Network description of dynamical systems: The clustering coefficent

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Title:

“Network description of dynamical systems: The clustering coefficient”

AUTHOR:

Àlex Arcas Cuerda

Master’s Thesis

Master’s degree in Physics of Complex Systems at the

UNIVERSITAT DE LES ILLES BALEARS

Academic year 2018-2019

Date: 18/09/2019

UIB Master’s Thesis Supervisor: Dr. Emilio Hernández-García

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Abstract

In this MSc thesis, we focus on studying the different types of clustering for a flow network. A flow network is a network representation of a dynamical system. The methodology to obtain these kind of networks is relatively new and it still has many aspects to study.

The motivation of this work was to verify a hypothesis dropped in Ref.[4] which theorized about the meaning of the undirected clustering for a flow network. More specifically, it was hypothesized that the undirected clustering of a flow network characterizes the stable manifolds of the dynamical system underlying it.

First, we expose the different concepts of the theoretical framework involved: elements of dynamical systems theory and of network theory, mainly the different definitions of clustering. At the successive sections we show the results of our computations for the Lorenz model. In the process, we prove the motivation hypothesis to be incorrect. We continue by studying the different types of directed clustering that constitute the undirected clustering, connecting them to the properties of the dynamical system. Finally, we discuss the possible meanings of the quantities studied.

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Acknowledgements

Before entering the paper, I would like to dedicate some words to those who had helped me to the process of elaboration of this paper.

First, special thanks to my supervisor Emilio. Despite his busy schedule we have always found time to meet and discuss about all the different matters arising during the whole process. Being always really clear and guiding me with especial care. Even in those days when he was travelling, he took the time to reply my emails or answer my doubts. We also want to thanks Manuel Mat´ıas who contributed with a really useful database from Ref.[8].

A database which we have used through the whole work.

Not only in the process of writing the paper but during the whole year, my classmates have helped to go through all the course. Essentially, the year would not have been as wonderful without them. Then they have also enlighten me with some smart ways to avoid some computational problems I had while doing the computations of the paper.

Finally, as a son, I must thank my parents and family as I have arrived here because of them and thanks to their unconditional help whenever I have needed.

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1 Introduction

Fluid transport has been a field widely studied in which great efforts have been committed since the start of modern science. Directly related to the study of fluid motion is the study of dynamical systems, another huge field of science in which many big names have focused. Both fields have used and worked on many of the same ideas such as non-linear dynamics or chaos theory. Despite their age, these are fields of science that still very active and do not seem to have an end in a close future as many open questions remain[1]-[2].

In such a task as it is the study of system dynamics, tons of different approaches have been used.

Most of them try to improve our knowledge about the different systems or to solve the different questions arising. In the past few years though, a completely new methodology was proposed, the flow network approach. This methodology proposes to construct a network from the fluid dynamics and apply network theory on it. Finally, map the obtained results with some structures or properties of the original system.

This methodology has been recently used to analyse important fluid properties such as fluid transport and fluid mixing with great results, relating network theory concepts with dynamical systems ones[3]-[5].

The process of constructing a flow network can actually be generalized to an arbitrary dynamical system. In general, the Perron-Frobenius operator is used to make a discrete approximation of the motion of the system. This operator maps the motion of a set of virtual particles from one initial volume to a final one. This characterization with its discrete scheme is, presumably, well fitted for a computational approach as many other techniques in science nowadays[6].

A common field of interest and research in dynamical systems theory that can be used to approximate turbulent flows is chaos theory. Chaos has been a really prolific field for many research groups and many efforts have been put into it. Efforts and works that have resulted in many innovative ideas that had helped to understand a vast range of systems. One of the most important works in chaos theory was the paper written by Lorenz, Ref.[7]. This paper had a great impact on the field becoming probably the most iconic one. The paper dealed with a dynamical system that modeled the atmospheric dynamics hence it was directly realted to the study of fluid motion. This particular system even got really popular in the mass media associating what they called thebutterfly effect to much quotidian stuff. Understanding such systems have been a great headache for the science community and new approaches such as the flow network one are always welcomed[7]-[9].

