OverlapBlockState

class graph_tool.inference.OverlapBlockState(g, b=None, clabel=None, pclabel=None, bfield=None, vweight=None, gen_overlap=True, **kwargs)

Bases: BlockState

The overlapping (mixed-membership) stochastic block model state of a given graph.

Parameters:

gGraph

Graph to be modelled.

bEdgePropertyMap (optional, default: None)

Initial block labels on the half-edges, i.e two values per edge.

clabelEdgePropertyMap (optional, default: None)

Constraint labels on the half-edges. If supplied, vertices with different label values will not be clustered in the same group.

pclabelEdgePropertyMap (optional, default: None)

Partition constraint labels on the half-edges. This has the same interpretation as clabel, but will be used to compute the partition description length.

bfieldEdgePropertyMap (optional, default: None)

Local field acting as a prior for the node partition. This should be a vector property map of type vector<double>, and contain the log-probability for each half-edge to be placed in each group.

vweightEdgePropertyMap (optional, default: None)

Half-edge multiplicities.

gen_overlapbool (optional, default: True)

If True, the generalized overlap partition prior will be used. Otherwise, a simplified (but faster) prior will be used instead.

**kwargs(keywork parameters) (optional, default: (none))

Parameters to be passed to BlockState.

Methods

add_vertex(v, r)	Add vertex v to block r.
collect_edge_marginals([p, update])	Collect the edge marginal histogram, which counts the number of times the endpoints of each node have been assigned to a given block pair.
collect_partition_histogram([h, update, unlabel])	Collect a histogram of partitions.
collect_vertex_marginals([p, b, unlabel, update])	Collect the vertex marginal histogram, which counts the number of times a node was assigned to a given block.
copy(**kwargs)	Copies the block state.
draw(**kwargs)	Convenience wrapper to graph_draw() that draws the state of the graph as colors on the vertices and edges.
entropy([adjacency, dl, partition_dl, ...])	Calculate the description length (a.k.a.
exhaustive_sweep([entropy_args, callback, ...])	Perform an exhaustive loop over all possible network partitions.
get_B()	Returns the total number of nonempty groups.
get_Be()	Returns the effective number of groups, defined as \(e^{H}\), with \(H=-\sum_r\frac{n_r}{N}\ln \frac{n_r}{N}\), where \(n_r\) is the number of nodes in group r.
get_E()	Returns the total number of edges.
get_N()	Returns the total number of nodes.
get_bclabel([clabel])	Returns a VertexPropertyMap corresponding to constraint labels for the block graph.
get_bg()	Returns the block graph.
get_blocks()	Returns the property map which contains the block labels for each vertex.
get_bpclabel()	Returns a VertexPropertyMap corresponding to partition constraint labels for the block graph.
get_entropy_args()	Return the current default values for the parameters of the function entropy(), together with other operations that depend on them.
get_er()	Returns the vertex property map of the block graph which contains the number \(e_r\) of half-edges incident on block \(r\).
get_ers()	Returns the edge property map of the block graph which contains the \(e_{rs}\) matrix entries.
get_majority_groups()	Returns a scalar-valued vertex property map with the majority block membership of each node.
get_matrix()	Returns the block matrix (as a sparse csr_matrix), which contains the number of edges between each block pair.
get_move_prob(v, s[, c, d, reverse])	Compute the log-probability of a move proposal for vertex v to block s according to sampling parameters c and d, as obtained with graph_tool.inference.BlockState.sample_vertex_move().
get_nr()	Returns the vertex property map of the block graph which contains the block sizes \(n_r\).
get_overlap_groups()	Returns the mixed membership of each vertex.
get_state()	Alias to get_blocks().
gibbs_sweep([beta, niter, entropy_args, ...])	Perform niter sweeps of a rejection-free Gibbs MCMC to sample network partitions.
mcmc_sweep([beta, c, d, niter, ...])	Perform niter sweeps of a Metropolis-Hastings acceptance-rejection MCMC to sample network partitions.
merge_groups(r, s)	Merges group r with s.
modify_edge(u, v, dm)	Changes the multiplicity of edge ;math:(u, v) by a integer difference dm.
modify_edge_dS(u, v, dm[, entropy_args])	Computes the difference in the description length if edge \((u, v)\) would have its multiplicity changed by an integer difference dm, taking into account the entropy_args parameters as described in entropy().
move_vertex(v, s[, parallel])	Move vertex v to block s.
multiflip_mcmc_sweep([niter, beta, c, ...])	Perform niter sweeps of a Metropolis-Hastings acceptance-rejection MCMC with multiple simultaneous moves (i.e. merges and splits) to sample network partitions.
multilevel_mcmc_sweep([niter, beta, ...])	Perform niter sweeps of a multilevel agglomerative acceptance-rejection pseudo-MCMC (i.e. detailed balance is not preserved) to sample network partitions, that uses a bisection search on the number of groups, together with group merges and singe-node moves.
remove_vertex(v)	Remove vertex v from its current group.
reset_entropy_args()	Reset the current default values for the parameters of the function entropy(), together with other operations that depend on them.
sample_graph([canonical, multigraph, ...])	Sample a new graph from the fitted model.
sample_vertex_move(v[, c, d])	Sample block membership proposal of vertex v according to real-valued sampling parameters c and d: For \(c\to 0\) the blocks are sampled according to the local neighborhood and their connections; for \(c\to\infty\) the blocks are sampled randomly.
set_state(b[, parallel])	Sets the internal partition of the state.
update_entropy_args(**kwargs)	Update the default values for the parameters of the function entropy() from the keyword arguments, in a stateful way, together with other operations that depend on them.
virtual_merge(r, s, **kwargs)	Computes the entropy difference if group r is merged with block s.
virtual_vertex_move(v, s, **kwargs)	Computes the entropy difference if vertex v is moved to block s.

