What is in a simplicial complex? A metaplex-based approach to its structure and dynamics

Citation: Miranda, M.; Estrada-Rodriguez, G.; Estrada, E. What Is in a Simplicial Complex? A Metaplex-Based Approach to Its Structure and Dynamics. Entropy 2023,25, 1599. https://doi.org/ 10.3390/e25121599 Academic Editors: Ginestra Bianconi, Rubén J. Sánchez-García, Anthony Baptista and Hanlin Sun Received: 5 October 2023 Revised: 24 November 2023 Accepted: 28 November 2023 Published: 29 November 2023 Copyright: © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). entropy Article What Is in a Simplicial Complex? A Metaplex-Based Approach to Its Structure and Dynamics Manuel Miranda 1, Gissell Estrada-Rodriguez 2and Ernesto Estrada 1,* 1Institute of Cross-Disciplinary Physics and Complex Systems, IFISC (UIB-CSIC), 07122 Palma de Mallorca, Spain; [email protected] 2Departament de Matemàtica, Universitat Politècnica de Catalunya, 08034 Barcelona, Spain; [email protected] *Correspondence: [email protected] Abstract: Geometric realization of simplicial complexes makes them a unique representation of complex systems. The existence of local continuous spaces at the simplices level with global discrete connectivity between simplices makes the analysis of dynamical systems on simplicial complexes a challenging problem. In this work, we provide some examples of complex systems in which this representation would be a more appropriate model of real-world phenomena. Here, we generalize the concept of metaplexes to embrace that of geometric simplicial complexes, which also includes the definition of dynamical systems on them. A metaplex is formed by regions of a continuous space of any dimension interconnected by sinks and sources that works controlled by discrete (graph) operators. The definition of simplicial metaplexes given here allows the description of the diffusion dynamics of this system in a way that solves the existing problems with previous models. We make a detailed analysis of the generalities and possible extensions of this model beyond simplicial complexes, e.g., from polytopal and cell complexes to manifold complexes, and apply it to a real-world simplicial complex representing the visual cortex of a macaque. Keywords: simplicial complex; geometric realization; metaplexes; diffusion; brain networks; higher-order networks 1. Introduction The complexities of many systems, generically known as complex systems, are manifested not only in their structures and behavior [ 1 , 2 ], but also in the difficulties of defining precisely what they are (see [ 3 ] and references therein) and, moreover, by the several ways of representing them [ 4 ]. Due to their networked nature, the use of graphs has been ubiquitous in representing complex systems [ 5 , 6 ]. However, other representations, including temporal networks [ 7 ], multiplexes and multilayers [ 8 , 9 ], hypergraphs [ 10 , 11 ], and simplicial complexes [ 12 – 14 ], have been claimed as (more) appropriate than simple graphs for (certain) classes of systems [ 15 ]. Certainly, the use of these representations constitutes an advancement in our understanding of the structure and dynamics of complex systems. But, unfortunately, their abuse may also represent a drawback to the necessary understanding of complex systems in the real world. Currently, we are living on the crest of a wave on the use of hypergraphs and simplicial complexes to study man-made and natural complex systems. Nowadays, the term “higherorder networks” has been coined to group these representations [ 16 – 22 ]. To motivate the necessity of using these higher-order structures let us consider the relations of coauthorship of scientific papers. These systems have been extensively studied as networks, representing authors and coauthorship as the nodes and edges of the graph, respectively [ 23 , 24 ]. However, sometimes coauthorship goes beyond the pairwise relations represented in a graph, and k -cliques (of different sizes) of coauthors may coexist in the same system [ 25 ]. This situation can be represented by a hypergraph in which pairs, triples, etc., are grouped Entropy 2023,25, 1599. https://doi.org/10.3390/e25121599 https://www.mdpi.com/journal/entropy Entropy 2023,25, 1599 2 of 22 together in hyperedges [ 10 ]. It is frequent in coauthorship networks that one author is a single author of a paper, as well as a coauthor of other papers with one, two, or more coauthors [ 25 ]. This is also illustrated by a hypergraph in which a single node would participate in hyperedges for different cardinalities. Some confusion may have emerged in the field of higher-order networks due to the fact that a hypergraph like the one described before, where the set of hyperedges is closed under inclusion, is also known as an abstract simplicial complex [ 16 ]; that is, in an abstract simplicial complex, every nonempty subset of a hyperedge is also a hyperedge. Therefore, some of the claims on the use of simplicial