# Big mapping class groups are not extremely amenable

Yusen Long

###### Abstract

This paper uses the renowned Kechris-Pestov-Todorčević machinery to show that
(big) mapping class groups are not extremely amenable unless the underlying
surface is a sphere or a once-punctured sphere, or equivalently when the
mapping class group is trivial. The same techniques also show that the pure
mapping class groups, as well as compactly supported mapping class groups, of
a surface with genus at least one can never be extremely amenable.

†† _Date_ : July 2023.
The author is supported by CSC-202108170018.

Keywords: mapping class groups, extreme amenability, Polish groups.

2020 Mathematics subject classification: 57K20, 54H11.

##  1 Introduction

Let $\Sigma$ be an orientable surface. A theorem of Richards [Ric63] asserts
that it is characterised by the following parameters: genus and the space of
ends. Let $g(\Sigma)$ be genus of the surface $\Sigma$ and $n(\Sigma)$ be the
cardinality of its ends. The complexity of the surface is defined as
$c(\Sigma)\coloneqq 3g(\Sigma)+n(\Sigma)-3$. Such a surface is said to have
_finite type_ if $c(\Sigma)<\infty$, otherwise it is of _infinite type_.

Define the mapping class group of $\Sigma$ by
$\operatorname{\mathcal{MCG}}(\Sigma)\coloneqq\mathrm{Homeo}^{+}(\Sigma)/\\{\mathrm{isotopy}\\}$,
where $\mathrm{Homeo}^{+}(\Sigma)$ is the group of orientation preserving
automorphisms of the surface, equipped with compact-open topology, and
$\operatorname{\mathcal{MCG}}(\Sigma)$ is carrying the quotient topology. With
this topology, the group $\operatorname{\mathcal{MCG}}(\Sigma)$ is a _non-
archimedean Polish group_ (see Proposition 2.1), or equivalently it is a
closed subgroup of $S_{\infty}$, the symmetric group of $\mathbb{N}$, with
respect to the poitwise convergence topology (see for example [BK96, §1.5]). A
mapping class group is _big_ if the underlying surface is of infinite type.

From a model theoretic aspect, a non-archimedean Polish group can always be
realised as the automorphism group of some countable first-order relational
structure. In particular, one may ask the following question:

######  Question 1.1.

Given a countable first-order relational structure $\mathcal{F}$, how can one
detect if its automorphism group $\mathrm{Aut}(\mathcal{F})$ can be realised
as the (extended) mapping class group of an orientable surface?

This paper tends to give a partial answer to Question 1.1: the mapping class
groups of all but finitely many orientable surfaces can never be the
automorphism group of a countable first-order relational structure
$\mathcal{F}$ such that $\mathrm{Age}(\mathcal{F})$ has Ramsey property.

Recall that a topological group $G$ is said to have _fixed point on compacta
property_ or _extremely amenable_ if every continuous $G$-action on a compact
Hausdorff space admits a fixed point, or equivalently the universal minimal
$G$-flow $M(G)$ reduces to a singleton. However, it is worth noticing that,
other than the trivial group, no locally compact groups are extremely amenable
[Vee77], thus _a fortiori_ no discrete ones [Ell60].

In a celebrated paper [KPT05], Kechris, Pestov and Todorčević develop a
surprising correspondence (_abbrv._ KPT correspondence) between model theory,
combinatorics and topological dynamics: if $\mathcal{F}$ is a structure with
universe $\mathbb{N}$, then the non-archimedean Polish group
$\mathrm{Aut}(\mathcal{F})$ is extremely amenable if and only if the _age_
$\mathrm{Age}(\mathcal{F})$ has Ramsey property. So it is a natural question
to ask for a non-archimedean Polish group arising in the study of a geometry
object whether or not it is extremely amenable. In particular, it is worth
knowing whether big mapping class groups are extremely amenable or not. One
shall notice that big mapping class groups are not locally compact [AV20,
Theorem 4.2].

######  Theorem A.

Let $\Sigma$ be an orientable surface of finite or infinite topological type.
Then $\operatorname{\mathcal{MCG}}(\Sigma)$ is not extremely amenable unless
$\Sigma$ is a sphere or a once-punctured sphere, in which cases the mapping
class groups are trivial.

Denote by $\operatorname{End}(\Sigma)$ the end space of a surface $\Sigma$. It
is a compact space with a natural continuous
$\operatorname{\mathcal{MCG}}(\Sigma)$-action. In many cases, this action is
fixed-point free, which will witness the non extreme amenability of the
mapping class group. But this is not always the case. For example, the _Loch
Ness monster_ surface has infinite genus but only one end and the action of
its mapping class group on the end space is trivial. Other non trivial
examples are (non self-similar) surfaces with a unique maximal end [MR22].

The proof of Theorem A mainly relies on the description of extremely amenable
non-archimedean Polish groups provided in [KPT05] and can be divided into the
discussion of two cases, depending on whether the surface is zero-genus or
not. When the surface $\Sigma$ has zero genus, the Mann-Rafi ordering [MR20]
yields two disjoint curves of the same topological type unless $\Sigma$ is a
sphere or a once-punctured sphere. In this case, the existence of these two
curves further indicates that $\operatorname{\mathcal{MCG}}(\Sigma)$ is not
extremely amenable. If $g(\Sigma)>0$, then one may also find another pair of
curves with the same topological type, not necessarily disjoint, of which the
existence also implies the non extreme amenability of
$\operatorname{\mathcal{MCG}}(\Sigma)$.

Moreover, the proof of Theorem A remains valid for closed subgroup of
$\operatorname{\mathcal{MCG}}(\Sigma)$ containing a non-trivial mapping class
with finite orbit, including pure mapping class groups
$\operatorname{\mathcal{PMCG}}(\Sigma)$, _i.e._ the subgroup in
$\operatorname{\mathcal{MCG}}(\Sigma)$ that fixes pointwise every ends, and
the closure of compactly supported mapping class group
$\operatorname{\mathcal{MCG}}_{c}(\Sigma)$.

######  Theorem B.

Let $\Sigma$ be an orientable surface of finite or infinite type. If
$G<\operatorname{\mathcal{MCG}}(\Sigma)$ is a subgroup containing a mapping
class $\phi\in G$ such that for some simple closed curve $c$ on $\Sigma$, the
orbit $\\{\phi^{n}(c):n\in\mathbb{Z}\\}$ is finite, then $G$ is not extremely
amenable. In particular, the groups $\operatorname{\mathcal{PMCG}}(\Sigma)$
and $\operatorname{\mathcal{MCG}}_{c}(\Sigma)$ of a surface $\Sigma$ with
$g(\Sigma)\geq 1$ are not extremely amenable.

Note that the non extreme amenability of
$\operatorname{\mathcal{PMCG}}(\Sigma)$ can also be shown in a way that one
constructs a $\operatorname{\mathcal{PMCG}}(\Sigma)$-action on the circle
$S^{1}$ without fixed-point from the homomorphisms built in [APV20]. But this
method cannot be adapted to $\operatorname{\mathcal{MCG}}_{c}(\Sigma)$.

##  2 Non extreme amenability

Recall that the _extended mapping class group_
$\operatorname{\mathcal{MCG}}(\Sigma)^{\pm}$ is defined similarly as
$MCG(\Sigma)$ but using the group of all automorphisms
$\mathrm{Homeo}(\Sigma)$ in the stead of the orientation preserving one. The
result below is already known following several basic facts about mapping
class groups:

######  Proposition 2.1.

If the surface $\Sigma$ is of complexity at least two, then the mapping class
group $\operatorname{\mathcal{MCG}}(\Sigma)$ and the extended mapping class
group $\operatorname{\mathcal{MCG}}^{\pm}(\Sigma)$ are non-archimedean Polish
groups.

###### Proof.

Let $\mathcal{C}(\Sigma)$ be the curve graph of the surface $\Sigma$, where
vertices are isotopic classes of essential simple closed curves and an edge is
attached to two vertices if the two classes have disjoint representatives. It
is well-known that for a surface $\Sigma$ with complexity at least two, of
finite or infinite type, we have the isomorphism
$\operatorname{\mathcal{MCG}}^{\pm}(\Sigma)\simeq\mathrm{Aut}(\mathcal{C}(\Sigma))$
between topological groups [Iva97, Luo00, HMV18, BDR20]. Note that the
vertices in $\mathcal{C}(\Sigma)$ are countable. The graph
$\mathcal{C}(\Sigma)$ is actually a countable relational first-order
structure. Such an automorphism group is closed subgroup of $S_{\infty}$ and
thus a non-archimedean Polish group (see for example [Kec12, Part I,
§9.B(7)]). Hence $\operatorname{\mathcal{MCG}}^{\pm}(\Sigma)$ is a non-
archimedean Polish group. To show that $\operatorname{\mathcal{MCG}}(\Sigma)$
is also one, it suffices to demonstrate that
$\operatorname{\mathcal{MCG}}(\Sigma)$ is closed in
$\operatorname{\mathcal{MCG}}^{\pm}(\Sigma)$. Indeed, given a convergent
sequence of homeomorphisms $\Sigma\to\Sigma$ in $\mathrm{Homeo}(\Sigma)$, if
they are all orientation preserving, then so will be their limit. ∎

To further discuss the extreme amenability of a non-archimedean Polish group,
we need to introduce the following notions. Let $G$ be a group acting on a
space $X$ and $Y$ be a subspace of $X$. We call the subgroup $G_{(Y)}=\\{g\in
G:gy=y,\ \forall y\in Y\\}$ of $G$ the pointwise stabiliser of $Y$. Similarly,
the subgroup $G_{Y}=\\{g\in G:gY=Y\\}$ is called the setwise stabiliser of $Y$
in $G$.

The following observation is a natural consequence after KPT correspondence
(see [KPT05, Proposition 4.3]), which can find its root in [GW02].

######  Lemma 2.2.

If $G$ is an extremely amenable non-archimedean Polish group acting
continuously on the discrete space $\mathbb{N}$, then $G_{(F)}=G_{F}$ for any
finite subset $F\subset\mathbb{N}$.

The end space $\operatorname{End}(\Sigma)$ is a totally disconnected,
separable, metrisable topological space (and thus a closed subset of Cantor
set). Among the ends, there are _non planary ends_ , of which the collection
is denoted $\operatorname{End}_{\infty}(\Sigma)$, in the sense that every
neighbourhood in $\Sigma$ of a non planary end has at least genus $1$. The non
planary ends $\operatorname{End}_{\infty}(\Sigma)$ form a closed subset of
$\operatorname{End}(\Sigma)$.

######  Remark 2.1.

There is a slight _abus de langage_ here. The neighbourhood mentioned above
resides in the end compactification of $\Sigma$ instead of
$\operatorname{End}(\Sigma)$, but its intersection with $\Sigma$ becomes a
subsurface in $\Sigma$ and can still be regarded as a “neighbourhood” of an
end.

Let $D,D^{\prime}$ be two subsets of $\operatorname{End}(\Sigma)$. We say that
$D$ and $D^{\prime}$ are _isomorphic_ if $D$ is homeomorphic to $D^{\prime}$
and $D\cap\operatorname{End}_{\infty}(\Sigma)$ is homeomorphic
$D^{\prime}\cap\operatorname{End}_{\infty}(\Sigma)$ simultaneously.

Mann and Rafi give a way to order the ends of a surface by their similarity
[MR20]. Let $x,y\in\operatorname{End}(\Sigma)$ be two ends. Then we write
$x\preccurlyeq y$ if every clopen neighbourhood of $y$ contains a clopen
subset that is isomorphic to a neighbourhood of $x$. Two ends are said
_equivalent_ if both $x\preccurlyeq y$ and $y\preccurlyeq x$. They are said
_non-comparable_ if neither case happens.

It is worth noticing that each compact neighbourhood of an end of $\Sigma$
corresponds to one of its clopen neighbourhoods $D$ in
$\operatorname{End}(\Sigma)$ and this compact neighbourhood can be chosen to
be the end compactification of a subsurface in $\Sigma$ whose ends are the
union of $D$ with an additional isolated point.

Since the group $\mathrm{Homeo}^{+}(\Sigma)$ acts on the surface $\Sigma$
continuously, this action has a natural continuous extension onto the end
compactification of $\Sigma$. By taking the quotient, it is not hard to see
that $\operatorname{\mathcal{MCG}}(\Sigma)$ acts continuously by
homoemorphisms on $\operatorname{End}(\Sigma)$ and
$\operatorname{End}_{\infty}(\Sigma)$ is an invariant subspace of this group
action. Moreover, from the definition, this action preserves the ordering
given above, namely $gx\preccurlyeq gy$ for any
$g\in\operatorname{\mathcal{MCG}}(\Sigma)$ if and only if $x\preccurlyeq y$.
In particular, if there exists $g\in\operatorname{\mathcal{MCG}}(\Sigma)$ such
that $gx=y$, then necessarily $x\preccurlyeq y$.

Adopting the terminology from [FM11], the topological type of a simple closed
curve $c$ on $\Sigma$ is $\Sigma\setminus c$. Two curves $c$ and $c^{\prime}$
on $\Sigma$ are said to _have the same topological type_ if there is a
homeomorphism between $\Sigma\setminus c$ and $\Sigma\setminus c^{\prime}$.

######  Remark 2.2.

Since no distinction will be needed here, in the sequel, a curve on the
surface can mean either a topological embedding of $S^{1}$ or its isotopy
class, depending on the context.

An argument of the renowned _Alexander method_ (see for example [FM11, §2.3]
and [HMV19]) is that given two distinct simple closed curves (up to isotopy)
of the same topological type, one can always find a non-trivial mapping class
sending one curve to the other. Moreover, this mapping class $g$ can be chosen
to have finite order if the surface $\Sigma$ is of finite type. This is not
true in general for infinite-type surfaces, but only some special cases will
be needed for our purpose here and similar arguments yield the same result for
these special cases:

######  Lemma 2.3.

Let $\Sigma$ be a surface with genus $0$. Suppose that there exists two
disjoint simple closed curves $c,c^{\prime}$ on $\Sigma$ with the same
topological type. Assume that $c$ cuts $\operatorname{End}(\Sigma)=E\sqcup N$
and $c^{\prime}$ cuts $\operatorname{End}(\Sigma)=E^{\prime}\sqcup N^{\prime}$
in a way that $E\simeq E^{\prime}$ and that $E,E^{\prime}$ are disjoint. Then
there exists a non-trivial mapping class
$\phi\in\operatorname{\mathcal{MCG}}(\Sigma)$ such that $\phi(c)=c^{\prime}$
and $\phi^{2}=\operatorname{Id}$.

