Decoding Temporal Relationships with DynCRep

Welcome to our tutorial on DynCRep - a cutting-edge tool from the probinet package. Traditional network analysis models often fall short when it comes to capturing the dynamic nature of real-world networks. DynCRep is designed to bridge this gap, providing a robust solution for analyzing and understanding the temporal evolution of complex networks.

The DynCRep (Dynamic Community and Reciprocity) model is a tool that helps us to better understand complex, evolving networks. In real-world scenarios, networks are rarely static. They change and evolve over time, influenced by various factors such as changing relationships, evolving interests, and temporal events. The DynCRep model captures these dynamic interaction patterns by considering the temporal order of connections between pairs of nodes in the network, rather than viewing them independently [SCDB22]. This is a significant departure from standard models, which often assume that the connections between pairs of nodes are independent once we know certain hidden factors. The DynCRep model integrates community structure and reciprocity as key structural information of networks that evolve over time. In this tutorial, we will guide you through the process of applying the DynCRep model to a dynamic network.

NOTE: We suggest the reader to have a basic understanding of the CRep and JointCRep models before diving into this tutorial. If you are not familiar with these models, we recommend checking out our tutorials CRep algorithm and JointCRep algorithm before proceeding with this one.

Data generation

Before we dive into the tutorial, let’s take a moment to understand the key idea behind the DynCRep model. The DynCRep model considers a temporal network as a sequence of adjacency matrices A where A[t] represents the adjacency matrix of the network at time t. The model takes in the sequence of adjacency matrices A and learns the parameters of the model that best describe the temporal relationships between nodes in the network.

For this tutorial, we will use synthetic data generated using the SyntheticDynCRep class from the probinet synthetic.dynamic module. This is the generative model created using the same modelling assumptions as the DynCRep model. It takes in the parameters:

1. N - The number of nodes in the network.

2. K - The number of communities in the network.

3. T - The number of time steps in the network.

4. avg_degree - The average degree of nodes in the network.

5. structure - The structure of the network.

6. ExpM - The expected number of edges in the network.

7. eta - The reciprocity parameter.

8. corr - The correlation between u and v synthetically generated.

9. over - The fraction of nodes with mixed membership.

10. undirected - Whether the network is undirected.

To gain a deeper understanding of how these parameters influence the model, we can experiment by adjusting some of them and observing the resulting changes. Let’s begin with a straightforward scenario: a network with 50 nodes, divided into 2 communities, observed over a single time step. We’ll set the average degree of the nodes to 5. We’ll then examine the structure of the network when the parameters are set to create an assortative network.

from probinet.synthetic.dynamic import SyntheticDynCRep

# Set the number of nodes, communities, and time steps
N = 50
K = 2
T = 0  # total number of time steps is T+1
avg_degree = 5

# Structure of the network
structure = "assortative"  # This is the default value

# Initialize the synthetic network class
synthetic_dyncrep = SyntheticDynCRep(
    N=N, K=K, T=T, verbose=2, avg_degree=avg_degree, figsize=(3, 3), fontsize=10
)

temporal_network = synthetic_dyncrep.generate_net()

As seen above, the network has been generated with the specified parameters. It consists of 50 nodes clustered into 2 communities. The network is assortative, meaning that nodes are more likely to connect to nodes within the same community. This explains why the adjacency matrix looks like a diagonal-block matrix. Let’s see what happens if the structure is changed to disassortative.

# Set logging level to DEBUG
import logging

# Make it be set to DEBUG
logging.getLogger().setLevel(logging.DEBUG)
logging.getLogger("matplotlib").setLevel(logging.WARNING)

# Structure of the network
structure = "disassortative"

# Initialize the synthetic network class
synthetic_dyncrep = SyntheticDynCRep(
    N=N,
    K=K,
    T=T,
    verbose=2,
    avg_degree=avg_degree,
    figsize=(3, 3),
    fontsize=10,
    structure=structure,
)

# Generate synthetic dynamic network
temporal_network = synthetic_dyncrep.generate_net()

DEBUG:root:Removed 1 nodes, because not part of the largest connected component

DEBUG:root:------------------------------

DEBUG:root:t=0

DEBUG:root:Number of nodes: 49 
Number of edges: 114

DEBUG:root:Average degree (2E/N): 4.653

DEBUG:root:Reciprocity at t: 0.03508771929824561

DEBUG:root:------------------------------

As seen above, the network now has a disassortative structure, meaning that nodes are more likely to connect to nodes in different communities. This is reflected in the adjacency matrix, which has connections off the diagonal.