The other big field from which the flow network methodology draws ideas is network theory or graph theory. Graph theory is an old field in mathematics and it has been studied for a while although it is not until this past half a century that it has had a boost. As a direct consequence of the extremelyconnected society in which we are living, many systems tractable as networks have arisen. Social networks, internet networks, transport networks, brain networks and communication networks are just some examples of networks that had been computed and analysed in the network theory framework. While doing so, many struggles on understanding the principles of network theory have been taken and as a result, network theory has improved its richness in concepts, methodologies and applications[10]-[11].

In this work, we are trying to specifically analyse the clustering of a flow network and understand what it means for the respective dynamical system. The clustering is a widely used concept in network theory that gives an idea of the connectivity of a network. As the flow network is intrinsically directed, it makes sense to use all the tools we have from network theory for these kinds of networks. Actually, in Ref.[12], four different type of directed clusterings are identified, defined and discussed. Then, the main part of this paper is focused on studying these new directed clusterings. Before doing so, we study the undirected clustering as it is the most common clustering for a network. In fact, the undirected clustering encloses all the directed clusterings. Studying the directed clustering by itself is then, the best way to break down a bigger problem.

Previous steps have already been taken in this path of work. As an example, the cyclic clustering, one type of directed clustering, has already been studied and mapped to such an important concept in dynamical systems as periodic orbits are. But not only the cyclic clustering has already been proved to be a prolific study, the degree, the betweenness and the shortest paths are network concepts that had also been mapped from the flow network construction to the original dynamical system[3]-[5].

In the course of our study, we chose the Lorenz Attractor to do our computational experiments but any other dynamical system could be chosen. If we did so is because this is a chaotic system that includes a wide range of exotic behaviours. Some of which we expect to be caught by the directed clustering in some manner. If this happens, we would have found a new tool to address these kind of systems.

The start of this work came from the hypothesis dropped in Ref.[4] at the ‘Further Extensions’ section.

In the course of this work we proved this hypothesis to be wrong. But as we just commented, we did

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not stop at proving it wrong and continued to study the types of directed clustering. In the process, we extracted some interesting properties characterising the different directed clusterings as well as some understanding of how they behave. Then, we organized the paper in three main sections. Sec.2 in which we give the theoretical framework to understand the posterior work. Sec.3 in which we show the results we have extracted, then compare the results on the cyclic clustering with the ones that had already been done in previous literature and comment all the new results we obtained for the other types of clustering (undirected, out, mid and in clustering). Finally, Sec.4 where we sum up all the relevant results and where we do a brief discussion about their possible meanings and usefulness.

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2 Theoretical Framework

In order to understand the the flow network construction methodology and the clustering measures we are taking on it, we need some previous knowledge of network theory. Due to this, before beginning with the applications, we are giving some necessary concepts to be able to go through the paper. Although most of the ideas we are presenting are advanced concepts in their respective fields, we are not entering in too much detail if not needed. For further reading in any of the presented topics refer to the bibliography.

This section was divided in two subsections, one for the flow network methodology and another for the different definitions of clustering.

2.1 Flow network construction

In this subsection, we go through the necessary steps to construct the flow network. We start from a set of equations for a continous dynamical system and end with a discrete transport network. Once we have the network, we explain a transformation we used to obtain the backwards flow network, a network we are using in the applications[3].

2.1.1 Nodes

First thing to realize is that the dynamics of the system and the network we want to build from it have different structures. The equations of the dynamical system determine a continuous flow of particles while a network is composed of discrete elements and the connections between them. In the network theory framework these elements are called nodes and the connections between them links or edges.

In order to get from one to the other, first thing we do is discretizing our space. To do so, we discretize the space in small volumes that are going to be our network nodes. It is important to realize that in doing so, we are implicitly introducing a coarse graining of the dynamics. This is going to be of a high relevance while analysing the results. When referring to the network, we are not analysing the trajectory of one initial condition but a set of them.