###### Proof.

Figure 1 depicts how such a surface $\Sigma$ looks like. The simplest case is
when $E$ and $E^{\prime}$ reduces to singleton and $\Sigma$ is a twice-
punctured sphere. Now the desired
$\phi\in\operatorname{\mathcal{MCG}}(\Sigma)$ is just the symmetry sending $E$
to $E^{\prime}$. More precisely, cut $\Sigma$ into three parts $S\sqcup
S^{\prime}\sqcup\big{(}\Sigma\setminus(S\cup S^{\prime})\big{)}$ along $c$ and
$c^{\prime}$ so that $\partial S=c$ and $\partial S^{\prime}=c^{\prime}$. By
Richards’ theorem, there is a homeomorphism $\varphi\colon S\to S^{\prime}$
sending $\partial S$ to $\partial S^{\prime}$. Take a symmetry
$\sigma\in\operatorname{\mathcal{MCG}}(\Sigma\setminus(S\cup S^{\prime}))$
that interchanges the position of $c$ and $c^{\prime}$: $\sigma$ is the
continuous extension by identity of the symmetry
$\overline{\sigma}\in\operatorname{\mathcal{MCG}}(S_{0,3})$, where
$S_{0,3}\subset\Sigma\setminus(S\cup S^{\prime})$ is a subsurface of finite
type with two of its ends being $\partial S=c$ and $\partial
S^{\prime}=c^{\prime}$. Now $\phi$ is piecewise defined by $\sigma$ on
$\Sigma\setminus(S\cup S^{\prime})$, $\varphi$ on $S$ and $\varphi^{-1}$ on
$S^{\prime}$ (up to isotopy).

Figure 1: Surface of Lemma 2.3

∎

Let us first deal with the (non) extreme amenability of
$\operatorname{\mathcal{MCG}}(\Sigma)$ for surfaces $\Sigma$ with genus $0$ by
using the results above.

######  Lemma 2.4.

Let $\Sigma$ be an orientable surface with genus $0$ and complexity at least
$2$. If $\operatorname{\mathcal{MCG}}(\Sigma)$ is extremely amenable, then any
two distinct ends in $\operatorname{End}(\Sigma)$ are non-comparable.

###### Proof.

Suppose for contradiction that $\operatorname{\mathcal{MCG}}(\Sigma)$ is
extremely amenable but there exist two distinct
$x,y\in\operatorname{End}(\Sigma)$ such that $x\preccurlyeq y$. Since
$\operatorname{End}(\Sigma)$ is metrisable and _a fortiori_ Hausdorff, we can
take a clopen neighbourhood $N$ of $y$ so that $x\notin N$. By definition,
inside of $N$, there exists a homeomorphic copy of a clopen neighbourhood of
$x$ but excludes $x$. This implies the existence of
$x^{\prime}\in\operatorname{End}(\Sigma)$ which is distinct from $x$ but
equivalent to $x$. Moreover, one can take a clopen neighbourhood $D$ of $x$
and a clopen neighbourhood $D^{\prime}$ of $x^{\prime}$ in the way that
$D\simeq D^{\prime}$ but $D\cap D^{\prime}=\emptyset$. We should note that
$\operatorname{End}(\Sigma)\setminus D$ is also homeomorphic to
$\operatorname{End}(\Sigma)\setminus D^{\prime}$. Now associate $D$ to a
subsurface with boundary $S\subset\Sigma$ as described above so that
$\operatorname{End}(S)=D\sqcup\\{\ast\\}$ and find a subsurface
$S^{\prime}\subset\Sigma$ likewise for $D^{\prime}$. By Richards’ theorem, the
curves $\partial S$ and $\partial S^{\prime}$ have the same topological type
and satisfies the hypothesis of Lemma 2.3. But Lemma 2.3 indicates that there
exists a mapping class $\phi\in\operatorname{\mathcal{MCG}}(\Sigma)$ such that
if $F=\\{\partial S,\partial S^{\prime}\\}$, then
$\phi\in\operatorname{\mathcal{MCG}}(\Sigma)_{F}\setminus\operatorname{\mathcal{MCG}}(\Sigma)_{(F)}$.
This yields a contradiction to Lemma 2.2. ∎

Now one can show the following proposition:

######  Proposition 2.5.

Let $\Sigma$ be an orientable surface with genus $0$. Then
$\operatorname{\mathcal{MCG}}(\Sigma)$ is extremely amenable if and only if
$\Sigma$ is a sphere or a once-punctured sphere, in which case
$\operatorname{\mathcal{MCG}}(\Sigma)$ is the trivial group.

###### Proof.

If the surface has complexity less than $2$, then it has non-trivial discrete
group as mapping class group unless it is a sphere or a once-punctured sphere.
So by the virtue of Lemma 2.4, it remains to show that there is no surface
with genus $0$, complexity at least $2$ and pairwise non-comparable ends
unless it is a sphere or a once-punctured sphere, where the statement is
satisfied vacuously. Indeed, let $\Sigma$ be a such surface. Assume that there
exist distinct $x,y\in\operatorname{End}(\Sigma)$ and an element
$g\in\operatorname{\mathcal{MCG}}(\Sigma)$ so that $gx=y$. Then necessarily
$x\preccurlyeq y$ as remarked above, which contradicts to the non-comparing
assumption. Hence the action of $\operatorname{\mathcal{MCG}}(\Sigma)$ on
$\operatorname{End}(\Sigma)$ must be trivial. This implies that
$\operatorname{\mathcal{MCG}}(\Sigma)=\operatorname{\mathcal{PMCG}}(\Sigma)$.
By [PV18, Theorem 3], this means that
$\operatorname{\mathcal{MCG}}(\Sigma)=\operatorname{\mathcal{PMCG}}(\Sigma)$
are simultaneously residually finite and as a result, the surface $\Sigma$
must be of finite type. But as the ends of $\Sigma$ must be all non-
comparable, it only happens when $\Sigma$ is a sphere or a once punctured
sphere, in which cases $\operatorname{\mathcal{MCG}}(\Sigma)$ is trivial. ∎

Let $S^{1}_{1}$ be the surface of genus $1$ with one boundary component and
let $S_{1,1}$ be once-punctured torus. Then there is an element
$g\in\operatorname{\mathcal{MCG}}(S_{1}^{1})$ and a simple closed curve $c$ on
$S^{1}_{1}$ such that $\\{g^{n}(c):n\in\mathbb{Z}\\}$ is finite. Indeed, by
the inclusion homomorphism [FM11, Theorem 3.18], we have the short sequence

$1\to\mathbb{Z}\to\operatorname{\mathcal{MCG}}(S^{1}_{1})\to\mathrm{SL}_{2}(\mathbb{Z})\simeq\operatorname{\mathcal{MCG}}(S_{1,1})\to
1.$

Note that

$\phi\coloneqq\begin{pmatrix}0&-1\\\
1&0\end{pmatrix}\in\mathrm{SL}_{2}(\mathbb{Z})\simeq\operatorname{\mathcal{MCG}}(S_{1,1})$

is a torsion and that there is a simple closed curve $c$ on $S_{1,1}$, which
can also be regarded as a curve in $S^{1}_{1}$, such that
$\\{\phi^{n}(c):n\in\mathbb{Z}\\}$ is finite. As a result, the orbit of $c$
under the action of any pre-image
$g\in\operatorname{\mathcal{MCG}}(S^{1}_{1})$ of
$\phi\in\operatorname{\mathcal{MCG}}(S_{1,1})$ is finite.

With the observation above, one can now prove Theorem A:

###### Proof of Theorem A.

If the surface $\Sigma$ has genus $0$, then the extreme amenability of
$\operatorname{\mathcal{MCG}}(\Sigma)$ is determined by Proposition 2.5. For
the torus, the mapping class group is not extremely amenable since it is a
non-trivial discrete group. Suppose now that the surface $\Sigma$ has
complexity at least $2$ and non-zero genus. If $\Sigma$ contains an essential
(sub)surface that is homeomorphic to $S_{1}^{1}$. Then we take a
$g\in\operatorname{\mathcal{MCG}}(S^{1}_{1})$ and a curve $c$ on
$S^{1}_{1}\subset\Sigma$ so that $\\{g^{n}(c):n\in\mathbb{Z}\\}$ is finite. By
the virtue of the inclusion homomorphism [FM11, Theorem 3.18], one can extend
$g$ by identity to an element in $\operatorname{\mathcal{MCG}}(\Sigma)$. Now
the pointwise stabiliser of $F\coloneqq\\{g^{n}(c):n\in\mathbb{Z}\\}$ in
$\operatorname{\mathcal{MCG}}(\Sigma)$ is different from its setwise
stabiliser. Hence by Lemma 2.2, $\operatorname{\mathcal{MCG}}(\Sigma)$ is not
extremely amenable. ∎

Finally, there are still some details in Theorem B that need further
clarifications.

######  Lemma 2.6.

Let $G$ be a topological group and $H$ be a dense subgroup of $G$. Then $G$ is
extremely amenable if and only if $H$ is.

###### Proof.

Note that a topological group $G$ is extremely amenable if and only if it
admits a $G$-invariant multiplicative mean over $\mathrm{RUCB}(G)$, the space
of right-uniformly continuous functions on $G$, or equivalently the $G$-action
on its _Samuel compactification_ $\sigma G$ has a fixed point (see for example
[Pes06, §1.1]). But if $H$ is a dense subgroup of $G$, then
$\mathrm{RUCB}(H)\simeq\mathrm{RUCB}(G)$ and any continuous $H$-action on the
multiplicative means of $\mathrm{RUCB}(G)$ can be extended continuously to a
$G$-action on it. Conversely, the restriction of a continuous $G$-action on
$H$ yields an $H$-action. Hence $H$ and $G$ can only be simultaneously
extremely amenable. ∎

###### Proof of Theorem B.

For any subgroup $G<\operatorname{\mathcal{MCG}}(\Sigma)$ with complexity
$c(\Sigma)>2$ that contains a mapping class $\phi$ such that for some simple
closed curve $c$ on the surface, the orbit
$F\coloneqq\\{\phi^{n}(c):n\in\mathbb{Z}\\}$ is finite, it is clear that
$\phi$ belongs to $\overline{G}_{F}$ but not $\overline{G}_{(F)}$. Hence the
closure of such subgroup $G$ in $\operatorname{\mathcal{MCG}}(\Sigma)$ is not
extremely amenable. It follows from Lemma 2.6 that $G$ itself is neither
extremely amenable. ∎

##  3 Perspectives

In a recent paper, Disarlo, Koberda and González [DKN23] establish a model
theoretic connection between the mapping class groups and the curve graph of
non-sporadic finite-type surfaces, which is motivated by Ivanov’s
metaconjecture [Iva06]. Following their ideas, another way to ask Question 1.1
is the following:

######  Question 3.1.

Given a graph on countable vertices, how can one detect if it is the curve
graph of an orientable surface?

Another notion that is closely related to the extreme amenability is the
amenability of a topological group, _i.e._ every continuous group action on a
compact Hausdorff space admits an invariant probability measure over the
space. Although the non extreme amenability is already clear, it remains
unknown if there are amenable big mapping class groups. For every but finitely
many finite-type orientable surfaces, the mapping class group is non-amenable
because as a discrete group, it contains a non-abelian free subgroup on two
generators. If the surface is of infinite type, the amenability of its mapping
class group is less clear. For the surfaces $\Sigma$ of infinite type having a
non-displaceable subsurface $S$ of finite type, one can construct a _blown-up
projection complex_ from the curve graphs of the
$\operatorname{\mathcal{MCG}}(\Sigma)$-orbit of $S$, see for example [HQR22,
DD22]. Equipped with the combinatorial metric, the blown-up projection complex
is a separable geodesic Gromov-hyperbolic space on which the mapping class
group $\operatorname{\mathcal{MCG}}(\Sigma)$ acts continuously by isometries
and the $\operatorname{\mathcal{MCG}}(\Sigma)$-action is of general type (or
sometimes non-elementary). However, an amenable group can never have a
continuous action of general type on a separable geodesic Gromov-hyperbolic
space by isometries. This implies that these big mapping class groups are not
amenable.

## Acknowledgements

The author thanks George Domat, Sergio Domingo, Bruno Duchesne and Jesús
Hernandez Hernandez for helpful discussions.

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Université Paris-Saclay, Laboratoire de Mathématique d’Orsay, 91405, Orsay,
France

Email address<EMAIL_ADDRESS>


# Integrated Information, a Complexity Measure for optimal partitions

Otavio Citton and Nestor Caticha
Instituto de Fisica, Universidade de Sao Paulo
Sao Paulo, SP, Brazil Current address: Bernoulli Institute for Mathematics,
Computer Science and Artificial Intelligence, University of Groningen,
Nijenborgh 9, 9747 AG Groningen, The
Netherlandso.c.citton<EMAIL_ADDRESS>
###### Abstract

Motivated by the possible applications that a better understanding of
consciousness might bring, we follow Tononi’s idea and calculate analytically
a complexity index for two systems of Ising spins with parallel update
dynamics, the homogeneous and a modular infinite range models. Using the
information geometry formulation of integrated information theory, we
calculate the geometric integrated information index, $\phi_{G}(\Pi)$ for a
fixed partition $\Pi$ with $K$ components and
$\Phi=$max${}_{\Pi}\phi_{G}(\Pi)$ for $K=2$ or $3$. For systems in the deep
ferromagnetic phase, the optimal partition undergoes a transition such that
the smallest (largest) component is above (resp. below) its critical
temperature. The effects of partitioning are taken into account by introducing
site dilution.

Keywords: Statistical Mechanics; Complexity Index; Integrated Information;
Ising model.

##  1 Introduction

There is no general agreement on what is meant by the term consciousness. The
impossibility, so far, of physical explanations of the first person perception
of pain or other forms of awareness, may lead to either abandoning the
physical description of consciousness or dismissing it as either an illusion
or even non-existent, a folly of subjectivism. Taking as a given that the only
certainty we can have are the inner personal experiences, Tononi’s Integrated
Information Theory (IIT) [1] proposes to extract a quantitative signature
$\Phi$, that would betray the existence of an inner complexity that conscious
systems ought to have. As a working hypothesis, “time is what clocks measure”,
may not be fully satisfying but permitted great progress and the refinement of
ideas about time. Measuring or estimating IIT’s $\Phi$ has the potential of
showing the direction of refining the question of what is it that science can
say about consciousness. It may be that “consciousness is what $\Phi$
measures” is just a stepping stone to better questions, experiments and
predictions. Denying that a system lacks consciousness hinges on having a
definition to assess whether the system has it or not. There is a mountain to
climb: from one side, obtaining consciousness as the emergent property of a
physical system; from the opposite route, the analysis of how we can identify
whether it is conscious from its necessarily complex intrinsic properties. No
matter how high these two routes meet, they will not convince those that seek
a first person description of how we experience the world.

There is also no consensus on the mathematical definition of $\Phi$ and Tononi
and collaborators and other groups have pressed on, presenting refinements and
variations on the theme. In general, IIT proposes $\Phi$ as a measure of
casual influences of the different parts, the constituents of the system. This
idea stems from the postulated irreducibility of consciousness into some sum
over the parts. A physical system that satisfies these properties would be
called a Physical Substrate of Consciousness (PSC). See [2] and [3] for a
description of the current state of the Desiderata of IIT, which is rather a
declaration of what elements an acceptable theory ought to present, but not
yet the basis for a unique tractable mathematical theory and hence many
possibilities arise when trying to translate this declaration of purpose into
something that can be calculated theoretically, measured experimentally and
compared. The merit lies in presenting the first attempt to quantify and
therefore enable measuring consciousness. It is certainly an unfinished
chapter in the history of science.

