Edges and Weights

Module 5: Complex Networks

Toolkit page

This page is self-study material for Session 23. It continues NetworkX Basics.

The first chunk rebuilds the country graph of the previous page, with four nodes and a population attribute.

import networkx as nx

G = nx.Graph()
G.add_node("Spain", population=47_000_000)
G.add_node("Portugal", population=10_000_000)
G.add_node("France", population=67_000_000)
G.add_node("Italy", population=60_000_000)

Edges (Links)

Edges represent the connections between nodes in a graph. To add edges to the graph, you can use the add_edge(<node1>, <node2>) method. This will create an undirected edge between node1 and node2 (you refer to nodes by their IDs). You can see the list of edges in the graph using the edges() method.

# Add edges between the nodes (neighboring countries)
G.add_edge("Spain", "Portugal")
G.add_edge("Spain", "France")
# Show the edges in the graph
print(G.edges())

[('Spain', 'Portugal'), ('Spain', 'France')]

Because you are working on an undirected graph, the connection between two nodes does not have a direction. However, you can also create directed graphs where the edges have a direction (from one node to another) using DiGraph() instead of Graph(). In a directed graph, you use add_edge(<source>, <target>) to specify the direction of the edge.

Edge Attributes

Just like nodes, edges can also have attributes, such as weight, which might represent the strength of the connection. For example, in a social network, an edge might represent a friendship between two people, and the weight could represent how close they are.

You can add attributes to an edge by accessing it through G.edges[<node1>, <node2>] and assigning values to the attributes. For example, add a “distance” (between capitals) attribute to represent the distance between the countries:

# Add distance attribute to the edges
G.edges["Spain", "Portugal"]["distance"] = 600  # distance in kilometers
G.edges["Spain", "France"]["distance"] = 1000  # distance in kilometers

# Show the edges with their attributes
distance = nx.get_edge_attributes(G, 'distance')
for edge, dist in distance.items():
    print(f"{edge}: {dist} km")

('Spain', 'Portugal'): 600 km
('Spain', 'France'): 1000 km

Using get_edge_attributes(<graph>, <attribute>), you can retrieve the distance attribute for all edges in the graph as a dictionary.

Again, you can also add attributes to an edge at the same time as you create it using the add_edge() method with keyword arguments. For example:

# Add a new edge with attributes
G.add_edge("France", "Italy", distance=800)

# Show the edges with their attributes
distance = nx.get_edge_attributes(G, 'distance')
for edge, dist in distance.items():
    print(f"{edge}: {dist} km")

('Spain', 'Portugal'): 600 km
('Spain', 'France'): 1000 km
('France', 'Italy'): 800 km

Adding Nodes and Edges Together

You can also add nodes and edges together using the add_edge() method. If you add an edge between two nodes that do not exist in the graph, networkx will automatically create those nodes. For example:

# Add an edge between two nodes that do not exist
G.add_edge("USA", "Canada", distance=3000)
G.add_edge("Netherlands", "Germany", distance=500)
G.add_edge("Germany", "France", distance=900)

# Add the population attribute for the new nodes separately
G.nodes["USA"]["population"] = 331_000_000
G.nodes["Canada"]["population"] = 38_000_000
G.nodes["Netherlands"]["population"] = 17_000_000
G.nodes["Germany"]["population"] = 83_000_000

# Show the nodes and edges in the graph
print("Nodes:", G.nodes())
print("Edges:", G.edges())

Nodes: ['Spain', 'Portugal', 'France', 'Italy', 'USA', 'Canada', 'Netherlands', 'Germany']
Edges: [('Spain', 'Portugal'), ('Spain', 'France'), ('France', 'Italy'), ('France', 'Germany'), ('USA', 'Canada'), ('Netherlands', 'Germany')]

Weighted Edges

The most common edge attribute is weight, which represents the strength of the connection between two nodes. For example, in a social network, the weight of an edge could represent how close two people are. In a transportation network, the weight could represent the distance or travel time between two locations.

Weight is so common that networkx has a special method to add weighted edges: add_weighted_edges_from(). This method takes a list of tuples, where each tuple contains the source node, target node, and weight of the edge. For example:

# Initialize a directed graph
D_contracts = nx.DiGraph()

# Add weighted edges (contracts between companies with the amount of money as weight)
# Disclaimer: weights are made up
weighted_edges = [
    ("OpenAI", "Microsoft", 100_000_000),
    ("OpenAI", "Oracle", 50_000_000),
    ("Microsoft", "Nvidia", 20_000_000),
    ("Oracle", "Nvidia", 10_000_000),
    ("Nvidia", "OpenAI", 5_000_000),
]
D_contracts.add_weighted_edges_from(weighted_edges)

# Show the edges with their weights
weights = nx.get_edge_attributes(D_contracts, 'weight')
for edge, weight in weights.items():
    print(f"{edge}: ${weight}")

('OpenAI', 'Microsoft'): $100000000
('OpenAI', 'Oracle'): $50000000
('Microsoft', 'Nvidia'): $20000000
('Oracle', 'Nvidia'): $10000000
('Nvidia', 'OpenAI'): $5000000

What’s Next?

