NetworkX is a graph analysis library for Python. It has become the standard library for anything graphs in Python. In addition, it’s the basis for most libraries dealing with graph machine learning. Stellargraph in particular requires an understanding of NetworkX to construct graphs.

Below is an overview of the most important API methods. The official documentation is extensive but it remains often confusing to make things happen. Some simple questions (adding arrows, attaching data…) are usually answered in StackOverflow, so the guide below collects these simple but important questions.

General remarks

The library is flexible but these are my golden rules:

  • do not use objects to define nodes, rather use integers and set data on the node. Layout has issues with objects.
  • the API changed a lot over the versions, make sure when you find an answer somewhere that it matches your version. Often methods and answers do not apply because they relate to an older version.
    import networkx as nx
    import matplotlib.pyplot as plt
    from faker import Faker
    faker = Faker()
    %matplotlib inline
    

Creating graphs

There are various constructors to create graphs, among others:

    # default 
    G = nx.Graph()
    # an empty graph
    EG = nx.empty_graph(100)
    # a directed graph
    DG = nx.DiGraph()
    # a multi-directed graph
    MDG = nx.MultiDiGraph()
    # a complete graph
    CG = nx.complete_graph(10)
    # a path graph
    PG = nx.path_graph(5)
    # a complete bipartite graph
    CBG = nx.complete_bipartite_graph(5,3)
    # a grid graph
    GG = nx.grid_graph([2, 3, 5, 2])

Graph generators

Graph generators produce random graphs with particular properties which are of interest in the context of statistics of graphs. The best-known phenomenon is six degrees of separation which you can find on the internet, our brains, our social network and whatnot.

Erdos-Renyi

The Erdos-Renyi model is related to percolations and phase transitions but is in general the most generic random graph model.

The first parameter is the amount of nodes and the second a probability of being connected to another one.

    er = nx.erdos_renyi_graph(50, 0.15)
    nx.draw(er, edge_color='silver')

/Users/swa/conda/lib/python3.7/site-packages/networkx/drawing/nx_pylab.py:611: MatplotlibDeprecationWarning: isinstance(..., numbers.Number)
  if cb.is_numlike(alpha):

png

Watts-Strogratz

The Watts-Strogratz model produces small-world properties.

The first parameter is the amound of node then follows the default degree and thereafter the probability of rewiring and edge. So, the rewiring probability is like the mutation of an otherwise fixed-degree graph.

    ws = nx.watts_strogatz_graph(30, 2, 0.32)
    nx.draw(ws)

png

Barabasi-Albert

The Barabasi-Albert model reproduces random scale-free graphs which are akin to citation networks, the internet and pretty much everywhere in nature.

    ba = nx.barabasi_albert_graph(50, 5)
    nx.draw(ba)

png

You can easily extract the exponential distribution of degrees:

    g = nx.barabasi_albert_graph(2500, 5)
    degrees = list(nx.degree(g))
    l = [d[1] for d in degrees]
    plt.hist(l)
    plt.show()

png

Drawing graphs

The draw method without additional will present the graph with spring-layout algorithm.

    nx.draw(PG)

png

There are of course tons of settings and features and a good result is really dependent on your graph and what you’re looking for. If we take the bipartite graph for example it would be nice to see the two sets of nodes in different colors:

    from networkx.algorithms import bipartite
    X, Y = bipartite.sets(CBG)
    cols = ["red" if i in X else "blue" for i in CBG.nodes() ]

    nx.draw(CBG, with_labels=True, node_color= cols)

png

The grid graph on the other hand is better drawn with the Kamada-Kawai layout in order to see the grid sturcture:

    nx.draw_kamada_kawai(GG)

png

Nodes

If you start from scratch the easiest way to define a graph is via the add_edges_from method as shown here:

    G = nx.Graph()
    G.add_node("time")
    G.add_edges_from([
        ("time","space"), 
        ("gravitation","curvature"),
        ("gravitation","space"),
        ("time","curvature"),
        ])
    labels = {}
    pos = nx.layout.kamada_kawai_layout(G)
    nx.draw(G, pos=pos, with_labels= True)
    nx.draw_networkx_edge_labels(G,pos, 
        {
            ("time","space"): "interacts with",
            ("gravitation","curvature"): "is"
        },
        label_pos=0.4
    )

/Users/swa/conda/lib/python3.7/site-packages/networkx/drawing/nx_pylab.py:611: MatplotlibDeprecationWarning: isinstance(..., numbers.Number)
  if cb.is_numlike(alpha):





{('time',
  'space'): Text(0.39999997946599436, 0.6000000143526016, 'interacts with'),
 ('gravitation',
  'curvature'): Text(-0.3999999793235416, -0.6000000147443909, 'is')}

png

The nodes can however be arbitrary objects:

    class Person:
        def __init__(self, name):
            self.name = name
        @staticmethod
        def random():
            return Person(faker.name())

    g = nx.Graph()
    a = Person.random()
    b = Person.random()
    c = Person.random()
    g.add_edges_from([(a,b), (b,c), (c,a)])
    # to show the names you need to pass the labels
    nx.draw(g, labels = {n:n.name for n in g.nodes()}, with_labels=True)


/Users/swa/conda/lib/python3.7/site-packages/networkx/drawing/nx_pylab.py:611: MatplotlibDeprecationWarning: isinstance(..., numbers.Number)
  if cb.is_numlike(alpha):

png

As mentioned earlier, it’s better to use numbers for the nodes and set the data via the set_node_attributes methods as shown below.

