Fig 1.
Different types of triadic relations and their classification according to the strict and loose rule.
In both classifications, C and D are balanced. According to the strict rule, the triads A and B are equally unbalanced. According to the loose rule, A is more unbalanced than B.
Fig 2.
Geometrical degeneracy of triads.
The [+ + −] triad (Type A) has 3 different micro-states defined by the permutations of edge valences shown here. These micro-states constitute the geometrical degeneracy of the triads.
Table 1.
Energies and degeneracies associated with the four types of triadic relationships considered in this work.
Fig 3.
Schematic diagram illustrating the three terms in the proposed Hamiltonian of Eq (6).
Table 2.
Conversion table for the network of alliance relations in EVE.
Fig 4.
Evolution of the number of nodes in the network of alliance relations.
Daily evolution of the number of alliances in EVE Online from February 2015 through April 2016. We discriminate between alliances with sovereignty (SOV) and alliances with more than 200 members (+200).
Fig 5.
Properties of the network of EVE’s alliances.
Left: Daily values of the density of complete triads and the density of edges for two classes of alliances in EVE Online. Right: Daily values of the clustering coefficients of the network of positive (“+Net”) and negative relationships (“−Net”) for the SOV and +200 alliances in EVE Online.
Fig 6.
Network of relationships in the Middle East.
Graph with the relationships between 21 different “agents” in the Middle East as reported by The Economist in December 2015 [24]. We discriminate between enemies (red), allies (blue) and neutrals (no link). We use the following abbreviations to specify the nodes: Moderate Sunni Arab opposition (SYR OP); Syrian government (SYR GOV); Syrian Kurds (SYR KUR); Iraqi Kurds (IRQ KUR); Iraqi Shia militias (IRQ MIL); Iraqi government (IRQ GOV); Turkey (TUR); Israel (ISR); Russia (RUS); USA and E.U (US&EU); Saudi Arabia & Arab League (SAU); Iran (IRN); Al-Qaeda/Jabhat al-Nusra (AL-Q); DAESH or ISIS (DAESH). The node “US&EU” represents 8 agents (US, Australia, Belgium, Canada, France, Germany, Netherlands and United Kingdom).
Fig 7.
Comparison between a randomly signed network and EVE’s real network of alliances.
Daily changes of the occupation probabilities for the four types of triads for the relationships between the alliances in EVE Online. We compare the data with a randomly signed directed network. Blue: [+ + −]; green: [− − −]; red: [+ − −]; magenta: [+ + +]. The left (right) panel is for alliances with sovereignty (alliances with more than 200 members).
Fig 8.
Time series of the extracted triadic energies.
We display the daily values of the log occupation probabilities −ln pi/g(Ei) for the four types of triadic relationships between the +200 (left) and SOV (middle) alliances in EVE Online. These results are based on the data of Fig 7. The right panel displays the yearly values of the −ln pi/g(Ei) for the four types of triadic relationships in the international relationships during the Cold War era.
Table 3.
Extracted triadic energies.
Fig 9.
Time series of the entropy for the triadic relationships in EVE Online.
Daily values of the entropy ST for the occupation probability of the four types of triads in the relationships between the SOV (dashed line) and the +200 (solid line) alliances in EVE Online. The entropies associated with the networks of randomly assigned edges are ST = 8.152 ± 0.011K (SOV) and ST = 8.152 ± 0.066K (+200).
Fig 10.
Time series of the entropy for the triadic relationships among countries during the Cold War era.
Table 4.
Hamiltonian parameters from a theory-data comparison.