complexes for representing complex systems reduce to the particular case of specific hypergraphs, i.e., purely combinatorial objects, where the set of hyperedges is closed under inclusion. It seems that the use of simplicial complexes predates that of hypergraphs for studying complex systems. It was the British mathematician Ronald H. Atkin in as early as 1972 who proposed the abstract simplicial complex “as the vehicle for that sense of structure which is inherent in either the laws of physics or the behavior of social systems” [ 26 ] (see also [ 27 ]). Almost a decade later, another mathematician Stephen B. Seidman proposed the use of hypergraphs to “permit the study of structure induced by non-dyadic relationships” [ 28 ]. As resumed by Freeman and White [ 29 ], the work of Seidman showed that the hypergraphs “could be used to provide a similar–and perhaps simpler–representation of two mode data” than the one of Atkin. Nowadays, hypergraphs are a powerful tool to represent and analyze data (see, for instance, [30,31]). However, one important step forward was already made in Atkin’s 1972 paper when the author stated that “it is often helpful to think of a geometric realization of the complex” such that every simplex is a closed convex polyhedron in a suitable space [ 26 ]. In this sense, informally (see further for a formal definition), a simplicial complex is a space with a triangulation, that is, a topological space made of vertices, edges, triangles, tetrahedra, and higher-dimensional equivalents connected to each other by their edges, vertices, faces, and so on. The main distinctive characteristic of a geometric simplex relative to other discrete objects is that the first encloses a continuous space. For instance, we can represent the connection between two airports by an edge, which indicates that there are flights that depart from one of the airports and arrive at the other. We cannot trace the position of a flight between the two airports, such information simply does not exist in this representation. However, if we represent the trajectory of a flight in space, we are connecting the two vertices by a 1-simplex, which describes the continuous space traveled by a plane between the two airports. In the geometric simplex, we also kept that the relations are closed under inclusion. For instance, in a 2-simplex (triangle) every of the three edges (1-simplices) are also part of the simplex, as well as the vertices (0-simplices). Although the geometric simplicial complex could be very appealing, as claimed by Atkin [ 26 , 27 ] and more recently by others [ 21 , 22 , 25 , 32 – 34 ], a problem may arise when considering dynamics on them. If a dynamics simulating the spreading of information using an epidemiological model is defined on top of the geometric simplicial complex, we will find probabilities of getting infected for any region of the continuous space; that is, in a 2-simplex, not only the three individuals can be infected or receive information, but such information or probability of infection exists also for any point in the whole triangle delimited by the three individuals. This is hard to digest without invoking mystical ideas, and the problem is not limited to the propagation of information but to most dynamical systems that we can define on a geometric simplicial complex. There are, however, systems in which the use of geometric simplicial complexes could be necessary for understanding some of the dynamics occurring on them. These are the cases, for instance, of (i) protein residue networks, where the interaction between amino acids generates (hydrophobic and electrostatic) fields filling the inter-residue space; (ii) protein–protein interactions, where the proteins form complexes of individual proteins glued together by noncovalent interactions; (iii) landscape networks, where very close patches can allow the diffusion of species across the areas delimited by them; networks of Entropy 2023,25, 1599 3 of 22 fractures in rocks, where close enough fractures may allow the diffusion of material through a porous continuous space; neuronal networks, where volume transmission (see further) can spill over neurotransmitters to the extracellular space between neurons, among others. Here, we will focus on geometric simplicial complexes and will use the term simplicial complex for short. In the case of combinatorial relations, such as the ones between coauthors, which are closed under inclusion, we do not see any reason to call them a simplicial complex but a hypergraph as there is nothing geometric in such representation. 