Attributes

B	Nominal number of groups
eweight	EdgePropertyMap with edge multiplicities
mrm	VertexPropertyMap with with sums of in-degrees per group
mrp	VertexPropertyMap with with sums of out-degrees per group
mrs	EdgePropertyMap with with edge counts between groups
vweight	VertexPropertyMap with node multiplicities
wr	Group sizes

add_vertex(v, r)

Add vertex v to block r.

This optionally accepts a list of vertices and blocks to add.

Warning

This can leave the state in an inconsistent state if a vertex is added twice to the same group.

collect_edge_marginals(p=None, update=1)

Collect the edge marginal histogram, which counts the number of times the endpoints of each node have been assigned to a given block pair.

This should be called multiple times, e.g. after repeated runs of the graph_tool.inference.BlockState.mcmc_sweep() function.

Parameters:

pEdgePropertyMap (optional, default: None)

Edge property map with edge marginals to be updated. If not provided, an empty histogram will be created.

updatefloat (optional, default: 1)

Each call increases the current count by the amount given by this parameter.

Returns:

pEdgePropertyMap

Edge property map with updated edge marginals.

Examples

>>> np.random.seed(42)
>>> gt.seed_rng(42)
>>> g = gt.collection.data["polbooks"]
>>> state = gt.BlockState(g, B=4, deg_corr=True)
>>> pe = None
>>> state.mcmc_sweep(niter=1000)   # remove part of the transient
(...)
>>> for i in range(1000):
...     ret = state.mcmc_sweep(niter=10)
...     pe = state.collect_edge_marginals(pe)
>>> gt.bethe_entropy(g, pe)[0]
11.985597...

collect_partition_histogram(h=None, update=1, unlabel=True)

Collect a histogram of partitions.

This should be called multiple times, e.g. after repeated runs of the graph_tool.inference.BlockState.mcmc_sweep() function.

Parameters:

hPartitionHist (optional, default: None)

Partition histogram. If not provided, an empty histogram will be created.

updatefloat (optional, default: 1)

Each call increases the current count by the amount given by this parameter.

unlabelbool (optional, default: True)

If True, a canonical labelling of the groups will be used, so that each partition is uniquely represented.

Returns:

hPartitionHist (optional, default: None)

Updated Partition histogram.

Examples

>>> np.random.seed(42)
>>> gt.seed_rng(42)
>>> g = gt.collection.data["polbooks"]
>>> state = gt.BlockState(g, B=4, deg_corr=True)
>>> ph = None
>>> state.mcmc_sweep(niter=1000)   # remove part of the transient
(...)
>>> for i in range(1000):
...     ret = state.mcmc_sweep(niter=10)
...     ph = state.collect_partition_histogram(ph)
>>> gt.microstate_entropy(ph)
137.419856...

collect_vertex_marginals(p=None, b=None, unlabel=False, update=1)

Collect the vertex marginal histogram, which counts the number of times a node was assigned to a given block.

This should be called multiple times, e.g. after repeated runs of the graph_tool.inference.BlockState.mcmc_sweep() function.

Parameters:

pVertexPropertyMap (optional, default: None)

Vertex property map with vector-type values, storing the previous block membership counts. If not provided, an empty histogram will be created.

bVertexPropertyMap (optional, default: None)

Vertex property map with group partition. If not provided, the state’s partition will be used.

unlabelbool (optional, default: False)

If True, a canonical labelling of the groups will be used, so that each partition is uniquely represented.

updateint (optional, default: 1)

Each call increases the current count by the amount given by this parameter.

Returns:

pVertexPropertyMap

Vertex property map with vector-type values, storing the accumulated block membership counts.