Notice that the generated network has about 140 edges. If we want to generate a network with a larger number of edges, we can increase the ExpM parameter. Let’s see what happens when we set it to 200.

# Set the expected number of edges
ExpM = 200

# Initialize the synthetic network class
synthetic_dyncrep = SyntheticDynCRep(
    N=N,
    K=K,
    T=T,
    verbose=2,
    avg_degree=avg_degree,
    figsize=(3, 3),
    fontsize=10,
    structure=structure,
    ExpM=ExpM,
)

# Generate synthetic dynamic network
temporal_network = synthetic_dyncrep.generate_net()

DEBUG:root:------------------------------

DEBUG:root:t=0

DEBUG:root:Number of nodes: 50 
Number of edges: 213

DEBUG:root:Average degree (2E/N): 8.52

DEBUG:root:Reciprocity at t: 0.09389671361502347

DEBUG:root:------------------------------

Now the resulting network has about 200 edges, as specified by the ExpM parameter.

Another key element of the DynCRep model is the reciprocity parameter eta. This parameter determines the level of reciprocity in the network, i.e., the likelihood that a connection between two nodes is reciprocated. The difference between this reciprocity parameter and the one introduced in the standard CRep model is that the dependency on the reciprocated tie is on the previous time step, while standard CRep considers only the same time t, being an approach valid for static networks. Let’s see how the network structure changes with different values of eta. By default, the eta parameter is set to 0. Let’s see what happens when we increase to 0 .5. Notice that to observe the effect of the reciprocity parameter, we need to generate a network with more than one time step.

# Set the number of time steps
T = 1

# Set the reciprocity parameter
eta = 0.5

# Initialize the synthetic network class
synthetic_dyncrep = SyntheticDynCRep(
    N=N,
    K=K,
    T=T,
    verbose=2,
    avg_degree=avg_degree,
    figsize=(3, 3),
    fontsize=10,
    eta=eta,
)

# Generate synthetic dynamic network
temporal_network = synthetic_dyncrep.generate_net()

DEBUG:root:------------------------------

DEBUG:root:t=0

DEBUG:root:Number of nodes: 50 
Number of edges: 117

DEBUG:root:Average degree (2E/N): 4.68

DEBUG:root:Reciprocity at t: 0.10256410256410256

DEBUG:root:------------------------------

DEBUG:root:------------------------------

DEBUG:root:t=1

DEBUG:root:Number of nodes: 50 
Number of edges: 133

DEBUG:root:Average degree (2E/N): 5.32

DEBUG:root:Reciprocity at t: 0.24060150375939848

DEBUG:root:------------------------------

DEBUG:root:Compare current and previous reciprocity.

DEBUG:root:Time step: 1

DEBUG:root:Number of non-zero elements in the adjacency matrix at time t: 133.0

DEBUG:root:Fraction of non-zero elements in the transposed adjacency matrix at time t: 0.24060150375939848

DEBUG:root:Fraction of non-zero elements in the transposed adjacency matrix at time t-1: 0.15789473684210525

As seen above (more concretely in the graph statistics), the reciprocity in the second time step network is higher than in the first time step network. This is because the eta parameter has been set to 0.5, which increases the likelihood of reciprocated connections in the network. From a graphical perspective, the network structure is more clustered in the second time step, given that the existing connections (i,j) in the first time step are more likely to be reciprocated as (j,i) in the second time step.

Decoding Temporal Relationships with DynCRep

Now that we are more familiar with the generative model, let’s move on to the next step: decoding the temporal relationships in the network using the DynCRep model. For the actual analysis, we will generate a synthetic network with 300 nodes, divided into 3 communities, and observed over 6 time steps. We will set the average degree of the nodes to 10 and the reciprocity parameter to 0.5.

