This appendix presents definitions and statements related to the indicator co-ranking R(…, …) that are intended for characterisation of the level of intransitivity in large and, possibly, evolving systems. The main assumption of this section is that all elements in system
${\mathrm{S}}_{0}$ are connected and form an overall group
${\mathrm{G}}_{0}$, implying that there is a positive weight g_{i} = g(C_{i}) > 0 specified for every element C_{i} ∈
${\mathrm{G}}_{0}$. In this section, weights g_{i} are associated with the whole system
${\mathrm{G}}_{0}$ and thus is the same for all groups
${\mathrm{G}}_{q}\subseteq {\mathrm{G}}_{0}$, which are referred to in this section as sets
${\mathrm{S}}_{q}={\mathrm{G}}_{q}$, q = 1, 2, … to emphasise that the element weights are specified only for the whole system.

#### C.2. Properties of Current Rankings

The current ranking is a special case of conditional indicator-ranking, i.e., R^{*}(A) is
${R}_{{\mathrm{G}}_{q}}(\mathrm{A})$ with set
${\mathrm{G}}_{q}$ expanded to the whole system
${\mathrm{G}}_{0}$. The following propositions characterise properties of

**Proposition C1** If the primary preference is currently transitive, the secondary preference is equivalent to the primary preference:$A{\underset{\xaf}{\prec}}^{\prime \prime}B\iff {R}^{*}(A)\le {R}^{*}(B)\iff A\underset{\xaf}{\prec}B$ for any A and B.

The proof is similar to the proof of Proposition B1, but the case of

R^{*}(A) =

R^{*}(B) (

i.e., A

∼″B) is now impossible when A≺B. Indeed, all elements are presumed to be connected and present in the reference set in definition of the current ranking. Hence, the terms

R(A,B) =

−R(B,A) < 0 are present in the sums

Equation (C1) evaluated for

R^{*}(A) while including C

_{i} =B, and

R^{*}(B) while including C

_{i} =A.

**Proposition C2** If B is transitively preferred to A in a generally intransitive preference, then the preference of B over A is preserved by the current ranking: A≺≺B =⇒ R^{*}(A) < R^{*}(B) ⇔A≺″B

Although this proposition refers to a transitive preference A≺≺B in a preference ≺ that is generally intransitive and is thus different from the previous statements, its proof is similar. The relation A≺≺B requires that A≺B and there are no C

_{i} that satisfies C

_{i} A≺B C

_{i}. Hence,

R(A,C

_{i}) ≤

R(B,C

_{i}) and

R(A,B) =

−R(B,A) < 0 so that

R^{*}(A)

< R^{*}(B) as defined by

Equation (C1). Note that the inverse statements

R^{*}(A)

< R^{*}(B) =⇒ A≺≺B and

R^{*}(A)

< R^{*}(B) =⇒ A≺B are incorrect in intransitive systems.

Since the preference specified by the primary current ranking is transitive, the preferences specified by the primary and secondary current rankings must be equivalent according to Proposition C1:

Here, Proposition C1 is applied to secondary (as primary) and tertiary (as secondary) preferences. The conclusion is that there is no independent tertiary preferences as they coincide with the secondary preferences. Equivalence of the primary and secondary rankings, however, does not imply that these rankings are identical: generally R^{*}(A) ≠ R^{**}(A). Deviations of secondary ranking from primary ranking indicate intransitivity in evolutions of competitive systems as determined by the following theorem:

**Theorem C1** The following statements are correct for a system${\mathrm{G}}_{0}$ of connected elements (i.e., g(

C_{i}) > 0

for any C_{i} ∈

${\mathrm{G}}_{0}$):(a) If the primary preference is currently transitive, the primary and secondary current rankings coincide (i.e., R^{*}(C_{i}) = R^{**}(Ci) for all C_{i} ∈
${\mathrm{G}}_{0}$).

(b) If the primary and secondary rankings coincide (i.e., R^{*}(C_{i}) = R^{**}(Ci) for all elements C_{i} ∈
${\mathrm{G}}_{0}$), then the secondary preference is a coarsening of the primary preference (i.e., C_{i} ≺″C_{j} =⇒ C_{i} ≺C_{j}).

(c) In particular, if the primary and secondary rankings coincide and are strict (i.e., current rankings of different elements are different: R^{*}(C_{i}) ≠ R^{*}(C_{i}) for any C_{i} ≁ C_{j}), then the primary preference is currently transitive.

**Proof**. Statement (a) immediately follows from the equivalence of primary and secondary preferences, as stated in Proposition C1, while statements (b) and (c) require detailed consideration.