Despite being incomplete in its logical structure, which is usual in an
incipient theory, several applications, see [4] for an extensive review, point
to the utility from a clinical perspective of being able to correlate clinical
characterization to numerical estimates of complexity. For example, in [5] the
Perturbational Complexity Index (PCI) related to IIT, is shown to serve as a
quantitative characterization of the neural correlates of consciousness, a
marker that could identify and measure the level of consciousness of comatose,
anesthetized, sleeping or fully awake subjects. A timid approach to the
subject would claim that even if $\Phi$ has no relation to consciousness, it
derives its importance from being a measure of the complexity of the dynamics.

The information geometry based framework to study complexity measures as
“distances” between probability manifolds introduced by Oizumi et al. [6] to
embody IIT ideas, is central to this paper. The “geometric integrated
information” $\Phi_{G}$ is the relative entropy of a full joint probability
distribution relative to the product of marginals of disconnected subsets for
a fixed partition. In all definitions of $\Phi$, a concept that is always
present is that it is a measure of how different a system is to a partially
disconnected version of itself, and the information geometry notion of
distance (or divergence) seems natural to describe such a measure. Different
measures of distance can be used, such as in [7] and [8] which use the
Wasserstein distance. There are, however, technical difficulties in the
calculation of either $\Phi$ or $\Phi_{G}$ which have limited such
characterizations to theoretical models which may fail to qualify as complex.
The first application to a system in the thermodynamic limit, by Aguilera and
Di Paolo [8], deals with a kinetic Ising model to calculate a complexity index
based on the Wasserstein metric, which yields results different from the
geometric approach we follow here.

In this paper we present a calculation of the geometric complexity index based
on the ideas of Oizumi, Tsuchiya and Amari [6]. We study the two versions of
the infinite range parallel update Ising model. We can calculate the geometric
complexity index for a general partition with a given number of disconnected
components, then $\Phi_{G}$ is defined as the maximum value, per spin, over
all those partitions. We present results for partitions into two or three
components for the homogeneous infinite range model. We also look at the model
with a modular structure with two groups of spins, in a simple attempt to make
an analogy with the separation of a brain into hemispheres. It has a natural
partition into two components which is not necessarily the partition that
maximizes integrated information. In the thermodynamic limit of both models,
$\Phi_{G}$ presents phase transitions.

A detailed comparison between some versions of $\Phi$ was made by Mediano et
al. [9]. Although they have calculated those measures for a not so complex and
small system, namely an autoregressive Gaussian model, it is still interesting
to note the large variability of behaviors. Each definition of $\Phi$ seems to
capture different aspects of the model, and the problem of choosing a single
definition that allows us to quantify consciousness, if at all possible,
remains open. This holds another motivation for studying different measures
for a system in the thermodynamic limit, since the results may give insights
on what it is measuring and if it has some property that would be interesting
as a signature of consciousness.

##  2 Geometric Integrated Information

Consider a system with $N$ interacting classical Ising spins, indexed on a set
$\Lambda\subset{\mathbb{Z}}$, $|\Lambda|=N$. We denote by
$X=\\{x_{i}|{i\in\Lambda}\\}$ and $Y=\\{y_{i}|{i\in\Lambda}\\}$ the variables
that characterize the system’s state at two consecutive steps of a discrete
time dynamics. The spatio-temporal interactions between its elements are
described by a distribution $P\left(X,Y\right)$111Conditional information
about the details of the model is not shown for notational simplicity. , the
full model. Let the distribution $Q\left(X,Y\right)$ be a member of a
probability manifold $\cal M$, defined by imposing some constraints, e.g. some
of the interactions are disconnected. The difference between the full model
and the disconnected model can be quantified by the Kullback-Leibler
divergence [6]:

$D_{\text{KL}}\left[P\|Q\right]=\sum_{X,Y}P\left(X,Y\right)\log\frac{P\left(X,Y\right)}{Q\left(X,Y\right)},$
(1)

a measure of the strength of the influence in the complete system between the
disconnected elements. Let $\Pi$ be a partition of the set $\Lambda$ into non-
overlapping subsets and denote by $I,J$ components of partition $\Pi$. A
variable $X_{I}=\\{x_{i}|i\in I\\}$ stands for the set of variables on a given
component $I$ of $\Pi$.

Integrated information aims to quantify the amount of “synergistic” influences
the whole system exerts on its future in excess of what the independent parts
of the system do. Synergy is an unusual word in the context of Statistical
Mechanics of phase transitions, and it probably refers to interactions between
the system’s degrees of freedom and their effect on emergent or collective
properties. In the information geometry framework, this can be achieved by
considering the following disconnection:

$Q\left(Y_{I}|X\right)=Q\left(Y_{I}|X_{I}\right),$ (2)

for every subset $I\in\Pi$. This defines the manifold

$\displaystyle\mathcal{M}_{G}$ $\displaystyle:=$
$\displaystyle\Big{\\{}Q(X,Y)\big{|}Q(Y_{I}|X)=Q(Y_{I}|X_{I}),\forall
I\in\Pi\Big{\\}},$ (3)

where the disconnected system lives.

The geometric integrated information [6] associated to a particular partition
is then defined as

$\phi_{G}(\Pi)=\min_{Q\in\mathcal{M}_{G}}\sum_{X,Y}P\left(X,Y\right)\log\frac{P\left(X,Y\right)}{Q\left(X,Y\right)}.$
(4)

This is a complexity index relative to the particular partition under
consideration.

We start by showing a specific model calculation with the simplest partition,
a bipartition $\Pi=\\{I,J\\}$ with non-overlapping components
$I\subset\Lambda$ and $J$, the complement of $I$ in $\Lambda$. We define
$\mathcal{M}_{IJ}$ the probability manifold whose elements satisfy the
integrated information disconnection constraint (2) for this bipartition.
Every distribution $Q_{IJ}\in\mathcal{M}_{IJ}$ must decompose as follows:

$Q_{IJ}\left(X,Y\right)=Q_{IJ}\left(X\right)Q_{IJ}\left(Y_{I}|X_{I}\right)Q_{IJ}\left(Y_{J}|X_{J}Y_{I}\right).$
(5)

The minimum of the KL divergence subject to such constraints is obtained
first, by supposing it exits and then, the minimization of the Lagrangian

$\displaystyle\mathcal{L}=D_{\text{KL}}\left[P\|Q_{IJ}\right]+\lambda\left(\sum_{X}Q_{IJ}\left(X\right)-1\right)+\sum_{X_{I}}\mu\left(X_{I}\right)\left(\sum_{Y_{I}}Q_{IJ}\left(Y_{I}|X_{I}\right)-1\right)+$
$\displaystyle+\sum_{X_{J},Y_{I}}\nu\left(X_{J},Y_{I}\right)\left(\sum_{Y_{J}}Q_{IJ}\left(Y_{J}|X_{J}Y_{I}\right)-1\right),$
(6)

is solved by

$\displaystyle Q_{IJ}^{*}\left(X\right)$ $\displaystyle=$ $\displaystyle
P\left(X\right),$ (7) $\displaystyle Q_{IJ}^{*}\left(Y_{I}|X_{I}\right)$
$\displaystyle=$ $\displaystyle P\left(Y_{I}|X_{I}\right),$ (8) $\displaystyle
Q_{IJ}^{*}\left(Y_{J}|X_{J},Y_{I}\right)$ $\displaystyle=$ $\displaystyle
P\left(Y_{J}|X_{J},Y_{I}\right).$ (9)

The integrated information $\phi_{G}(\Pi)$ for a bipartition $\Pi=\\{I,J\\}$
is:

$\phi_{IJ}=\sum_{X,Y}P\left(X,Y\right)\log\frac{P\left(Y|X\right)P\left(Y_{I}|X_{J}\right)}{P\left(Y|X_{J}\right)P\left(Y_{I}|X_{I}\right)},$
(10)

and the geometric integrated information per spin

$\Phi_{G}=\max_{\Pi}\lim_{N\rightarrow\infty}\frac{1}{N}\phi_{IJ}.$ (11)

##  3 Long range kinetic Ising Model

To proceed we choose a particular version of the infinite range Ising model.
It is formulated with an intrinsic discrete dynamics and defined through the
interaction at two consecutive times, hence it fits perfectly the information
geometry description of the integrated information index.

The Hamiltonian is

$H\left(X,Y\right)=-\sum_{i,j\in\Lambda}J_{ij}x_{j}y_{i},$ (12)

and on the assumption that it is conserved, maximum entropy leads to the
distribution

$P\left(X,Y|{J}\right)=\frac{1}{Z}\exp\left\\{\beta\sum_{i,j}J_{ij}x_{j}y_{i}\right\\},$
(13)

where $\beta$ is an inverse temperature for the system and $Z$, the partition
function, ensures normalization. At this point the interaction $J_{ij}$ is
simply $J_{0}/N$, for the fully connected system, but may take other values in
order to implement a disconnection.

From the product rule, the transition probability distribution is

$P\left(Y|X;J\right)=\frac{P\left(X,Y|J\right)}{P\left(X|J\right)}=\frac{P\left(X,Y|J\right)}{\sum_{Y}P\left(X,Y|J\right)}=\frac{\exp\left\\{\beta\sum_{i,j}J_{ij}x_{j}y_{i}\right\\}}{\prod_{i}2\cosh\left(\beta\sum_{j}J_{ij}x_{j}\right)},$
(14)

a product of the transition distributions for each element,

$P\left(Y|X;J\right)=\prod_{i=1}^{N}\frac{\exp\left\\{\beta
h_{i}y_{i}\right\\}}{2\cosh\left(\beta
h_{i}\right)}=\prod_{i=1}^{N}P\left(y_{i}|X;J\right),$ (15)

where the local field on site $i$ is $h_{i}=\sum_{j=1}^{N}J_{ij}x_{j}$.

We only consider the case with zero external field, since $h\neq 0$ only
diminishes complexity. The partition function is

$Z(\beta,J_{0},N)=\sum_{X,Y}\exp\left\\{\beta\frac{J_{0}}{N}\sum_{ij}x_{i}y_{j}\right\\}.$
(16)

Introducing a $\delta\left(\tilde{m}-\frac{1}{N}\sum_{i}x_{i}\right)$
$\delta\left(\tilde{n}-\frac{1}{N}\sum_{i}y_{i}\right)$ into the partition
function:

$Z(\beta,J_{0},N)=\int d\tilde{m}\frac{d\hat{m}}{2\pi}\int
d\tilde{n}\frac{d\hat{n}}{2\pi}e^{\beta
J_{0}N\tilde{m}\tilde{n}}e^{i\hat{m}\tilde{m}}e^{i\hat{n}\tilde{n}}e^{N\log{\mathcal{Z}}}.$
(17)

We have introduced the single site partition function
$\mathcal{Z}=\sum_{x,y}e^{-\mathcal{H}}$, the canonical partition function
associated with the Hamiltonian
$\mathcal{H}=-\frac{i}{N}\left(\hat{m}x+\hat{n}y\right)$. The partition
function can be written as

$\displaystyle Z(\beta,J_{0},N)$ $\displaystyle=$ $\displaystyle\int e^{-\beta
N\mathcal{F}(\tilde{m},\tilde{n};\beta,J_{0})}d\tilde{m}d\tilde{n}$ (18)
$\displaystyle\beta F(\beta,J_{0})$ $\displaystyle=$
$\displaystyle-\lim_{N\rightarrow\infty}\frac{1}{N}\log Z(\beta,J_{0},N).$
(19)

$F(\beta,J_{0})$ is the free energy of the system per spin and we call
$\mathcal{F}(\tilde{m},\tilde{n};\beta,J_{0})$ the free energy functional,
with some abuse of language. Then

$\displaystyle F(\beta,J_{0})$ $\displaystyle=$
$\displaystyle\stackunder{min}{$\tilde{m},\tilde{n}$}\,\,\mathcal{F}(\tilde{m},\tilde{n};\beta,J_{0})$
(20) $\displaystyle\left(m(\beta,J_{0},),n(\beta,J_{0})\right)$
$\displaystyle=$
$\displaystyle\stackunder{argmin}{$\tilde{m},\tilde{n}$}\,\,\mathcal{F}(\tilde{m},\tilde{n};\beta,J_{0}).$
(21)

For large $N$ the saddle point integration leads to

$Z(\beta,J_{0},N)=e^{\beta
J_{0}Nmn}e^{i\hat{m}m}e^{i\hat{n}n}e^{N\log{\mathcal{Z}}}$ (22)

where the values of $\tilde{m},\tilde{n}$ and $\hat{m},\hat{n}$ were
substituted by the saddle point equation solutions. By imposing that the
derivative of the exponent with respect to each one of them is equal to zero,
we find:

$\displaystyle\hat{m}$ $\displaystyle=$ $\displaystyle iN\beta J_{0}n,$ (23)
$\displaystyle\hat{n}$ $\displaystyle=$ $\displaystyle iN\beta J_{0}m,$ (24)

the saddle point values of $\hat{m}$ and $\hat{n}$ depend on the values of $n$
and $m$ respectively. Evaluating the $\hat{m},\hat{n}$ integrals by constant
phase integration:

$\displaystyle\mathcal{H}$ $\displaystyle=$ $\displaystyle(\beta
J_{0}n)x+(\beta J_{0}m)y,$ (25) $\displaystyle\mathcal{Z}$ $\displaystyle=$
$\displaystyle\exp\left(\log 2\cosh(\beta J_{0}n)+\log 2\cosh(\beta
J_{0}m)\right).$ (26)

The saddle point equations for the expected values are simply

$\displaystyle m$ $\displaystyle=$ $\displaystyle\tanh\left(\beta
J_{0}n\right),$ (27) $\displaystyle n$ $\displaystyle=$
$\displaystyle\tanh\left(\beta J_{0}m\right),$ (28)

and their stable solutions yield $m=n$. The free energy functional per spin is

$\displaystyle\mathcal{F}(\tilde{m},\tilde{n};\beta,J_{0})$ $\displaystyle=$
$\displaystyle J_{0}\tilde{m}\tilde{n}-\frac{1}{\beta}\log 4\cosh(\beta
J_{0}\tilde{n})\cosh(\beta J_{0}\tilde{m}).$ (29)

When the order parameters are the solutions of the saddle point equations, the
free energy functional is at is minimum and yields the thermodynamic free
energy

$F(\beta)=J_{0}m^{2}-\frac{2}{\beta}\log(2\cosh(\beta J_{0}m)).$ (30)

The entropy of the system can be calculated directly from the Shannon
expression evaluated at the saddle point solutions, or obtained using the
thermodynamic relation, taking into account that $m(\beta)=\tanh\left(\beta
J_{0}m(\beta)\right)$

$\displaystyle S[P(X,Y)]$ $\displaystyle=$ $\displaystyle-N\frac{\partial
F}{\partial T}$ (31) $\displaystyle=$ $\displaystyle 2N\left(-\beta
J_{0}m(\beta)^{2}+\log 2\cosh(\beta J_{0}m(\beta))\right),$ (32)

For $-1\leq\tilde{m}\leq 1$, define the function

$\displaystyle\mathcal{S}(\tilde{m},\beta,J_{0},N)$ $\displaystyle=$
$\displaystyle 2N\left(-\beta J_{0}\tilde{m}^{2}+\log 2\cosh(\beta
J_{0}\tilde{m})\right),$ (33)

such that when $\tilde{m}=m(\beta)$, it gives the value of the entropy of the
full model $S(\beta,J_{0},N):=S[P(X,Y)]$, but it is defined for any value of
$\tilde{m}$.