Examples

>>> np.random.seed(42)
>>> gt.seed_rng(42)
>>> g = gt.collection.data["polbooks"]
>>> state = gt.BlockState(g, B=4, deg_corr=True)
>>> pv = None
>>> state.mcmc_sweep(niter=1000)   # remove part of the transient
(...)
>>> for i in range(1000):
...     ret = state.mcmc_sweep(niter=10)
...     pv = state.collect_vertex_marginals(pv)
>>> gt.mf_entropy(g, pv)
16.771713...
>>> gt.graph_draw(g, pos=g.vp["pos"], vertex_shape="pie",
...               vertex_pie_fractions=pv, output="polbooks_blocks_soft_B4.svg")
<...>

“Soft” block partition of a political books network with \(B=4\).

copy(**kwargs)

Copies the block state. The parameters override the state properties, and have the same meaning as in the constructor.

draw(**kwargs)

Convenience wrapper to graph_draw() that draws the state of the graph as colors on the vertices and edges.

entropy(adjacency=True, dl=True, partition_dl=True, degree_dl=True, degree_dl_kind='distributed', edges_dl=True, dense=False, multigraph=True, deg_entropy=True, parallel_entropy=True, beta_dl=1.0, Bfield=True, propagate=True, constants=True)

Calculate the description length (a.k.a. negative joint log-likelihood) associated with the current block partition.

Warning

The default arguments of this function are overriden by those obtained from get_entropy_args(). To update the defaults in a stateful way, update_entropy_args() should be called.

Parameters:

adjacencybool (optional, default: True)

If True, the adjacency term of the description length will be included.

dlbool (optional, default: True)

If True, the description length for the parameters will be included.

partition_dlbool (optional, default: True)

If True, and dl == True the partition description length will be included.

degree_dlbool (optional, default: True)

If True, and dl == True the degree sequence description length will be included (for degree-corrected models).

degree_dl_kindstr (optional, default: "distributed")

This specifies the prior used for the degree sequence. It must be one of: "uniform" or "distributed" (default).

edges_dlbool (optional, default: True)

If True, and dl == True the edge matrix description length will be included.

densebool (optional, default: False)

If True, the “dense” variant of the entropy will be computed.

multigraphbool (optional, default: True)

If True, the multigraph entropy will be used.

deg_entropybool (optional, default: True)

If True, the degree entropy term that is independent of the network partition will be included (for degree-corrected models).

parallel_entropybool (optional, default: True)

If True, the entropy term relating to the multiplcities of parallel edges, which is independent of the network partition, will be included (for muligraph models).

beta_dldouble (optional, default: 1.)

Prior inverse temperature.

Bfieldbool (optional, default: False)

If True, the Bfield parameter passed to the construtor will be taken into account.

Notes

The “entropy” of the state is the negative log-likelihood of the microcanonical SBM, that includes the generated graph \(\boldsymbol{A}\) and the model parameters \(\boldsymbol{\theta}\),

\[\begin{split}\Sigma &= - \ln P(\boldsymbol{A},\boldsymbol{\theta}) \\ &= - \ln P(\boldsymbol{A}|\boldsymbol{\theta}) - \ln P(\boldsymbol{\theta}).\end{split}\]

This value is also called the description length of the data, and it corresponds to the amount of information required to describe it (in nats).

For the non-degree-corrected blockmodel (deg_corr == False), the model parameters are \(\boldsymbol{\theta} = \{\boldsymbol{e}, \boldsymbol{b}\}\), where \(\boldsymbol{e}\) is the matrix of edge counts between blocks, and \(\boldsymbol{b}\) is the partition of the nodes into blocks. For the degree-corrected blockmodel (deg_corr == True), we have an additional set of parameters, namely the degree sequence \(\boldsymbol{k}\).

For the non-degree-corrected blockmodel, the model likelihood is

\[\begin{split}P(\boldsymbol{A}|\boldsymbol{e},\boldsymbol{b}) &= \frac{\prod_{r<s}e_{rs}!\prod_re_{rr}!!}{\prod_rn_r^{e_r}}\times \frac{1}{\prod_{i<j}A_{ij}!\prod_iA_{ii}!!},\\ P(\boldsymbol{A}|\boldsymbol{e},\boldsymbol{b}) &= \frac{\prod_{rs}e_{rs}!}{\prod_rn_r^{e_r}}\times \frac{1}{\prod_{ij}A_{ij}!},\end{split}\]

for undirected and directed graphs, respectively, where \(e_{rs}\) is the number of edges from block \(r\) to \(s\) (or the number of half-edges for the undirected case when \(r=s\)), and \(n_r\) is the number of vertices in block \(r\).

For the degree-corrected variant the equivalent expressions are

\[\begin{split}P(\boldsymbol{A}|\boldsymbol{e},\boldsymbol{b},\boldsymbol{k}) &= \frac{\prod_{r<s}e_{rs}!\prod_re_{rr}!!}{\prod_re_r!}\times \frac{\prod_ik_i!}{\prod_{i<j}A_{ij}!\prod_iA_{ii}!!},\\ P(\boldsymbol{A}|\boldsymbol{e},\boldsymbol{b},\boldsymbol{k}) &= \frac{\prod_{rs}e_{rs}!}{\prod_re_r^+!\prod_re_r^-!}\times \frac{\prod_ik_i^+!\prod_ik_i^-!}{\prod_{ij}A_{ij}!},\end{split}\]

where \(e_r = \sum_se_{rs}\) is the number of half-edges incident on block \(r\), and \(e^+_r = \sum_se_{rs}\) and \(e^-_r = \sum_se_{sr}\) are the numbers of out- and in-edges adjacent to block \(r\), respectively.