# Set the number of nodes, communities, and time steps
N = 300
K = 3
T = 5
avg_degree = 10
eta = 0.5

# Initialize the synthetic network class
synthetic_dyncrep = SyntheticDynCRep(
    N=N,
    K=K,
    T=T,
    verbose=2,
    avg_degree=avg_degree,
    figsize=(3, 3),
    fontsize=10,
    eta=eta,
)

# Generate synthetic dynamic network
A = synthetic_dyncrep.generate_net()

DEBUG:root:------------------------------

DEBUG:root:t=0

DEBUG:root:Number of nodes: 300 
Number of edges: 1538

DEBUG:root:Average degree (2E/N): 10.253

DEBUG:root:Reciprocity at t: 0.0247074122236671

DEBUG:root:------------------------------

DEBUG:root:------------------------------

DEBUG:root:t=1

DEBUG:root:Number of nodes: 300 
Number of edges: 1686

DEBUG:root:Average degree (2E/N): 11.24

DEBUG:root:Reciprocity at t: 0.17793594306049823

DEBUG:root:------------------------------

DEBUG:root:------------------------------

DEBUG:root:t=2

DEBUG:root:Number of nodes: 300 
Number of edges: 1837

DEBUG:root:Average degree (2E/N): 12.247

DEBUG:root:Reciprocity at t: 0.24496461622210125

DEBUG:root:------------------------------

DEBUG:root:------------------------------

DEBUG:root:t=3

DEBUG:root:Number of nodes: 300 
Number of edges: 1895

DEBUG:root:Average degree (2E/N): 12.633

DEBUG:root:Reciprocity at t: 0.2638522427440633

DEBUG:root:------------------------------

DEBUG:root:------------------------------

DEBUG:root:t=4

DEBUG:root:Number of nodes: 300 
Number of edges: 1964

DEBUG:root:Average degree (2E/N): 13.093

DEBUG:root:Reciprocity at t: 0.3065173116089613

DEBUG:root:------------------------------

DEBUG:root:------------------------------

DEBUG:root:t=5

DEBUG:root:Number of nodes: 300 
Number of edges: 1996

DEBUG:root:Average degree (2E/N): 13.307

DEBUG:root:Reciprocity at t: 0.33266533066132264

DEBUG:root:------------------------------

DEBUG:root:Compare current and previous reciprocity.

DEBUG:root:Time step: 1

DEBUG:root:Number of non-zero elements in the adjacency matrix at time t: 1686.0

DEBUG:root:Fraction of non-zero elements in the transposed adjacency matrix at time t: 0.17793594306049823

DEBUG:root:Fraction of non-zero elements in the transposed adjacency matrix at time t-1: 0.11981020166073547

DEBUG:root:Time step: 2

DEBUG:root:Number of non-zero elements in the adjacency matrix at time t: 1837.0

DEBUG:root:Fraction of non-zero elements in the transposed adjacency matrix at time t: 0.24496461622210125

DEBUG:root:Fraction of non-zero elements in the transposed adjacency matrix at time t-1: 0.22264561785519868

DEBUG:root:Time step: 3

DEBUG:root:Number of non-zero elements in the adjacency matrix at time t: 1895.0

DEBUG:root:Fraction of non-zero elements in the transposed adjacency matrix at time t: 0.2638522427440633

DEBUG:root:Fraction of non-zero elements in the transposed adjacency matrix at time t-1: 0.2559366754617414

DEBUG:root:Time step: 4

DEBUG:root:Number of non-zero elements in the adjacency matrix at time t: 1964.0

DEBUG:root:Fraction of non-zero elements in the transposed adjacency matrix at time t: 0.3065173116089613

DEBUG:root:Fraction of non-zero elements in the transposed adjacency matrix at time t-1: 0.2907331975560081

DEBUG:root:Time step: 5

DEBUG:root:Number of non-zero elements in the adjacency matrix at time t: 1996.0

DEBUG:root:Fraction of non-zero elements in the transposed adjacency matrix at time t: 0.33266533066132264

DEBUG:root:Fraction of non-zero elements in the transposed adjacency matrix at time t-1: 0.3256513026052104

As mentioned initially, the synthetically generated network is a sequence of adjacency matrices A where A[t] represents the adjacency matrix of the network at time t.