In this work where we are studying a three-dimensional system, we are discretizing the space in cubic blocks of equal volume where each one represents a node of the network. Then, if we discretize our space to study in{Bi; i= 1, ..., N}blocks, our network is going to containN nodes, each one referring to the respective volume in the space.

The discretization does not necessarily contain equal cubic volumes. In fact, an arbitrary set of sizes and shapes can be freely chosen. Taking the cubes exactly equal in shape and size is mainly to simplify posterior discussions. If all the cubes are equal, all the nodes contain the same set of initial conditions being only distinguishable by their positions. For our purposes, the best would be to have blocks as small as possible and with as many initial particles as possible.

As a final note for network theorists, we have to remember that each node in the network has an attribute fixing its position in the three dimensional space. Then our network is embedded into an Euclidean space. This is important to remember as usual networks do not have this characteristic.

2.1.2 Links and weights

Next step is connecting our network nodes with links. In doing so, we first choose weighted directed links as it is the most natural choice. The resulting network would be a directed weighted network. Choosing the weighted links also enables us to compare the significance or strength of different particles paths.

In this paper though, we just work with the unweighted version of the network. A version we get after applying a transformation we are explaining in the next section.

The idea behind the links construction is to follow the trajectories of ideal particles during a certain amount of time. To be consistent with the construction of our equal volume nodes, we follow the trajectories of an equal number of particles starting from each block of space. This sets up the initial profile of particle densities homogeneous. If we do not start with this homogeneous initial conditions, the final network would have a bias which does not interest us. Then, we integrate all the trajectories for a timeτsaving the initial and final position of each particle. The specific path that a particle follows is irrelevant for our construction.

This construction can be expressed formally using a flow map Φ^τ_t₀ such that

~x(t0+τ) = Φ^τ_t₀(x~0), (1)

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where we are integrating the trajectory of a particle at an initial position x~0 and initial time t0 over a velocity field v(~x, t) during a time τ. The velocity field is given by the dynamical system motion equations ˙~x=v(~x, t), which is seen from a Lagrangian point of view [1].

As we are actually integrating a set of initial conditions hence following a set of particle trajectories, we have to apply the map to our volume blocks instead of individual particles. The result can be written as

A(t0+τ) = Φ^τ_t₀(A). (2)

Eq.2 contains the information of all the final and initial positions of the particles departed from our volume blocks. This information is all we need to construct our flow network.

2.1.3 Network adjacency matrix

Once we have applied the flow map to the volume blocks{Bi}, we get the flow of particles between them.

In other words, the flow of particles between nodes in our network. The flow map applied to all the blocks can be explicitly characterized by the Perron-Frobenius transport operator from fluid dynamics.

This operator defines the time evolution of particle densities in the space. If we are presenting the Perron-Frobenius operator is because it can be directly interpreted as a network adjacency matrix.

There are many approaches to approximate the Perron-Frobenius operator. One of them is to inter- pret the operator in a probabilistic fashion. This interpretation is useful for us as it is easy to compute computationally. In this interpretation, our operator P(t₀, τ)_ij can be understood as, the probability of a particle starting from an uniformly distributed position inB_i at timet₀ to flow for a timeτ and end insideB_j (see Fig.1). We can approximate this probability mathematically such that

P(t₀, τ)_ij ≈n^o of part. fromB_i toB_j Ni

, (3)

where Ni is the number of particles leaving Bi at time t0. This expression for P(t0, τ)ij is only an approximation for the Perron-Frobenius operator. To be exact, we have to be in the limit of Ni → ∞ and to have the particles uniformly distributed across all B_i. Using Eq.(3), we can also define the probability of a particle being at a positionj such that,p_j(t₀+τ) =P

ip_i(t₀)P(t₀, τ)_ij.

Figure 1: Sketch for the adjacency matrix construction methodology proposed in Eq.(3) (from Ref.[3]).

Our adjacency matrix P(t0, τ)ijdepends on the integration timeτand the initial timet0if the velocity field for the particles is time-dependent. Generally, each dynamical system has an infinite number of flow networks, one for each possible time of integration. At the end, the network may only present different behaviours for different time scales of the dynamical system.

