TopolearnMatrix form connects graphs to linear algebra and machine learning, and is how fast numeric code processes them.
A graph with n nodes can be written as an n x n adjacency matrix A, where A[i, j] is the weight of edge i-j (or 1), and 0 otherwise.
nx.to_numpy_array(G, nodelist=order, weight="weight")gives a dense NumPy array. Fine for up to a few thousand nodes.nx.to_scipy_sparse_array(G, nodelist=order)gives a SciPy sparse array, the right choice for large sparse graphs.nx.from_numpy_array(A)andnx.from_scipy_sparse_array(S)go back; nodes are numbered 0 to n-1, so relabel afterwards if needed.
Pass nodelist= to control row order; otherwise it follows G.nodes insertion order, which is easy to get wrong when matching rows to nodes.
Matrix powers count walks: (A @ A)[i, j] is the number of 2-step walks from i to j. The Laplacian L = D - A (nx.laplacian_matrix(G)) underlies spectral methods. The number of zero eigenvalues of L equals the number of connected components, and the second-smallest eigenvalue, the algebraic connectivity (nx.algebraic_connectivity), measures how well connected the graph is. Its eigenvector (Fiedler vector) gives a classic way to split a graph in two.
nx.adjacency_spectrum, nx.laplacian_spectrum and nx.normalized_laplacian_matrix are available too. These need NumPy and SciPy installed.
Example
import networkx as nx
import numpy as np
G = nx.Graph([("a", "b"), ("b", "c"), ("c", "a"), ("d", "e")])
order = ["a", "b", "c", "d", "e"]
A = nx.to_numpy_array(G, nodelist=order)
print(A[0].tolist()) # [0.0, 1.0, 1.0, 0.0, 0.0]
print(int((A @ A)[0, 0])) # 2 (a->b->a and a->c->a)
S = nx.to_scipy_sparse_array(G, nodelist=order)
print(type(S).__name__, S.nnz) # csr_array 8
L = nx.laplacian_matrix(G, nodelist=order).toarray()
eig = np.linalg.eigvalsh(L)
print(int(np.sum(np.isclose(eig, 0)))) # 2 (= number of components)
H = nx.from_numpy_array(A)
print(sorted(H.edges)[:2]) # [(0, 1), (0, 2)] (integer labels)Remember
- Dense arrays for small graphs, SciPy sparse arrays for large ones
- Always pass
nodelist=to fix the row order - Laplacian spectra reveal components and connectivity
Watch out for
- Building a dense adjacency matrix for a million-node graph and running out of memory.