Taking turns at doing X and in parallel trading X for Y. This may correspond to a relationship evolving with time from one RM to the other. The generalization of our model to N social actions, presented in the next section, helps represent any familiar composite relationship.PLOS ONE | DOI:10.1371/journal.pone.0120882 March 31,9 /A Generic Model of Dyadic Social order ABT-737 RelationshipsGeneralization to N social actionsIn real social relationships, the number of occurring social actions is expected to be larger than two, which motivates the generalization of our results to any number N of social actions. This is our third result. For the generalization that follows, we let X and Y be elements of a larger set S of N social actions Si: S = Siji = 1,. . .,N, such that X,Y 2 S, for instance S1 X and S2 Y. Proposition 2: In the general case of N non-null social actions (S1,S2, . . . ,SN 2 S, N ! 2), one still needs exactly the six categories of Table 3 to describe all possible relationships arising ! from the setting A B. S =S =:::=S =;1 2 NS1 =S2 =:::=SN =;Idea of the proof: We show that the proof of exhaustiveness of the six categories of Table 3 carried out for N = 2 holds for any N ! 2. Namely, the same process allows to build the same six mutually disjoint categories of action fluxes, and these categories span the relationship space for any N ! 2. Proof: In the general case of N ! 2 non-null social actions, there are 2N+1 elementary inter2 actions and 2(N+1) -1 relationships. S1 �S2 ! Cases such as A B (where A performs several actions simultaneously) can be writtenSA ! B (where S4 is a bundle of actions). More generally, any number of actions can be bundled SSas in that example. Starting from a set of N social actions, the set S can include all subsets of that set. (The cardinality of S is then 2N.) Hence, any union of two or more subsets (such as S1 and S2 to give S4) gives another subset that is an element of S. Then, because there are still two agents and thus at most two different social actions per elementary interaction, the elementary interactions have the same forms as for N = 2, with additional notations for the social actions. As an illustration, Table 4 shows the sixteen elementary interactions that result from the ! case N = 3, i.e. the model A B. X=Y=Z=; For any N ! 2, looking at an elementary interaction between two individuals, one can still only differentiate between (i) identical or different actions, (ii) interchangeable or non-interchangeable roles, (iii) null or non-null actions. Hence, with more than two actions, this differentiation process leads to the same six disjoint categories, except with more PX-478 site alternative notations than in Table 3. For example, for N = 3, category 1 (EM) gets one more alternative notation than for N = 2, Z Z Z namely A ! B. Category 3 (MP) gets two alternative notations: [A ! B and A ! B], andZ X X X=Y=Z=;Table 4. Sixteen elementary interactions for N = 3 social actions. A!B X A!B X A!B X AX Z Y XA!B Y A!B Y A!B Y AY Z YXA!B Z A!B Z A!B Z AZ Z YXX A! B Y A! B Z A! BBBBA !B ;;This table shows the sixteen elementary interactions arising from our model with N = 3 non-null social ! actions X,Y,Z between two agents A and B, that is, A B. We use simplified notations for theX=Y=Z=; X=Y=Z=;interactions involving one empty flux. doi:10.1371/journal.pone.0120882.tPLOS ONE | DOI:10.1371/journal.pone.0120882 March 31,10 /A Generic Model of Dyadic Social Relationships[A ! B and A ! B]. Category 4 (AR) gets f.Taking turns at doing X and in parallel trading X for Y. This may correspond to a relationship evolving with time from one RM to the other. The generalization of our model to N social actions, presented in the next section, helps represent any familiar composite relationship.PLOS ONE | DOI:10.1371/journal.pone.0120882 March 31,9 /A Generic Model of Dyadic Social RelationshipsGeneralization to N social actionsIn real social relationships, the number of occurring social actions is expected to be larger than two, which motivates the generalization of our results to any number N of social actions. This is our third result. For the generalization that follows, we let X and Y be elements of a larger set S of N social actions Si: S = Siji = 1,. . .,N, such that X,Y 2 S, for instance S1 X and S2 Y. Proposition 2: In the general case of N non-null social actions (S1,S2, . . . ,SN 2 S, N ! 2), one still needs exactly the six categories of Table 3 to describe all possible relationships arising ! from the setting A B. S =S =:::=S =;1 2 NS1 =S2 =:::=SN =;Idea of the proof: We show that the proof of exhaustiveness of the six categories of Table 3 carried out for N = 2 holds for any N ! 2. Namely, the same process allows to build the same six mutually disjoint categories of action fluxes, and these categories span the relationship space for any N ! 2. Proof: In the general case of N ! 2 non-null social actions, there are 2N+1 elementary inter2 actions and 2(N+1) -1 relationships. S1 �S2 ! Cases such as A B (where A performs several actions simultaneously) can be writtenSA ! B (where S4 is a bundle of actions). More generally, any number of actions can be bundled SSas in that example. Starting from a set of N social actions, the set S can include all subsets of that set. (The cardinality of S is then 2N.) Hence, any union of two or more subsets (such as S1 and S2 to give S4) gives another subset that is an element of S. Then, because there are still two agents and thus at most two different social actions per elementary interaction, the elementary interactions have the same forms as for N = 2, with additional notations for the social actions. As an illustration, Table 4 shows the sixteen elementary interactions that result from the ! case N = 3, i.e. the model A B. X=Y=Z=; For any N ! 2, looking at an elementary interaction between two individuals, one can still only differentiate between (i) identical or different actions, (ii) interchangeable or non-interchangeable roles, (iii) null or non-null actions. Hence, with more than two actions, this differentiation process leads to the same six disjoint categories, except with more alternative notations than in Table 3. For example, for N = 3, category 1 (EM) gets one more alternative notation than for N = 2, Z Z Z namely A ! B. Category 3 (MP) gets two alternative notations: [A ! B and A ! B], andZ X X X=Y=Z=;Table 4. Sixteen elementary interactions for N = 3 social actions. A!B X A!B X A!B X AX Z Y XA!B Y A!B Y A!B Y AY Z YXA!B Z A!B Z A!B Z AZ Z YXX A! B Y A! B Z A! BBBBA !B ;;This table shows the sixteen elementary interactions arising from our model with N = 3 non-null social ! actions X,Y,Z between two agents A and B, that is, A B. We use simplified notations for theX=Y=Z=; X=Y=Z=;interactions involving one empty flux. doi:10.1371/journal.pone.0120882.tPLOS ONE | DOI:10.1371/journal.pone.0120882 March 31,10 /A Generic Model of Dyadic Social Relationships[A ! B and A ! B]. Category 4 (AR) gets f.