Let us sort all elements into

k sets of decreasing primary ranking

${R}_{1}^{*}>{R}_{2}^{*}>\dots {R}_{k}^{*}$ ;

i.e.,

q^{th} set

${\mathrm{G}}_{q}$ contains

n_{q} ≥ 1 elements that have primary current ranking

${R}_{q}^{*}$. The elements C

_{1},

…,C

_{i},

…,C

_{n} are thus ordered according to decreasing primary ranking.

Figure C1 shows the structure of the

n × n matrices

R_{ij} =

R(C

_{i},C

_{j}) and

${{R}^{\u2033}}_{ij}={R}^{\u2033}({\mathrm{C}}_{i},{\mathrm{C}}_{j})$, which correspond to the primary and secondary co-rankings. The co-ranking of different sets is denoted by

${\overline{R}}_{qp}=\overline{R}({\mathrm{G}}_{q},{\mathrm{G}}_{p})$ for primary preferences and by

${{\overline{R}}^{\u2033}}_{qp}={\overline{R}}^{\u2033}({\mathrm{G}}_{q},{\mathrm{G}}_{p})$ for the secondary preferences. The average primary current ranking of a set is denoted by

${\overline{R}}^{*}{}_{q}={\overline{R}}^{*}({\mathrm{G}}_{q})=\overline{R}({\mathrm{G}}_{q},{\mathrm{G}}_{0})$, while the average secondary current ranking is denoted by

${\overline{R}}^{**}{}_{q}={\overline{R}}^{**}({\mathrm{G}}_{q})={\overline{R}}^{\u2033}({\mathrm{G}}_{q},{\mathrm{G}}_{0})$. These quantities are specified by

Equations (24),

(25) and

(C1)–

(C4). Obviously

${\overline{R}}^{*}{}_{q}={\overline{R}}_{q}^{**}$ for all

q due to equivalence of the primary and secondary current rankings as stated in the theorem (

i.e.,

R^{*}(C

_{i})=

R^{**}(C

_{i}) for all C

_{i} ∈

${\mathrm{G}}_{0}$ ).

**Figure C1.**
Co-ranking matrix with elements ordered according to their current rankings.

**Figure C1.**
Co-ranking matrix with elements ordered according to their current rankings.

The first (leading) set

${\mathrm{G}}_{1}$ is considered first. The co-rankings

$\overline{R}({\mathrm{G}}_{1},{\mathrm{G}}_{1})$ and

${\overline{R}}^{\u2033}({\mathrm{G}}_{1},{\mathrm{G}}_{1})$, which are determined by the sums over the dark square region a1-b1-b2-a2 (

Figure C1), are zeros as the matrices

R_{ij} and

${{\overline{R}}^{\u2033}}_{ij}$ are antisymmetric. Since

${\overline{R}}_{1}^{*}=\overline{R}({\mathrm{G}}_{1},{\mathrm{G}}_{0})$ is the same as

${\overline{R}}_{1}^{**}={\overline{R}}^{\u2033}({\mathrm{G}}_{1},{\mathrm{G}}_{0})$ while

$\overline{R}({\mathrm{G}}_{1},{\mathrm{G}}_{1})={\overline{R}}^{\u2033}({\mathrm{G}}_{1},{\mathrm{G}}_{1})=0$, the co-rankings

$\overline{R}({\mathrm{G}}_{1},{\mathrm{G}}_{0}-{\mathrm{G}}_{1})$ and

${\overline{R}}^{\u2033}({\mathrm{G}}_{1},{\mathrm{G}}_{0}-{\mathrm{G}}_{1})$ evaluated in terms the corresponding sums over the rectangle a2-b2-b4-a4 must be the same. Hence

R_{ij} = 1 in this rectangle since in any other case the sums

$\overline{R}({\mathrm{G}}_{1},{\mathrm{G}}_{0}-{\mathrm{G}}_{1})$ cannot coincide with

${\overline{R}}^{\u2033}({\mathrm{G}}_{1},{\mathrm{G}}_{0}-{\mathrm{G}}_{1})$.

Since the co-ranking matrices are antisymmetric, R_{ij} = −1 and
${{R}^{\u2033}}_{ij}=-1$ in the rectangle b1-c1-c2-b2. We take into account that
$\overline{R}({\mathrm{G}}_{2},{\mathrm{G}}_{2})={\overline{R}}^{\u2033}({\mathrm{G}}_{2},{\mathrm{G}}_{2})=0$ (the sums over the dark squares are zeros) and reiterate our previous consideration for rectangle b3-c3-c4-b4, where
${{R}^{\u2033}}_{ij}=1$ and the sums
$\overline{R}({\mathrm{G}}_{2},{\mathrm{G}}_{0}-{\mathrm{G}}_{1}-{\mathrm{G}}_{2})$ and
${\overline{R}}^{\u2033}({\mathrm{G}}_{2},{\mathrm{G}}_{0}-{\mathrm{G}}_{1}-{\mathrm{G}}_{2})$ must be the same. Hence, R_{ij} = 1 in this rectangle. Continuing this consideration for the remaining sets q = 3, 4…, k proves that
${R}_{ij}={{R}^{\u2033}}_{ij}$ provided i and j belong to different sets.