2. On the Problem of Representation We illustrate the representation problem of a complex system by means of the example of a neuronal system. For the sake of simplicity, we consider three neurons having synapses among them. There are two different types of transmission mechanisms through a pair of neurons. The first is the most common one and it is known as wiring transmission (WT) [ 35 ]. This refers to the mode of intercellular communication in which the existence of a virtual wire connecting the cell source of the signal (message) with the cell target of the signal exists. It is typical of the electrical synapses but also of the chemical ones. However, in the last case, a second type of inter-neuron transmission may occur: volume transmission (VT) [ 35 – 39 ]. It refers to the mode of intercellular communication that occurs through the extracellular fluid (ECF) and in the cerebrospinal fluid (CSF) of the brain; that is, during chemical synapses, amounts of neurotransmitters are spilled over to the ECF near the perisynaptic region, such that VT signals move from source to target cells via energy gradients leading to diffusion and convection. If we are interested in analyzing the WT between pairs of neurons in the system, we are in the presence of a scenario like the one illustrated in Figure 1a. In this scenario, it is enough to use a representation of the system as a (weighted, directed) graph. A (weighted) graph G=(V,E,φ,W) is formed by a set of vertices V={v0, . . . , vk} and a set of edges E=vi,vjvi,vj∈V [ 6 ]. Then, a set of edge weights W may represent a characteristic feature of the synaptic connection, which are mapped onto the edges by the mapping: φ:W→E . Note that the vertex index starts from 0 instead of 1. This choice is to make the vertex of the graph to follow the same index as the vertex of the simplicial complexes, which will be defined later. It is plausible that the synapses between neurons occur in a simultaneous way including not only pairwise interactions between neurons but also by triples, quadruples, etc., as illustrated in Figure 1b. In this scenario, the previous graph-theoretic representation is no longer appropriate. To capture these new types of interactions we need an extension of the graph known as hypergraph [ 40 ]. In the simple hypergraph H= (V , E) , V continues to be the set of nodes, but now, E is the set of hyperedges. A hyperedge is any subset of nodes in V. Thus, the set of hyperedges can be seen as the subset of the power set of V,E⊂ P(V). We remark that the hypergraph, like the graph, is a discrete system; that is, although we have drawn the hyperedges as colored contours that include the set of vertices in the hyperedge, they do not mean any kind of “physical” space. These hyperedges indicate only that a relation (binary in graphs or k -ary in hypergraphs) exists among the group of vertices. Therefore, in diffusive dynamics taking place on the hypergraph, we cannot find the diffusive particles on the (hyper)edges of the (hyper)graph. The dynamics occur as if the particles are annihilated at a given vertex and created at the others. So far we have mentioned only the WT among the neurons of the system, but when VT is considered the situation changes dramatically. Let us consider that some amount of the chemicals transmitted between two neurons is spilled over through the ECF in the perisynaptic region and that such neurotransmitter is recaptured by another neuron. We are no longer in the presence of a discrete relation between the three neurons. The intercellular region, bounded by the three neurons, forms a continuous space, which can be approximated by the triangle between the three neurons. VT is always accompanied by WT. This means that not all the neurotransmitters are spilled over to the perisynaptic region, but some of them are diffused through the wiring connection between the two neurons. Entropy 2023,25, 1599 4 of 22 Therefore, we should be able to trace back the concentration of the neurotransmitter through the path connecting the two neurons. In other words, we have a one-dimensional space connecting the pairs of neurons and a two-dimensional continuous space between the triple of neurons. Similarly, we can extend this idea to consider three-dimensional regions of continuous space. This new scenario cannot be appropriately described by the graph or by the hypergraph. A new type of representation, like the one illustrated in Figure 1c is needed. It is known as a simplex [41]. Figure 1. Schematic illustration of three different representations of the interaction between three neurons. ( a ) A graph representation of the system when wiring transmission (WT) alone is considered between pairs of neurons. ( b ) A hypergraph representation of the system when simultaneous WT may occur not only in a pairwise way but among k -ary groups. ( c ) In the case of chemical synapses, WT may coexist with volume transmission (VT) produced by the spillover of neurotransmitters to the extracellular space. In this case, the representation of the system as a simplex is more appropriate (see text for details). Formally, a simplex is defined as follows: Definition 1. A set of points s={a0,a1, . . . , ak} in Rn ( k≥n≥ 2) is geometrically independent if the set {a1−a0,a2−a0, . . . , ak−a0,} is linearly independent in Rn . By definition, any