Recap: Introduction to Networks

DS701 Session 23 — Mon Nov 23, 2026

Today’s plan

  • 5 min — knowledge-check review
  • 20 min — highlights and Q&A (answer or pass — answering always earns credit)
  • 60 min — in-class activity (small groups)
  • wrap-up and cold-call check-ins

Knowledge-Check Review

KC 1: Degree, and why \(2e/n\)

An undirected graph has \(n\) nodes and \(e\) edges. State what the degree of a node is, and explain in one or two sentences why the average degree is \(2e/n\). Then say how the notion of degree changes for a directed graph.

KC 2: Clustering vs. short paths

The lecture defined a small-world network as one with high clustering and* short average path length. Explain why those two properties seem to be in tension (think of the ring lattice), and what Watts and Strogatz showed about how little it takes to get both at once.*

KC 3: Why compare to a random graph?

In the football example the lecture built a \(G(n,p)\) random graph with the same number of nodes and edges as the real network and compared the two. Why is a random graph a useful baseline, and name two ways the lecture showed real networks typically differ from \(G(n,p)\) — say what you would look at to see each difference.

Highlights

A graph is nodes + edges — and the adjacency matrix is the graph

\(E = \{(1,2),(1,3),(2,3),(3,4),(3,5)\}\)

Number the nodes; \(A_{ij} = 1\) if \((i,j) \in E\), else \(0\):

\[ A = \begin{bmatrix} 0 & 1 & 1 & 0 & 0\\ 1 & 0 & 1 & 0 & 0\\ 1 & 1 & 0 & 1 & 1\\ 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 \end{bmatrix} \qquad \text{row sums} = \begin{bmatrix} 2\\2\\4\\1\\1 \end{bmatrix} \]

  • Edge list and \(A\) carry the same information — nothing is lost either way.
  • Degree = row sum. Total \(= 10 = 2e\).
  • Undirected \(\Leftrightarrow\) \(A\) symmetric. Directed: row \(i\) = out-edges, column \(j\) = in-edges — node 1 has in-degree 0, out-degree 2; node 3 has 2 and 2.
  • Directed paths run head-to-tail: 1 → 3 → 5 exists, 5 → 1 does not. Reachability becomes one-way.
  • Weighted: put the weight where the 1 was.

The first questions to ask of any network

  1. How big, how dense? \(n\) nodes, \(e\) edges, average degree \(2e/n\) (undirected).
  2. Who is connected to whom? Degree of each node — its distribution is the first summary.
  3. Is it connected? If not, list the connected components and work on the largest. (Directed: strongly connected components — mutual directed reachability.)
  4. How far apart are things? Shortest paths \(d(i,j)\); the diameter is the largest one, the average shortest path length \(L = \binom{N}{2}^{-1}\sum_{i<j} d(i,j)\) the typical one — both only make sense within a component.
  5. Do neighbours know each other? The clustering coefficient — next slide.

Every one of these is one NetworkX call: G.degree, nx.connected_components, nx.shortest_path_length, nx.diameter, nx.average_clustering. The skill is knowing which to ask, and what a surprising answer would look like.

The baseline: \(G(n,p)\)

  • \(n\) nodes; each of the \(\binom{n}{2}\) pairs gets an edge independently with probability \(p\).
  • Expected degree \(\langle k\rangle = p(n-1) \approx np\). Degrees are concentrated around \(np\) — no hubs.
  • Clustering coefficient \(= p\): two of your neighbours are just two nodes.
  • To match a real network with \(n\) nodes and \(e\) edges: \(p = e / \binom{n}{2}\).

It is not a model of anything real. It is what “no structure” looks like at this size and density — so anything the real graph does differently is worth explaining.

Real networks I: heavy tails

Degree distribution as a CCDF: \(P(X > x)\), the fraction of nodes with degree above \(x\).

  • Power law / Pareto: \(P(X > x) = (k/x)^{\alpha}\) — a straight line on log–log axes, slope \(-\alpha\).
  • Meaning: most nodes have few neighbours; a small but real set of nodes have very many. Hubs.
  • \(G(n,p)\) with the same mean would show no such tail — its log–log plot falls off a cliff.
  • Not universal: the power grid (e) is straight on lin–log axes — an exponential tail.

Real networks II: clustering, and the small world

Clustering coefficient — the probability that two of your neighbours are connected.

\[ C^{(1)} = \frac{\sum_i \#\text{triangles at } i}{\sum_i \#\text{neighbour pairs at } i} \]

Five-node example: \(\dfrac{1+1+1}{1+1+6} = \dfrac{3}{8}\).

\(C^{(2)}\) averages the per-node ratios instead (nx.average_clustering): \(\dfrac{1}{5}\bigl(1+1+\tfrac16\bigr) = 0.433\).

Random graph: \(C = p\). Real networks: much higher.

Watts–Strogatz: a few rewired edges collapse \(L\) while \(C\) holds. Small world = high clustering and short paths. Milgram’s 6.2 hops; Granovetter’s weak ties; Kevin Bacon is not special.

Putting it together: football vs. random

  • Same \(n\), same \(e\). Spring layout: conferences clump; the random graph is a blob.
  • Clustering coefficient: real \(\gg\) random. Diameter and average path: both short.
  • Degree histogram: a spike at 11 games — a scheduling rule. “Different from random” here means more regular, not heavier-tailed. The baseline tells you that there is structure; you still have to say what.

Representation is a modelling choice

The same relationship data can be turned into several graphs — and they answer questions differently.

  • Directed or undirected? “Follows” is directed; symmetrising it throws away who is followed vs. who follows. In-degree and out-degree rank people differently.
  • Weighted or unweighted? Number of games played, strength of a tie, a distance. Shortest paths and “closest” nodes change when weights count.
  • From a similarity matrix? Threshold last month’s distance matrix (L02) at \(\tau\): an edge where similarity \(> \tau\). Loose \(\tau\): one giant component. Tight \(\tau\): many islands. The components you find depend on a number you chose.

There is no representation-free “the network”. Today’s activity: same data, three representations, three sets of answers — then you decide which is right for the question. Next session: centrality and PageRank — the representation choice bites even harder there.

In-Class Activity

Activity: build the graph, then change what “the graph” means

Goal: build small networks in NetworkX from edge lists — adjacency matrix, degree, components, shortest paths, diameter, clustering, degree distributions against a \(G(n,p)\) baseline and a heavy-tailed one — then re-represent the same data as directed / weighted / thresholded graphs and watch the answers move.

  • Work in groups of 2–3. Open your section’s notebook.
  • Parts marked (autograded) are submitted to Gradescope; the rest is participation.
  • Staff will circulate — be ready to explain any part of your work.
  • Dependencies: numpy, pandas, matplotlib, scipy, networkx.

Section A1

Section B1

The activity notebook goes live on the day of the lecture. Colab is optional — you can also open it on GitHub and run it locally.

Going deeper

Full lecture notes: Introduction to Networks