If dense == True, the likelihood for the non-degree-corrected model becomes instead

\[\begin{split}P(\boldsymbol{A}|\boldsymbol{e},\boldsymbol{b}) &= \left[\prod_{r<s}{n_rn_s\choose e_{rs}}\prod_r{{n_r\choose 2}\choose e_{rr}/2}\right]^{-1},\\ P(\boldsymbol{A}|\boldsymbol{e},\boldsymbol{b}) &= \left[\prod_{rs}{n_rn_s\choose e_{rs}}\right]^{-1}\end{split}\]

if multigraph == False, otherwise we replace \({n\choose m}\to\left(\!\!{n\choose m}\!\!\right)\) above, where \(\left(\!\!{n\choose m}\!\!\right) = {n+m-1\choose m}\). A “dense” entropy for the degree-corrected model is not available, and if requested will raise a NotImplementedError.

If dl == True, the description length \(\mathcal{L} = -\ln P(\boldsymbol{\theta})\) of the model parameters will be returned as well. The terms \(P(\boldsymbol{e})\) and \(P(\boldsymbol{b})\) are described in described as follows.

For an undirected graph, the number of distinct \(e_{rs}\) matrices is given by,

\[\Omega_m = \left(\!\!{B(B+1)/2 \choose E}\!\!\right)\]

and for a directed graph,

\[\Omega_m = \left(\!\!{B^2 \choose E}\!\!\right)\]

where \(\left(\!{n \choose k}\!\right) = {n+k-1\choose k}\) is the number of \(k\) combinations with repetitions from a set of size \(n\). Hence, we have the description length of the edge counts

\[-\ln P(\boldsymbol{e}) = \ln \Omega_m.\]

For the node partition \(\boldsymbol{b}\) we assume a two-level Bayesian hierarchy, where first the group size histogram is generated, and conditioned on it the partition, which leads to a description length:

\[-\ln P(\boldsymbol{b}) = \ln {N - 1 \choose B - 1} + \ln N! - \sum_r \ln n_r!.\]

where \(n_r\) is the number of nodes in block \(r\).

The total information necessary to describe the model is then,

\[-\ln P(\boldsymbol{e}, \boldsymbol{b}) = -\ln P(\boldsymbol{e}) - \ln P(\boldsymbol{b}).\]

If nr is None, it is assumed \(n_r=N/B\). If nr is False, the partition term \(-\ln P(\boldsymbol{b})\) is omitted entirely.

For the degree-corrected model we need to specify the prior \(P(\boldsymbol{k})\) for the degree sequence as well. Here there are two options:

1. degree_dl_kind == "uniform"

   \[P(\boldsymbol{k}|\boldsymbol{e},\boldsymbol{b}) = \prod_r\left(\!\!{n_r\choose e_r}\!\!\right)^{-1}.\]

   This corresponds to a noninformative prior, where the degrees are sampled from a uniform distribution.

2. degree_dl_kind == "distributed" (default)

   \[P(\boldsymbol{k}|\boldsymbol{e},\boldsymbol{b}) = \prod_r\frac{\prod_k\eta_k^r!}{n_r!} \prod_r q(e_r, n_r)^{-1}\]

   with \(\eta_k^r\) being the number of nodes with degree \(k\) in group \(r\), and \(q(m,n)\) being the number of partitions of integer \(m\) into at most \(n\) parts. This is given by the recurrence

   \[q(m, n) = q(m, n-1) + q(m-n, n),\]

   with boundary conditions \(q(m,n) = q(m,m)\) for \(n>m\), \(q(m,0) = 0\) for \(m>0\), and \(q(0,0) = 1\).

   This corresponds to a prior for the degree sequence conditioned on the degree counts, which are themselves sampled from a uniform hyperprior. This option should be preferred in most cases.

For the directed case, the above expressions are duplicated for the in- and out-degrees.

References

[peixoto-nonparametric-2017]

Tiago P. Peixoto, “Nonparametric Bayesian inference of the microcanonical stochastic block model”, Phys. Rev. E 95 012317 (2017), DOI: 10.1103/PhysRevE.95.012317 [sci-hub, @tor], arXiv: 1610.02703

[peixoto-hierarchical-2014]

Tiago P. Peixoto, “Hierarchical block structures and high-resolution model selection in large networks “, Phys. Rev. X 4, 011047 (2014), DOI: 10.1103/PhysRevX.4.011047 [sci-hub, @tor], arXiv: 1310.4377.

exhaustive_sweep(entropy_args={}, callback=None, density=None, vertices=None, initial_partition=None, max_iter=None)

Perform an exhaustive loop over all possible network partitions.