Then for the errors in the methodology, although one could compute the trajectories of the particles precisely, we introduce an error while discretizing the space. Any behaviour of the system with a spatial scale lower than our block size will not be catch by our network due to our selection of length scale through the discretization (volume block side length).

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2.1.4 Backwards Flow Network

In dynamical systems theory, one standard procedure is to run the system dynamics backwards as a way to test the stability of space structures. For example, imagine that the origin of our system is a stable fixed point, if we do a time inversion transformation such that{t→ −t}for the whole system dynamics, the stable fixed point at the origin becomes an unstable one.

In this work, we are using this idea but with the backwards flow network and the original one (without the time inversion). To construct the backwards flow network, instead of building the network from running the dynamics backwards, we start from the original flow network and apply a time inversion transformation. This transformation is presented in Ref.[3] and is expressed mathematically as;

P(t₀+τ,−τ)_ij = P(t0, τ)ji

PN

k=1P(t₀, τ)_jk (4)

where we just transpose the original adjacency matrix applying a renormalization to each column. More visually, the transposed matrix represents the adjacency matrix of the same network but with all the links pointing in the opposite direction. This means that the flow of particles for this networks is going backwards (time direction has been inverted). Note that in Eq.3 we use the number of initial particles in the departure block. If going backwards, the departure block becomes the arrival one and vice versa.

This general transformation also implies making a renormalization when constructing the backwards flow network as Eq.4 shows.

2.2 Network theory. Different types of clustering

In this subsection, we present necessary concepts from the network theory framework. We focus on the clustering as it is the main point of our whole work. Understanding the clustering and its different types is imprescindible for the study we are addressing. Here, we expose the different types of directed clustering that exist for unweighted networks or at least, the ones we have chosen from the literature, Ref.[12].

Before addressing the directed clustering though, we give the definition for the undirected clustering as well as some explanation. Although we work with a directed network, the undirected clustering may be relevant and can be equally computed. Here, definitions are given for the unweighted case, in which the matrix elementsAij of the adjacency matrix are either null or one.

2.2.1 Undirected clustering

The undirected clustering is the clustering for any undirected network or the clustering computed without taking into account the directionality of the links for directed networks. The undirected clustering can be defined as the average probability that two neighbours of a node are themselves neighbours, neighbours meaning two connected nodes. To compute it, one just has to compute the density of triangles in the network. Being a triangle a group of three nodes that are all connected between them (neighbours between them).

Before defining the undirected clustering formally, we must introduce new degree definitions that appear for directed networks. For those, the degree of a node has different ways to be determined as a consequence of the directionality of the links. In this case, we have two different possibilities of connecting the links, outwards or inwards. The out-degree, k^out_i , defines the number of links pointing outwards of the nodei and the in-degree, k_iⁱⁿ, defines the number of links pointing inwards to the nodei. We are using two extra definitions for simplicity. The total degree, k_i^tot, defined as the sum of both previous degrees,k_i^tot =k^out_i +k_iⁱⁿ. And finally, the degree of bidirectional links, k_i^↔, where bidirectional links are those links that connect two nodes in both ways. Two nodes connected both ways do not form a triangle but they do form a closed path. In fact same happens to self-loops, but as we comment later, we are deleting them from our network hence we do not account for them.

Therefore, working with a directed network, the undirected clustering can be expressed mathematically as;

C_i = (A+A^T)³_ii

2 [k^tot_i (k^tot_i −1)−2k^↔_i ], (5) Ci being the undirected clustering for node i, Aand A^T the adjacency matrix and its transposed,k^tot_i the total degree of nodeiandk^↔_i the bidirectional degree of nodei. Then for the average clustering we

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have;

C= 1 N

N

X

i

C_i, (6)

where N is the number of nodes in the network.

