Correspondence between these categories and the RMs is presented later. 1. Non-null identical actions result in a simple relationship symbolized by A ! B, or A ! B. X Y It does not matter whether the social action is denoted by X or Y. We choose A ! B as the X representative relationship in that situation. 2. Two agents identically doing Quinoline-Val-Asp-DifluorophenoxymethylketoneMedChemExpress Q-VD-OPh nothing toward each other are in a null relationship written A ! B. ;; X X YPLOS ONE | DOI:10.1371/journal.pone.0120882 March 31,6 /A Generic Model of Dyadic Social RelationshipsTable 3. Six categories of action fluxes. Category Fluxes characteristics Identical actions 1 2 Non-null actions Null actions Different actions 3 4 5 6 Non-null actions, exchangeable roles Non-null actions, non-exchangeable roles One null action, exchangeable roles One null action, non-exchangeable roles [A ! B and A ! B] Y X A!B YX [A! B and A X X X YRepresentative relationship A!B X A !B ;; XAlternative notations A!B YYRMTEM NullMP A!B XYARYB]Y [A! B and AB]YCS B AsocialA! BXA! B, or AYXB, or AExhaustive categorization of relationships in the model of two agents A and B that can each do X, Y or nothing (;). In elementary interactions, agents can do the same thing or not (i.e. actions can be identical or different) and actions can be null (;) or not (X or Y). Within the relationship, agents can be able to exchange roles or not. doi:10.1371/journal.pone.0120882.t3. Agents performing different non-null actions and able to exchange roles are in a composite X Y representative relationship [A ! B and A ! B].Y X4. In the previous case, if agents cannot exchange roles, the relationship is simple and consists X Y in just A ! B (or just A ! B: again, the notation used for the actions does not matter).Y X5. When one individual does nothing and the other performs a non-null action in an elemenX tary interaction, and roles are exchangeable in the relationship, it is symbolized by [A ! B X and A B] (or the same interactions with the notation Y instead of X).X 6. In the previous case, if roles cannot be exchanged, the relationship consists in only A ! B (or the same with Y instead of X, or with the action flux going from B to A).Proof of exhaustivenessOur first result is to prove the proposition that the six categories of action fluxes given in Table 3 are exhaustive. ! Proposition 1: To describe all relationships arising from the model A B, one needs exX=Y=;X=Y=;actly the six categories of action fluxes defined in Table 3. Proof: On the one hand, the six categories are mutually disjoint. STI-571MedChemExpress Imatinib (Mesylate) Indeed, the fluxes characteristics defining the categories do not overlap. For example, two actions cannot be identical and different at the same time. This shows that no less than these six categories could suffice to ! characterize relationships arising from the setting A B. X=Y=;X=Y=;On the other hand, we noted during the building of the six categories that in some cases, the notation X or Y does not matter, giving rise to alternative notations for some categories. Taking into account these arbitrary choices of notation, the six categories of Table 3 cover the nine elementary interactions of Table 2, as is seen by comparing these two tables. Hence, any relationship built on these nine elementary interactions can be expressed in terms of the six categories,PLOS ONE | DOI:10.1371/journal.pone.0120882 March 31,7 /A Generic Model of Dyadic Social Relationshipssingly or in combination. This shows that no more than these six categories are necessary to ! ch.Correspondence between these categories and the RMs is presented later. 1. Non-null identical actions result in a simple relationship symbolized by A ! B, or A ! B. X Y It does not matter whether the social action is denoted by X or Y. We choose A ! B as the X representative relationship in that situation. 2. Two agents identically doing nothing toward each other are in a null relationship written A ! B. ;; X X YPLOS ONE | DOI:10.1371/journal.pone.0120882 March 31,6 /A Generic Model of Dyadic Social RelationshipsTable 3. Six categories of action fluxes. Category Fluxes characteristics Identical actions 1 2 Non-null actions Null actions Different actions 3 4 5 6 Non-null actions, exchangeable roles Non-null actions, non-exchangeable roles One null action, exchangeable roles One null action, non-exchangeable roles [A ! B and A ! B] Y X A!B YX [A! B and A X X X YRepresentative relationship A!B X A !B ;; XAlternative notations A!B YYRMTEM NullMP A!B XYARYB]Y [A! B and AB]YCS B AsocialA! BXA! B, or AYXB, or AExhaustive categorization of relationships in the model of two agents A and B that can each do X, Y or nothing (;). In elementary interactions, agents can do the same thing or not (i.e. actions can be identical or different) and actions can be null (;) or not (X or Y). Within the relationship, agents can be able to exchange roles or not. doi:10.1371/journal.pone.0120882.t3. Agents performing different non-null actions and able to exchange roles are in a composite X Y representative relationship [A ! B and A ! B].Y X4. In the previous case, if agents cannot exchange roles, the relationship is simple and consists X Y in just A ! B (or just A ! B: again, the notation used for the actions does not matter).Y X5. When one individual does nothing and the other performs a non-null action in an elemenX tary interaction, and roles are exchangeable in the relationship, it is symbolized by [A ! B X and A B] (or the same interactions with the notation Y instead of X).X 6. In the previous case, if roles cannot be exchanged, the relationship consists in only A ! B (or the same with Y instead of X, or with the action flux going from B to A).Proof of exhaustivenessOur first result is to prove the proposition that the six categories of action fluxes given in Table 3 are exhaustive. ! Proposition 1: To describe all relationships arising from the model A B, one needs exX=Y=;X=Y=;actly the six categories of action fluxes defined in Table 3. Proof: On the one hand, the six categories are mutually disjoint. Indeed, the fluxes characteristics defining the categories do not overlap. For example, two actions cannot be identical and different at the same time. This shows that no less than these six categories could suffice to ! characterize relationships arising from the setting A B. X=Y=;X=Y=;On the other hand, we noted during the building of the six categories that in some cases, the notation X or Y does not matter, giving rise to alternative notations for some categories. Taking into account these arbitrary choices of notation, the six categories of Table 3 cover the nine elementary interactions of Table 2, as is seen by comparing these two tables. Hence, any relationship built on these nine elementary interactions can be expressed in terms of the six categories,PLOS ONE | DOI:10.1371/journal.pone.0120882 March 31,7 /A Generic Model of Dyadic Social Relationshipssingly or in combination. This shows that no more than these six categories are necessary to ! ch.