Assortativity 동류성, 유유상종

Birds of a feather flock together. Do similar nodes link to each other, or do opposites attract? Rewire a network and watch the assortativity coefficient r respond.

Created by Claude Opus 5.5 (Anthropic), based on lecture slides by Sang Hoon Lee for an introductory network science course.

degree assortativity r

0.00

neutral

similar degreesvery different

Click any node to see its k_nn.

Rewiring rule

Coupling J 0.00

Disassortative: hubs grab small nodesAssortative: hubs link to hubs

Model Nodes Speed

Each step picks two edges (a–b, c–d) and reconnects the four endpoints. Every node keeps its degree, so the degree distribution never changes; only who links to whom does.

Example networks

From Noh, Phys. Rev. E 76, 026116 (2007): Poisson networks after Metropolis rewiring

Real networks (degree-disassortative)

Toy networks with a clear sign

Pick an example to load it. You can then rewire it with the controls above; degrees stay fixed, so you can push any example toward either regime.

⟨k_nn(k)⟩ vs k. Rising: assortative. Falling: disassortative.

Degrees at the two ends of every edge. r is the Pearson correlation of this cloud.

Average degree of a node's neighbours

k_nn(i) = 1k_i Σ_j a_ij k_j

Add up the degrees of node i's neighbours, then divide by how many neighbours it has. Click a node in the network to compute it.

How r is computed (Newman 2003)

With e_jk the fraction of edges joining nodes of excess degree j and k, and q_k the excess-degree distribution:

r = Σ_jk jk (e_jk − q_jq_k)σ_q²

This equals the Pearson correlation between the degrees at the two ends of an edge. Using excess degree k−1 or plain degree k gives the same value, because a correlation ignores a constant shift. The dashed line in the left plot is ⟨k²⟩/⟨k⟩, what k_nn would be with no degree correlation at all.

Created by Claude Opus 5.5 (Anthropic). Content and examples are based on lecture slides on assortativity by Sang Hoon Lee for an introductory network science course.

Based on M. E. J. Newman, “Mixing patterns in networks,” Phys. Rev. E 67, 026126 (2003), and Menczer, Fortunato & Davis, A First Course in Network Science, ch. 2. The Metropolis rewiring rule follows J. D. Noh, “Percolation transition in networks with degree-degree correlation,” Phys. Rev. E 76, 026116 (2007); the biased-swap rule follows R. Xulvi-Brunet & I. M. Sokolov, Phys. Rev. E 70, 066102 (2004). Real-network edge lists are the versions bundled with NetworkX. In NetworkX: nx.degree_assortativity_coefficient(G) and nx.attribute_assortativity_coefficient(G, attr).