Multilayer networks as embodied consciousness interactions. A formal model approach

Abstract

An algebraic interpretation of multigraph networks is introduced in relation to conscious experience, brain and body. These multigraphs have the ability to merge by an associative binary operator $\odot$, accounting for biological composition. We also study a mathematical formulation of splitting layers, resulting in a formal analysis of the transition from conscious to non-conscious activity. From this construction, we recover core structures for conscious experience, dynamical content and causal constraints that conscious interactions may impose. An important result is the prediction of structural topological changes after conscious interactions. These results may inspire further use of formal mathematics to describe and predict new features of conscious experience while aligning well with formal tries to mathematize phenomenology, phenomenological tradition and applications to artificial consciousness.

Appendix A

In this supplementary section, we briefly discuss our multilayer implementation in light of its algebraic structure through Category Theory and provide some algebraic definitions used in the main text. We first introduce some content on algebraic structures used in the paper, some formal definitions on Multilayer Networks and finally categorical concepts for a formal account of multilayer networks (see Signorelli et al. 2021b, Fong and Spivak 2019, Kivela et al. 2014 for other introductions).

1.1 Monoids A.1

We start by defining a monoid as a particular algebraic structure:

Definition 16

A monoid is a tuple $(M,\oplus ,\epsilon )$ where M is a set and for every $m,n,p \in M$:

• $\oplus$ is a total associative binary operation on M, that is

  $$ (m\oplus n)\oplus p=m\oplus (n\oplus p) $$

• $\epsilon$ is the identity for $\oplus$, that is

  $$ \epsilon \oplus m=m=m\oplus \epsilon $$

A monoid is said to be commutative if $\oplus$ is commutative, that is, if

$$ m\oplus n=n\oplus m $$

1.2 Tensor product of monoids A.2

We define the tensor product of (commutative) monoids starting from the well-known concept of cartesian product of sets: given two sets F and S their cartesian product is defined as

$$ F\times S=\{(f,s)/f\in F, s\in S\} $$

We can think of the sets of first and second dishes when ordering in a restaurant: F would be all available first dishes and S the second ones, then $F\times S$ represent all possible combinations one can order.

Now the cartesian product of two given monoids $(M_1,\ominus ,\epsilon _1)$ and $(M_2,\oslash ,\epsilon _2)$ is defined through the componentwise operation $\star$ given by

$$ (m,n)\star (m',n')=(m\ominus m',n\oslash n') $$

for $m,m'\in M_1$ and $n,n'\in M_2$. Now $M_1\times M_2$ with $\star$ is a two-fold monoid.

Finally, the tensor product of monoids, written $M_1\otimes {M_2}$, is the quotient of their cartesian product by the relations

$$ (m,n)\star (m',n)\sim (m\ominus m',n)\qquad (m,n)\star (m,n')\sim (m,n\oslash n') $$

$$ (\epsilon _1,n)\sim (m,\epsilon _2) $$

1.3 Partial monoids A.3

Definition 17

A partial monoid is a monoid $(M,\oplus ,\epsilon )$ where $\oplus$ is a partial operation. A partial monoid is said to be commutative if $\oplus$ has this property whenever defined.

1.4 Concepts on multilayer networks A.4

The following is a sketch of some of the concepts involved in the theory of Multilayer Networks belonging to Kivela et al. (2014), they are mentioned and used in the paper and given here in a more formal presentation. If a node u has different aspects $a_1,\ldots ,a_d$ in layers into the sets $(L_{1},\ldots ,L_{d})$ respectively, a widely used notation is:

$$(u,\textbf{a})\equiv (u,a_{1},\ldots ,a_{d})$$

We note here that in this paper we avoided this notation in favour of a more appropriate way to express the features we want to describe.