A similar calculation leads to the conditional entropy in equilibrium

$\displaystyle\frac{1}{\beta N}S[P(Y|X)]$ $\displaystyle=$ $\displaystyle-
J_{0}m^{2}+\frac{1}{\beta}\log 2\cosh(\beta J_{0}m).$ (34)

From equations (32) and (34) we see that the entropy of the joint distribution
is simply twice the entropy of the conditional on the past distribution. This
is interesting since $S[P(X,Y)]=S[P(Y|X)]+S[P(X)]$, it follows that
$S[P(X)]=S[P(Y|X)]$ and since the Hamiltonian is symmetric in $X$ and $Y$,
using $S[P(X,Y)]=S[P(X|Y)]+S[P(Y)]$, we conclude that in equilibrium
$S[P(Y|X)]=S[P(Y)]$, which means that, for calculating the entropy, the past
is not informative about the future. Of course, the past is informative about
the future before reaching equilibrium.

##  4 Implementing disconnections by Site Dilution

To evaluate $\phi_{IJ}$, expression (10), we need the disconnected transition
probabilities distributions. We deal with this problem by introducing a method
to calculate them: disconnect by site dilution. Given a bipartition of
$\Lambda$ into subsets $I,J$, introduce the set of dilution variables
$\bm{\eta}=\\{\eta_{i}\\}_{i\in\Lambda}$, with $\eta_{i}=1$ if $i\in I$ and
$0$ if $i\in J$. There is a one to one correspondence between bipartitions and
$\bm{\eta}$ configurations, so $\phi_{\bm{\eta}}=\phi_{IJ}$. The interaction
matrix of the disconnected system is obtained with the auxiliary $\bm{\eta}$
variables. For example, for an interaction matrix
$J_{ij}\eta_{i}(1-\eta_{j})$, only when $i\in I$ and $j\in J$, $i$ and $j$
interact.

The transition probabilities densities needed in equation (10) are:

$\displaystyle P\left(Y|X;J,\bm{\eta}\right)$ $\displaystyle=$
$\displaystyle\frac{\exp\left\\{\beta\sum_{i,j}J_{ij}x_{j}y_{i}\right\\}}{\prod_{i}2\cosh\left(\beta\sum_{j}J_{ij}x_{j}\right)},$
(35) $\displaystyle P\left(Y_{I}|X_{I};J,\bm{\eta}\right)$ $\displaystyle=$
$\displaystyle\frac{\exp\left\\{\beta\sum_{i,j}J_{ij}\eta_{i}\eta_{j}x_{j}y_{i}\right\\}}{\prod_{i}2\cosh\left(\beta\sum_{j}J_{ij}\eta_{i}\eta_{j}x_{j}\right)},$
(36) $\displaystyle P\left(Y_{I}|X_{J};J,\bm{\eta}\right)$ $\displaystyle=$
$\displaystyle\frac{\exp\left\\{\beta\sum_{i,j}J_{ij}\eta_{i}\left(1-\eta_{j}\right)x_{j}y_{i}\right\\}}{\prod_{i}2\cosh\left(\beta\sum_{j}J_{ij}\eta_{i}\left(1-\eta_{j}\right)x_{j}\right)},$
(37) $\displaystyle P\left(Y|X_{J};J,\bm{\eta}\right)$ $\displaystyle=$
$\displaystyle\frac{\exp\left\\{\beta\sum_{i,j}J_{ij}\left(1-\eta_{j}\right)x_{j}y_{i}\right\\}}{\prod_{i}2\cosh\left(\beta\sum_{j}J_{ij}\left(1-\eta_{j}\right)x_{j}\right)}.$
(38)

Our expression for $\phi_{\bm{\eta}}$ can be shown by explicit calculation to
equal a difference of conditional entropies

$\phi_{\bm{\eta}}=S[P(Y_{I}|X_{I})]+S[P(Y_{J}|X_{J})]-S[P(Y|X)].$ (39)

This means that, for this type of simple model, the geometric $\phi_{G}$ is
equal to the stochastic interaction complexity (see [6] and [10]), and can be
interpreted as the amount of information gained when the components $I$ and
$J$ are allowed to communicate with each other, in order to predict the future
state. A word of warning about notation: $S[P(Y_{I}|X_{I})]$ is not the
entropy of an Ising model defined on component $I\subset\Lambda$, but rather
of the distribution in expression (36), which is defined in the whole
$\Lambda$.

With $m$ the magnetization of the full system and
$\gamma=\sum_{i\in\Lambda}\eta_{i}/N$, the fraction of spins in set $I$, the
conditional entropies in equilibrium are

$\displaystyle S[P(Y_{I}|X_{I})]$ $\displaystyle=$ $\displaystyle-\beta
J_{0}\gamma^{2}Nm^{2}+N\gamma\log 2\cosh(\beta J_{0}\gamma m),$ (40)
$\displaystyle S[P(Y_{J}|X_{J})]$ $\displaystyle=$ $\displaystyle-\beta
J_{0}(1-\gamma)^{2}Nm^{2}+N(1-\gamma)\log 2\cosh(\beta J_{0}(1-\gamma)m,)$
(41) $\displaystyle S[P(Y|X)]$ $\displaystyle=$ $\displaystyle-\beta
J_{0}Nm^{2}+N\log 2\cosh(\beta J_{0}m),$ (42)
$\displaystyle\frac{1}{N}\phi_{\bm{\eta}}$ $\displaystyle=$ $\displaystyle
2\beta J_{0}\gamma(1-\gamma)m^{2}+\log\frac{(\cosh^{\gamma}(\beta J_{0}\gamma
m))(\cosh^{1-\gamma}(\beta J_{0}(1-\gamma)m))}{\cosh(\beta J_{0}m)}.$

It is important to stress that the value of $m$ is the magnetization of the
$N$ units full model with interaction $J_{0}/N$ and entropy given by
$S(\beta;J_{0},N)$. Hence, using definition (33) we can write

$\displaystyle\phi_{\bm{\eta}}$ $\displaystyle=$
$\displaystyle\frac{\gamma}{2}\mathcal{S}(m,\gamma\beta;J_{0},N)+\frac{1-\gamma}{2}\mathcal{S}(m,(1-\gamma)\beta;J_{0},N)-\frac{1}{2}S(\beta;J_{0},N).$
(43)

Due to the lack of spatial geometry of the original spin system, the influence
of the partition is only through its size and not on the particular sets
$(I,J)$. For a general partition $\Pi$ into $K$ non overlapping subsets of
fractional sizes $\\{\gamma_{a}\\}$, with $\sum_{a=1}^{K}\gamma_{a}=1$

$\displaystyle\phi_{\Pi}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{a=1}^{K}\gamma_{a}\mathcal{S}(m,\gamma_{a}\beta;J_{0},N)-\frac{1}{2}S(\beta;J_{0},N).$
(44)

###  4.1 Bipartition

(a)

(b)

Figure 1: Fig 1(a): $\phi_{IJ}(\beta,\gamma)$. The dark circles show the
maximum values of $\phi_{IJ}$ per spin for fixed $\beta$. The floor shows its
projection into the $(\beta,\gamma)$ plane. Fig 1(b): The optimal bipartition
in the $(\beta,\gamma)$ plane. Symmetry breaks for $\beta>2$, with the optimal
partition, given by the inverse of $\beta_{B}(\gamma)$. The small component
remains in the (gray) paramagnetic region, below $\beta_{c}(\gamma)$ and the
large component is in a ferromagnetic state, above $\beta_{c}(\gamma)$.

In figure 1(a) we show $\phi_{IJ}$ for a general bipartition. In the
paramagnetic phase, the index is zero, which shows that the paramagnetic phase
for the full system, is similar to that of the disconnected system.
Statistically, the full model in the paramagnetic phase behaves as if
effectively disconnected. As soon as $\beta^{-1}$ becomes smaller than
$\beta^{-1}_{c}=1$, the critical temperature of the full system measured in
units of $J_{0}$, $\phi_{IJ}$ is different from zero and has a single maximum,
at $\gamma=1/2$. However, as $\beta$ goes above ${2}$, both disconnected
systems with $\gamma=1/2$ can present ferromagnetic order, and the value of
$\gamma$ where $\phi_{IJ}$ is maximum, bifurcates away from $1/2$. This
reflects the fact that the small component of the partition is paramagnetic
and the large component is ferromagnetic. If both were ordered, the KL
divergence to the full model would be smaller. At that temperature, both
components can’t be disordered. Hence, the disconnection into equal sets
doesn’t yield the largest integrated information loss to the full system. For
a given $\beta$, the bipartition into $\gamma$ and $1-\gamma$ is optimal for
(see figure 1(b)):

$\displaystyle\gamma$ $\displaystyle=$ $\displaystyle\frac{1}{2}+{\cal
A}(\beta-\beta_{c})^{\frac{1}{2}},$ (45)

with values $\beta_{c}=2$ and ${\cal A}\approx 0.400$ for $\beta>2$ and ${\cal
A}=0$ for $\beta<0$. Details are shown in [11].

###  4.2 Tripartition

(a)

(b)

Figure 2: Fig 2(a):
$\phi_{IJK}(\gamma,\beta)=\text{max}_{\gamma_{2}}\phi_{IJK}(\gamma_{1},\gamma_{2},\gamma_{3},\beta)$
for a partition into three subsets. The floor shows, as a function of $\beta$
the values of $\gamma_{1-3}$ that yield the maximum value. Before the
bifurcation at $\beta=3$, all $\gamma_{i}=1/3$. For larger $\beta$, one subset
increases ($\gamma_{1}$) and the other two decrease, but remain equal. Fig
2(b): The optimal tripartition in the $(\beta,\gamma)$ plane. There is a
discontinuous partially symmetry breaking transition at $\beta>3$. The small
components, with fractions less than $1/3$, remain in the (gray) paramagnetic
region, below $\beta_{c}(\gamma)$ and the large component is in a
ferromagnetic state, above $\beta_{c}(\gamma)$.

Panel 2 shows $\phi_{\Pi}$ when $\Pi$ has three components: $\phi_{IJK}$. Call
the fractional sizes $\gamma_{a}$, and choose their labels such that
$\gamma_{1}\geq\gamma_{2}\geq\gamma_{3}$, with $\sum_{i}\gamma_{i}=1$. For a
fixed $\gamma_{a}\geq\frac{1}{3}$, we calculate the value of $\gamma_{b}$ in
the interval $[0,1-\gamma_{a}]$ that maximizes $\phi_{\Pi}$ given by equation
44. Then $\gamma_{1}=$max$(\gamma_{a},\gamma_{b})$, $\gamma_{2}=$max$($min
$(\gamma_{a},\gamma_{b}),1-\gamma_{a}-\gamma_{b})$ and
$\gamma_{3}=1-\gamma_{1}-\gamma_{2}$. Again, in the paramagnetic region
$\phi_{IJK}=0$. For $1\leq\beta\leq 3$, the system is equi-partitioned
$\gamma_{1}=\gamma_{2}=\gamma_{3}=1/3$. For $\beta>3$, a component with
$\gamma_{1}\geq 1/3$ can be ordered, leading to a symmetry breaking transition
with $\gamma_{2}=\gamma_{3}=(1-\gamma_{1})/2<1/3$. The symmetry is only
partially broken since there is still a remnant symmetry, with the two small
components equal in size.

Partitioning the system into three subsets allows for a larger $\phi_{\Pi}$
than for partitions into two subsets. This is to be expected, since the KL
divergences from a ferromagnetic state to a partitioned system increases with
the size of the paramagnetic component of the partitioned system. It is
expected, then that in general $\Phi_{G}$ grows with the number of components
of $\Pi$.

##  5 Modular Ising model

In the previous examples there is no natural partitioning of the system into
components. It is possible that this index of complexity finds its utility
when applied to neural systems which present a natural partition. For example,
see figure 3, a system might be naturally divided into broad groups such as
the hemispheres of a brain, or into a larger set of components, such as
Brodmann areas. Of course, we are not ready to work with realistic models of a
brain. We thus study the parallel updating version of a modular Ising model,
similar to one studied by [12] in the context of super-paramagnetic
clustering.

Figure 3: A system with a natural separation into modules $1$ and $2$ can be
divided, for calculation of $\phi_{IJ}$ into arbitrary subsets $I$ and $J$,
giving rise to four types of units.

The system has $N$ Ising units, and again, the state of the system is given at
time $t-1$ by $X=\left\\{x_{i}\right\\}_{i=1}^{N}$, and by
$Y=\\{y_{i}\\}_{i=1}^{N}$ at $t$. The Hamiltonian is:

$H_{M}(X,Y)=-\frac{J_{1}}{N}\sum_{i,j=1}^{M}x_{j}y_{i}-\frac{J_{2}}{N}\sum_{i,j=M+1}^{N}x_{j}y_{i}-\frac{J_{21}}{N}\sum_{i=M+1}^{N}\sum_{j=1}^{M}x_{j}y_{i}-\frac{J_{12}}{N}\sum_{i=1}^{M}\sum_{j=M+1}^{N}x_{j}y_{i}.$
(46)

The first $M$ units belong to group $1$. They interact with their future
through $J_{1}/N$ and with the future of the other $N-M$ units, belonging to
group $2$, through $J_{21}/N$. The $N-M$ elements interact with their future
through $J_{2}/N$ and with group $1$ units future through $J_{12}/N$. The
equilibrium is described by a Gibbs distribution. Call
$\tilde{\gamma}_{1}=M/N,\tilde{\gamma}_{2}=1-\tilde{\gamma}_{1}$,
$\mathbf{m}={m_{1}\choose m_{2}}$, $\mathbf{n}={n_{1}\choose n_{2}}$ and

$\displaystyle\mathbf{J}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}J_{1}&J_{12}\\\ J_{21}&J_{2}\end{bmatrix}.$ (47)

Using the same methods as in the previous section, we obtain for the free
energy functional:

${\mathcal{F}_{M}}(\mathbf{m,n};\beta,\mathbf{J},\mathbf{\tilde{\gamma}})=\mathbf{n}^{T}\cdot\mathbf{J}\cdot\mathbf{m}-\frac{1}{\beta}\sum_{i=1,2}\tilde{\gamma}_{i}\log
4\cosh\left[\beta\left(\mathbf{n}^{T}\cdot\mathbf{J}\right)_{i}\right]\cosh\left[\beta\left(\mathbf{J}\cdot\mathbf{m}\right)_{i}\right],$
(48)

and the free energy

$\displaystyle{{F}_{M}}(\beta,\mathbf{J},\mathbf{\tilde{\gamma}})$
$\displaystyle=$ $\displaystyle-\lim_{N\rightarrow\infty}\frac{1}{\beta N}\log
Z,$ (49)

and the saddle point equations, for $i=1,2$:

$\displaystyle n_{i}$ $\displaystyle=$
$\displaystyle\tilde{\gamma}_{i}\tanh\left[\beta(\mathbf{J}\cdot\mathbf{m})_{i}\right]$
(50) $\displaystyle m_{i}$ $\displaystyle=$
$\displaystyle\tilde{\gamma}_{i}\tanh\left[\beta(\mathbf{n}^{T}\cdot\mathbf{J})_{i}\right]$
(51)

Again we obtain the entropy function from the derivative of the free-energy

${\mathcal{S}_{M}}(\mathbf{m,n};\beta,\mathbf{J},\mathbf{\tilde{\gamma}})=-\beta\mathbf{n}^{T}\cdot\mathbf{J}\cdot\mathbf{m}+\sum_{i=1,2}\tilde{\gamma}_{i}\log
4\cosh\left[\beta\left(\mathbf{n}^{T}\cdot\mathbf{J}\right)_{i}\right]\cosh\left[\beta\left(\mathbf{J}\cdot\mathbf{m}\right)_{i}\right].$
(53)

While the modular Ising has a natural partition into groups $1$ and $2$, the
partition considered for the integrated information can, in principle, be
different. We consider the partitions $I$ and $J$ of size $\gamma N$ and
$(1-\gamma)N$ respectively. Introduce the mixing parameters: $\alpha$, the
fraction of elements of module $1$ in partition $I$ and
$\bar{\alpha}=\frac{\gamma-\alpha\tilde{\gamma}}{1-\tilde{\gamma}}$, the
fraction of elements of module $2$ in partition $I$. When
$\gamma=\tilde{\gamma}=1/2$, the natural partition occurs for
$\bar{\alpha}=1-\alpha$ and $\alpha=0$ or $1$.