singleton is geometrically independent. Lemma 1. Let s={a0,a1, . . . , ak}⊆Rn be geometrically independent. Then, there is a unique k-dimensional hyperplane in Rnthat contains s. Remark 1. Any set s={a0,a1} is geometrically independent in Rn . A set s={a0,a1,a2}⊂Rn , for n≥ 2, is geometrically independent if and only if they are the vertices of a triangle. A set s={a0,a1,a2,a3}⊂Rn , for n≥ 3, is geometrically independent if and only if these points are the vertices of a tetrahedron. Generally, a set s={a0 , . . . , am} ⊂ Rn , for n≥m , is geometrically independent if these points are the vertices of a polytope. Entropy 2023,25, 1599 5 of 22 Definition 2. Let s ={a0,a1, . . . , ak}⊆Rnbe geometrically independent. Then, the set sk=(a0,a1, . . . , ak) =(k ∑ i=0 λiai: k ∑ i=0 λi=1, λi∈R,λi>0, i=0, . . . , k),(1) is called a k-simplex with vertices given by ai. Lemma 2. Let s={a0,a1, . . . , ak}⊆Rn be geometrically independent. Then, the simplex sk is a convex subset of Rn. A face τof a simplex skis a linear space spanned by proper subsets of vertices of sk. Of course, we can extend the scheme of the three neurons to an entire web of interneuron interactions. In this case, we will have a set of simplices which are mutually connected. This system is then known as a simplicial complex, and it is formally defined below. Definition 3. Let K be a collection of finitely many simplices. Then, K is a simplicial complex if the following conditions are satisfied: 1. If sk∈ K and τis a face of sk, then this implies that τ∈ K. 2. If sk,τ∈ K, and sk∩τ6=∅, then this implies that sk∩τis a common face of skand τ. These properties are important in the field of algebraic topology, as they allow the definition of a homology and use powerful tools such as the Hodge decomposition theorem [ 42 ]. Hence, the concept of abstract simplicial complex [ 43 ] (p. 153) was created to denote any family of sets with such property, without requiring any geometrical property. In the study of complex systems, it is frequent to use the term simplicial complex to denote hypergraphs whose hyperedges are closed by inclusion [ 44 ]. In such cases, these objects refer to abstract simplicial complexes where geometric properties are not explicitly considered. 3. What Is a Metaplex? In the representations of complex systems as graphs and hypergraphs, the emphasis is placed on the patterns of connectivity between the entities of the system. In these representations, the nodes are reduced to a simple, structureless, point. However, the internal structure of complex entities, e.g., neurons, individuals, landscape patches, etc., plays a fundamental role in the evolution of the dynamics of these systems. With this in mind, some of the current authors have developed the concept of “metaplex” [ 45 ]. From a graph-theoretic point of view, a metaplex can be seen as a graph in which each vertex has a continuous structure, instead of being a structureless point. Additionally, a metaplex can be seen from the perspective of Differential Geometry as a collection of domains in which there are different regions that are coupled to other domains in the collection. We now proceed to define this concept in a rigorous way. Definition 4. A metaplex is a 5-tuple Υ= (V , E , ω , I , F) , where (V , E) is a graph, ω= {Ωi|i= 0, . . . , k} is a collection of domains, Ωi⊂Rni , where ni∈N for each i , I:V→ω and F={Fij |i , j= 0, . . . , k ; i6=j} are bounded analytic maps between these domains. This is, for 0≤i6=j≤k, Fij :Ωi→Ωj,kFijk∞<∞. Here, the underlying graph of the metaplex is the set (V= [ 0, . . . , k] , E={[i , j]:Fij 6= ∅}) . Then, the tuple (ω , I) consists of the set of domains (Ωi) without repetitions, and I assigns to each node i∈[ 0, . . . , k] the element on the set ω corresponding to the original Ωi . We have modified here the previous definition of metaplex [ 45 ] to remark on an important topological characteristic of this mathematical object: sinks and sources. Entropy 2023,25, 1599 6 of 22 Definition 5. Let Υ= (V , E , ω , I , F) be a metaplex. Then, for each 0 ≤i , j≤k we call dom(Fij)⊆Ωi a sink of the domain Ωi connected to Im(Fij)⊆Ωj , which is a source of the domain Ωj. These pairwise connected regions relate the different domains, coupling the two (continuous) spaces in a similar way as edges relate the discrete nodes in a graph. 