Parameters:

entropy_argsdict (optional, default: {})

Entropy arguments, with the same meaning and defaults as in graph_tool.inference.BlockState.entropy().

callbackcallable object (optional, default: None)

Function to be called for each partition, with three arguments (S, S_min, b_min) corresponding to the the current entropy value, the minimum entropy value so far, and the corresponding partition, respectively. If not provided, and hist is None an iterator over the same values will be returned instead.

densitytuple (optional, default: None)

If provided, it should contain a tuple with values (S_min, S_max, n_bins), which will be used to obtain the density of states via a histogram of size n_bins. This parameter is ignored unless callback is None.

verticesiterable of ints (optional, default: None)

If provided, this should be a list of vertices which will be moved. Otherwise, all vertices will.

initial_partitioniterable of ints (optional, default: None)

If provided, this will provide the initial partition for the iteration.

max_iterint (optional, default: None)

If provided, this will limit the total number of iterations.

Returns:

statesiterator over (S, S_min, b_min)

If callback is None and hist is None, the function will return an iterator over (S, S_min, b_min) corresponding to the the current entropy value, the minimum entropy value so far, and the corresponding partition, respectively.

Ss, countspair of numpy.ndarray

If callback is None and hist is not None, the function will return the values of each bin (Ss) and the state count of each bin (counts).

b_minVertexPropertyMap

If callback is not None or hist is not None, the function will also return partition with smallest entropy.

Notes

This algorithm has an \(O(B^N)\) complexity, where \(B\) is the number of groups, and \(N\) is the number of vertices.

get_B()

Returns the total number of nonempty groups.

get_Be()

Returns the effective number of groups, defined as \(e^{H}\), with \(H=-\sum_r\frac{n_r}{N}\ln \frac{n_r}{N}\), where \(n_r\) is the number of nodes in group r.

get_E()

Returns the total number of edges.

get_N()

Returns the total number of nodes.

get_bclabel(clabel=None)

Returns a VertexPropertyMap corresponding to constraint labels for the block graph.

get_bg()

Returns the block graph.

get_blocks()

Returns the property map which contains the block labels for each vertex.

get_bpclabel()

Returns a VertexPropertyMap corresponding to partition constraint labels for the block graph.

get_entropy_args()

Return the current default values for the parameters of the function entropy(), together with other operations that depend on them.

get_er()

Returns the vertex property map of the block graph which contains the number \(e_r\) of half-edges incident on block \(r\). If the graph is directed, a pair of property maps is returned, with the number of out-edges \(e^+_r\) and in-edges \(e^-_r\), respectively.

get_ers()

Returns the edge property map of the block graph which contains the \(e_{rs}\) matrix entries. For undirected graphs, the diagonal values (self-loops) contain \(e_{rr}/2\).

get_majority_groups()

Returns a scalar-valued vertex property map with the majority block membership of each node.

get_matrix()

Returns the block matrix (as a sparse csr_matrix), which contains the number of edges between each block pair.

Warning

This corresponds to the adjacency matrix of the block graph, which by convention includes twice the amount of edges in the diagonal entries if the graph is undirected.

Examples

>>> g = gt.collection.data["polbooks"]
>>> state = gt.BlockState(g, B=5, deg_corr=True)
>>> state.mcmc_sweep(niter=1000)
(...)
>>> m = state.get_matrix()
>>> figure()
<...>
>>> matshow(m.todense())
<...>
>>> savefig("bloc_mat.svg")

A 5x5 block matrix.

get_move_prob(v, s, c=1.0, d=0.1, reverse=False)

Compute the log-probability of a move proposal for vertex v to block s according to sampling parameters c and d, as obtained with graph_tool.inference.BlockState.sample_vertex_move(). If reverse == True, the reverse probability of moving the node back from block s to its current one is obtained.

get_nr()

Returns the vertex property map of the block graph which contains the block sizes \(n_r\).

get_overlap_groups()

Returns the mixed membership of each vertex.

Returns:

bvVertexPropertyMap

A vector-valued vertex property map containing the block memberships of each node.

bc_inVertexPropertyMap

The labelled in-degrees of each node, i.e. how many in-edges belong to each group, in the same order as the bv property above.

bc_outVertexPropertyMap

The labelled out-degrees of each node, i.e. how many out-edges belong to each group, in the same order as the bv property above.

bc_totalVertexPropertyMap

The labelled total degrees of each node, i.e. how many incident edges belong to each group, in the same order as the bv property above.

get_state()

Alias to get_blocks().

gibbs_sweep(beta=1.0, niter=1, entropy_args={}, allow_new_group=True, sequential=True, deterministic=False, vertices=None, verbose=False, **kwargs)

Perform niter sweeps of a rejection-free Gibbs MCMC to sample network partitions.

Parameters:

betafloat (optional, default: 1.)