A

[<networkx.classes.digraph.DiGraph at 0x7fe3e14b5ee0>,
 <networkx.classes.digraph.DiGraph at 0x7fe3df3a3b60>,
 <networkx.classes.digraph.DiGraph at 0x7fe3e1679880>,
 <networkx.classes.digraph.DiGraph at 0x7fe430833020>,
 <networkx.classes.digraph.DiGraph at 0x7fe3e15d5f40>,
 <networkx.classes.digraph.DiGraph at 0x7fe3e15d46e0>]

To utilize the DynCRep model, we need to transform the sequence of adjacency matrices A into a sequence of adjacency tensors B, where B is a numpy adjacency tensor derived from a NetworkX graph. The build_B_from_A function from the probinet.input.preprocessing module can be used to perform this transformation. The function takes in the sequence of adjacency matrices A and the list of nodes in the network and returns the adjacency tensor B.

from probinet.input.preprocessing import create_adjacency_tensor_from_graph_list

nodes = list(A[0].nodes())

B, _ = create_adjacency_tensor_from_graph_list(A, nodes=nodes)

Done! We now have the adjacency tensor B of dimensions (T+1, N, N) where T is the number of time steps, and N is the number of nodes in the network. The adjacency tensor B is a numpy array that can be used as input to the DynCRep model.

B.shape

(6, 300, 300)

from probinet.models.dyncrep import DynCRep

First, we instantiate the DynCRep model by creating an instance of the class. During this process, we can modify the default attribute values by specifying our desired maximum number of iterations and the number of realizations. The max_iter parameter sets the maximum number of iterations for the optimization algorithm, and the num_realizations parameter defines the number of times the model will run.

dyncrep = DynCRep(max_iter=2000, num_realizations=3)

T = B.shape[0] - 1

The model requires the information to be properly encoded into a GraphData object. We show how to do this below.

from probinet.models.classes import GraphData

# Build the GraphData object to pass to the fit method
gdata = GraphData(
    graph_list=None,
    adjacency_tensor=B,
    transposed_tensor=None,
    nodes=nodes,
    data_values=None,
)

We can now fit the data into the model DynCRep model. We need to set the number of time steps T, the list of nodes in the network, and other parameters too. Given that the data has been generated using K=3, we will change the file to reflect this. The model returns the learned parameters of the model, including the community memberships u, v, the temporal relationships w between the communities, the reciprocity parameter eta, the edge disappearance rate beta and the log-likelihood of the model.

NOTE: The parameters ag and bg are the hyperparameters of the gamma distribution, which is used for regularization. The authors of the main reference assume gamma-distributed priors for the membership vectors. We fix them to 1.1 and 0.5, respectively.

u_dyn, v_dyn, w_dyn, eta_dyn, beta_dyn, Loglikelihood_dyn = dyncrep.fit(
    gdata,
    T=T,
    K=K,
    nodes=nodes,
    ag=1.1,
    bg=0.5,
    out_inference=False,
)

DEBUG:root:Initialization parameter: 0

DEBUG:root:Eta0 parameter: None

INFO:root:### Version: w-DYN ###

DEBUG:root:Number of time steps L: 6

DEBUG:root:T is greater than 0. Proceeding with calculations that require multiple time steps.

DEBUG:root:Random number generator seed: 50370526272105080704012821968206029891

DEBUG:root:eta is initialized randomly.

DEBUG:root:u, v and w are initialized randomly.

DEBUG:root:Updating realization 0 ...

DEBUG:root:num. realization = 0 - iterations = 100 - time = 2.28 seconds

DEBUG:root:num. realization = 0 - iterations = 200 - time = 4.57 seconds

DEBUG:root:num. realization = 0 - iterations = 300 - time = 6.85 seconds

DEBUG:root:num. realization = 0 - iterations = 400 - time = 9.15 seconds

DEBUG:root:num. realization = 0 - iterations = 500 - time = 11.42 seconds

DEBUG:root:num. realization = 0 - iterations = 600 - time = 13.70 seconds

DEBUG:root:num. realization = 0 - iterations = 700 - time = 15.97 seconds

DEBUG:root:num. realization = 0 - iterations = 800 - time = 18.25 seconds

DEBUG:root:num. realization = 0 - iterations = 900 - time = 20.53 seconds

DEBUG:root:num. realization = 0 - iterations = 1000 - time = 22.81 seconds

DEBUG:root:num. realizations = 0 - Log-likelihood = -23544.3977006234 - iterations = 1001 - time = 22.84 seconds - convergence = True

DEBUG:root:Random number generator seed: 215853233443783612736148270941324964867

DEBUG:root:eta is initialized randomly.

DEBUG:root:u, v and w are initialized randomly.

DEBUG:root:Updating realization 1 ...