Edges

Arrows can only be shown if the graph is directed. NetworkX is essentially a graph analysis library and much less a graph visualization toolbox.

    G=nx.DiGraph()
    G.add_edge(1,2)  
    pos = nx.circular_layout(G)    
    nx.draw(G, pos, with_labels = True , arrowsize=25)   
    plt.show()

png

Data can be assigned to an edge on creation

    G=nx.DiGraph()
    a = Person.random()
    b = Person.random()
    G.add_node(0, data = a)
    G.add_node(1, data = b)
    G.add_edge(0, 1, label="knows")
    labelDic = {n:G.nodes[n]["data"].name for n in G.nodes()}
    edgeDic = {e:G.get_edge_data(*e)["label"] for e in G.edges}
    kpos = nx.layout.kamada_kawai_layout(G)
    nx.draw(G,kpos,  labels = labelDic, with_labels=True, arrowsize=25)
    nx.draw_networkx_edge_labels(G, kpos, edge_labels= edgeDic, label_pos=0.4)




{(0, 1): Text(-0.19999999999999996, -8.74227765734758e-09, 'knows')}

png

Analysis

There many analysis oriented methods in NetworkX, below are just a few hints to get you started.

Let’s assemble a little network to demonstrate the methods.

    gr = nx.DiGraph()

    gr.add_node(1, data = {'label': 'Space' })
    gr.add_node(2, data = {'label': 'Time' })
    gr.add_node(3, data = {'label': 'Gravitation' })
    gr.add_node(4, data = {'label': 'Geometry' })
    gr.add_node(5, data = {'label': 'SU(2)' })
    gr.add_node(6, data = {'label': 'Spin' })
    gr.add_node(7, data = {'label': 'GL(n)' })
    edge_array = [(1,2), (2,3), (3,1), (3,4), (2,5), (5,6), (1,7)]
    gr.add_edges_from(edge_array)
    import random
    for e in edge_array:
    #     nx.set_edge_attributes(gr, {e: {'data':{'weight': round(random.random(),2)}}})
        gr.add_edge(*e, weight=round(random.random(),2))


    labelDic = {n:gr.nodes[n]["data"]["label"] for n in gr.nodes()}
    edgeDic = {e:gr.edges[e]["weight"] for e in G.edges}
    kpos = nx.layout.kamada_kawai_layout(G)
    nx.draw(G,kpos,  labels = labelDic, with_labels=True, arrowsize=25)
    o=nx.draw_networkx_edge_labels(G, kpos, edge_labels= edgeDic, label_pos=0.4)

png

Getting the adjacency matrix gives a sparse matrix. You need to use the todense method to see the dense matrix. There is also a to_numpy_matrix method which makes it easy to integrate with numpy mechanics.

    nx.adjacency_matrix(gr).todense()




matrix([[0, 1, 0, 0, 0, 0, 1],
        [0, 0, 1, 0, 1, 0, 0],
        [1, 0, 0, 1, 0, 0, 0],
        [0, 0, 0, 0, 0, 0, 0],
        [0, 0, 0, 0, 0, 1, 0],
        [0, 0, 0, 0, 0, 0, 0],
        [0, 0, 0, 0, 0, 0, 0]], dtype=int64)

The spectrum of this adjacency matrix can be directly obtained via the adjacency_spectrum method:

    nx.adjacency_spectrum(gr)




array([-0.5+0.8660254j, -0.5-0.8660254j,  1. +0.j       ,  0. +0.j       ,
        0. +0.j       ,  0. +0.j       ,  0. +0.j       ])

The Laplacian matrix (see definition here) is only defined for undirected graphs but is just a method away:

    nx.laplacian_matrix(gr.to_undirected()).todense()




matrix([[ 3, -1, -1,  0,  0,  0, -1],
        [-1,  3, -1,  0, -1,  0,  0],
        [-1, -1,  3, -1,  0,  0,  0],
        [ 0,  0, -1,  1,  0,  0,  0],
        [ 0, -1,  0,  0,  2, -1,  0],
        [ 0,  0,  0,  0, -1,  1,  0],
        [-1,  0,  0,  0,  0,  0,  1]], dtype=int64)

If you need to use the edge data in the adjacency matrix this goes via the attr_matrix:

    nx.attr_matrix(gr, edge_attr="weight")