4. Diffusion on Graphs, Simplicial Complexes and Metaplexes 4.1. Diffusion on Graphs Diffusive processes are ubiquitous in man-made and natural complex systems. Due to the networked nature of these systems, it is frequent to analyze the diffusion dynamics on a graph [ 46 ]. For the sake of completion, we state here the main aspects of the diffusion dynamics on simple graphs. Let L=K−A be the Laplacian matrix of the graph, where K is a diagonal matrix of vertex degrees and A is the adjacency matrix of the graph. Notice that if ∇ is the vertex-edge oriented incidence matrix (gradient) of the graph, then L=∇∇T . Then, the change in the concentration of an item at the nodes of the graph is accounted for by the vector ˙ x(t)and described by the diffusion equation on graphs: ˙ x(t)=−γLx(t),x(0)=x0, (2) where γ is the diffusivity coefficient, hereafter taken as unity. The solution of this equation is given by the heat kernel e−tL , such that x(t)=e−tLx0 . Let µ0≤µ1≤ ··· ≤ µk be the eigenvalues of L and ψj the column eigenvector of L corresponding to µj . Then, the solution of the diffusion equation on the graph can be written as: x(t)=e−tµ0ψT 0x0ψ0+e−tµ1ψT 1x0ψ1+···+e−tµkψT kx0ψk. (3) It is easy to see that µ0= 0 and that its multiplicity are equal to the number of connected components of the graph. Therefore, in a connected graph, when t→∞: x(t)→ψT 0x0ψ0=1 k k ∑ i=0 x0(i). (4) This characteristic feature of diffusion is very important for the further analysis of these dynamics on simplicial complexes; that is, standard diffusion is a conservative consensus process in which the entities of the system end up at an equilibrium state, which is the average of their initial states. The rate of convergence depends on the algebraic connectivity of the graph: µ1. An example is provided in Figure 2. (a) 0 100 200 300 400 Time 0 0.2 0.4 0.6 0.8 1 Time evolution of the density node 1 node 2 node 3 node 4 node 5 (b) Figure 2. Illustration of a simple graph ( a ) and a diffusion process taken place on it with a random initial condition (b). Entropy 2023,25, 1599 7 of 22 4.2. Diffusion on Simplicial Complexes A way to consider diffusive dynamics on a simplicial complex is to represent it by means of its incidence matrices Bm for 1 ≤m≤M . Here, M is the dimension of the simplicial complex, namely the highest dimension among the simplices composing the complex. The matrix B1 is the usual incidence matrix of a graph, with dimension equal to the number of nodes times the number of edges, which is constructed as follows. Let i<j be two nodes of the simplicial complex so that the edge [i , j] is also in the simplicial complex. Then, (B1)i,[i,j]=− 1, (B1)j,[i,j]= 1 and (B1)l,[i,j]= 0 for l any other node in the simplicial complex. Note that the choice of the sign depends on the labels assigned to the nodes, so the same graph may be represented by different incidence matrices. In general, the incidence matrix of order m , Bm , has a dimension equal to the number of (m− 1 ) -simplices in the simplicial complex multiplied by the number of m -simplices. The entry that corresponds to the (m− 1 ) -simplex [i1 , . . . , im−1] and the m -simplex [i1 , . . . , im−1 , j] is equal to the sign of the permutation that orders the set {i1 , . . . , im−1 , j} into an increasing set. Then, we can define the m -th order Laplacian as Lm=B> mBm+Bm+1B> m+1 for 0 ≤ m≤M , assuming the notation B0=∅=BM+1 . The case L0=B> 1B1 is the usual graph Laplacian. These m -th order Laplacians act over the m -dimensional simplices in the complex that are connected via m− 1 or m+ 1-dimensional simplices. Thus, we can define the total Laplacian L as the block diagonal matrix consisting of the m -th Laplacians, L = (Lm)K m=0 , which acts over vectors that take values on each of the simplices forming the complex. These operators were studied in [ 47 ], where the authors remarked on some relevant properties, such as the positive semidefiniteness, but also showed several problems. The first one is that the number of repetitions of the zero eigenvalues of the m -th Laplacian equals the Betti number βm . For the usual Laplacian L0 , the associated Betti number β0 equals the number of connected components in the graph, thus assuring the existence of a non-trivial steady state for the equation ˙ x(t) = −L0x(t) . On the other hand, there are simplicial complexes for which βm= 0 for some m , which means that the equation ˙ x(t) = −Lmx(t) only accepts the trivial solution x= 0 for large times whatever is the initial condition (see Figure 3). This problem was patched in the same article [ 47 ] by proposing the use of L−λ1v1vT 1 as diffusion operator, where λ1 is the first non-zero eigenvalue and v1its associated eigenvector. Nevertheless, modeling diffusion in this way gives rise to other problems. For instance, even for positive initial conditions, the system may evolve by taking negative values; that is, there are negative concentrations of the entities of the system at certain times, which should not happen in a model of a diffusive process. For instance, in the simplex shown in Figure 4, we ran the equation ˙ x(t) = −Lx(t) with initial condition u( 0 ) = u0= 1 in the node V1 , in the edge ( 1, 2 ) and in the triangle ( 1, 2, 3 ) , and 0 elsewhere. Figure 3shows the results, where it can be observed that the concentration in the edge ( 2, 3 ) and in the triangle ( 3, 4, 5 ) take negative values at the beginning of the simulation. Also worrying is the fact that there is not a global steady state for diffusion in the simplicial complex, but the nodes reach an independent consensus state from the