This divides the units into four groups, of sizes:

$\begin{pmatrix}|1\wedge I|&|2\wedge I|\\\ |1\wedge J|&|2\wedge
J|\end{pmatrix}=\begin{pmatrix}\alpha\tilde{\gamma}&\bar{\alpha}(1-\tilde{\gamma})\\\
(1-\alpha)\tilde{\gamma}&(1-\bar{\alpha})(1-\tilde{\gamma})\end{pmatrix}.$
(54)

The expression for $\phi_{\bm{{\eta}}}$, as shown in [11] is

$\displaystyle\phi_{\bm{\eta}}/N=$
$\displaystyle\beta\left(\mathbf{n}_{J}^{T}\cdot\mathbf{J}\cdot\mathbf{m}_{I}+\mathbf{n}_{I}^{T}\cdot\mathbf{J}\cdot\mathbf{m}_{J}\right)+$
$\displaystyle-\tilde{\gamma}\log
2\cosh\left[\beta\left(\mathbf{J}\cdot\mathbf{m}\right)_{1}\right]-(1-\tilde{\gamma})\log
2\cosh\left[\beta\left(\mathbf{J}\cdot\mathbf{m}\right)_{2}\right]+$
$\displaystyle+\alpha\tilde{\gamma}\log
2\cosh\left[\beta\left(\mathbf{J}\cdot\mathbf{m}_{I}\right)_{1}\right]+\bar{\alpha}(1-\tilde{\gamma})\log
2\cosh\left[\beta\left(\mathbf{J}\cdot\mathbf{m}_{I}\right)_{2}\right]+$
$\displaystyle+(1-\alpha)\tilde{\gamma}\log
2\cosh\left[\beta\left(\mathbf{J}\cdot\mathbf{m}_{J}\right)_{1}\right]+(1-\bar{\alpha})(1-\tilde{\gamma})\log
2\cosh\left[\beta\left(\mathbf{J}\cdot\mathbf{m}_{J}\right)_{2}\right]$ (55)

The vectors $\mathbf{m}_{I}={\alpha m_{1}\choose\bar{\alpha}m_{2}}$ and
$\mathbf{n}_{I}={\alpha n_{1}\choose\bar{\alpha}n_{2}}$ represent the partial
magnetizations for the fraction of the system that belongs to the component
$I$ and, analogously, $\mathbf{m}_{J}$ and $\mathbf{n}_{J}$ are the partial
magnetizations for the elements that belong to the component $J$, such that
$\mathbf{m}=\mathbf{m}_{I}+\mathbf{m}_{J}$ and
$\mathbf{n}=\mathbf{n}_{I}+\mathbf{n}_{J}$. The measure of complexity
$\phi_{\mathbf{\eta}}$ is not symmetric in the present and past variables, but
with the approach to equilibrium their expected values become the same.

(a) $J_{1}=J_{2}=1$, $J_{12}=J_{21}=2$. There is no mixing for strong cross
couplings. Similar behavior for $\gamma=\tilde{\gamma}=1/2$ (shown) and for
$1/4$ (not shown)

(b) $2=J_{21}>J_{1}=J_{2}=1>J_{12}=.5$. For asymmetric cross couplings and
$\gamma=\tilde{\gamma}=1/4$ there is discontinuous transitions

(c) $2=J_{21}>J_{1}=J_{2}=1>J_{12}=.5$. For asymmetric cross couplings and
$\gamma=\tilde{\gamma}=1/2$ there are two transitions.

(d) $J_{1}=J_{2}=1>J_{12}=J_{21}=0.4$. Weak cross couplings. Mixing symmetry
breaking has a preference, $\gamma=\tilde{\gamma}=0.45$.

(e) $J_{1}=J_{2}=1>J_{12}=J_{21}=0.4$. Weak cross couplings. Symmetry breaking
in the mixing, $\gamma=\tilde{\gamma}=1/2$.

(f) Connections that influence group $1$ are small and group $2$ are large:
$J_{1}=J_{12}=.5$ and $J_{2}=J_{21}=2$. $\tilde{\gamma}=\gamma=1/4$

Figure 4:  The dark circles follow as a function of $\beta$, the maximum of
$\phi$, with respect to $\alpha$. The light circles in the floor show the
projection of the maxima to the $(\alpha,\beta)$ plane.

This expression is analyzed in panel 4, which presents $\phi_{\bm{\eta}}$ as a
function of the mixing parameter $\alpha$ and the inverse temperature $\beta$,
for fixed $\gamma$ and $\tilde{\gamma}$. We show results in 4(a) to 4(e) when
the intra-group connections are equal $J_{1}=J_{2}$ and the inter-group
connection are either both higher $J_{12},J_{21}>J_{1}$, both smaller
$J_{12},J_{21}<J_{1}$, or one smaller and one higher than the intra-group
$J_{12}>J_{1}>J_{21}$.

For large inter-group connections, the partition along the natural partition
causes the largest difference, and thus there is no mixing: $\alpha=0$ or $1$
and $\bar{\alpha}=1$ or $0$ for $\tilde{\gamma}=\gamma=1/2$, see 4(a). All
members of one module are in the same partition. However, for
$\tilde{\gamma}=\gamma=1/4$, which is not shown, there is a small and a large
partition, as well as a small and a large module. Then $\alpha=0$ and
$\bar{\alpha}=1/3$ can’t be zero and is as small as possible. The inter-group
connections are large and the partition that maximizes $\phi$ separates the
modules as much as possible.

Figure 4(b) shows the case when the inter-group couplings take values larger
and smaller than $J_{1},J_{2}$ and the partitions are not equal in size,
$\tilde{\gamma}=\tilde{\gamma}\neq 1/2$. Within the ferromagnetic region, but
still low $\beta$, the partition separates the modules as much as possible as
in the previous case. However, for sufficiently high $\beta$ there is a
discontinuous transition into a mixing state. But when the partitions are
symmetric, $\tilde{\gamma}=\tilde{\gamma}=1/2$, see 4(c), in addition to the
discontinuous transition to the $\alpha=\bar{\alpha}=1/2$ state, there is a
second continuous transition into a $\alpha\neq 1/2$ state. The bifurcated
curve shows the evolution of the values of $\alpha$ and $\bar{\alpha}$. Again
the reason for the bifurcation is that all the disconnected sets would enter
the ferromagnetic phase, and the KL is reduced by preventing that, by reducing
the size of two of the four subsets.

For small inter-group connections the mixing for small $\beta>1$ goes to one
half, for lower temperatures and then it changes. As $\beta$ increases for the
asymmetric partitions $\tilde{\gamma}=\gamma\neq 1/2$, 4(d), changes into
larger mixing of the small component. However, it bifurcates symmetrically
when $\tilde{\gamma}=\gamma=1/2$, 4(e), as in the case of the homogeneous
system.

Finally, 4(f) shows the case $J_{12}=J_{1}<J_{2}=J_{21}$, where the couplings
to the future of group $1$ is smaller than the corresponding couplings to
group $2$ and $\tilde{\gamma}=\gamma=1/4$. $\alpha$ changes abruptly from $0$
to $1$, while $\bar{\alpha}$ goes from $1/3$ to $0$ which is the natural
partition. The transition occurs when the small partition $I$, containing only
members of group $2$ becomes ferromagnetic. Then it is more disrupting to
disconnect the inter-group couplings.

##  6 Conclusions

The main contribution of the IIT program is laying a road map to overcoming
the barrier between a physical approach to a deduction of consciousness and
the characterization of the degree to which an information processing system
may be deemed conscious. While the first seems to be currently a vaguely posed
problem, the second may lead to operational improvements of what it is meant
by the many different aspects of consciousness, and even illuminating the
first. Sadi Carnot motivated, specially by his father’s failures to prove that
no perpetuum mobile exists, just postulated their impossibility. The rewards
were immense. Decades later, the probabilistic approach of Statistical
Mechanics furnished a rational explanation for Carnot’s leap, with the
refinement of questions by the introduction of concepts like probability,
entropy and later, the quantification of information. The study of complexity
indices for different systems may lead to better questions about
consciousness, and even if not, the improvement of the computational
techniques and understanding of complex systems is worthy of the efforts.

In this work, we have analyzed a complexity index, inspired by the geometrical
approach in integrated information theory, for physical systems, in an attempt
to better understand how it can be used to assess complexity. Our main
experimental motivation was the application by Casali et al. [5] of related
indices as markers of consciousness in patients whose brains are subject to
different clinical conditions. From a theoretical perspective, we follow the
information geometric framework developed by Oizumi et al. [6]. By restricting
the analysis to partitions into $2$ and $3$ sets, we present the calculation
of the geometric integrated information index in two models. An optimal
partition is one in which the loss in information is greatest by removing the
coupling between the components. A phase transition in the size of the optimal
partition was found. As the temperature is lowered in the ferromagnetic phase
(below $T_{c}/2$ for bipartitions), symmetry is broken and an asymmetric
bipartion becomes optimal. Going from the paramagnetic phase to the
ferromagnetic phase $\phi$ increases. In the case of tripartitions, the
symmetry is partially broken and the two small sets remain of the same size.

We have addressed the question about what partition should be considered. For
the homogeneous model, it seems natural to consider the partition which at a
given value of $\beta$, maximizes the KL divergence to the full system.
However, interesting systems, serious candidates that deserve to be deemed
conscious, are far from homogeneous. Information processing systems, arising
from evolution, have a modular architecture of specialized macroscopic units
and may suggest a particular partition as natural. There is a high degree of
localization for different cognitive tasks in the brain, and it seems
interesting to consider the partition of the system into something resembling
the functional partition. There is no general theory to decide in this
classification of different areas, in part because of the high connectivity
among them. Thus, we look at the modular model which implements the idea of
functional partitioning and explore the possibility of a different
partitioning for IIT purposes, where the similarity of the functional and the
IIT partition is measured by the mixing parameter $\alpha$. We defined $\Phi$
as the maximum over all partitions, while fixing $K$ the number of components
of the partition. This invites the question of about its behavior as a
function of the number of components; the restricted scenario of $K=2,3$
supports that this may be monotonic in $K$.

It was shown, in [6], that the manifold of interest for the calculation of
$\Phi_{G}$, $\mathcal{M}_{G}$, contains a submanifold, $\mathcal{M}_{S}$, of
the disconnected distributions whose components evolve independent of each
other. The complexity measure calculated using the later manifold was called
stochastic interaction, $SI$ [10], and is always equal to the same difference
of entropies we found in equation (39). We know from the hierarchical relation
$\mathcal{M}_{S}\subset\mathcal{M}_{G}$, that $\phi_{G}\leq SI$. For the
models analyzed here we have an equality, but the necessary conditions for
this are still unclear, and an analysis of how the integrated information
index and the stochastic interactions complexity measures differs from one
another may yield interesting results.

The definition of $\Phi$ is dynamic and encompasses two consecutive states in
time, so our equilibrium analysis only captures a small part of its phase
space and more will be learned by studying the system out of equilibrium,
since it is difficult to make the case for consciousness in a system in
equilibrium.

## Acknowledgments

We thank Leonardo S. Barbosa, Marcus V. Baldo and O. Kinouchi for discussions.
This work was partially supported by CAPES as part of project
88887.612147/2021-00.

## References

  * [1] Giulio Tononi.  An information integration theory of consciousness.  BMC Neuroscience, 5(1):42, November 2004.
  * [2] Leonardo S. Barbosa, William Marshall, Sabrina Streipert, Larissa Albantakis, and Giulio Tononi.  A measure for intrinsic information.  Scientific Reports, 10(1), 2020.
  * [3] Leonardo S. Barbosa, William Marshall, Larissa Albantakis, and Giulio Tononi.  Mechanism integrated information.  Entropy, 23(3), 2021.
  * [4] Simone Sarasso, Adenauer Girardi Casali, Silvia Casarotto, Mario Rosanova, Corrado Sinigaglia, and Marcello Massimini.  Consciousness and complexity: a consilience of evidence.  Neuroscience of Consciousness, 08 2021.  niab023.
  * [5] Adenauer G Casali, Olivia Gosseries, Mario Rosanova, Mélanie Boly, Simone Sarasso, Karina R Casali, Silvia Casarotto, Marie-Aurélie Bruno, Steven Laureys, Giulio Tononi, and Marcello Massimini.  A theoretically based index of consciousness independent of sensory processing and behavior.  Sci Transl Med, 5(198):198ra105, August 2013.
  * [6] Masafumi Oizumi, Naotsugu Tsuchiya, and Shun-ichi Amari.  Unified framework for information integration based on information geometry.  Proceedings of the National Academy of Sciences, 113(51):14817–14822, 2016.
  * [7] Oizumi M., Albantakis L, and Tononi G.  From the phenomenology to the mechanisms of consciousness: Integrated information theory 3.0.  PLoS Computational Biology, pages 10(5), e1003588, 2014.
  * [8] Aguilera M. and E. Di Paolo.  Integrated information in the thermodynamic limit.  Neural Networks, 114:136–146, 2019.
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##  Appendix A Supplementary Material

###  A.1 $\gamma$ bifurcation

Here we show equation 45. With $m$ given by $m=\tanh\beta m$, write
$0=\frac{1}{N}\frac{\partial\Phi}{\partial\gamma}=I_{1}+I_{2}+I_{3}$ with

$\displaystyle I_{1}$ $\displaystyle=$ $\displaystyle 2\beta(1-2\gamma)m^{2}$
(56) $\displaystyle I_{2}$ $\displaystyle=$
$\displaystyle\log\cosh{\beta\gamma m}-\log\cosh{\beta(1-\gamma)m}$ (57)
$\displaystyle I_{3}$ $\displaystyle=$ $\displaystyle
m\beta\left(\gamma\tanh(\beta\gamma
m)-(1-\gamma)\tanh(\beta(1-\gamma)m)\right)$ (58)

