Wikipedia Ising Networks

• Wikipedia Ising Networks are directed article graphs that model opinion dynamics using Ising-type spin variables and asynchronous Monte Carlo updates.
• The model employs seeding and weighted link methods (OPA and OPS voting) to capture both local influence and global polarization patterns.
• It provides actionable insights into political and cultural contests on Wikipedia by quantifying node-level deviations and global opinions.

Wikipedia Ising Networks are directed article networks in which Wikipedia pages are treated as nodes, links $j \to i$ are induced by article citations or hyperlinks, and each node carries an Ising-type opinion variable whose evolution is determined by the opinions arriving through in-links. In the formulation of Ermann et al., the network is built separately for a given language edition, seeded with a small number of fixed competing opinions, and evolved by asynchronous Monte Carlo updates until it reaches a spin-polarized steady state. The resulting framework is used to quantify node-level and global opinion preferences for political leaders, countries, and social concepts across very large Wikipedia graphs, including the English edition of March 2025 with $N=6\,969\,712$ articles (Ermann et al., 29 Jul 2025).

1. Directed article graph and voting kernels

The basic object is a directed network with nodes $i=1,\dots,N$ representing Wikipedia articles in one language edition. A directed edge $j \to i$ exists if page $j$ cites or links to page $i$, with multiple links counted only once. The adjacency matrix is therefore

$$A_{ij} = \begin{cases} 1 & \text{if } j \to i \text{ in the raw dump},\ 0 & \text{otherwise,} \end{cases}$$

and the column sums $k_j=\sum_i A_{ij}$ are the out-degrees of node $j$ (Ermann et al., 29 Jul 2025).

Two related link-weight operators are used. The first is the raw adjacency itself,

$V_{ij}=A_{ij},$

called OPA-voting. The second is based on a column-stochastic Google-style matrix,

$N=6\,969\,712$0

together with a variant $N=6\,969\,712$1 defined by $N=6\,969\,712$2 for dangling $N=6\,969\,712$3. Choosing

$N=6\,969\,712$4

gives OPS-voting. The difference is structural: OPA uses unnormalized incoming support, whereas OPS weights incoming influence by the source article’s out-degree and suppresses dangling-node broadcast.

The spin variables take values $N=6\,969\,712$5 in the two-opinion setting. The formal Hamiltonian is written as

$N=6\,969\,712$6

with $N=6\,969\,712$7 in practice. Because the network is directed and the operational rule is update-based rather than derived from equilibrium sampling, this Hamiltonian mainly supplies an Ising-type interpretation of alignment on the citation graph.

2. Seeding, asynchronous dynamics, and finite temperature

Opinion competition begins by fixing a small set $N=6\,969\,712$8 of nodes at $N=6\,969\,712$9 and a disjoint set $i=1,\dots,N$0 at $i=1,\dots,N$1. All other nodes are initialized in a neutral white state $i=1,\dots,N$2. White nodes have no initial opinion and no effect on the vote; once a white node first flips to $i=1,\dots,N$3, it never returns to $i=1,\dots,N$4 (Ermann et al., 29 Jul 2025).

A single asynchronous sweep proceeds by taking a random permutation of all non-fixed nodes and updating them one by one. For a node $i=1,\dots,N$5, the local field is

$i=1,\dots,N$6

with $i=1,\dots,N$7. At $i=1,\dots,N$8, the update rule is deterministic: $i=1,\dots,N$9 This is a majority-vote rule defined on incoming weighted links. Repeating the sweep up to $j \to i$0 is typically sufficient; by then only $j \to i$1 of spins still flip.

The model also admits finite-temperature Glauber dynamics. One defines

$j \to i$2

with $j \to i$3, introduces $j \to i$4, and assigns

$j \to i$5

When node $j \to i$6 is selected, it is set to $j \to i$7 with probability $j \to i$8 and to $j \to i$9 otherwise. As $j$0, this recovers the deterministic majority rule; as $j$1, the update becomes random. Empirically, the polarized phase remains stable for small fluctuations and melts near a critical temperature $j$2 (Ermann et al., 29 Jul 2025).

3. Pathway dependence, node polarization, and global order parameters

At $j$3, the final polarized configuration depends on the random update order. For a fixed seed set, different asynchronous pathways can reach different steady patterns even though each individual run stabilizes after roughly twenty sweeps. The model therefore averages over many independent realizations with identical seeds but different random orders; the reported values use $j$4 realizations for English Wikipedia 2025, and $j$5 in some smaller-network studies (Ermann et al., 29 Jul 2025).