If

i and

j belong to the same set, then

${{R}^{\u2033}}_{ij}=0$ according to

Equation (C5) and either

R_{ij} = 0 or competition is intransitive within the set (if the primary preferences within the set

q were transitive then, according to Proposition C2, C

_{i}≻C

_{j} demands

R^{*}(C

_{i})

> R^{*}(C

_{j}), which contradicts C

_{i},C

_{j} ∈

${\mathrm{G}}_{q}$ ). The secondary preference represents a coarsening of the primary preference since

R_{ij}^{00} > 0 demands

R_{ij} > 0 (when

j and

i belong to different sets), while

${{R}^{\u2033}}_{ij}=0$ may correspond to

R_{ij} > 0,

R_{ij} = 0 or

R_{ij} < 0 (when

j and

i belong to a common set).

If current rankings of all elements are different, that is all n_{q} = 1 for all q, then
${R}_{ij}={{R}^{\u2033}}_{ij}$ since all i ≠ j always belong to different sets. This means that the primary preference coincides with the secondary preference based on primary current ranking and is transitive. ■

#### C.3. Maps of Current Rankings

The statements proven in the previous subsection suggest a relatively simple method of analysing intransitivity in large systems. This method, which is based on ranking maps (i.e., plots of primary current ranking R^{*} of the elements against their secondary current ranking R^{**}), indicates the presence, intensity, extent and localisation of intransitivity by deviations from the line R^{*} = R^{**}. We wish to stay under the conditions of Theorem C1(c), and avoid complexities related to statement b of this theorem since, in this case, equivalence between the primary and secondary rankings ensures transitivity. However, in the case of g(C_{i}) = 1, which perhaps is most common in practice, coincidences of primary rankings for different elements R^{*}(C_{i}) = R^{*}(C_{j}) for i ≠ j are likely due to the limited number of values spaced by 1/n that these rankings can take. The practical solution for this problems is simple—consider g(C_{i}) = (1 + ε_{i})g_{0}(C_{i}), where ε_{i} represent small random values and g_{0}(C_{i}) are the original weights. Presence of these small values does not significantly alter the maps but makes coincidences R^{*}(C_{i}) = R^{*}(C_{j}) impossible unless all properties of the elements C_{i} and C_{j} are identical, which is sufficient for our purposes.

Two (or more) sets (or groups) are said to be subject to a preference when all possible selections of elements from these sets are compliant with the preference. For example
${\mathrm{G}}_{q}\succ \succ {\mathrm{G}}_{p}$ implies that sets
${\mathrm{G}}_{q}$ and
${\mathrm{G}}_{p}$ are subject to preference ≻≻ so that C_{i}_{(}_{q}_{)} ≻≻C_{j}_{(}_{p}_{)} for any C_{i}_{(}_{q}_{)} ∈
${\mathrm{G}}_{q}$ and any C_{j}_{(}_{p}_{)} ∈
${\mathrm{G}}_{p}$. According to these notations, the index i(q) runs over all elements of set
${\mathrm{G}}_{q}$. In this subsection, we consider partition of all elements in the system into range sets, where each set is represented by a range of primary current ranking (and consequently by a range of the secondary current ranking). The range sets are non-overlapping and jointly cover all elements.

We now turn to consideration of the

R^{*} versus

R^{**} maps. Consider an intransitive preference and its transitive closure. In this closure, the elements C

_{1},

…,C

_{n} are divided into

k transitively ordered sets

${\mathrm{G}}_{1}\succ \succ {\mathrm{G}}_{2}\succ \succ \dots \succ \succ {\mathrm{G}}_{k}$ of elements that are transitively equivalent within each set. That is for any C

_{i}_{(}_{q}_{)} ∈

${\mathrm{G}}_{q}$ and any C

_{j}_{(}_{p}_{)} ∈

${\mathrm{G}}_{p}$Proposition C2 indicates that

R^{*}(C

_{i}_{(}_{q}_{)})

> R^{*}(C

_{j}_{(}_{p}_{)}) and

R^{**}(C

_{i}_{(}_{q}_{)})