The transition occurs near $\beta=2$ and $\gamma=\frac{1}{2}$, so define
$\beta=2(1+\epsilon)$, $\gamma=(1+\theta)/2$, so that
$1-\gamma=\frac{1-\theta}{2}$. Then $\beta\gamma=a+b$ and
$\beta(1-\gamma)=a-b$, with $a=1+\epsilon\sim{\cal O}(1)$ and
$b=\theta+\epsilon\theta\sim{\cal O}(\theta)$ is small near the transition.
Substitution leads to

$\displaystyle I_{1}$ $\displaystyle=$ $\displaystyle-4bm^{2}.$ (59)

Write for short $t_{a}=\tanh(am)$ and $t_{b}=\tanh(bm)$ and expand $t_{b}$:
$\tanh(x)=x-x^{3}/3$, and for $b\neq 0$

$\displaystyle I_{2}$ $\displaystyle\approx$ $\displaystyle
2t_{a}bm-\frac{2m^{3}b^{3}}{3}t_{a}(1-t_{a}^{2})+{\cal O}(b^{4})$ (60)

$\displaystyle I_{3}$ $\displaystyle\approx$ $\displaystyle
2m\left(t_{a}+am(1-t_{a}^{2})\right)b-2m^{3}(1-t_{a}^{2})\left(t_{a}-amt_{a}^{2}+\frac{1}{3}am\right)b^{3}+{\cal
O}(b^{4})$ (61)

The maximum with $\gamma$:

$\displaystyle 0=\frac{1}{N}\frac{\partial\Phi}{\partial\gamma}=$
$\displaystyle\left(-4m^{2}+2t_{a}m+2m\left(t_{a}+am(1-t_{a}^{2})\right)\right)b+$
$\displaystyle-2m^{3}(1-t_{a}^{2})\left(\frac{4}{3}t_{a}-amt_{a}^{2}+\frac{1}{3}am\right)b^{3}.$
(62)

For $b=a\theta=\frac{\beta}{2}\theta\neq 0$, $\theta=2(\gamma-1/2)$

$\displaystyle\gamma$ $\displaystyle=$ $\displaystyle\frac{1}{2}+{\cal
A}\left(\beta-\beta_{c}\right)^{\frac{1}{2}}$ (63)

where

$\displaystyle\beta_{c}$ $\displaystyle=$
$\displaystyle\frac{4(m-t_{a})}{m(1-t_{a}^{2})}=2$ (64) $\displaystyle{\cal
A}$ $\displaystyle=$
$\displaystyle\left(\frac{1}{2\beta^{2}}\frac{1}{\left(\frac{4}{3}t_{a}-amt_{a}^{2}+\frac{1}{3}am\right)}\right)^{\frac{1}{2}}$
(65)

To prove 64 use that $m=\tanh\beta m$ is the magnetization of the full system,
and $\beta=2(1+\epsilon)=2a$, then $t_{a}=\tanh(am)=\tanh(\beta m/2)$

$\displaystyle\tanh(\beta m)=m$ $\displaystyle=$
$\displaystyle\frac{2t_{a}}{1+t_{a}^{2}}=t_{a}+t_{a}\frac{1-t_{a}^{2}}{1+t_{a}^{2}}$
(66) $\displaystyle\frac{4(m-t_{a})}{(1-t_{a}^{2})}$ $\displaystyle=$
$\displaystyle\frac{4t_{a}}{1+t_{a}^{2}}=2m$ (67)

proving 64. Numerically, at $\beta=2$, ${\cal A}=0.40000$.

###  A.2  $\phi_{\eta}$ for the modular Ising model

Here are presented the steps that lead to equation 55, $\phi_{\bm{\eta}}$ for
the Modular Ising Model. Writing $\phi_{\bm{\eta}}$ as a difference of
conditional entropies explicitly

$\phi_{\bm{\eta}}=\sum_{X,Y}P\left(X,Y|\mathbf{J}\right)\left(\beta\sum_{i=1}^{N}y_{i}h_{i|D}-\sum_{i=1}^{N}\log
2\cosh(\beta h_{i})+\sum_{i=1}^{N}\log 2\cosh(\beta h_{i|S})\right),$ (68)

where we introduce the quantities $h_{i|S}$ and its complementary $h_{i|D}$,
defined as

$\displaystyle h_{i|S}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{l
l}\sum_{j}J_{ij}\eta_{j}x_{j},&\text{for}\quad i\in I\\\ \\\
\sum_{j}J_{ij}\left(1-\eta_{j}\right)x_{j},&\text{for}\quad i\in
J\end{array}\right.,$ (72) $\displaystyle h_{i|D}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{l
l}\sum_{j}J_{ij}\left(1-\eta_{j}\right)x_{j},&\text{for}\quad i\in I\\\ \\\
\sum_{j}J_{ij}\eta_{j}x_{j},&\text{for}\quad i\in J\end{array}\right.,$ (76)

as the fields generated on $i$ by the elements in the same partition and in
the different partition, respectively. We immediately note that the total
field is $h_{i}=h_{i|S}+h_{i|D}$. Consider the 3 averages separately and call
then $\phi_{A}$, $\phi_{B}$ and $\phi_{C}$, given by

$\displaystyle\phi_{A}$ $\displaystyle=$
$\displaystyle\beta\sum_{X,Y}P\left(X,Y|\mathbf{J}\right)\sum_{i=1}^{N}y_{i}h_{i|D},$
(77) $\displaystyle\phi_{B}$ $\displaystyle=$
$\displaystyle-\sum_{X,Y}P\left(X,Y|\mathbf{J}\right)\sum_{i=1}^{N}\log
2\cosh(\beta h_{i}),$ (78) $\displaystyle\phi_{C}$ $\displaystyle=$
$\displaystyle\sum_{X,Y}P\left(X,Y|\mathbf{J}\right)\sum_{i=1}^{N}\log
2\cosh(\beta h_{i|S}).$ (79)

The procedure we follow for the calculation of these averages is to consider
auxiliary distributions. The distributions we are interested are those that
are parameterized in such a way that the original distribution for the modular
Ising model, can be recovered by a particular choice of the parameters. To
calculate $\phi_{A}$, $\phi_{B}$ and $\phi_{C}$, consider, respectively,

$\displaystyle P_{A}\left(X,Y|\mathbf{J};\bm{\eta},\beta_{S},\beta_{D}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{Z_{A}}\exp\left\\{\sum_{i}\left(\beta_{S}h_{i|S}+\beta_{D}h_{i|D}\right)y_{i}\right\\},$
(80) $\displaystyle P_{B}\left(X|\mathbf{J};\bm{\eta},\beta,\lambda\right)$
$\displaystyle=$ $\displaystyle\frac{1}{Z_{B}}\exp\left\\{\lambda\sum_{i}\log
2\cosh\left(\beta\sum_{j}J_{ij}x_{j}\right)\right\\},$ (81) $\displaystyle
P_{C}(X,Y|\mathbf{J};\bm{\eta},\beta,\lambda_{S})$ $\displaystyle=$
$\displaystyle\frac{1}{Z_{C}}\exp\left\\{\beta\sum_{i,j}J_{ij}x_{j}y_{i}+\lambda_{S}\sum_{i}\log
2\cosh\left(\beta\sum_{j}J_{ij}\theta_{ij}^{S}x_{j}\right)\right\\}.$ (82)

We note that by setting $\beta_{S}=\beta_{D}=\beta$, $\lambda=1$ and
$\lambda_{S}=0$ they all become the equilibrium distribution for the modular
Ising model222With exception of $P_{B}$ which, for $\lambda=0$, becomes the
marginalized distribution $P(X|\mathbf{J})=\sum_{Y}P(X,Y|\mathbf{J})$, which
is not a problem, since $\phi_{B}$ only depends on the variables $X$., and the
desired averages can be easily calculated as derivatives of the free energies:

$\displaystyle\phi_{A}$ $\displaystyle=$
$\displaystyle\beta\left.\frac{\partial}{\partial\beta_{D}}\log
Z_{A}\right|_{\beta_{S}=\beta_{D}=\beta},$ (83) $\displaystyle\phi_{B}$
$\displaystyle=$ $\displaystyle-\left.\frac{\partial}{\partial\lambda}\log
Z_{B}\right|_{\lambda=1},$ (84) $\displaystyle\phi_{C}$ $\displaystyle=$
$\displaystyle\left.\frac{\partial}{\partial\lambda_{S}}\log
Z_{C}\right|_{\lambda_{S}=0}.$ (85)

####  A.2.1 $\log Z_{A}$ calculation

$Z_{A}=\sum_{X,Y}\exp\left\\{\sum_{i,j}J_{ij}x_{j}y_{i}\left(\beta_{S}\theta_{ij}^{S}+\beta_{D}\theta_{ij}^{D}\right)\right\\},$
(86)

we introduce here the projectors
$\theta_{ij}^{S}=\eta_{i}\eta_{j}+(1-\eta_{i})(1-\eta_{j})$ and
$\theta_{ij}^{D}=\eta_{i}(1-\eta_{j})+(1-\eta_{i})\eta_{j}$, that take into
account the cases where $i$ and $j$ belongs to the same component and to
different components, respectively.

Introducing integrals over Dirac delta distributions and using its Fourier
representation

$\displaystyle Z_{A}=$
$\displaystyle\int\frac{dm_{1}^{I}d\hat{m}_{1}^{I}}{2\pi/N}\int\frac{dm_{1}^{J}d\hat{m}_{1}^{J}}{2\pi/N}\int\frac{dm_{2}^{I}d\hat{m}_{2}^{I}}{2\pi/N}\int\frac{dm_{2}^{J}d\hat{m}_{2}^{J}}{2\pi/N}\int\frac{dn_{1}^{I}d\hat{n}_{1}^{I}}{2\pi/N}\int\frac{dn_{1}^{J}d\hat{n}_{1}^{J}}{2\pi/N}\int\frac{dn_{2}^{I}d\hat{n}_{2}^{I}}{2\pi/N}\int\frac{dn_{2}^{J}d\hat{n}_{2}^{J}}{2\pi/N}\times$
$\displaystyle\times\exp\bigg{\\{}\sum_{i=1}^{M}\log\zeta_{i}^{(1)}+\sum_{i=M+1}^{N}\log\zeta_{i}^{(2)}+\beta_{S}J_{1}N\left(m_{1}^{I}n_{1}^{I}+m_{1}^{J}n_{1}^{J}\right)+\beta_{D}J_{1}N\left(m_{1}^{I}n_{1}^{J}+m_{1}^{J}n_{1}^{I}\right)+$
$\displaystyle+\beta_{S}J_{2}N\left(m_{2}^{I}n_{2}^{I}+m_{2}^{J}n_{2}^{J}\right)+\beta_{D}J_{2}N\left(m_{2}^{I}n_{2}^{J}+m_{2}^{J}n_{2}^{I}\right)+\beta_{S}J_{12}N\left(m_{2}^{I}n_{1}^{I}+m_{2}^{J}n_{1}^{J}\right)+$
$\displaystyle+\beta_{D}J_{12}N\left(m_{2}^{I}n_{1}^{J}+m_{2}^{J}n_{1}^{I}\right)+\beta_{S}J_{21}N\left(m_{1}^{I}n_{2}^{I}+m_{1}^{J}n_{2}^{J}\right)+\beta_{D}J_{21}N\left(m_{1}^{I}n_{2}^{J}+m_{1}^{J}n_{2}^{I}\right)+$
$\displaystyle+iN\left(m_{1}^{I}\hat{m}_{1}^{I}+m_{1}^{J}\hat{m}_{1}^{J}+m_{2}^{I}\hat{m}_{2}^{I}+m_{2}^{J}\hat{m}_{2}^{J}+n_{1}^{I}\hat{n}_{1}^{I}+n_{1}^{J}\hat{n}_{1}^{J}+n_{2}^{I}\hat{n}_{2}^{I}+n_{2}^{J}\hat{n}_{2}^{J}\right)\bigg{\\}},$
(87)

where

$\displaystyle\zeta_{i}^{(1)}$ $\displaystyle=$
$\displaystyle\sum_{x_{i},y_{i}}e^{-i\left(\hat{m}_{1}^{I}\eta_{i}+\hat{m}_{1}^{J}(1-\eta_{i})\right)x_{i}-i\left(\hat{n}_{1}^{I}\eta_{i}+\hat{n}_{1}^{J}(1-\eta_{i})\right)y_{i}},$
(88) $\displaystyle\zeta_{i}^{(2)}$ $\displaystyle=$
$\displaystyle\sum_{x_{i},y_{i}}e^{-i\left(\hat{m}_{2}^{I}\eta_{i}+\hat{m}_{2}^{J}(1-\eta_{i})\right)x_{i}-i\left(\hat{n}_{2}^{I}\eta_{i}+\hat{n}_{2}^{J}(1-\eta_{i})\right)y_{i}}.$
(89)

Note that the integrand is an exponential with exponent proportional to $N$,
thus we can use the saddle point integration method and obtain

$\displaystyle\log Z_{A}=$
$\displaystyle-\beta_{S}J_{1}N\left(m_{1}^{I}n_{1}^{I}+m_{1}^{J}n_{1}^{J}\right)-\beta_{D}J_{1}N\left(m_{1}^{I}n_{1}^{J}+m_{1}^{J}n_{1}^{I}\right)-\beta_{S}J_{2}N\left(m_{2}^{I}n_{2}^{I}+m_{2}^{J}n_{2}^{J}\right)+$
$\displaystyle-\beta_{D}J_{2}N\left(m_{2}^{I}n_{2}^{J}+m_{2}^{J}n_{2}^{I}\right)-\beta_{S}J_{12}N\left(m_{2}^{I}n_{1}^{I}+m_{2}^{J}n_{1}^{J}\right)-\beta_{D}J_{12}N\left(m_{2}^{I}n_{1}^{J}+m_{2}^{J}n_{1}^{I}\right)+$
$\displaystyle-\beta_{S}J_{21}N\left(m_{1}^{I}n_{2}^{I}+m_{1}^{J}n_{2}^{J}\right)-\beta_{D}J_{21}N\left(m_{1}^{I}n_{2}^{J}+m_{1}^{J}n_{2}^{I}\right)+$
$\displaystyle+\alpha\tilde{\gamma}N\log
2\cosh\left[J_{1}\left(\beta_{S}n_{1}^{I}+\beta_{D}n_{1}^{J}\right)+J_{21}\left(\beta_{S}n_{2}^{I}+\beta_{D}n_{2}^{J}\right)\right]+$
$\displaystyle+(1-\alpha)\tilde{\gamma}N\log
2\cosh\left[J_{1}\left(\beta_{S}n_{1}^{J}+\beta_{D}n_{1}^{I}\right)+J_{21}\left(\beta_{S}n_{2}^{J}+\beta_{D}n_{2}^{I}\right)\right]+$
$\displaystyle+\alpha\tilde{\gamma}N\log
2\cosh\left[J_{1}\left(\beta_{S}m_{1}^{I}+\beta_{D}m_{1}^{J}\right)+J_{12}\left(\beta_{S}m_{2}^{I}+\beta_{D}m_{2}^{J}\right)\right]+$
$\displaystyle+(1-\alpha)\tilde{\gamma}N\log
2\cosh\left[J_{1}\left(\beta_{S}m_{1}^{J}+\beta_{D}m_{1}^{I}\right)+J_{12}\left(\beta_{S}m_{2}^{J}+\beta_{D}m_{2}^{I}\right)\right]+$
$\displaystyle+(\gamma-\alpha\tilde{\gamma})N\log
2\cosh\left[J_{2}\left(\beta_{S}n_{2}^{I}+\beta_{D}n_{2}^{J}\right)+J_{12}\left(\beta_{S}n_{1}^{I}+\beta_{D}n_{1}^{J}\right)\right]+$
$\displaystyle+(1-\gamma-(1-\alpha)\tilde{\gamma})N\log
2\cosh\left[J_{2}\left(\beta_{S}n_{2}^{J}+\beta_{D}n_{2}^{I}\right)+J_{12}\left(\beta_{S}n_{1}^{J}+\beta_{D}n_{1}^{I}\right)\right]+$
$\displaystyle+(\gamma-\alpha\tilde{\gamma})N\log
2\cosh\left[J_{2}\left(\beta_{S}m_{2}^{I}+\beta_{D}m_{2}^{J}\right)+J_{21}\left(\beta_{S}m_{1}^{I}+\beta_{D}m_{1}^{J}\right)\right]+$
$\displaystyle+(1-\gamma-(1-\alpha)\tilde{\gamma})N\log
2\cosh\left[J_{2}\left(\beta_{S}m_{2}^{J}+\beta_{D}m_{2}^{I}\right)+J_{21}\left(\beta_{S}m_{1}^{J}+\beta_{D}m_{1}^{I}\right)\right],$
(90)

where we define $\alpha$ as the fraction of elements of the group 1 that
belongs to the partition $I$. The fractional sizes for the other components
can be obtained by imposing the constraint that the fractional size of the
component $I$ must be $\gamma$ and the whole system must sum up to 1.