The set of edges is partitioned into intra-layer edges and inter-layer edges. Intra-layer edges correspond to the interactions inside layers (e.g. yellow and light blue edges, Fig. 1A) and belong to sets

$$E_{A}=\{((u,\textbf{a}),(v,\mathbf {a'}))\in E_{M}| \textbf{a}=\mathbf {a'}\}$$

1.5 Categorizing multilayers A.5

Category theory is a branch of highly abstract mathematics. One of the main features is its power to generalize relationships across different mathematical structures. This generalization is due to its general definition of objects as mathematical types, morphisms as maps or transformations from objects/types to other objects/types, and the composition operation, which may respect associative and identity law. It can be consulted (Maclane, 1965) as a source of the main categorical concepts.

In order to give a categorical interpretation of our multilayer networks of Section 2.2, we make use of the notions introduced in Baez et al. (2018). Following that reference, we consider a functor that produces many-coloured networks out of non-directed multigraphs.

Denote by $\mathcal {S}$ the permutation groupoid, that is, a skeleton of the groupoid of finite sets and bijections. $\mathcal {S}$ is a (strict) symmetric monoidal category.

Definition 18

A one-coloured network model is a lax symmetric monoidal functor $F:\mathcal {S}\rightarrow Mon$ where Mon is the category of monoids.

Example 19

Let MG(n) be the set of multigraphs on $\textbf{n}=\{1,\ldots ,n\}$. Then, a network model is a functor $MG:\mathcal {S}\rightarrow Mon$ with values $(MG(n),+)$ where $+$ is a multiset sum, that is, the addition of multiplicities of edges sharing the same vertices.

The category of network models over a fixed colour set $\mathcal {C}$ is denoted by $NetMod_{\mathcal {C}}$ (see Baez et al. 2018). $NetMod_{\mathcal {C}}$ is a symmetric monoidal category and therefore we can have tensor functors such as those introduced above.

Definition 20

A network model for multigraphs with coloured edges is a functor

$$ MG^{\otimes \mathcal {C}}:\mathcal {S}\rightarrow Mon $$

where $MG^{\otimes \mathcal {C}}(n)$ is a product of $\left| \mathcal {C}\right|$ copies of the monoid MG(n), being $\left| \mathcal {C}\right|$ the cardinal of the set $\mathcal {C}$.

It is crucial that, in these tensor models, a network is not only a $\mathcal {C}$-tuple of multigraphs: it may also be viewed as a multigraph with as many edges of each colour between any pair of distinct vertices as they appear in every layer. The fact that the functor $MG^{\otimes \mathcal {C}}$ is indeed a network model comes from the disjoint union operation:

$$ \begin{array}{c} MG^{\otimes \mathcal {C}}(n)\times MG^{\otimes \mathcal {C}}(m)\overset{\sqcup }{\longrightarrow }MG^{\otimes \mathcal {C}}(n+m)\end{array} $$

The particular kind of network model considered for the rotation multigraphs of Section 5 depends precisely on the rotation angle and, subsequently, a variable on time. However, it does not satisfy one of the requirements introduced in Baez et al. (2018), namely, being lax monoidal (see Baez et al. 2018). This is due to the fact that the operation analogous to $\sqcup$ in non-rotation models, turns out to be partial in the rotation ones (i.e. subject to the condition $\left| \alpha (t)\right| +\left| \beta (t)\right| \ge {{\pi }/{2}}$).

Definition 21

Given $t\in T$ and PMon the category of partial monoids, a network rotation model is a functor

$$ RMG^{t}:\mathcal {S}\rightarrow PMon $$

Example 22

A rotation network model for multigraphs with coloured edges is a functor

$$ [RMG^{t}]^{\otimes \mathcal {C}}:\mathcal {S}\rightarrow PMon $$

where $[RMG^{t}]^{\otimes \mathcal {C}}(n)$ is a product of $\left| \mathcal {C}\right|$ partial monochrome monoids $RMG^{t}(n)$ (recall that $\mathcal {C}$ can be a multiset and then $\left| \mathcal {C}\right|$ refers to the multiset cardinal).

