Recording Which Things Are Connected

Botie has a reading for each pump, but a pump’s behaviour can depend on what reaches it through the pipe network. Independent rows do not describe that network.

A graph adds two fundamental pieces: nodes for the things and edges for the relationships. We will turn a small network into an adjacency matrix and check exactly what each row means.

Choose a pump row and count its neighbours. When changing an undirected connection, change the matching entry on the other side of the matrix as well.

We need two tables for different jobs. The feature matrix gives one row of measurements per node. The adjacency matrix records which nodes are connected, using that same node order for its rows and columns.

An undirected connection appears in both directions. For a directed edge, we need an agreed orientation. Here the row names the source and the column names the destination. That convention will determine which matrix multiplication sends information the way we intend.

Changing node order requires changing both adjacency axes and feature rows together.

For an unweighted graph without self-loops, row sums count outgoing neighbours. In an undirected graph they give degree. For weighted edges the same sum is total edge weight, which need not equal neighbour count.

A = [[0,1,0], [1,0,1], [0,1,0]]
Row sums = [1,2,1]

A node with no edges still exists and still needs a feature row.

A dense adjacency matrix makes a small network easy to inspect but uses space for every possible pair. An edge list stores only existing edges as source-destination pairs. Sparse matrices support many of the same algebraic operations without storing every zero.

Duplicate entries require a policy. Counting the same physical pipe twice may accidentally double its influence when building a weighted matrix.

The NumPy exercises use dense arrays for clarity, not as a recommendation for large networks.

import numpy as np
pump_row = np.array([0, 1, 1, 1])
print(np.flatnonzero(pump_row).tolist())
print(int(pump_row.sum()))

[1, 2, 3]
3

A physical pipe, a friendship, a citation, and a similarity edge imply different things. Decide whether messages may travel with or against a directed edge and whether edge weights express capacity, affinity, or another quantity.

A connection alone does not establish causation. Later learning can use relationships predictively even when their causal interpretation is unknown.

Botie’s workshop first checks degrees before any information is passed through the network.

An undirected edge says two pumps are connected. A directed edge can instead describe permitted flow from one pump to another. Decide which relation the model needs before filling an adjacency matrix.

State whether rows identify senders or receivers. Matrix multiplication does not infer this convention. With sender rows, multiplying a row-vector distribution on the right moves mass along outgoing edges; gathering incoming node features uses the transposed adjacency.

An edge list stores endpoint indices instead of a dense table full of zeros. The indices must refer to a saved node vocabulary. Reordering a feature table without reindexing the edges sends messages to the wrong pumps.

Repeated edges need a policy: accumulate their weights for a multigraph or deduplicate them for a simple graph. Do not let the storage conversion choose that meaning accidentally.

import numpy as np
source = np.array([0,0,1])
target = np.array([1,1,2])
A = np.zeros((3,3), dtype=int)
np.add.at(A, (source,target), 1)
print(A.tolist())
print(A.sum(axis=1).tolist())

[[0, 2, 0], [0, 0, 1], [0, 0, 0]]
[2, 1, 0]

A node with no edges should not disappear merely because an edge list never mentions it. Store the full node vocabulary independently. Its features can still support a node-level prediction or a later connection.

For a large sparse network, avoid constructing a dense square matrix solely for convenience. The edge list or a sparse matrix can express the same connectivity with storage proportional to observed edges. Keep the tiny dense version as an inspectable teaching case.