> R^{**}(C

_{j}_{(}_{p}_{)}) when

q < p, that is the ranges of current rankings of different sets do not overlap. Hence sets

${\mathrm{G}}_{1},\dots ,{\mathrm{G}}_{k}$ represent a set of range sets. This implies equivalence of primary and secondary set co-rankings

for all

p and

q. Note that

$\overline{R}({\mathrm{G}}_{p},{\mathrm{G}}_{q})={\overline{R}}^{\u2033}({\mathrm{G}}_{p},{\mathrm{G}}_{q})=0$ for any

q. This also implies that

for any

q. The property expressed by

Equation (C9) is reflected in the following proposition

**Proposition C3** The primary and secondary average current rankings of range sets coincide if and only if these sets are subject to a transitive primary preference (although this preference may remain intransitive within each set).

First we note that current primary set rankings of different range sets cannot coincide. The rest of the proof is similar to that of Theorem C1, where the leading set
${\mathrm{G}}_{1}$ is considered first. Since C_{i}_{(1)} ≻″C_{i}_{(}_{q}_{)} for all q > 1, the equality
${\overline{R}}^{*}({\mathrm{G}}_{1})={\overline{R}}^{**}({\mathrm{G}}_{1})$ is achieved if and only if C_{i}_{(1)} ≻C_{i}_{(}_{q}_{)} for all q > 1. After applying this consideration sequentially to sets
${\mathrm{G}}_{2},\dots ,{\mathrm{G}}_{k}$, we conclude that C_{i}_{(}_{p}_{)} ≻C_{i}_{(}_{q}_{)} for p < q. Finally we note that
${\mathrm{G}}_{1}\succ {\mathrm{G}}_{2}\succ \dots \succ {\mathrm{G}}_{K}$ requires
${\mathrm{G}}_{1}\succ \succ {\mathrm{G}}_{2}\succ \succ \dots \succ \succ {\mathrm{G}}_{K}$ since the preference ≻′ defined by C_{i}_{(}_{p}_{)} ≻′C_{i}_{(}_{q}_{)} for p < q and C_{i}_{(}_{p}_{)} ∼′C_{i}_{(}_{q}_{)} for p = q is transitive and is a coarsening of the primary preference and therefore is a coarsening of its transitive closure.

Figure C2 demonstrates a possible structure of the ranking map, where primary current ranking is plotted versus secondary current ranking. The elements are ordered according to their current rankings. The map in

Figure C2 indicates that the preference is generally intransitive (since the map deviates from the line specified by

R* =

R^{**}). The range sets, shown in the figure, are transitively ordered so that

${\mathrm{G}}_{1}\succ \succ {\mathrm{G}}_{2}\succ \succ {\mathrm{G}}_{3}\succ \succ {\mathrm{G}}_{4}\succ \succ {\mathrm{G}}_{5}$. The large dots indicate average set ranking, which is compliant with

Equation (C9). The preferences are transitive within

${\mathrm{G}}_{3}$ and

${\mathrm{G}}_{5}$ and intransitive within

${\mathrm{G}}_{1},{\mathrm{G}}_{2}$ and

${\mathrm{G}}_{4}$. Small deviation from the line specified by

R* =

R^{**} within set

${\mathrm{G}}_{4}$ indicate that intransitivity is present but not frequent within this set. Two subsets

${\mathrm{G}}_{1a}$ and

${\mathrm{G}}_{1b}$ are distinguished within set

${\mathrm{G}}_{1}$. The preferences between these sets are close to be transitive but some intransitive interference between subsets is present as indicated by angle

γ > 0. The small dots show current set rankings of

${\mathrm{G}}_{1a}$ and

${\mathrm{G}}_{1b}$, which are not compliant with

Equation (C9).

**Figure C2.**
Current ranking map (the thick red line shows primary vs secondary current ranking). The map is divided into range sets (groups)
${\mathrm{G}}_{1},\dots ,{\mathrm{G}}_{5}$ (group current rankings of the sets are shown by large dots) and subsets
${\mathrm{G}}_{1a}$ and
${\mathrm{G}}_{1b}$ (group current rankings of the subsets are shown by small dots). The black dashed line corresponds to R^{∗} = R^{∗∗}.

**Figure C2.**
Current ranking map (the thick red line shows primary vs secondary current ranking). The map is divided into range sets (groups)
${\mathrm{G}}_{1},\dots ,{\mathrm{G}}_{5}$ (group current rankings of the sets are shown by large dots) and subsets
${\mathrm{G}}_{1a}$ and
${\mathrm{G}}_{1b}$ (group current rankings of the subsets are shown by small dots). The black dashed line corresponds to R^{∗} = R^{∗∗}.