Inverse temperature.

niterint (optional, default: 1)

Number of sweeps to perform. During each sweep, a move attempt is made for each node.

entropy_argsdict (optional, default: {})

Entropy arguments, with the same meaning and defaults as in graph_tool.inference.BlockState.entropy().

allow_new_groupbool (optional, default: True)

Allow the number of groups to increase and decrease.

sequentialbool (optional, default: True)

If sequential == True each vertex move attempt is made sequentially, where vertices are visited in random order. Otherwise the moves are attempted by sampling vertices randomly, so that the same vertex can be moved more than once, before other vertices had the chance to move.

deterministicbool (optional, default: False)

If sequential == True and deterministic == True the vertices will be visited in deterministic order.

verticeslist of ints (optional, default: None)

If provided, this should be a list of vertices which will be moved. Otherwise, all vertices will.

verbosebool (optional, default: False)

If verbose == True, detailed information will be displayed.

Returns:

dSfloat

Entropy difference after the sweeps.

nattemptsint

Number of vertex moves attempted.

nmovesint

Number of vertices moved.

Notes

This algorithm has an \(O(E\times B)\) complexity, where \(B\) is the number of groups, and \(E\) is the number of edges.

mcmc_sweep(beta=1.0, c=0.5, d=0.01, niter=1, entropy_args={}, allow_vacate=True, sequential=True, deterministic=False, vertices=None, verbose=False, **kwargs)

Perform niter sweeps of a Metropolis-Hastings acceptance-rejection MCMC to sample network partitions.

Parameters:

betafloat (optional, default: 1.)

Inverse temperature.

cfloat (optional, default: .5)

Sampling parameter c for move proposals: For \(c\to 0\) the blocks are sampled according to the local neighborhood of a given node and their block connections; for \(c\to\infty\) the blocks are sampled randomly. Note that only for \(c > 0\) the MCMC is guaranteed to be ergodic.

dfloat (optional, default: .01)

Probability of selecting a new (i.e. empty) group for a given move.

niterint (optional, default: 1)

Number of sweeps to perform. During each sweep, a move attempt is made for each node.

entropy_argsdict (optional, default: {})

Entropy arguments, with the same meaning and defaults as in graph_tool.inference.BlockState.entropy().

allow_vacatebool (optional, default: True)

Allow groups to be vacated.

sequentialbool (optional, default: True)

If sequential == True each vertex move attempt is made sequentially, where vertices are visited in random order. Otherwise the moves are attempted by sampling vertices randomly, so that the same vertex can be moved more than once, before other vertices had the chance to move.

deterministicbool (optional, default: False)

If sequential == True and deterministic == True the vertices will be visited in deterministic order.

verticeslist of ints (optional, default: None)

If provided, this should be a list of vertices which will be moved. Otherwise, all vertices will.

verbosebool (optional, default: False)

If verbose == True, detailed information will be displayed.

Returns:

dSfloat

Entropy difference after the sweeps.

nattemptsint

Number of vertex moves attempted.

nmovesint

Number of vertices moved.

Notes

This algorithm has an \(O(E)\) complexity, where \(E\) is the number of edges (independent of the number of groups).

References

[peixoto-efficient-2014]

Tiago P. Peixoto, “Efficient Monte Carlo and greedy heuristic for the inference of stochastic block models”, Phys. Rev. E 89, 012804 (2014), DOI: 10.1103/PhysRevE.89.012804 [sci-hub, @tor], arXiv: 1310.4378

merge_groups(r, s)

Merges group r with s.

modify_edge(u, v, dm)

Changes the multiplicity of edge ;math:(u, v) by a integer difference dm.

modify_edge_dS(u, v, dm, entropy_args={})

Computes the difference in the description length if edge \((u, v)\) would have its multiplicity changed by an integer difference dm, taking into account the entropy_args parameters as described in entropy().

move_vertex(v, s, parallel=False)

Move vertex v to block s.

This optionally accepts a list of vertices and blocks to move simultaneously. In this case, if parallel is True, the updates are done in parallel.

multiflip_mcmc_sweep(niter=1, beta=1.0, c=0.5, psingle=None, psplit=1, pmerge=1, pmergesplit=1, pmovelabel=0, d=0.01, gibbs_sweeps=10, vertices=None, parallel_thres=1000, entropy_args={}, accept_stats=None, verbose=False, **kwargs)

Perform niter sweeps of a Metropolis-Hastings acceptance-rejection MCMC with multiple simultaneous moves (i.e. merges and splits) to sample network partitions.

Parameters:

niterint (optional, default: 1)

Number of sweeps to perform. During each sweep, a move attempt is made for each node, on average.

betafloat (optional, default: 1.)