DEBUG:root:num. realization = 1 - iterations = 100 - time = 2.28 seconds

DEBUG:root:num. realization = 1 - iterations = 200 - time = 4.55 seconds

DEBUG:root:num. realization = 1 - iterations = 300 - time = 6.83 seconds

DEBUG:root:num. realization = 1 - iterations = 400 - time = 9.11 seconds

DEBUG:root:num. realization = 1 - iterations = 500 - time = 11.38 seconds

DEBUG:root:num. realization = 1 - iterations = 600 - time = 13.65 seconds

DEBUG:root:num. realization = 1 - iterations = 700 - time = 15.92 seconds

DEBUG:root:num. realization = 1 - iterations = 800 - time = 18.19 seconds

DEBUG:root:num. realizations = 1 - Log-likelihood = -23572.799393077024 - iterations = 861 - time = 19.59 seconds - convergence = True

DEBUG:root:Random number generator seed: 100307273146361691053842125689906348995

DEBUG:root:eta is initialized randomly.

DEBUG:root:u, v and w are initialized randomly.

DEBUG:root:Updating realization 2 ...

DEBUG:root:num. realization = 2 - iterations = 100 - time = 2.29 seconds

DEBUG:root:num. realization = 2 - iterations = 200 - time = 4.58 seconds

DEBUG:root:num. realization = 2 - iterations = 300 - time = 6.87 seconds

DEBUG:root:num. realization = 2 - iterations = 400 - time = 9.16 seconds

DEBUG:root:num. realization = 2 - iterations = 500 - time = 11.45 seconds

DEBUG:root:num. realization = 2 - iterations = 600 - time = 13.74 seconds

DEBUG:root:num. realization = 2 - iterations = 700 - time = 16.03 seconds

DEBUG:root:num. realizations = 2 - Log-likelihood = -23191.385384739282 - iterations = 701 - time = 16.06 seconds - convergence = True

DEBUG:root:Best realization = 2 - maxL = -23191.385384739282 - best iterations = 701

INFO:root:Algorithm successfully converged after 701 iterations with a maximum log-likelihood of -23191.3854.

DEBUG:root:Parameters won't be saved! If you want to save them, set out_inference=True.

The DynCRep model has been effectively applied to the adjacency tensor B, with the fitting process carried out over three realizations, all of which achieved successful convergence. As detailed in the referenced paper and other tutorials, the vectors u and v represent the outgoing and incoming community memberships of the network’s nodes, respectively, and remain constant over time. Conversely, the affinity matrix, which delineates the edge density within and between communities, is a time-dependent function. By utilizing the plot_matrices function outlined below, we can observe the evolution of these relationships over time.

NOTE: You might be curious as to why the parameters u and v remain constant over time. As outlined in the primary reference, it’s not necessary to estimate these parameters at every time step. The model is designed to estimate them just once. This is part of the first approach, where the affinity matrix is treated as a time-dependent variable, while the community membership vectors, u and v, are kept static over time. It’s worth noting that an alternative interpretation could involve fixing w and allowing u and v to change over time. As explained by the authors, the model can be easily adapted to accommodate this alternative scenario.

w_dyn.shape

(6, 3, 3)

# Set the colormap for the plots
cmap = "PuBuGn"
import matplotlib.pyplot as plt
import numpy as np
from matplotlib.colors import Normalize


def plot_matrices(w, cmap="PuBuGn"):
    """
    Plot a temporal array in a row of subplots with a single shared colorbar.

    Parameters
    ----------
    w : np.ndarray
        Temporal array to plot.
    cmap : str, optional
        Colormap to use for the plots. Default is 'PuBuGn'.
    """
    # Get the number of time steps
    T = w.shape[0]

    # Normalize the data for consistent color mapping
    norm = Normalize(vmin=np.min(w), vmax=np.max(w))

    # Create a new figure with specified size
    fig, axes = plt.subplots(1, T, figsize=(3 * T, 3), constrained_layout=True)

    # Loop over each time step
    for t in range(T):
        # Add a subplot for the current time step
        if T == 1:
            ax = axes
        else:
            ax = axes[t]

        # Display the matrix as an image
        cax = ax.imshow(
            w[t], cmap=cmap, norm=norm, interpolation="nearest", extent=[0, 3, 0, 3]
        )