The saddle point equations are

$\displaystyle m_{1}^{I}$ $\displaystyle=$
$\displaystyle\alpha\tilde{\gamma}\tanh\left[J_{1}\left(\beta_{S}n_{1}^{I}+\beta_{D}n_{1}^{J}\right)+iJ_{21}\left(\beta_{S}n_{2}^{I}+\beta_{D}n_{2}^{J}\right)\right],$
(91) $\displaystyle m_{1}^{J}$ $\displaystyle=$
$\displaystyle(1-\alpha)\tilde{\gamma}\tanh\left[J_{1}\left(\beta_{S}n_{1}^{J}+\beta_{D}n_{1}^{I}\right)+iJ_{21}\left(\beta_{S}n_{2}^{J}+\beta_{D}n_{2}^{I}\right)\right],$
(92) $\displaystyle m_{2}^{I}$ $\displaystyle=$
$\displaystyle\bar{\alpha}(1-\tilde{\gamma})\tanh\left[J_{2}\left(\beta_{S}n_{2}^{I}+\beta_{D}n_{2}^{J}\right)+iJ_{12}\left(\beta_{S}n_{1}^{I}+\beta_{D}n_{1}^{J}\right)\right],$
(93) $\displaystyle m_{2}^{J}$ $\displaystyle=$
$\displaystyle(1-\bar{\alpha})\alpha(1-\tilde{\gamma})\tanh\left[J_{2}\left(\beta_{S}n_{2}^{J}+\beta_{D}n_{2}^{I}\right)+iJ_{12}\left(\beta_{S}n_{1}^{J}+\beta_{D}n_{1}^{I}\right)\right],$
(94) $\displaystyle n_{1}^{I}$ $\displaystyle=$
$\displaystyle\alpha\tilde{\gamma}\tanh\left[J_{1}\left(\beta_{S}m_{1}^{I}+\beta_{D}m_{1}^{J}\right)+iJ_{12}\left(\beta_{S}m_{2}^{I}+\beta_{D}m_{2}^{J}\right)\right],$
(95) $\displaystyle n_{1}^{J}$ $\displaystyle=$
$\displaystyle(1-\alpha)\tilde{\gamma}\tanh\left[J_{1}\left(\beta_{S}m_{1}^{J}+\beta_{D}m_{1}^{I}\right)+iJ_{12}\left(\beta_{S}m_{2}^{J}+\beta_{D}m_{2}^{I}\right)\right],$
(96) $\displaystyle n_{2}^{I}$ $\displaystyle=$
$\displaystyle\bar{\alpha}(1-\tilde{\gamma})\tanh\left[J_{2}\left(\beta_{S}m_{2}^{I}+\beta_{D}m_{2}^{J}\right)+iJ_{21}\left(\beta_{S}m_{1}^{I}+\beta_{D}m_{1}^{J}\right)\right],$
(97) $\displaystyle n_{2}^{J}$ $\displaystyle=$
$\displaystyle(1-\bar{\alpha})(1-\tilde{\gamma})\tanh\left[J_{2}\left(\beta_{S}m_{2}^{J}+\beta_{D}m_{2}^{I}\right)+iJ_{21}\left(\beta_{S}m_{1}^{J}+\beta_{D}m_{1}^{I}\right)\right],$
(98)

where we define

$\bar{\alpha}=\frac{\gamma-\alpha\tilde{\gamma}}{1-\tilde{\gamma}},$ (99)

as the fraction of elements of the group 2 that belongs to the component $I$.

Taking the derivative

$\displaystyle\frac{\partial}{\partial\beta_{D}}\log
Z_{A}=NJ_{1}\left(m_{1}^{I}n_{1}^{J}+m_{1}^{J}n_{1}^{I}\right)+NJ_{21}\left(m_{1}^{I}n_{2}^{J}+m_{1}^{J}n_{2}^{I}\right)+$
$\displaystyle+NJ_{12}\left(m_{2}^{I}n_{1}^{J}+m_{2}^{J}n_{1}^{I}\right)+NJ_{2}\left(m_{2}^{I}n_{2}^{J}+m_{2}^{J}n_{2}^{I}\right),$
(100)

where we used the equations of state to substitute the $\tanh$ for the
correspondent order parameter.

Taking $\beta_{S}=\beta_{D}=\beta$ we have:

$\displaystyle m_{1}^{I}=\alpha m_{1},$ $\displaystyle\qquad n_{1}^{I}=\alpha
n_{1},$ (101) $\displaystyle m_{1}^{J}=(1-\alpha)m_{1},$ $\displaystyle\qquad
n_{1}^{J}=(1-\alpha)n_{1},$ (102) $\displaystyle m_{2}^{I}=\bar{\alpha}m_{2},$
$\displaystyle\qquad n_{2}^{I}=\bar{\alpha}n_{2},$ (103) $\displaystyle
m_{2}^{J}=(1-\bar{\alpha})m_{2},$ $\displaystyle\qquad
n_{2}^{J}=(1-\bar{\alpha})n_{2},$ (104)

where $m_{1}$, $m_{2}$, $n_{1}$ and $n_{2}$ are now the order parameter of the
modular Ising model.

And the value of $\phi_{A}$

$\displaystyle\phi_{A}=$ $\displaystyle 2\alpha(1-\alpha)N\beta
J_{1}m_{1}n_{1}+\left[\alpha(1-\bar{\alpha})+\bar{\alpha}(1-\alpha)\right]N\beta
J_{21}m_{1}n_{2}+$
$\displaystyle+\left[\bar{\alpha}(1-\alpha)+\alpha(1-\bar{\alpha})\right]N\beta
J_{12}m_{2}n_{1}+2\bar{\alpha}(1-\bar{\alpha})N\beta J_{2}m_{2}n_{2}.$ (106)

####  A.2.2 $\log Z_{B}$ calculation

$Z_{B}=\sum_{X}\exp\left\\{\lambda\sum_{i}\log
2\cosh\left(\beta\sum_{j}J_{ij}x_{j}\right)\right\\}.$ (107)

Introducing integrals over delta distributions
$\delta\left(w_{i}-i\beta\sum_{j}J_{ij}x_{j}\right)$ and its Fourier
representation

$Z_{B}=\sum_{X}\int\prod_{i}\frac{dw_{i}d\hat{w}_{i}}{2\pi}\exp\left\\{\lambda\sum_{i}\log
2\cosh\left(-iw_{i}\right)+i\sum_{i}\hat{w}_{i}w_{i}+\beta\sum_{i}\hat{w}_{i}\sum_{j}J_{ij}x_{j}\right\\}.$
(108)

Introducing more integrals over delta distributions and using its Fourier
representation for the order parameters:

$\displaystyle Z_{B}=$
$\displaystyle\int\frac{dm_{1}d\hat{m}_{1}}{2\pi/N}\int\frac{dm_{2}d\hat{m}_{2}}{2\pi/N}\int\frac{dn_{1}d\hat{n}_{1}}{2\pi/N}\int\frac{dn_{2}d\hat{n}_{2}}{2\pi/N}\times$
$\displaystyle\times\exp\bigg{\\{}\beta J_{1}Nm_{1}n_{1}+\beta
J_{21}Nm_{1}n_{2}+\beta J_{12}Nm_{2}n_{1}+\beta J_{2}Nm_{2}n_{2}+$
$\displaystyle+iN\left(m_{1}\hat{m}_{1}+m_{2}\hat{m}_{2}+n_{1}\hat{n}_{1}+n_{2}\hat{n}_{2}\right)+\tilde{\gamma}N\log\mathcal{Z}_{1}+(1-\tilde{\gamma})N\log\mathcal{Z}_{2}\bigg{\\}},$
(109)

where

$\displaystyle\mathcal{Z}_{1}$ $\displaystyle=$ $\displaystyle
2\cosh(-i\hat{m}_{1})\left[2\cosh(-i\hat{u}_{1})\right]^{\lambda},$ (110)
$\displaystyle\mathcal{Z}_{2}$ $\displaystyle=$ $\displaystyle
2\cosh(-i\hat{m}_{2})\left[2\cosh(-i\hat{u}_{2})\right]^{\lambda}.$ (111)

Using the saddle point method of integration

$\displaystyle\log Z_{B}=$ $\displaystyle-\beta J_{1}Nm_{1}n_{1}-\beta
J_{21}Nm_{1}n_{2}-\beta J_{12}Nm_{2}n_{1}-\beta J_{2}Nm_{2}n_{2}+$
$\displaystyle+\tilde{\gamma}N\log
2\cosh[\beta\left(J_{1}u_{1}+J_{21}u_{2}\right)]+\lambda\tilde{\gamma}N\log
2\cosh[\beta\left(J_{1}m_{1}+J_{21}m_{2}\right)]+$
$\displaystyle+(1-\tilde{\gamma})N\log
2\cosh[\beta\left(J_{12}u_{1}+J_{2}u_{2}\right)]+\lambda(1-\tilde{\gamma})N\log
2\cosh[\beta\left(J_{21}m_{1}+J_{2}m_{2}\right)].$ (112)

The saddle point equations are:

$\displaystyle m_{1}$ $\displaystyle=$
$\displaystyle\tilde{\gamma}\tanh\left[\beta\left(J_{1}u_{1}+J_{21}u_{2}\right)\right],$
(113) $\displaystyle m_{2}$ $\displaystyle=$
$\displaystyle(1-\tilde{\gamma})\tanh\left[\beta\left(J_{12}u_{1}+J_{2}u_{2}\right)\right],$
(114) $\displaystyle u_{1}$ $\displaystyle=$
$\displaystyle\tilde{\gamma}\tanh\left[\beta\left(J_{1}m_{1}+J_{21}m_{2}\right)\right],$
(115) $\displaystyle u_{2}$ $\displaystyle=$
$\displaystyle(1-\tilde{\gamma})\tanh\left[\beta\left(J_{21}m_{1}+J_{2}m_{2}\right)\right].$
(116)

Taking the derivative with respect to $\lambda$, we obtain:

$\phi_{B}=-\tilde{\gamma}N\log
2\cosh\left[\beta\left(J_{1}m_{1}+J_{21}m_{2}\right)\right]-(1-\tilde{\gamma})N\log
2\cosh\left[\beta\left(J_{21}m_{1}+J_{2}m_{2}\right)\right].$ (117)

####  A.2.3 $\log Z_{C}$ calculation

$Z_{C}=\sum_{X,Y}\exp\left\\{\beta\sum_{i,j}J_{ij}x_{j}y_{i}+\lambda_{S}\sum_{i}\log
2\cosh\left(\beta\sum_{j}J_{ij}\theta_{ij}^{S}x_{j}\right)\right\\}.$ (118)

Again we introduce integrals over
$\delta\left(w_{i}-i\beta\sum_{j}J_{ij}\theta_{ij}^{S}x_{j}\right)$, and use
its Fourier representation

$\displaystyle
Z_{C}=\sum_{X,Y}\int\prod_{i}\frac{dw_{i}d\hat{w}_{i}}{2\pi}\exp\left\\{\beta\sum_{i,j}J_{ij}x_{j}y_{i}+\lambda_{S}\sum_{i}\log
2\cosh\left(-iw_{i}\right)+\right.$
$\displaystyle+\left.i\sum_{i}\hat{w}_{i}w_{i}+\beta\sum_{i}\hat{w}_{i}\sum_{j}J_{ij}\theta_{ij}^{S}x_{j}\right\\},$
(119)

where, again we used the projector
$\theta_{ij}^{S}=\eta_{i}\eta_{j}+(1-\eta_{i})(1-\eta_{j})$ that selects all
the interactions between the same component.