Inverse temperature.

cfloat (optional, default: .5)

Sampling parameter c for move proposals: For \(c\to 0\) the blocks are sampled according to the local neighborhood of a given node and their block connections; for \(c\to\infty\) the blocks are sampled randomly. Note that only for \(c > 0\) the MCMC is guaranteed to be ergodic.

psinglefloat (optional, default: None)

Relative probability of proposing a single node move. If None, it will be selected as the number of nodes in the graph.

psplitfloat (optional, default: 1)

Relative probability of proposing a group split.

pmergefloat (optional, default: 1)

Relative probability of proposing a group merge.

pmergesplitfloat (optional, default: 1)

Relative probability of proposing a marge-split move.

pmovelabelfloat (optional, default: 0)

Relative probability of proposing a group label move.

dfloat (optional, default: 1)

Probability of selecting a new (i.e. empty) group for a given single-node move.

gibbs_sweepsint (optional, default: 10)

Number of sweeps of Gibbs sampling to be performed (i.e. each node is attempted once per sweep) to refine a split proposal.

verticesiterable of int` (optional, default: None):

If provided, this specified the list of vertices to be considerd. Otherwise, all vertices are considered.

parallel_thresint (optional, default: 1000)

If the number of nodes to be moved fall below this thershold, the operation is performed serially. Otherwise, the moves are done in parallel.

entropy_argsdict (optional, default: {})

Entropy arguments, with the same meaning and defaults as in graph_tool.inference.BlockState.entropy().

accept_statsdict (optional, default: None)

If provided, this dictionary will be updated with the proposal and acceptance counts for each kind of move.

verbosebool (optional, default: False)

If verbose == True, detailed information will be displayed.

Returns:

dSfloat

Entropy difference after the sweeps.

nattemptsint

Number of vertex moves attempted.

nmovesint

Number of vertices moved.

Notes

This algorithm has an \(O(E)\) complexity, where \(E\) is the number of edges (independent of the number of groups).

References

[peixoto-merge-split-2020]

Tiago P. Peixoto, “Merge-split Markov chain Monte Carlo for community detection”, Phys. Rev. E 102, 012305 (2020), DOI: 10.1103/PhysRevE.102.012305 [sci-hub, @tor], arXiv: 2003.07070

multilevel_mcmc_sweep(niter=1, beta=inf, bisection=True, random_bisection=True, refine=False, c=0.5, d=0.01, r=0.9, dB_epsilon=0.01, merge_sweeps=10, mh_sweeps=10, init_r=0.99, init_min_iter=8, init_min_iter_beta=4, init_beta=1.0, gibbs=False, B_min=1, B_max=18446744073709551615, b_min=None, b_max=None, M=None, sample_group_thres=1000, cache_states=True, force_accept=False, parallel=True, parallel_thres=1000, parallel_merge=True, vertices=None, entropy_args={}, verbose=False, **kwargs)

Perform niter sweeps of a multilevel agglomerative acceptance-rejection pseudo-MCMC (i.e. detailed balance is not preserved) to sample network partitions, that uses a bisection search on the number of groups, together with group merges and singe-node moves.

Parameters:

niterint (optional, default: 1)

Number of sweeps to perform. During each sweep, a move attempt is made for each node, on average.

betafloat (optional, default: numpy.inf)

Inverse temperature.

bisectionbool (optional, default: True)

If True, the algorithm will perform a bisection search over the number of groups. Otherwise, a simpler agglomerative approach will be used instead.

random_bisectionbool (optional, default: True)

If True, bisections are done at intervals chosen uniformly at random. Otherwise a Fibonacci sequence is used.

refinebool (optional, default: True)

If True, a strictly agglomerative run will be appended to the end of the algorithm.

cfloat (optional, default: .5)

Sampling parameter c for move proposals: For \(c\to 0\) the blocks are sampled according to the local neighborhood of a given node and their block connections; for \(c\to\infty\) the blocks are sampled randomly. Note that only for \(c > 0\) the MCMC is guaranteed to be ergodic.

dfloat (optional, default: .01)

Probability of selecting a new (i.e. empty) group for a given single-node move.

rfloat (optional, default: 0.9)

Group shrink ratio. The number of groups is reduced by this fraction at each merge sweep.

merge_sweepsint (optional, default: 10)

Number of sweeps spent to find good merge proposals.

mh_sweepsint (optional, default: 10)

Number of single-node Metropolis-Hastings sweeps between merge splits.

init_rdouble (optional, default: 0.99)

Stopping criterion for the intialization phase, after each node is put in their own group, to set the initial upper bound of the bisection search. A number of single-node Metropolis-Hastings sweeps is done until the number of groups is shrunk by a factor that is larger than this parameter.

init_min_iterint (optional, default: 8)

Minimum number of iterations at the intialization phase.

init_min_iter_betaint (optional, default: 4)

Minimum number of iterations at the intialization phase with beta given by init_beta.

init_betafloat (optional, default: 1.)

Inverse temperature to be used for the first init_min_iter_beta sweeps of the initialization phase.

gibbsbool (optional, default: False)

If True, the single node moves use (slower) Gibbs sampling, rather than Metropolis-Hastings.

B_minint (optional, default: 1)

Minimum number of groups to be considered in the search.

b_minVertexPropertyMap (optional, default: None)

If provided, this will be used for the partition corresponding to B_min.

B_maxint (optional, default: numpy.iinfo(numpy.uint64).max)

Maximum number of groups to be considered in the search.

b_maxVertexPropertyMap (optional, default: None)

If provided, this will be used for the partition corresponding to B_max.