The next page draws these graphs.

Drawing Graphs

--- title: "Edges and Weights" subtitle: "Module 5: Complex Networks" format: html --- ::: {.callout-note} ## Toolkit page This page is self-study material for Session 23. It continues [NetworkX Basics](networkx-basics.qmd). ::: The first chunk rebuilds the country graph of the previous page, with four nodes and a population attribute. ```{python} import networkx as nx G = nx.Graph() G.add_node("Spain", population=47_000_000) G.add_node("Portugal", population=10_000_000) G.add_node("France", population=67_000_000) G.add_node("Italy", population=60_000_000) ``` ## Edges (Links) Edges represent the connections between nodes in a graph. To add edges to the graph, you can use the `add_edge(<node1>, <node2>)` method. This will create an undirected edge between `node1` and `node2` (you refer to nodes by their IDs). You can see the list of edges in the graph using the `edges()` method. ```{python} #| echo: true #| message: true # Add edges between the nodes (neighboring countries) G.add_edge("Spain", "Portugal") G.add_edge("Spain", "France") # Show the edges in the graph print(G.edges()) ``` Because you are working on an **undirected** graph, the connection between two nodes does not have a direction. However, you can also create directed graphs where the edges have a direction (from one node to another) using `DiGraph()` instead of `Graph()`. In a directed graph, you use `add_edge(<source>, <target>)` to specify the direction of the edge. ### Edge Attributes Just like nodes, edges can also have attributes, such as weight, which might represent the strength of the connection. For example, in a social network, an edge might represent a friendship between two people, and the weight could represent how close they are. You can add attributes to an edge by accessing it through `G.edges[<node1>, <node2>]` and assigning values to the attributes. For example, add a "distance" (between capitals) attribute to represent the distance between the countries: ```{python} #| echo: true #| message: true # Add distance attribute to the edges G.edges["Spain", "Portugal"]["distance"] = 600 # distance in kilometers G.edges["Spain", "France"]["distance"] = 1000 # distance in kilometers # Show the edges with their attributes distance = nx.get_edge_attributes(G, 'distance') for edge, dist in distance.items(): print(f"{edge}: {dist} km") ``` Using `get_edge_attributes(<graph>, <attribute>)`, you can retrieve the distance attribute for all edges in the graph as a dictionary. Again, you can also add attributes to an edge at the same time as you create it using the `add_edge()` method with keyword arguments. For example: ```{python} #| echo: true #| message: true # Add a new edge with attributes G.add_edge("France", "Italy", distance=800) # Show the edges with their attributes distance = nx.get_edge_attributes(G, 'distance') for edge, dist in distance.items(): print(f"{edge}: {dist} km") ``` ### Adding Nodes and Edges Together You can also add nodes and edges together using the `add_edge()` method. If you add an edge between two nodes that do not exist in the graph, `networkx` will automatically create those nodes. For example: ```{python} #| echo: true #| message: true # Add an edge between two nodes that do not exist G.add_edge("USA", "Canada", distance=3000) G.add_edge("Netherlands", "Germany", distance=500) G.add_edge("Germany", "France", distance=900) # Add the population attribute for the new nodes separately G.nodes["USA"]["population"] = 331_000_000 G.nodes["Canada"]["population"] = 38_000_000 G.nodes["Netherlands"]["population"] = 17_000_000 G.nodes["Germany"]["population"] = 83_000_000 # Show the nodes and edges in the graph print("Nodes:", G.nodes()) print("Edges:", G.edges()) ``` ### Weighted Edges The most common edge attribute is weight, which represents the strength of the connection between two nodes. For example, in a social network, the weight of an edge could represent how close two people are. In a transportation network, the weight could represent the distance or travel time between two locations. Weight is so common that `networkx` has a special method to add weighted edges: `add_weighted_edges_from()`. This method takes a list of tuples, where each tuple contains the source node, target node, and weight of the edge. For example: ```{python} #| echo: true #| message: true # Initialize a directed graph D_contracts = nx.DiGraph() # Add weighted edges (contracts between companies with the amount of money as weight) # Disclaimer: weights are made up weighted_edges = [ ("OpenAI", "Microsoft", 100_000_000), ("OpenAI", "Oracle", 50_000_000), ("Microsoft", "Nvidia", 20_000_000), ("Oracle", "Nvidia", 10_000_000), ("Nvidia", "OpenAI", 5_000_000), ] D_contracts.add_weighted_edges_from(weighted_edges) # Show the edges with their weights weights = nx.get_edge_attributes(D_contracts, 'weight') for edge, weight in weights.items(): print(f"{edge}: ${weight}") ``` ## What's Next? The next page draws these graphs. [Drawing Graphs](networkx-drawing.qmd){.btn .btn-primary}