steady states reached by edges and triangles, which is equal to 0 in this case. This should not happen in connected simplicial complexes like the one studied here. Again, we emphasize that in the standard diffusion process, the steady state is a global consensus of the system given by the average of the initial condition. If we were modeling the diffusion of temperature in the previous example, the outcome would be that the system arrives at a state in which one part has a temperature and another part has a different one, never reaching a common temperature. Entropy 2023,25, 1599 8 of 22 Figure 3. Evolution of the diffusion equation on the simplicial complex Figure 4using operator L. Figure 4. Example of the simplicial complex used by [47]. Moreover, the higher-order Laplacians only reflect changes between simplices of the same order, so the values of the concentration over the simplices of higher and/or lower order connected to them do not affect the value of the concentration of the original simplex. As an example, consider two 0 simplices (nodes) i and j connected by the 1-simplex (link) l= [i , j] . Let x= (xi , xj , xl) be a vector of concentrations on this simplicial complex satisfying equation ˙ x(t) = − L x(t) . Then, changes in the values of xi produce changes in the values of xj thanks to the dynamical equation, but will not affect xl in any way, although the node iis connected to the link land the changes in node jhappen through the link l. Entropy 2023,25, 1599 9 of 22 4.3. Diffusion on Metaplexes To describe a diffusion on a metaplex, let u(x , t) be the density of the diffusive particle at time t for x in a domain in Rn . Then, a dynamical system on a metaplex [ 45 ] is defined as follows. Definition 6. A dynamical system on a metaplex is a tuple (H={Hi:L2(Ωi)→L2(Ωi)} , T={Tij :L2(dom(Fij)) →L2(Im(Fij))}) such that, for any u0= ((u0)i)k i=0∈(L2(Ωi))k i=0 , the initial value problem ∂tui(t) = Hi(ui(t)),ui(0) = (u0)iis well-posed. Each of the elements in the tuple is related to one of the continuous or discrete points of view that metaplexes join together. Each of the operators in H is a continuous differential operator describing the process on the space Ωi , such as the Laplacian. On the other hand, T can be seen as the matrix representing a discrete operator describing the process among the nodes Ωi, such as the graph Laplacian; thus, it is compact and linear. We can see the previous definition as a system of coupled differential equations for the density ui(x , t) in the node vi∈V , as in [ 45 ], provided that we extend all functions Fij as zero outside of the sink–source, ∂tui(x,t) = Hi(ui(x,t)) − k ∑ j=0 Tij(ui(x,t)) + k ∑ j=0 Tji(uj(F−1 ji (x),t)),x∈Ωi. (5) The conditions on the sets of operators H and T allow us to know that analytic solutions should exist by the Cauchy–Kovalevski theorem [48]. These definitions allow extending both the concept of dynamical systems to graphs and to continuous domains. A dynamical system on a graph is equivalent to a dynamical system on a metaplex where all the domains Ωi are points, Fij are the weights of the edges, and the dynamical system consists of Hi being the identity on the points with Tij being the entries of the matrix operator on the graph. The case of a dynamical system on a continuous domain is even simpler, as it would consist only of one domain Ω1=Ω with operator H1=H and, as there are no other domains to which it is connected, there are neither Fij nor Tij. Moreover, it is important to note some differences that arise when combining these two points of view. First, it is the change between T and a transition matrix in a graph due to the sink–source relation. Similarly to the adjacency matrix of the underlying graph of the metaplex, we can construct a transition matrix related to the dynamical system with T . This matrix has as i , j -entry, i6=j , equal to the sum of the values Tij in the i , j -source of the domain Ωj , while the diagonal entries are equal to the negative of the sum of all the values Tij in all the sinks of the domain Ωi . Nevertheless, there are different situations that lead to the same transition matrix (see Figure 5). For example, consider two domains connected by identical sink–sources, which would lead to a transition matrix equal to the graph Laplacian of two nodes simply connected. This is the same no matter whether the sink and source for each domain are located in the same place or distant from each other, but this leads to different behavior (see [ 45 ] for examples on this difference). Thus, not only the dynamical object T is necessary to obtain the behavior of the dynamical process, but also the topology generated by (ω,F). Second, there exists the possibility that different domains may have different dimensions. Thus, using a metaplex to model classical diffusion leads to the use of the Laplacian in each domain Ωi , but such Laplacians do not behave equally if they are in a 1-dimensional space or in