For each node $j$6, let $j$7 and $j$8 be the numbers of realizations ending with $j$9 and $i$0. The node polarization is

$i$1

Nodes that stayed white almost always, approximately $i$2–$i$3 of the network, are dropped from the polarization averages. The global polarization is then

$i$4

A positive $i$5 indicates a global tilt toward the $i$6 seed and a negative $i$7 a tilt toward the $i$8 seed.

A related diagnostic is the deviation

$i$9

which measures whether article $$A_{ij} = \begin{cases} 1 & \text{if } j \to i \text{ in the raw dump},\ 0 & \text{otherwise,} \end{cases}$$0 is more positive or more negative than the global average. In geographical or topical projections, $$A_{ij} = \begin{cases} 1 & \text{if } j \to i \text{ in the raw dump},\ 0 & \text{otherwise,} \end{cases}$$1 is used to identify which countries, leaders, or concepts are unusually aligned with one side of the seeded contest. At finite temperature, one may also monitor the number of spin flips per sweep; the jump in that quantity near $$A_{ij} = \begin{cases} 1 & \text{if } j \to i \text{ in the raw dump},\ 0 & \text{otherwise,} \end{cases}$$2 marks the loss of polarization stability.

4. Two-opinion contests on Wikipedia editions

The two-opinion version has been used for several contests on the English, Russian, and Chinese editions of Wikipedia of March 2025, typically with OPS-voting (Ermann et al., 29 Jul 2025).

For Socialism/Communism (red) versus Capitalism/Imperialism (blue) on English Wikipedia 2025, the global polarization is $$A_{ij} = \begin{cases} 1 & \text{if } j \to i \text{ in the raw dump},\ 0 & \text{otherwise,} \end{cases}$$3, implying that $$A_{ij} = \begin{cases} 1 & \text{if } j \to i \text{ in the raw dump},\ 0 & \text{otherwise,} \end{cases}$$4 of nodes end blue. The world map of $$A_{ij} = \begin{cases} 1 & \text{if } j \to i \text{ in the raw dump},\ 0 & \text{otherwise,} \end{cases}$$5 shows Europe slightly blue, much of the Global South strongly blue, but Russia and the former USSR tilting red.

For Apple Inc. (red) versus Microsoft (blue), $$A_{ij} = \begin{cases} 1 & \text{if } j \to i \text{ in the raw dump},\ 0 & \text{otherwise,} \end{cases}$$6, so Microsoft narrowly wins globally. In the reported geographical decomposition, India and Nigeria tilt to Apple, whereas Russia, Georgia, and Ukraine tilt to Microsoft.

For Trump (blue) versus Putin (red), the English edition gives $$A_{ij} = \begin{cases} 1 & \text{if } j \to i \text{ in the raw dump},\ 0 & \text{otherwise,} \end{cases}$$7, corresponding to Putin winning $$A_{ij} = \begin{cases} 1 & \text{if } j \to i \text{ in the raw dump},\ 0 & \text{otherwise,} \end{cases}$$8 against Trump’s $$A_{ij} = \begin{cases} 1 & \text{if } j \to i \text{ in the raw dump},\ 0 & \text{otherwise,} \end{cases}$$9. Among G20 leaders, only Canada and Mexico tilt Trump; all others tilt Putin. The English-language geographical pattern places North America, Latin America, Western Europe, and Japan on the Trump side, while Russia, the former USSR, and Turkey tilt Putin. In the Russian edition, almost all regions tilt Putin, with $k_j=\sum_i A_{ij}$0. In the Chinese edition, $k_j=\sum_i A_{ij}$1, so almost all tilt Trump; the paper attributes this to heavy China–US trade and censored Chinese-mainland access.

Finite-temperature behavior was examined on English Wikipedia 2024 for Socialism versus Capitalism under OPS-voting. At $k_j=\sum_i A_{ij}$2 and $k_j=\sum_i A_{ij}$3, the fraction $k_j=\sum_i A_{ij}$4 is bimodal, approximately $k_j=\sum_i A_{ij}$5 or $k_j=\sum_i A_{ij}$6. Near $k_j=\sum_i A_{ij}$7, the distribution becomes broadly uniform, which is identified as the critical melting point. For $k_j=\sum_i A_{ij}$8, $k_j=\sum_i A_{ij}$9, and the number of flips jumps near $j$0.

5. Three-opinion generalization and global color fractions

The model extends to a three-state Potts-like RGB competition by fixing seed sets $j$1, $j$2, and $j$3, while all other nodes begin white. For a node $j$4, one computes

$j$5

If one color strictly maximizes $j$6, the node takes that color; otherwise it remains unchanged. As in the two-state case, white does not reappear once a node has adopted a non-white state (Ermann et al., 29 Jul 2025).