One interesting property of Eq.5 is that it reduces to the undirected definition of the clustering if A is symmetric. In this casek^tot_i = 2ki andk^↔_i =ki. Then;

Ci= (2Asym)³_ii

2 [2ki(2ki−1)−2ki] = (Asym)³_ii

ki(ki−1) (7)

wherek_iis the undirected degree. For us, we can thinkA_symas the adjacency matrix for the unweighted version of our flow network. To compute it, we just need to symmetrize our original matrix using A_sym= (A+A^T)/2.

2.2.2 Directed clustering

While for the undirected networks the clustering is a well known and understood parameter, the clustering for the directed networks is a more obscure matter. The directed clustering does not have an univocal definition. Even the most accepted definition divides it into different types of directed clustering. When analysing the ways in which the directed links can be connected, one finds that there are eight different possible connection combinations. In fact, there are only four different types of directed clustering since the combinations are topologically identical by pairs. This directly implies a much wider range of behaviours than for the typical undirected networks.

Here we are using the most widely accepted definitions in the literature for the directed clustering.

These definitions focus on the directionality of the edges as they are precisely conceived for directed networks. From now on, we will have four new definitions for our directed clustering with which to work.

Then, the definitions for the directed clustering are:

• Cyclic clustering: Is the ratio between the number of cyclic triangles formed by node iand the total number of cyclic triangles that node i can possibly form. Cyclic triangles are those where three nodes connect with a closed cycle. This clustering has two possible configurations, clockwise or anti-clockwise (Fig.2).

C_i^cyc= A³_ii

k_iⁱⁿk^out_i −k^↔_i (8)

Figure 2: Scheme for both cyclic clustering possibilities (from Ref.[12]).

• Out clustering: Is the ratio between the number of triangles in which node i has both links pointing outwards and the total number of those triangles that node i can form. There are two possible ways of forming this clustering, one for each direction of the connection between the neighbours (Fig.3).

C_i^out= (A²A^T)ii

k_i^out(k^out_i −1) (9)

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Figure 3: Scheme for both out clustering possibilities (from Ref.[12]).

• In clustering: Is the ratio between the number of triangles in which nodeihas both links pointing inwards and the total number of those triangles that nodeican form. There are two possible ways of forming this clustering, one for each direction of the connection between the neighbours (Fig.4).

C_iⁱⁿ= (A^TA²)_ii

k_iⁱⁿ(kⁱⁿ_i −1) (10)

Figure 4: Scheme for both in clustering possibilities (from Ref.[12]).

• Middle clustering: Is the ratio between the number of triangles in which node i has one link pointing outwards and one inwards, and the total number of those triangles that nodeican form.

There are two possible ways of forming this clustering, one for each direction of the connection between the neighbours (Fig.5).

C_i^mid= (AA^TA)ii

k_iⁱⁿk^out_i −k^↔_i (11)

Figure 5: Scheme for both middle clustering possibilities (from Ref.[12]).

Mathematically, one can show that the undirected clustering is actually the sum of the four different types of directed clustering. Therefore in our notation,

C_i=C_i^cyc+C_i^out+C_i^mid+C_iⁱⁿ. (12) This relation is important as it gives a relation between the different clusterings. Due to this, after studying the undirected clustering we have a condition that helps us to understand the different directed clustering behaviours.

In the following sections we are not only using the local clustering defined for a specific node ibut the average above all of the network nodes. To refer these average clustering we just drop the sub-index ias we did for the undirected one.

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3 Application to the Lorenz attractor

We have finally given all the theoretical concepts needed to follow the present work. In this section, we move on and expose our results as well as comment on them. The section is divided in four different subsections. In the first one we explain the Lorenz attractor and why we chose to work with it. The second subsection provides the technical details we used to construct our flow network. The third subsection studies the undirected network of our flow network. Finally, the fourth subsection is where we present our results for the study of the directed clustering.

3.1 The Lorenz attractor

The Lorenz Attractor was first presented by Lorenz in Ref.[7] where he described the strange properties of these atmospheric differential equations he had studied, Eq.(13). He showed the system was highly unpredictable although it presented a purely deterministic scheme. Nowadays, the Lorenz Attractor is a well-known chaotic attractor as well as the first example of the so called butterfly effect. A name popularized by the press due it is characteristic shape, Fig.6.