a 3-dimensional one, so the results obtained by using metaplexes are more complex than those obtained by the union of the partial results in each of the considered spaces. For instance, in the field of (continuous) partial differential equations, there are articles showing how mixing domains of different dimensions may lead to different results. An example of this situation can be found in [ 49 ], where it is shown that the differential operator arising from the diffusion Laplacian when “stretching” a high-dimensional do- Entropy 2023,25, 1599 16 of 22 diffusion using the simplicial metaplex always shows positive values of the density and converges to the consensus state, as theoretically predicted. Another important difference between the models is that, while the diffusion on the simplicial metaplex is conservative, the process ruled by the higher-order Laplacian is not, and total density in the steady state is not the same as the sum of densities in the initial state. Finally, we observe that the convergence time is much longer in the case of the simplicial metaplex, as the process realistically performs continuous diffusion dynamics inside the simplices and a discrete inter-simplex one. Figure 10. Each line represents the evolution of the concentration of neurotransmitters inside each of the macaque metaplex elements. Figure 11. Spatial distribution of the concentration of neurotransmitters along two triangles around the areas 7a, FEF, 46, and DP in the macaque visual cortex after 10,000 time steps. Entropy 2023,25, 1599 17 of 22 Figure 12. Spatial distribution of the concentration of neurotransmitters along two triangles around the areas MSTd, 46, FST, PITd and FEF in the macaque visual cortex after 10,000 timesteps. 0 100 200 300 400 500 600 700 800 900 1000 Time -1 -0.5 0 0.5 1 1.5 2 Density Node 1 Node 4 Edge 6-27 Triangle 1-2-3 Node 2 Node 5 Edge 6-29 Triangle 1-2-5 Node 3 Edge 6-25 Edge 6-30 Triangle 1-6-24 0 100 200 300 400 500 600 700 800 900 1000 Time -1 -0.5 0 0.5 1 1.5 2 Density Node 1 Node 2 Node 3 Node 4 Node 5 Edge 6-25 Edge 6-27 Edge 6-29 Edge 6-30 Triangle 1-2-3 Triangle 1-2-5 Triangle 1-6-24 Figure 13. Time evolution of the density of diffusive particles on each of the simplices in the macaque visual cortex. The left figure shows the results of the evolution under the higher-order Laplacian in the simplicial complex, while the right figure plots the evolution using the simplicial metaplexes presented in this article. Lower figures show the density along selected nodes, edges, and triangles. Note that both left figures present non-positive values of the concentration. Another fundamental difference between the way in which higher-order Laplacian and the current approach treat diffusion on simplices is manifested in the following. The diffusion based on higher-order Laplacians treats the whole space inside the simplices as a “patch” in which the concentration of diffusive particles is exactly the same at every part of this space; that is, if we visualize the concentration inside a 2-simplex using this Entropy 2023,25, 1599 18 of 22 approach we will see a homogeneous color across the whole triangle (the same for a line). As mentioned before, the current approach gives a much more realistic picture in which the concentrations are not distributed homogeneously across the simplices, as we can see in the previous figures. Let us now focus on the areas V2, V3, V3A, V4, and V4A (see Figure 14), which have been recently identified as “more active for stimuli containing figures compared to ground, regardless of whether figures were defined by texture, motion, luminance, or disparity” [ 58 ]. As Area V4A is not present in the network from [ 61 ], we changed it to Area V4t, which is a nearby area, so that the results from this proof of concept application should be comparable. In Figure 14, we can observe that the potential presence of diffusive particles is not equal around all 5 regions. In particular, diffusive particles are more concentrated in the communication between Regions V3, V4, and V4t. This suggests that volume transmission could play a relevant role in the coactivation of these three areas under certain stimuli. Figure 14. Spatial distribution of the concentration of neurotransmitters in and around the areas V2, V3, V3a, V4, and V4t of the macaque visual cortex after 7500 time steps. Although at the center of the 2-simplex V2, V3, and V3A there is significantly less concentration of the diffusive particles than in the regions close to the 1-simplices, the concentration is still relatively high, as there is about 20% less concentration at the center than in the borders. However, there are situations in which the 2-simplices are almost empty. This can be observed, for instance, in Figure 15 in which we illustrate several triangles in the macaque metaplex where the concentration of the diffusive particles is very low almost everywhere in the 2-simplices, but particularly at their