Graphs from correlation matrix

Prerequisites for annotating the produced graph

The prerequisites for annotating the produced graph are the same as the ones described at the ## Prerequisites for annotating the produced dendrogram section of the Clustering from correlation matrix unit:

  • Make sure you have used the correlated_reactions function after split of the reversible reactions to separate forward and reverse reactions

  • Create a dictionary from the dictionary_map_reverse_reaction_id_to_pathway function

Construct a graph

The construct_graph function creates a networkx graph from a linear correlation matrix or from both a linear correlation and a non-linear copula dependencies matrix. In this graph reactions are nodes and correlation values are edges. Users can also provide reaction-pathway mapping information in the group_map dictionary parameter and this will be saved in the graph for potential vizualization.

  • linear_correlation_matrix is a numpy 2D array corresponding to the linear correlation matrix

  • non_linear_correlation_matrix is a numpy 2D array (optional) corresonding to the non-linear copula dependencies matrix

  • reactions is a list of reaction names (ordered like matrix indices)

  • remove_unconnected_nodes is a boolean variable that if True, removes isolated nodes

  • correction is a boolean variable if True, absolute values of correlations (edges) are used

  • group_map is a dictionary mapping reaction names to group names (pathways)

G100_full, pos100_full = construct_graph(
    linear_correlation_matrix = linear_correlation_matrix_100_full,
    non_linear_correlation_matrix = non_linear_correlation_matrix_100_full,
    reactions = extended_reactions_100,
    remove_unconnected_nodes = False,
    correction = False,
    group_map = group_map_100_full)

The compute_nodes_centrality_metrics function computes centrality measures for nodes (reactions) in the graph network: (A) Weighted degree centrality (normalized by number of nodes), (B) Betweenness centrality, © Clustering coefficient. Users should provide the full correlation matrices as input and not a subset based on certain reactions/pathways

  • G is a NetworkX graph with nodes and weighted edges; edges should have ‘weight’ and ‘source’ attributes

centrality_dict_100 = compute_nodes_centrality_metrics(
    G = G100_full)

print(centrality_dict_100.get("betweenness").get("PFK"))
print(centrality_dict_100.get("degree").get("PFK"))
print(centrality_dict_100.get("clustering").get("PFK"))

0.00809659410427305
0.22683071469001767
0.6179079858733398

Now, suppose we had created a second graph called G0_full from a different sampling dataset (corresponding to asking at least 0% of the biomass maximum value as an objective). The compare_node_centralities function compares node centralities between two centrality dictionaries for shared nodes and returns the difference by substracting the corresponding metric score of graph 2 (second argument) from graph 1 (first argument).

  • centrality_dict_1 is a dictionary mapping node IDs to centrality values (first graph)

  • centrality_dict_2 is a dictionary mapping node IDs to centrality values (second graph)

sorted_betwenness_nodes = compare_node_centralities(
    centrality_dict_1 = centrality_dict_100.get("betweenness"), 
    centrality_dict_2 = centrality_dict_0.get("betweenness"))

sorted_weighted_degree_nodes = compare_node_centralities(
    centrality_dict_1 = centrality_dict_100.get("degree"), 
    centrality_dict_2 = centrality_dict_0.get("degree"))

print(dict(sorted_betwenness_nodes).get("CYTBD"))
print(dict(sorted_weighted_degree_nodes).get("CYTBD"))

0.1663975459581592
0.09272427760787

Both centrality metrics agree that the CYTBD node, has a higher centrality in the G100_full graph. Users are encouraged to examine the nodes with most extreme differences between the given graphs to get insights on the model’s behavior.

Visualize the constructed graph

The plot_graph function a correlation-based networkx graph with multiple visual features including node annotations, edge styles, and clique-based shadowing. Here we are gonna plot a subset based on the 2 common pathays, we used in the previous sections: Glycolysis and Pentose Phosphate Pathway. The centrality metrics, provided in the centralities parameter should have occured from a full network and thus not be restricted to calculations from a sub-graph

  • G is a NetworkX graph with nodes and weighted edges; edges should have ‘weight’ and ‘source’ attributes

  • pos is a dictionary of node positions

  • remove_clique_edges is a boolean variable that if True, edges within positive cliques and between negative cliques are removed and replaced with shadow areas (green for positive and red for negative)

  • include_matrix2_in_cliques is a boolean variable that if False, only matrix1 (linear edges) is used for positive clique detection. If True cliques may be computed from a combination of linear corelations (matrix1) and copula dependencies (matrix2)

  • min_clique_size is a minimum size for cliques to be considered (default = 5)

  • shadow_edges visualization mode for clique shadows:

    • None : no shadows

    • positive: show shadows around positive cliques

    • negative: show shadows between cliques with negative edges

    • mixed: show both types of shadows

  • centralities is a dictionary containing precomputed centrality metrics

plot_graph(
    G = G100_glycolysis_ppp, 
    pos = pos100_glycolysis_ppp, 
    remove_clique_edges = True, 
    include_matrix2_in_cliques = False, 
    min_clique_size = 5, 
    shadow_edges = "mixed", 
    centralities = centrality_dict_100)

graph_plot