Figure 6: Representation of the Lorentz attractor (from Ref.[9]).

As explained, the Lorenz attractor is one of the most popular systems in chaos theory. The system displays many characteristic behaviours of chaos. Despite of the relevance, our main reason to chose the Lorenz attractor is to be able to prove/unprove the hypothesis postulated in Ref.[4]. The hypothesis said that; “the structures displayed by the undirected clustering of the flow network for the Lorenz attractor are stable manifolds of the non-vanishing unstable fixed points embedded in the attractor”. If this argument results to be true, next step would be to check if it is a general property for some type dynamical systems.

In order to prove or unprove if the undirected clustering structures represent the stable manifolds, we constructed the flow network for the Lorenz system in the chaotic regime. In this regime, the system exhibits a chaotic attractor, the Lorenz attractor or strange attractor as Lorenz first discribed it in Ref.[7].

The set of differential equations for the Lorenz system are dx

dt =σ(y−x) dy

dt =x(ρ−z)−y (13)

dx

dt =xy−βz

where we are using the typical parametersσ= 10,β= 8/3 andρ= 28. These parameters set our system in the chaotic regime.

There are many interesting features displayed by the Lorenz attractor, in this work though, we are just going to refer to those that are of some relevance to understand the present work. In such a task, we should know that the Lorenz attractor is symmetric under the transformation (x, y) → (−x,−y).

As a direct consequence, we can output results for one of the attractor wings and extrapolate them to the symmetric counterpart. Even most important are their stationary points and how they behave.

Studying its nullclines we find that it has three stationary points: the origin (0,0,0) and two non-zero

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Figure 7: Representation of the Lorenz Attractor (orange) and its stationary points (green) with their respective one dimensional manifolds, unstable (red) and stable (blue) (from Ref.[9]).

C_±≈(±7.348,±7.348,27.000). Linearising we find that all three of them are saddle points. Then all of them contain unstable and stable directions.

Then, studying the stability of the eigenvectors for the stationary points, we find that for the non-null stationary points there is one stable direction going into the the points and two unstable ones in the plane of the attractor. These last two eject the particles that fall into the non-null stationary point to the actual attractor, Fig.7.

Other important structures besides the stationary points of the attractor are the Unstable Periodic Orbits (UPOs from now on). These are periodic trajectories embedded in the chaotic attractor. They can also be refered to non-chaotic systems but is not that usual. UPOs are called this way because any particle following one falls into an aperiodic trajectory (typical trajectories for chaotic systems) under the effect of any perturbation. For the Lorenz attractor the typical way to name them is writing as many

‘L’s as cycles the particle does around the left wing of the attractor and same for the right wing with

‘R’s. For example, the UPO LLRRR is an unstable periodic orbit that does two cycles to the left wing of the attractor and three to the right one to finally end at the starting position. As consequence of their definitions and the chaotic systems they belong to, these trajectories are extremely hard to compute. In this work we used Ref.[8] to help us compute them.

3.2 Network Construction

In this subsection we address the technical procedure we followed to construct the flow network for the Lorenz Attractor. We also explain the transformations we applied to the flow network in order to get something easier to work with.

First, we discretized the space in cubic blocks of 1.59 units per side. We chose this exact number as it is the length used in Ref.[4], enabling us to test our results against theirs. Then, for the size of the system, we used two different cubic sets. First one extending its big diagonal from{−20ˆx,−20ˆy,5ˆz} → {20ˆx,20ˆy,45ˆz}and containing a total of 15.625 blocks. Second, the double-sized one, extending its big diagonal from{−40ˆx,−40ˆy,−15ˆz} → {40ˆx,40ˆy,65ˆz}and containing a total of 64.000 blocks.

