beta)=1. Note also that the recovery probably :math:\gamma plays no role on the reconstruction algorithm, since its term in the likelihood does not involve :math:\boldsymbol A (and hence, gets cancelled out in the denominator :math:P(\boldsymbol\sigma|\gamma)=P(\gamma|\boldsymbol\sigma)P(\boldsymbol\sigma)/P(\gamma)). This means the above posterior only depends on the infection events :math:0\to 1, and thus is also valid without any modifications to all epidemic variants SI, SIR, SEIR, etc, since the infection events occur with the same probability for all these models. In the example below is shown how to perform reconstruction from an epidemic process. .. testsetup:: dynamics import os ... ... @@ -16,47 +85,75 @@ Reconstruction from dynamics .. testcode:: dynamics g = gt.collection.data["dolphins"] ss = [] for i in range(1000): si_state = gt.SIState(g, beta=1) s = [] for j in range(10): si_state.iterate_sync() s.append(si_state.get_state().copy()) s = gt.group_vector_property(s) ss.append(s) # We will first simulate the dynamics with a given network g = gt.collection.data["dolphins"] u = g.copy() u.clear_edges() ss = [u.own_property(s) for s in ss] # The algorithm accepts multiple independent time-series for the # reconstruction. We will generate 100 SI cascades starting from a # random node each time, and uniform infection probability 0.7. rstate = gt.EpidemicsBlockState(u, s=ss, beta=None, r=1e-6, global_beta=.99, state_args=dict(B=1), nested=False) ss = [] for i in range(100): si_state = gt.SIState(g, beta=.7) s = [si_state.get_state().copy()] for j in range(10): si_state.iterate_sync() s.append(si_state.get_state().copy()) # Each time series should be represented as a single vector-valued # vertex property map with the states for each note at each time. s = gt.group_vector_property(s) ss.append(s) # Now we collect the marginals for exactly 100,000 sweeps, at # intervals of 10 sweeps: # Prepare the initial state of the reconstruction as an empty graph u = g.copy() u.clear_edges() ss = [u.own_property(s) for s in ss] # time series properties need to be 'owned' by graph u # Create reconstruction state rstate = gt.EpidemicsBlockState(u, s=ss, beta=None, r=1e-6, global_beta=.1, state_args=dict(B=1), nested=False, aE=g.num_edges()) # Now we collect the marginals for exactly 100,000 sweeps, at # intervals of 10 sweeps: gm = None bm = None def collect_marginals(s): global gm, bm gm = s.collect_marginal(gm) b = gt.perfect_prop_hash([s.bstate.b]) betas = [] def collect_marginals(s): global gm, bm gm = s.collect_marginal(gm) b = gt.perfect_prop_hash([s.bstate.b]) bm = s.bstate.collect_vertex_marginals(bm, b=b) betas.append(s.params["global_beta"]) gt.mcmc_equilibrate(rstate, force_niter=10000, mcmc_args=dict(niter=10, xstep=0), callback=collect_marginals) print("Posterior similarity: ", gt.similarity(g, gm, g.new_ep("double", 1), gm.ep.eprob)) print("Inferred infection probability: %g ± %g" % (mean(betas), std(betas))) gt.graph_draw(gm, gm.own_property(g.vp.pos), vertex_shape="pie", vertex_color="black", vertex_pie_fractions=gm.own_property(bm), vertex_pen_width=1, edge_pen_width=gt.prop_to_size(gm.ep.eprob, 0, 5), eorder=gm.ep.eprob, output="dolphins-posterior.svg") The reconstruction can accurately recover the hidden network and the infection probability: .. testoutput:: dynamics gt.mcmc_equilibrate(rstate, force_niter=10000, mcmc_args=dict(niter=10, xstep=0, p=0, h=0), callback=collect_marginals) Posterior similarity: 0.978108... Inferred infection probability: 0.687213 ± 0.054126 b = bm.new_vp("int", vals=[bm[v].a.argmax() for v in bm.vertices()]) The figure below shows the reconstructed network and the inferred community structure. .. figure:: dolphins-posterior.* :align: center :width: 400px graph_draw(gm, gm.own_property(g.vp.pos), vertex_shape="square", vertex_color="black", vertex_fill_color=b, vertex_pen_width=1, edge_pen_width=prop_to_size(gm.ep.eprob, 0, 5), eorder=gm.ep.eprob, output="dolphins") eprob = u.ep.eprob print("Posterior probability of edge (11, 36):", eprob[u.edge(11, 36)]) print("Posterior probability of non-edge (15, 73):", eprob[u.edge(15, 73)]) print("Estimated average local clustering: %g ± %g" % (np.mean(cs), np.std(cs))) Reconstructed network of associations between 62 dolphins, from the dynamics of a SI epidemic model, using the degree-corrected SBM as a latent prior. The edge thickness corresponds to the marginal posterior probability of each edge, and the node pie charts to the marginal posterior distribution of the node partition.
 ... ... @@ -169,7 +169,8 @@ i.e. the posterior probability that a node belongs to a given group: def collect_marginals(s): global pv pv = s.collect_vertex_marginals(pv) b = gt.perfect_prop_hash([s.b]) pv = s.collect_vertex_marginals(pv, b=b) # Now we collect the marginals for exactly 100,000 sweeps, at # intervals of 10 sweeps: ... ...