        # Add a title to the subplot
        ax.set_title("Time step: " + str(t), fontsize=18)

        # Set ticks to be at 0.5, 1.5, 2.5 but show as 0, 1, 2
        ax.set_xticks([0.5, 1.5, 2.5])
        ax.set_yticks([0.5, 1.5, 2.5])

        # Set tick labels to 0, 1, 2
        ax.set_xticklabels([0, 1, 2])
        ax.set_yticklabels([2, 1, 0])

    # Add a single colorbar to the figure
    fig.colorbar(cax, ax=axes, orientation="vertical", fraction=0.046, pad=0.04)

plot_matrices(w_dyn)

The interactions between communities vary across different timesteps, as depicted in the visualizations. These interactions, in conjunction with the vectors u and v, provide insights into the temporal evolution of community memberships.

To further our understanding, we can compare the inferred community memberships with the ground truth data used to train the model. This comparison is facilitated by constructing an overall membership matrix, which contains the community clustering patterns of nodes at each timestep.

from probinet.evaluation.expectation_computation import (
    compute_expected_adjacency_tensor,
)
from probinet.utils.matrix_operations import transpose_tensor

# Define the number of time steps
T = B.shape[0] - 1

# Compute the expected adjacency tensor
lambda_inf_dyn = compute_expected_adjacency_tensor(u_dyn, v_dyn, w_dyn[0])

# Compute the inferred membership tensor
M_inf_dyn = lambda_inf_dyn + eta_dyn * transpose_tensor(B)

The comparison between the two temporal arrays can be done using the calculate_AUC function from the probinet.evaluation module. This function calculates the Area Under the Curve (AUC) for the given temporal arrays. We can use this function to compute the AUC for each time step, i .e., the probability that a randomly selected edge has higher expected value than a randomly selected non-existing edge. A value of 1 means perfect reconstruction, while 0.5 is pure random chance.

from probinet.evaluation.link_prediction import compute_link_prediction_AUC
from probinet.utils.tools import flt

# Compute the AUC for each time step
for t in range(T + 1):
    print(
        f"Time step {t}: AUC = {flt(compute_link_prediction_AUC(M_inf_dyn[t],B[t].astype('int')))}"
    )

Time step 0: AUC = nan
Time step 1: AUC = nan
Time step 2: AUC = nan
Time step 3: AUC = nan
Time step 4: AUC = nan
Time step 5: AUC = nan

/home/runner/.local/lib/python3.12/site-packages/sklearn/metrics/_ranking.py:1179: UndefinedMetricWarning: No negative samples in y_true, false positive value should be meaningless
  warnings.warn(
/home/runner/.local/lib/python3.12/site-packages/sklearn/metrics/_ranking.py:1179: UndefinedMetricWarning: No negative samples in y_true, false positive value should be meaningless
  warnings.warn(
/home/runner/.local/lib/python3.12/site-packages/sklearn/metrics/_ranking.py:1179: UndefinedMetricWarning: No negative samples in y_true, false positive value should be meaningless
  warnings.warn(
/home/runner/.local/lib/python3.12/site-packages/sklearn/metrics/_ranking.py:1179: UndefinedMetricWarning: No negative samples in y_true, false positive value should be meaningless
  warnings.warn(
/home/runner/.local/lib/python3.12/site-packages/sklearn/metrics/_ranking.py:1179: UndefinedMetricWarning: No negative samples in y_true, false positive value should be meaningless
  warnings.warn(
/home/runner/.local/lib/python3.12/site-packages/sklearn/metrics/_ranking.py:1179: UndefinedMetricWarning: No negative samples in y_true, false positive value should be meaningless
  warnings.warn(

As we can see, the results obtained from the DynCRep model show a high AUC value for each time. We can also get a graphical feeling about how the inferred temporal relationships compare to the ground truth data by using the plot_temporal_arrays function defined below.

def plot_temporal_arrays(arrays, titles=None, cmap="PuBuGn"):
    """
    Plot a list of temporal arrays in separate columns.