Mint (optional, default: None)

Maximum number of groups to select for the multilevel move. If None is provided, then all groups are always elected.

cache_statesbool (optional, default: True)

If True, intermediary states will be cached during the bisection search.

force_acceptbool (optional, default: False)

If True, new state will be accepted even if it does not improve the objective function.

parallelbool (optional, default: True)

If True, the algorithm will run in parallel (if enabled during compilation).

parallel_thresint (optional, default: 1000)

If the number of nodes to be moved fall below this thershold, the operation is performed serially. Otherwise, the moves are done in parallel.

verticesiterable of int` (optional, default: None):

If provided, this specified the list of vertices to be considerd. Otherwise, all vertices are considered.

entropy_argsdict (optional, default: {})

Entropy arguments, with the same meaning and defaults as in graph_tool.inference.BlockState.entropy().

verbosebool (optional, default: False)

If verbose == True, detailed information will be displayed.

Returns:

dSfloat

Entropy difference after the sweeps.

nattemptsint

Number of vertex moves attempted.

nmovesint

Number of vertices moved.

Notes

This algorithm has an \(O(E\ln^2 N)\) complexity, where \(E\) is the number of edges and \(N\) is the number of nodes (independently of the number of groups).

Parallel implementation.

If enabled during compilation, this algorithm will run in parallel using OpenMP. See the parallel algorithms section for information about how to control several aspects of parallelization.

References

[peixoto-efficient-2014]

Tiago P. Peixoto, “Efficient Monte Carlo and greedy heuristic for the inference of stochastic block models”, Phys. Rev. E 89, 012804 (2014), DOI: 10.1103/PhysRevE.89.012804 [sci-hub, @tor], arXiv: 1310.4378

remove_vertex(v)

Remove vertex v from its current group.

This optionally accepts a list of vertices to remove.

Warning

This will leave the state in an inconsistent state before the vertex is returned to some other group, or if the same vertex is removed twice.

reset_entropy_args()

Reset the current default values for the parameters of the function entropy(), together with other operations that depend on them.

sample_graph(canonical=False, multigraph=True, self_loops=True, sample_params=False, max_ent=False, n_iter=1000)

Sample a new graph from the fitted model.

Parameters:

canonicalbool (optional, default: False)

If canonical == True, the graph will be sampled from the maximum-likelihood estimate of the canonical stochastic block model. Otherwise, it will be sampled from the microcanonical model.

multigraphbool (optional, default: True)

If True, parallel edges will be allowed.

self-loopsbool (optional, default: True)

If True, self-loops will be allowed.

sample_paramsbool (optional, default: True)

If True, and canonical == True and max_ent == False, the count parameters (edges between groups and node degrees) will be sampled from their posterior distribution conditioned on the actual state. Otherwise, their maximum-likelihood values will be used.

max_entbool (optional, default: False)

If True, maximum-entropy model variants will be used.

n_iterint (optional, default: 1000)

Number of iterations used (only relevant if canonical == False and max_ent == True).

Returns:

gGraph

Generated graph.

Notes

This function is just a convenience wrapper to generate_sbm(). However, if max_ent==True and canonical == False it wraps random_rewire() instead.

Examples

>>> g = gt.collection.data["polbooks"]
>>> state = gt.minimize_blockmodel_dl(g, multilevel_mcmc_args=dict(B_max=3))
>>> u = state.sample_graph(canonical=True, self_loops=False, multigraph=False)
>>> ustate = gt.BlockState(u, b=state.b)
>>> state.draw(pos=g.vp.pos, output="polbooks-sbm.svg")
<...>
>>> ustate.draw(pos=u.own_property(g.vp.pos), output="polbooks-sbm-sampled.svg")
<...>

Left: Political books network. Right: Sample from the degree-corrected SBM fitted to the original network.

sample_vertex_move(v, c=1.0, d=0.1)

Sample block membership proposal of vertex v according to real-valued sampling parameters c and d: For \(c\to 0\) the blocks are sampled according to the local neighborhood and their connections; for \(c\to\infty\) the blocks are sampled randomly. With a probability d, a new (empty) group is sampled.

set_state(b, parallel=False)

Sets the internal partition of the state. If parallel is True, the update is done in parallel.

update_entropy_args(**kwargs)

Update the default values for the parameters of the function entropy() from the keyword arguments, in a stateful way, together with other operations that depend on them.

Values updated in this manner are preserved by the copying or pickling of the state.

virtual_merge(r, s, **kwargs)

Computes the entropy difference if group r is merged with block s. The remaining parameters are the same as in graph_tool.inference.BlockState.entropy().

virtual_vertex_move(v, s, **kwargs)

Computes the entropy difference if vertex v is moved to block s. The remaining parameters are the same as in graph_tool.inference.BlockState.entropy().

B

Nominal number of groups

eweight

EdgePropertyMap with edge multiplicities

mrm

VertexPropertyMap with with sums of in-degrees per group

mrp

VertexPropertyMap with with sums of out-degrees per group

mrs

EdgePropertyMap with with edge counts between groups

vweight

VertexPropertyMap with node multiplicities

wr

Group sizes