 ... ... @@ -53,6 +53,7 @@ table.docutils.figure { margin-left:auto; margin-right:auto; display: table; } /* stupid workaround to hide ugly c++ signature stuff from sphinx*/ ... ... @@ -172,4 +173,17 @@ div.sphinxsidebar div.bodywrapper { margin: 0 0 0 310px; } div.sidebar { margin: 1em 0 0.5em 1em; border: 1px solid #eee; padding: 7px 7px 0 7px; background-color: #eee; width: 40%; float: right; } div.sidebar p { margin: 0px 7px 7px 7px; } \ No newline at end of file
 ... ... @@ -13,7 +13,7 @@ font_size=14 rcParams["backend"] = "PDF" rcParams["figure.figsize"] = (4, 3) rcParams["font.family"] = "Serif" rcParams["font.serif"] = ["Times"] #rcParams["font.serif"] = ["Times"] rcParams["font.size"] = font_size rcParams["axes.labelsize"] = font_size rcParams["xtick.labelsize"] = font_size - 2 ... ... @@ -36,7 +36,7 @@ rcParams["ps.usedistiller"] = "xpdf" rcParams["pdf.compression"] = 9 rcParams["ps.useafm"] = True rcParams["path.simplify"] = True rcParams["text.latex.preamble"] = [r"\usepackage{times}", rcParams["text.latex.preamble"] = [#r"\usepackage{times}", #r"\usepackage{euler}", r"\usepackage{amssymb}", r"\usepackage{amsmath}"] ... ...
 ... ... @@ -2245,8 +2245,7 @@ class Graph(object): -------- >>> g = gt.collection.data["pgp-strong-2009"] >>> g.get_out_degrees([42, 666]) array([20, 38]) array([20, 39], dtype=uint64) """ return libcore.get_degree_list(self.__graph, numpy.asarray(vs, dtype="uint64"), ... ... @@ -2261,8 +2260,7 @@ class Graph(object): -------- >>> g = gt.collection.data["pgp-strong-2009"] >>> g.get_in_degrees([42, 666]) array([20, 39]) array([20, 38], dtype=uint64) """ return libcore.get_degree_list(self.__graph, numpy.asarray(vs, dtype="uint64"), ... ...
 ... ... @@ -620,7 +620,7 @@ def eigenvector(g, weight=None, vprop=None, epsilon=1e-6, max_iter=None): >>> w.a = np.random.random(len(w.a)) * 42 >>> ee, x = gt.eigenvector(g, w) >>> print(ee) 724.302745922... 724.302745... >>> gt.graph_draw(g, pos=g.vp["pos"], vertex_fill_color=x, ... vertex_size=gt.prop_to_size(x, mi=5, ma=15), ... vcmap=matplotlib.cm.gist_heat, ... ...
 ... ... @@ -165,10 +165,10 @@ def global_clustering(g, weight=None): c = 3 \times \frac{\text{number of triangles}} {\text{number of connected triples}} If weights are given, the following definition is used If weights are given, the following definition is used: .. math:: c = \frac{\operatorname{Tr}{{\boldsymbol A}^3}}{\sum_{i\ne j}[{\boldsymbol A}^2]_{ij}}, c = \frac{\mathrm{Tr}{{\boldsymbol A}^3}}{\sum_{i\ne j}[{\boldsymbol A}^2]_{ij}}, where :math:\boldsymbol A is the weighted adjacency matrix, and it is assumed that the weights are normalized, i.e. :math:A_{ij} \le 1. ... ...
 ... ... @@ -25,27 +25,33 @@ dl_import("from . import libgraph_tool_inference as libinference") from numpy import sqrt def latent_multigraph(g, epsilon=1e-8, max_niter=10000): r""" def latent_multigraph(g, epsilon=1e-8, max_niter=0): r"""Infer latent Poisson multigraph model given an "erased" simple graph. Parameters ---------- g : :class:~graph_tool.Graph Graph to be used. epsilon : float (optional, default: 1e-8) Convergence criterion. max_niter : int (optional, default: 0) Maximum number of iterations allowed (if 0, no maximum is assumed). Returns ------- Notes ----- u : :class:~graph_tool.Graph Latent graph. w : :class:~graph_tool.EdgePropertyMap` Edge property map with inferred multiplicity parameter. Examples -------- >>> g = gt.collection.data["football"] >>> gt.modularity(g, g.vp.value_tsevans) 0.5744393497... References ---------- >>> g = gt.collection.data["as-22july06"] >>> gt.scalar_assortativity(g, "out") (-0.198384..., 0.001338...) >>> u, w = gt.latent_multigraph(g) >>> scalar_assortativity(u, "out", eweight=w) (-0.048426..., 0.034526...) """ g = g.copy() ... ...
 ... ... @@ -1394,7 +1394,7 @@ def vertex_percolation(g, vertices, second=False): Text(...) >>> ylabel("Size of largest component") Text(...) >>> legend(loc="lower right") >>> legend(loc="upper left") <...> >>> savefig("vertex-percolation.svg") ... ...
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