central parts. The first useful insight of this result is that we can use the current method in a self-consistent way; that is, we can start by considering every triangle in the network as a 2simplex. However, after performing diffusive dynamics in this simplicial metaplex, maybe using a lower resolution to guarantee faster convergence, we can detect such triangles which are almost empty. Then, using a better resolution, we can perform further diffusive dynamics by considering that such triangles are not 2-simplices but holes. Of course, this detection of “holes” is more far reaching than its simple use in improving computational efficiency. For instance, it can be used in the framework of topological data analysis (TDA) [ 62 , 63 ], where there is an interest in finding persistent holes in data [12,64–67] . Even more, the current approach can be extended to detect holes not only on simplicial complexes but on more general polytopal or CW complexes, which is an interesting line of research nowadays [54]. Entropy 2023,25, 1599 19 of 22 Figure 15. Spatial distribution of the concentration in 8 triangles of the macaque metaplex after 10,000 timesteps. Secondly, the results obtained through this toy model approach can also be interpreted as indicating the importance of these two methods of transmitting information (VT and WT). Starting with a homogeneously diffusing condition, the structure of the SC appears to favor higher concentrations of neurotransmitters in certain triangles. This suggests that VT may play a crucial role in the connection between such triads of brain regions. Conversely, empty triangles may indicate that considering only WT would be the relevant means of transmitting information between those neurons. 7. Conclusions To advance our understanding of complex systems, we need the appropriate level of sophistication in their representation. Higher-order representation is an important step forward in this direction. In particular, simplicial complexes allow us to capture unique features that combine the geometric and topological features of some complex systems. However, this representation is not appropriate for any kind of complex system, particularly when what we are interested in is the dynamics taking place on it. In those cases where the simplicial complex representation is needed, the proper existence of local continuous spaces interconnected by discrete relations challenges our models to describe dynamics on them. We have considered here an extension of the concept of metaplexes previously defined to account for geometric simplicial complexes and (diffusive) dynamics on them. Using these simplicial metaplexes we have solved the problem of diffusion on these representations of complex systems and resolved some of the problems found with previous models. The diffusion model on simplicial metaplexes allows us to uniquely trace the concentration of diffusive particles across the continuous spaces of the simplices in any dimension while maintaining the discrete navigation between the simplices. The formal definition of simplicial metaplexes and of dynamical systems on them allow several further avenues for the study of complex systems. For instance, although not limited to them we can mention: (i) extension to other dynamics, e.g., synchronization, reaction–diffusion; (ii) extension to other types of continuous spaces beyond simplices, e.g., polytopal and CW complexes, manifolds, etc.; (iii) changing the sinks and sources from being the common faces between simplices to be particular regions inside them, allowing transitions between simplices of different dimensions. Author Contributions: Conceptualization E.E., writing E.E. with inputs from M.M. and G.E.-R.; methodology, E.E., M.M. and G.E.-R.; simulations, M.M.; review and editing, E.E., M.M. and G.E.-R.; software, M.M. and G.E.-R. All authors have read and agreed to the published version of the manuscript. Entropy 2023,25, 1599 20 of 22 Funding: M.M. and E.E. acknowledge support from Project OLGRA (PID2019-107603GB-I00) funded by the Spanish Ministry of Science and Innovation, as well as by the Maria de Maeztu project CEX2021001164-M funded by the MCIN/AEI/10.13039/501100011033. G.E.-R acknowledges support from the project macroKNIGHTs (PID2022-143012NA-100) funded by the Spanish Ministry of Science and Innovation. Data Availability Statement: The codes used to generate the results in this article can be found in [ 59 ]. More codes generated or used during this study are available from the corresponding author upon request. Conflicts of Interest: The authors declare no conflict of interest. References 1. Siegenfeld, A.F.; Bar-Yam, Y. An introduction to complex systems science and its applications. Complexity 2020 ,2020, 6105872. [CrossRef] 2. Thurner, S.; Hanel, R.; Klimek, P. Introduction to the Theory of Complex Systems; Oxford University Press: Oxford, UK, 2018. 3. Estrada, E. 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