Introducing integrals over delta distribution for each parameter

$\displaystyle Z_{C}=$
$\displaystyle\int\frac{dm_{1}d\hat{m}_{1}}{2\pi/N}\int\frac{dm_{2}d\hat{m}_{2}}{2\pi/N}\int\frac{dn_{1}d\hat{n}_{1}}{2\pi/N}\int\frac{dn_{2}d\hat{n}_{2}}{2\pi/N}\int\frac{dm_{1}^{I}d\hat{m}_{1}^{I}}{2\pi/N}\int\frac{dm_{1}^{J}d\hat{m}_{1}^{J}}{2\pi/N}\times$
$\displaystyle\times\int\frac{dm_{2}^{I}d\hat{m}_{2}^{I}}{2\pi/N}\int\frac{dm_{2}^{J}d\hat{m}_{2}^{J}}{2\pi/N}\int\frac{du_{1}^{I}d\hat{u}_{1}^{I}}{2\pi/N}\int\frac{du_{1}^{J}d\hat{u}_{1}^{J}}{2\pi/N}\int\frac{du_{1}^{I}d\hat{u}_{1}^{I}}{2\pi/N}\int\frac{du_{1}^{J}d\hat{u}_{1}^{J}}{2\pi/N}\times$
$\displaystyle\times\exp\Bigg{\\{}\beta J_{1}Nm_{1}n_{1}+\beta
J_{21}Nm_{1}n_{2}+\beta J_{12}Nm_{2}n_{1}+\beta J_{2}Nm_{2}n_{2}+\beta
J_{1}N\left(m_{1}^{I}u_{1}^{I}+m_{1}^{J}u_{1}^{J}\right)+$
$\displaystyle+\beta
J_{21}N\left(m_{1}^{I}u_{2}^{I}+m_{1}^{J}u_{2}^{J}\right)+\beta
J_{12}N\left(m_{2}^{I}u_{1}^{I}+m_{2}^{J}u_{1}^{J}\right)+\beta
J_{2}N\left(m_{2}^{I}u_{2}^{I}+m_{2}^{J}u_{2}^{J}\right)+$
$\displaystyle+iN\big{(}m_{1}\hat{m}_{1}+m_{2}\hat{m}_{2}+n_{1}\hat{n}_{1}+n_{2}\hat{n}_{2}+m_{1}^{I}\hat{m}_{1}^{I}+m_{1}^{J}\hat{m}_{1}^{J}+m_{2}^{I}\hat{m}_{2}^{I}+m_{2}^{J}\hat{m}_{2}^{J}+$
$\displaystyle+u_{1}^{I}\hat{u}_{1}^{I}+u_{1}^{J}\hat{u}_{1}^{J}+u_{2}^{J}\hat{u}_{2}^{J}+u_{2}^{J}\hat{u}_{2}^{J}\big{)}+\sum_{i=1}^{M}\log\zeta_{i}^{(1)}+\sum_{i=M+1}^{N}\log\zeta_{i}^{(2)}\Bigg{\\}},$
(120)

where

$\displaystyle\zeta_{i}^{(1)}=$ $\displaystyle
2\cosh\left(-i\hat{n}_{1}\right)\left[2\cosh\left(-i\left(\eta_{i}\hat{u}_{1}^{I}+(1-\eta_{i})\hat{u}_{1}^{J}\right)\right)\right]^{\lambda_{S}}\times$
$\displaystyle\times
2\cosh\left(-i\left(\hat{m}_{1}+\eta_{i}\hat{m}_{1}^{I}+(1-\eta_{i})\hat{m}_{1}^{J}\right)\right),$
(121) $\displaystyle\zeta_{i}^{(2)}=$ $\displaystyle
2\cosh\left(-i\hat{n}_{2}\right)\left[2\cosh\left(-i\left(\eta_{i}\hat{u}_{2}^{I}+(1-\eta_{i})\hat{u}_{2}^{J}\right)\right)\right]^{\lambda_{S}}\times$
$\displaystyle\times
2\cosh\left(-i\left(\hat{m}_{2}+\eta_{i}\hat{m}_{2}^{I}+(1-\eta_{i})\hat{m}_{2}^{J}\right)\right).$
(122)

The saddle point integration method gives us

$\displaystyle\log Z_{C}=$ $\displaystyle\beta J_{1}Nm_{1}n_{1}+\beta
J_{21}Nm_{1}n_{2}+\beta J_{12}Nm_{2}n_{1}+\beta J_{2}Nm_{2}n_{2}+$
$\displaystyle+\beta
J_{1}N\left(m_{1}^{I}u_{1}^{I}+m_{1}^{J}u_{1}^{J}\right)+\beta
J_{21}N\left(m_{1}^{I}u_{2}^{I}+m_{1}^{J}u_{2}^{J}\right)+$
$\displaystyle+\beta
J_{12}N\left(m_{2}^{I}u_{1}^{I}+m_{2}^{J}u_{1}^{J}\right)+\beta
J_{2}N\left(m_{2}^{I}u_{2}^{I}+m_{2}^{J}u_{2}^{J}\right)+$
$\displaystyle+iN\big{(}m_{1}\hat{m}_{1}+m_{2}\hat{m}_{2}+n_{1}\hat{n}_{1}+n_{2}\hat{n}_{2}+m_{1}^{I}\hat{m}_{1}^{I}+m_{1}^{J}\hat{m}_{1}^{J}+$
$\displaystyle+m_{2}^{I}\hat{m}_{2}^{I}+m_{2}^{J}\hat{m}_{2}^{J}+u_{1}^{I}\hat{u}_{1}^{I}+u_{1}^{J}\hat{u}_{1}^{J}+u_{2}^{J}\hat{u}_{2}^{J}+u_{2}^{J}\hat{u}_{2}^{J}\big{)}+$
$\displaystyle+\tilde{\gamma}N\log
2\cosh\left(-i\hat{n}_{1}\right)+(1-\tilde{\gamma})N\log
2\cosh\left(-i\hat{n}_{2}\right)+$ $\displaystyle+\sum_{i=1}^{M}\log
2\cosh\left(-i\left(\hat{m}_{1}+\eta_{i}\hat{m}_{1}^{I}+(1-\eta_{i})\hat{m}_{1}^{J}\right)\right)+$
$\displaystyle+\lambda_{S}\sum_{i=1}^{M}\log
2\cosh\left(-i\left(\eta_{i}\hat{u}_{1}^{I}+(1-\eta_{i})\hat{u}_{1}^{J}\right)\right)+$
$\displaystyle+\sum_{i=M+1}^{N}\log
2\cosh\left(-i\left(\hat{m}_{2}+\eta_{i}\hat{m}_{2}^{I}+(1-\eta_{i})\hat{m}_{2}^{J}\right)\right)+$
$\displaystyle+\lambda_{S}\sum_{i=M+1}^{N}\log
2\cosh\left(-i\left(\eta_{i}\hat{u}_{2}^{I}+(1-\eta_{i})\hat{u}_{2}^{J}\right)\right).$
(123)

With the saddle point equations

$\displaystyle\hat{m}_{1}=i\beta\left(J_{1}n_{1}+J_{21}n_{2}\right),$
$\displaystyle\qquad\hat{n}_{1}=i\beta\left(J_{1}m_{1}+J_{12}m_{2}\right),$
(124) $\displaystyle\hat{m}_{2}=i\beta\left(J_{12}n_{1}+J_{2}n_{2}\right),$
$\displaystyle\qquad\hat{n}_{2}=i\beta\left(J_{21}m_{1}+J_{2}m_{2}\right),$
(125)
$\displaystyle\hat{m}_{1}^{I}=i\beta\left(J_{1}u_{1}^{I}+J_{21}u_{2}^{I}\right),$
$\displaystyle\qquad\hat{u}_{1}^{I}=i\beta\left(J_{1}m_{1}^{I}+J_{12}m_{2}^{I}\right),$
(126)
$\displaystyle\hat{m}_{1}^{J}=i\beta\left(J_{1}u_{1}^{J}+J_{21}u_{2}^{J}\right),$
$\displaystyle\qquad\hat{u}_{1}^{J}=i\beta\left(J_{1}m_{1}^{J}+J_{12}m_{2}^{J}\right),$
(127)
$\displaystyle\hat{m}_{2}^{I}=i\beta\left(J_{12}u_{1}^{I}+J_{2}u_{2}^{I}\right),$
$\displaystyle\qquad\hat{u}_{2}^{I}=i\beta\left(J_{21}m_{1}^{I}+J_{2}m_{2}^{I}\right),$
(128)
$\displaystyle\hat{m}_{2}^{J}=i\beta\left(J_{12}u_{1}^{J}+J_{2}u_{2}^{J}\right),$
$\displaystyle\qquad\hat{u}_{2}^{J}=i\beta\left(J_{21}m_{1}^{J}+J_{2}m_{2}^{J}\right),$
(129) $\displaystyle n_{1}$ $\displaystyle=$
$\displaystyle\tilde{\gamma}\tanh\left(-i\hat{n}_{1}\right),$ (130)
$\displaystyle n_{2}$ $\displaystyle=$
$\displaystyle(1-\tilde{\gamma})\tanh\left(-i\hat{n}_{2}\right),$ (131)
$\displaystyle m_{1}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{M}\tanh\left[-i\left(\hat{m}_{1}+\eta_{i}\hat{m}_{1}^{I}+(1-\eta_{i})\hat{m}_{1}^{J}\right)\right],$
(132) $\displaystyle m_{1}^{I}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{M}\eta_{i}\tanh\left[-i\left(\hat{m}_{1}+\eta_{i}\hat{m}_{1}^{I}+(1-\eta_{i})\hat{m}_{1}^{J}\right)\right],$
(133) $\displaystyle m_{1}^{J}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{M}(1-\eta_{i})\tanh\left[-i\left(\hat{m}_{1}+\eta_{i}\hat{m}_{1}^{I}+(1-\eta_{i})\hat{m}_{1}^{J}\right)\right],$
(134) $\displaystyle m_{2}$ $\displaystyle=$
$\displaystyle\sum_{i=M+1}^{N}\tanh\left[-i\left(\hat{m}_{2}+\eta_{i}\hat{m}_{2}^{I}+(1-\eta_{i})\hat{m}_{2}^{J}\right)\right],$
(135) $\displaystyle m_{2}^{I}$ $\displaystyle=$
$\displaystyle\sum_{i=M+1}^{N}\eta_{i}\tanh\left[-i\left(\hat{m}_{2}+\eta_{i}\hat{m}_{2}^{I}+(1-\eta_{i})\hat{m}_{2}^{J}\right)\right],$
(136) $\displaystyle m_{2}^{J}$ $\displaystyle=$
$\displaystyle\sum_{i=M+1}^{N}(1-\eta_{i})\tanh\left[-i\left(\hat{m}_{2}+\eta_{i}\hat{m}_{2}^{I}+(1-\eta_{i})\hat{m}_{2}^{J}\right)\right],$
(137) $\displaystyle u_{1}^{I}$ $\displaystyle=$
$\displaystyle\lambda_{S}\sum_{i=1}^{M}\eta_{i}\tanh\left[-i\left(\eta_{i}\hat{u}_{1}^{I}+(1-\eta_{i})\hat{u}_{1}^{J}\right)\right],$
(138) $\displaystyle u_{1}^{J}$ $\displaystyle=$
$\displaystyle\lambda_{S}\sum_{i=1}^{M}(1-\eta_{i})\tanh\left[-i\left(\eta_{i}\hat{u}_{1}^{I}+(1-\eta_{i})\hat{u}_{1}^{J}\right)\right],$
(139) $\displaystyle u_{2}^{I}$ $\displaystyle=$
$\displaystyle\lambda_{S}\sum_{i=M+1}^{N}\eta_{i}\tanh\left[-i\left(\eta_{i}\hat{u}_{2}^{I}+(1-\eta_{i})\hat{u}_{2}^{J}\right)\right],$
(140) $\displaystyle u_{2}^{J}$ $\displaystyle=$
$\displaystyle\lambda_{S}\sum_{i=M+1}^{N}(1-\eta_{i})\tanh\left[-i\left(\eta_{i}\hat{u}_{2}^{I}+(1-\eta_{i})\hat{u}_{2}^{J}\right)\right].$
(141)

Now we are able to take the derivative of $\log Z_{C}$ with respect to
$\lambda_{S}$:

$\displaystyle\frac{\partial}{\partial\lambda_{S}}\log Z_{C}=$
$\displaystyle\sum_{i=1}^{M}\log
2\cosh\left(-i\left(\eta_{i}\hat{u}_{1}^{I}+(1-\eta_{i})\hat{u}_{1}^{J}\right)\right)+$
$\displaystyle+\sum_{i=M+1}^{N}\log
2\cosh\left(-i\left(\eta_{i}\hat{u}_{2}^{I}+(1-\eta_{i})\hat{u}_{2}^{J}\right)\right).$
(142)

Taking $\lambda_{S}=0$, we notice that the parameters $u_{1}^{I}$,
$u_{1}^{J}$, $u_{2}^{I}$ and $u_{2}^{J}$ all are zero. Thus, in this case, the
expression for the order parameters become

$\displaystyle n_{1}$ $\displaystyle=$
$\displaystyle\tilde{\gamma}\tanh\left(-i\hat{n}_{1}\right),$ (143)
$\displaystyle n_{2}$ $\displaystyle=$
$\displaystyle(1-\tilde{\gamma})\tanh\left(-i\hat{n}_{2}\right),$ (144)
$\displaystyle m_{1}$ $\displaystyle=$
$\displaystyle\tilde{\gamma}\tanh\left(-i\hat{m}_{1}\right),$ (145)
$\displaystyle m_{2}$ $\displaystyle=$
$\displaystyle(1-\tilde{\gamma})\tanh\left(-i\hat{m}_{2}\right),$ (146)
$\displaystyle m_{1}^{I}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{M}\eta_{i}\tanh\left(-i\hat{m}_{1}\right),$ (147)
$\displaystyle m_{1}^{J}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{M}(1-\eta_{i})\tanh\left(-i\hat{m}_{1}\right),$
(148) $\displaystyle m_{2}^{I}$ $\displaystyle=$
$\displaystyle\sum_{i=M+1}^{N}\eta_{i}\tanh\left(-i\hat{m}_{2}\right),$ (149)
$\displaystyle m_{2}^{J}$ $\displaystyle=$
$\displaystyle\sum_{i=M+1}^{N}(1-\eta_{i})\tanh\left(-i\hat{m}_{2}\right),$
(150)

and we notice that $m_{1}$, $m_{2}$, $n_{1}$ and $n_{2}$ are the order
parameters of the full modular Ising model, and by using a similar notation we
used before for $\phi_{A}$, we can write this order parameters in terms of
same variable $\alpha$ that denotes the fraction of first $M$ elements that
belongs to the component $I$:

$\displaystyle m_{1}^{I}$ $\displaystyle=$ $\displaystyle\alpha m_{1},$ (152)
$\displaystyle m_{1}^{J}$ $\displaystyle=$ $\displaystyle(1-\alpha)m_{1},$
(153) $\displaystyle m_{2}^{I}$ $\displaystyle=$
$\displaystyle\bar{\alpha}m_{2},$ (154) $\displaystyle m_{2}^{J}$
$\displaystyle=$ $\displaystyle(1-\bar{\alpha})m_{2},$ (155)

where $m_{1}$, $m_{2}$, $n_{1}$ and $n_{2}$ are the order parameters for the
modular Ising model and, $\bar{\alpha}$ is, again, given by

$\bar{\alpha}=\frac{\gamma-\alpha\tilde{\gamma}}{1-\tilde{\gamma}}.$ (156)

Finally we have:

$\displaystyle\phi_{C}=$ $\displaystyle\alpha\tilde{\gamma}N\log
2\cosh\left[\beta\left(J_{1}\alpha
m_{1}+J_{12}\bar{\alpha}m_{2}\right)\right]+$
$\displaystyle+(1-\alpha)\tilde{\gamma}N\log
2\cosh\left[\beta\left(J_{1}(1-\alpha)m_{1}+J_{12}(1-\bar{\alpha})m_{2}\right)\right]+$
$\displaystyle+\bar{\alpha}(1-\tilde{\gamma})N\log
2\cosh\left[\beta\left(J_{21}\alpha
m_{1}+J_{2}\bar{\alpha}m_{2}\right)\right]+$
$\displaystyle+(1-\bar{\alpha})(1-\tilde{\gamma})N\log
2\cosh\left[\beta\left(J_{21}(1-\alpha)m_{1}+J_{2}(1-\bar{\alpha})m_{2}\right)\right].$
(157)

Defining the vectors

$\mathbf{m}_{I}=\begin{pmatrix}\alpha m_{1}\\\
\bar{\alpha}m_{2}\end{pmatrix},\quad\mathbf{m}_{J}=\begin{pmatrix}(1-\alpha)m_{1}\\\
(1-\bar{\alpha})m_{2}\end{pmatrix},\quad\mathbf{n}_{I}=\begin{pmatrix}\alpha
n_{1}\\\
\bar{\alpha}n_{2}\end{pmatrix},\quad\mathbf{n}_{J}=\begin{pmatrix}(1-\alpha)n_{1}\\\
(1-\bar{\alpha})n_{2}\end{pmatrix},$ (158)

we obtain equation (55).