(matrix([[0.  , 0.75, 0.  , 0.  , 0.  , 0.  , 0.96],
         [0.  , 0.  , 0.76, 0.  , 0.53, 0.  , 0.  ],
         [0.35, 0.  , 0.  , 0.13, 0.  , 0.  , 0.  ],
         [0.  , 0.  , 0.  , 0.  , 0.  , 0.  , 0.  ],
         [0.  , 0.  , 0.  , 0.  , 0.  , 0.06, 0.  ],
         [0.  , 0.  , 0.  , 0.  , 0.  , 0.  , 0.  ],
         [0.  , 0.  , 0.  , 0.  , 0.  , 0.  , 0.  ]]), [1, 2, 3, 4, 5, 6, 7])

Simple things like degrees are simple to access:

    list(gr.degree)




[(1, 3), (2, 3), (3, 3), (4, 1), (5, 2), (6, 1), (7, 1)]

The shortest path between two vertices is just as simple but please note that there are dozens of variations in the library:

    nx.shortest_path(gr, 1, 6, weight="weight")




[1, 2, 5, 6]

Things like the radius of a graph is defined for undirected graphs:

    nx.radius(gr.to_undirected())




2




    nx.find_cores(gr)




{1: 2, 2: 2, 3: 2, 4: 1, 5: 1, 6: 1, 7: 1}

Centrality is also a whole world on its own. If you wish to visualize the betweenness centrality you can use something like:

    cent = nx.centrality.betweenness_centrality(gr)
    nx.draw(gr, node_size=[v * 1500 for v in cent.values()], edge_color='silver')

png

Getting the connected components of a graph.

    nx.is_connected(gr.to_undirected())
    comps = nx.components.connected_components(gr.to_undirected())
    for c in comps:
        print(c)

{1, 2, 3, 4, 5, 6, 7}

Clique

A clique is a complete subgraph of a particular size. Large, dense subgraphs are useful for example in the analysis of protein-protein interaction graphs, specifically in the prediction of protein complexes.

    def show_clique(graph, k = 4):
        '''
        Draws the first clique of the specified size.
        '''
        cliques = list(nx.algorithms.find_cliques(graph))
        kclique = [clq for clq in cliques if len(clq) == k]

        if len(kclique)>0:        
            print(kclique[0])
            cols = ["red" if i in kclique[0] else "white" for i in graph.nodes() ]
            nx.draw(graph, with_labels=True, node_color= cols, edge_color="silver")
            return nx.subgraph(graph, kclique[0])
        else:
            print("No clique of size %s."%k)
            return nx.Graph()

Taking the Barabasi graph above and checking for isomorphism with the complete graph of the same size we can check that the found result is indeed a clique of the requested size.

    subg = show_clique(ba,5)
    nx.is_isomorphic(subg, nx.complete_graph(5))

[5, 1, 7, 9, 6]





True

png

    red = nx.random_lobster(100, 0.9, 0.9)
    nx.draw(ba)

png

    petersen = nx.petersen_graph()
    nx.draw(petersen)

png

    G=nx.karate_club_graph()

    cent = nx.centrality.betweenness_centrality(G)
    nx.draw(G, node_size=[v * 1500 for v in cent.values()], edge_color='silver')

png

Graph visualization

As described above, if you want pretty images you should use packages outside NetworkX. The dot and GraphML formats are pretty standard and saving a graph to a particular format is really easy.

For example, here we save the karate-club to GraphML and used yEd to layout things together with a centrality resizing of the nodes.

    G=nx.karate_club_graph()
    nx.write_graphml(G, "./karate.graphml")

For Gephi you can use the GML format:

    nx.write_gml(G, "./karate.gml")

Get and set data

In the context of machine learning and real-world data graphs it’s important that nodes and edges carry data. The way it works in NetworkX can be a bit tricky, so let’s make it clear here how it functions.

Node get/set goes like this:

    G = nx.Graph()
    G.add_node(12, payload = {'id': 44, 'name': 'Swa' })
    print(G.nodes[12]['payload'])
    print(G.nodes[12]['payload']['name'])

{'id': 44, 'name': 'Swa'}
Swa

One can also set the data after the node is added:

    G = nx.Graph()
    G.add_node(12)
    nx.set_node_attributes(G, {12:{'payload':{'id': 44, 'name': 'Swa' }}})
    print(G.nodes[12]['payload'])
    print(G.nodes[12]['payload']['name'])

{'id': 44, 'name': 'Swa'}
Swa

Edge get/set is like so:

    G = nx.Graph()
    G.add_edge(12,15, payload={'label': 'stuff'})
    print(G.get_edge_data(12,15))
    print(G.get_edge_data(12,15)['payload']['label'])

{'payload': {'label': 'stuff'}}
stuff

One can also set the data after the edge is added:

    G = nx.Graph()
    G.add_edge(12,15)
    nx.set_edge_attributes(G, {(12,15): {'payload':{'label': 'stuff'}}})
    print(G.get_edge_data(12,15))
    print(G.get_edge_data(12,15)['payload']['label'])

{'payload': {'label': 'stuff'}}
stuff