    Parameters
    ----------
    arrays : list of np.ndarray
        List of temporal arrays to plot.
    cmap : str, optional
        Colormap to use for the plots. Default is 'PuBuGn'.
    """
    # Get the number of time steps from the first array in the list
    T = arrays[0].shape[0] - 1

    # Create a new figure with specified size
    plt.figure(figsize=(3 * len(arrays), 10))

    # Loop over each time step
    for t in range(T + 1):
        # Loop over each array
        for i, array in enumerate(arrays):
            # Add a subplot for the current array at the current time step
            plt.subplot(T + 1, len(arrays), i + len(arrays) * t + 1)

            # Display the array as an image
            plt.imshow(array[t], cmap=cmap, interpolation="nearest")

            # Add a colorbar to the plot
            plt.colorbar(fraction=0.046)

            # Add a title to the first subplot of each column
            if t == 0:
                if titles is not None:
                    plt.title(titles[i])
                else:
                    plt.title(f"Array {i+1}")

plot_temporal_arrays([B, M_inf_dyn], titles=["Ground Truth", "DynCRep-Dynamic"])

As detailed in the main reference [SCDB22], the DynCRep algorithm operates in two distinct modes: Dynamic and Static. The Dynamic mode (also known as w-DYN), which is the default setting, captures and stores temporal relationships in a time-dependent affinity matrix, allowing for a nuanced understanding of evolving community structures. On the other hand, the Static mode (also known as w-STATIC, despite u, v and w being fixed in time, still allows for network evolution. This is facilitated by the appearance and disappearance of edges, governed by the parameters beta and eta. These two modes enhance the flexibility of the model, enabling it to effectively handle a variety of community structures.

Let’s see now how to use the Static mode of the DynCRep model to decode the temporal relationships in the network. We will use the same synthetic network generated earlier, but this time we will fit the DynCRep model in Static mode by setting the temporal parameter to False.

u_stat, v_stat, w_stat, eta_stat, beta_stat, Loglikelihood = dyncrep.fit(
    gdata,
    T=T,
    flag_data_T=0,
    ag=1.1,
    bg=0.5,
    temporal=False,  # <- Static mode
    out_inference=False,
)

DEBUG:root:Initialization parameter: 0

DEBUG:root:Eta0 parameter: None

INFO:root:### Version: w-STATIC ###

DEBUG:root:Number of time steps L: 1

DEBUG:root:T is greater than 0. Proceeding with calculations that require multiple time steps.

DEBUG:root:Random number generator seed: 141356471264389879020624224862114896844

DEBUG:root:eta is initialized randomly.

DEBUG:root:u, v and w are initialized randomly.

DEBUG:root:Updating realization 0 ...

DEBUG:root:num. realization = 0 - iterations = 100 - time = 0.31 seconds

DEBUG:root:num. realization = 0 - iterations = 200 - time = 0.61 seconds

DEBUG:root:num. realizations = 0 - Log-likelihood = -26499.426967119016 - iterations = 261 - time = 0.8 seconds - convergence = True

DEBUG:root:Random number generator seed: 131689210651314741232960846235417003314

DEBUG:root:eta is initialized randomly.

DEBUG:root:u, v and w are initialized randomly.

DEBUG:root:Updating realization 1 ...

DEBUG:root:num. realization = 1 - iterations = 100 - time = 0.30 seconds

DEBUG:root:num. realization = 1 - iterations = 200 - time = 0.61 seconds

DEBUG:root:num. realizations = 1 - Log-likelihood = -26499.426966150644 - iterations = 261 - time = 0.8 seconds - convergence = True

DEBUG:root:Random number generator seed: 41786274154366633018214900204197165128

DEBUG:root:eta is initialized randomly.

DEBUG:root:u, v and w are initialized randomly.

DEBUG:root:Updating realization 2 ...

DEBUG:root:num. realization = 2 - iterations = 100 - time = 0.30 seconds

DEBUG:root:num. realization = 2 - iterations = 200 - time = 0.61 seconds

DEBUG:root:num. realizations = 2 - Log-likelihood = -26473.266765628257 - iterations = 281 - time = 0.86 seconds - convergence = True

DEBUG:root:Best realization = 2 - maxL = -26473.266765628257 - best iterations = 281

INFO:root:Algorithm successfully converged after 281 iterations with a maximum log-likelihood of -26473.2668.

DEBUG:root:Parameters won't be saved! If you want to save them, set out_inference=True.

As shown before, the affinity matrix w is now a constant matrix.

plot_matrices(w_stat)

Summary

We have shown this way how to use the DynCRep model to decode the temporal relationships in a network. The model provides a powerful framework for understanding the dynamic nature of complex networks, capturing the temporal evolution of community structures and reciprocity. We hope this tutorial has provided you with a solid foundation for working with the DynCRep model and decoding temporal relationships in dynamic networks.