For each node $j$7, pathway averaging yields color fractions

$j$8

with rare white outcomes neglected. Global color fractions are

$j$9

and $V_{ij}=A_{ij},$0 measures local excess relative to the edition-wide mean.

In the Trump/Putin/Xi contest on 2025 Wikipedia editions under OPS-voting, the global fractions are:

• English: $V_{ij}=A_{ij},$1, with Putin strongest, especially in Russia and the former USSR; Trump next, especially in the USA and Anglosphere; and Xi smaller, concentrated in Southeast Asia.
• Russian: $V_{ij}=A_{ij},$2, so almost everything tilts Putin.
• Chinese: $V_{ij}=A_{ij},$3, so Xi dominates in China and Central Asia.

In the USA/Russia/China contest, the global fractions are:

• English: $V_{ij}=A_{ij},$4, so the USA dominates.
• Russian: $V_{ij}=A_{ij},$5, so Russia dominates.
• Chinese: $V_{ij}=A_{ij},$6, so China dominates.

The paper interprets the corresponding RGB map as showing that each edition “inflates” its own country’s soft power.

The same formalism has also been used for a societal-concepts triad on English Wikipedia 2025: Liberalism (blue), Communism (red), and Nationalism (green). The global fractions are

$V_{ij}=A_{ij},$7

so nationalism is globally strongest. Serbia, Turkey, and Bulgaria are strongly green; the United Kingdom, the United States, and the Benelux countries are strongly liberal; Nepal is red.

6. Related Wikipedia Ising models, nomenclature, and scope

Wikipedia Ising Networks belong to a broader family of Ising-inspired opinion models on directed Wikipedia graphs. A closely related predecessor is the INOF model of Ermann and Shepelyansky, applied to six 2017 language editions—EN, DE, ES, FR, IT, and RU—with sizes including EN at $V_{ij}=A_{ij},$8 nodes and $V_{ij}=A_{ij},$9 directed edges. INOF uses the normalized in-link weight

$N=6\,969\,712$00

drops the teleportation term for dangling columns, and updates non-fixed spins through

$N=6\,969\,712$01

with the deterministic rule $N=6\,969\,712$02, $N=6\,969\,712$03, and $N=6\,969\,712$04 unchanged. For the “Socialism vs. Capitalism” option, the reported global polarizations are $N=6\,969\,712$05 for EN, $N=6\,969\,712$06 for DE, $N=6\,969\,712$07 for ES, $N=6\,969\,712$08 for FR, $N=6\,969\,712$09 for IT, and $N=6\,969\,712$10 for RU; with the expanded “+Communism/Imperialism” option they become $N=6\,969\,712$11, $N=6\,969\,712$12, $N=6\,969\,712$13, $N=6\,969\,712$14, $N=6\,969\,712$15, and $N=6\,969\,712$16, respectively (Ermann et al., 2024).

Another adjacent construction is the Ising-PageRank model. There, each original node is doubled into red and blue copies, each original hyperlink is replaced by a $N=6\,969\,712$17 propagation block $N=6\,969\,712$18 or $N=6\,969\,712$19, and voting is determined by whether the red or blue PageRank component is larger. On English Wikipedia 2017, a purely blue elite of $N=6\,969\,712$20 nodes can shift an otherwise $N=6\,969\,712$21–$N=6\,969\,712$22 society by up to $N=6\,969\,712$23 for a PageRank elite or about $N=6\,969\,712$24 for a CheiRank elite, with

$N=6\,969\,712$25

and $N=6\,969\,712$26 for the PageRank elite (Frahm et al., 2018).

The terminology also overlaps with a different usage in condensed-matter network science. In that literature, IsingNets are weighted graphs whose nodes are Monte Carlo snapshots of the two-dimensional Ising model, connected by configuration-space similarity, and analyzed with percolation, persistent homology, and spectral methods to detect ferromagnetic, critical, and paramagnetic regimes (Sun et al., 2023). This is a different object from Wikipedia Ising Networks, where the nodes are Wikipedia articles rather than spin-configuration snapshots.

This suggests that “Wikipedia Ising Networks” is best understood as a family of related, non-identical models that share three structural elements: a directed Wikipedia article graph, a small set of fixed-opinion seeds, and asynchronous propagation of Ising-type states through incoming links. Across the variants, the central output is not an equilibrium phase diagram in the lattice sense but a node-resolved and edition-resolved polarization map on a very large directed knowledge network.