1.6 Distributivity A.6

The $\odot$ operation is distributive over $\times$. That is, we obtain:

$$\begin{aligned} (\prod _{i=1}^{d_{1}}G_{i})\odot (\prod _{j=1}^{d_{2}}H_{j})=\prod _{i,j}(G_{i}\odot H_{j})=G_{1}\odot H_{1}\times \ldots \times G_{d_{1}}\odot H_{d_{2}} \end{aligned}$$

(1)

whenever we calculate over expressions of multilayers containing monochrome layers such as

$$ G_{1} \times \ldots \times G_{d_{1}} \in MG^{{c}_1}(n_{1}) \times \ldots \times MG^{{c}_{d_{1}}}(n_{d_{1}}) $$

$$ H_{1} \times \ldots \times H_{d_{2}} \in MG^{{c'}_1}(m_{1}) \times \ldots \times MG^{{c'}_{d_{2}}}(m_{d_{2}}) $$

The expression (Eq. 1) contains all combinations in the form $G_{i}\odot H_{j}$ for $i=1,\ldots ,d_{1}$ and $j=1,\ldots ,d_{2}$ and lives in

$$ MG^{{c}_1\otimes {c'}_{1}}(n_{1}+m_{1}-p_{11}) \times \ldots \times MG^{{c}_{d_{1}}\otimes {c'}_{d_{2}}}(n_{d_{1}}+m_{d_{2}}-p_{d_{1}d_{2}}) $$

where $p_{ij}=\left| V(G_{i})\cap V(H_{j})\right|$.

1.7 Inclusion-exclusion principle A.7

Counting the number of vertices into a graph

$$[G_{1},\alpha _{1}]\varvec{\odot }\ldots \varvec{\odot }[G_{k},\alpha _{k}]$$

means considering the known as Inclusion-Exclusion Principle in the form:

$$\begin{aligned} \sum _{1 \le {i_1} \le k} \left| V(G_{i_1})\right| \!-\! \sum _{1 \le {i_1}< {i_2} \le k} \left| V(G_{i_1}) \cap V(G_{i_2})\right| \!+\! \sum _{1 \le {i_1}< {i_2} < {i_3} \le k} \left| V(G_{i_1}) \cap V(G_{i_2})\cap V(G_{i_3})\right| \!-\!\ldots \end{aligned}$$

$$\begin{aligned} \ldots + (-1)^{k+1} \left| \bigcap _{i=1}^k V(G_{i}) \right| \end{aligned}$$

where k corresponds to the number of graphs.

1.8 Decomposition in biobrane A.8

The decomposition of graphs in Biobrane is determined by the equation:

$$ {\varvec{\odot }}_{j=1}^{m}[{G}_{j},{\alpha }_{j}]={\varvec{\cup }}_{j=1}^{m}[{H}_{j},{\beta }_{j}] $$

where we impose that $E(G_{j})=E(H_{j})$ for $j=1,\ldots m$ but $|V(H_{j})|=n$ and, moreover, $\left| {\beta }_{j}\right| +\left| \beta _{l}\right| < {\pi /2}$ for $j,l=1,\ldots ,m$.

${H_{j}}$ are then defined exactly as $G_{j}$ but adding an isolated vertice for every missing one in $G_{j}$ up to n. $[G_{j},\alpha _j]$ are the monochrome graphs in Biobrane from which $[G,\alpha ]$ was constructed and $[H_{j},\beta _{j}]$ are defined from them satisfying the following conditions:

• every $[H_{j},\beta _{j}]$ is monochromatic

• every $[H_{j},\beta _{j}]$ has n vertices

• m is the number of layers used to build and decompose $[G,\alpha ]$ by applications of $\varvec{\odot }$ and $\varvec{\cup }$ respectively

• $\left| \beta _{j}\right| +\left| \beta _{l}\right| < {\pi /2}$ for $j,l=1,\ldots ,m$

• $m_1+ \ldots +m_d=m$.