Once we had the nodes defined by the discretization, we had to build the links, as explained in the previous section, we need to follow the trajectories of the particles for a certain amount of time. In our case, we uniformly distributed 1.000 particles in each box and integrated their trajectories over different periods of time. This part is the most expensive computationally as it implies the precise numeric integration of Eqs.(13) for 15 or 64 million particles (normal or double-sized). Being this computation for just one integration time. What we actually do is integrating for the maximum time and writing all the positions for the lower times, doing so, we just have to compute the trajectories once to obtain all the flow networks.

After computing all the final positions of the particles, we can proceed to construct our actual flow network. We apply Eq.(3) to obtain our adjacency matrix elementsA_ij =P_ij. Then, instead of directly using A, we are going to make some transformations to it as it is going to make it simpler to study.

Remark that at this point, our matrixAis defining a directed weighted network without any restriction.

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As we are interested in studying the trajectories of the particles and their dynamics, we are going to erase all the self-loops of the network. Self-loops complicate the analysis in the network framework and they are not that interesting in the dynamical framework as they represent particles withslowtrajectories most of the times. To do so, we just apply the following rule{Aii = 0;∀i}, erasing all the information in the diagonal. With this transformation we may be deleting self-loops made by particles that have a fast cycle, we expect this scenario to be a minor case.

Finally, we unweight our network applying the rule {A_ij = 1 if A_ij >0; ∀i, j}. Applying this rule makes all the weights of our network equal to the unity or null. From here, we can think of our network as an unweighted one. This procedure loses a lot of information of the system as it evens the strength of all the links independently of how many particles contributed to form it. This is probably the most harsh simplification we made to the network, it excludes us from outputting results that need a precise value for the clustering.

Now we already have our adjacency matrixA defining our unweighted directed network with which we are able to work in the network theory framework.

3.3 Undirected clustering

In this subsection we focus on studying the undirected clustering of the flow network and the spatial structures it forms. The final goal is to understand why these structures appear and if they have any property we can generalize for other systems.

We divided the study of the undirected clustering in four different parts. A first part where we do a general approach to the undirected clustering, showing the structures it forms and commenting on them.

Next, two parts in which we show the different strategies we used to try and understand the behaviours displayed. And a final one in which we present the theoretical conclusion we were driven to.

3.3.1 Spatial structures

As we explained, we know from Ref.[4] that some spatial structures appear at the Lorenz attractor flow network if we compute its undirected clustering. What we do not know is for what integration times this happens or even what thesetubes look like. As we have too little information, we start by computing the average undirected clustering for different integration times. We do it to visualize if we should expect different behaviours for different integration times.

Figure 8: Mean undirected clustering for the Lorenz flow network as function of the integration timeτ. Then, a first glance at Fig.8 reveals that we should expect different behaviours for different integration times or at least differences in their strength. We see a clear peak around 0.65 units of time and two lower peaks for bigger times. This first computation shows us that we must compute the local clustering for at least two different integration times, one for high clustering and another for low clustering. Doing so, we may provide some insight into why these differences.

The following step is to plot the undirected clustering per node and visualize the forming shapes in the space. We do so in Fig.9 where we can see two symmetric shapes crossing the attractor that look

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liketubes. We expect these structures to be the tubes refered in Ref.[4] and what they postulated to be, the stable manifolds of the non-vanishing unstable fixed points embedded in the attractor.

Figure 9: Undirected clustering for the nodes with higher clustering (>0.3) in their respective positions.

Both graphs also plot the LR symmetric UPO (black) to visualize the attractor position. Left;network constructed for an integration time of 0.65 units. Right;network constructed for an integration time of 0.9 units.

Looking at Fig.9 we notice the big difference between the networks for the two times. Left one seems really noisy and right one is much clear, only displaying both symmetric tubes and some other nodes at the walls of our lattice. The difference in density for the graphs makes sense because the left graph is done with the integration time of the peak in Fig.8 and the right one with an integration time of low clustering. Although we can not yet conclude anything, seems that the behaviours do not change drastically but their strength does.

To try and understand why these structures form, we are going to follow the trajectories of a handful of particles. This may help to understand why each node was coloured or not. If the particles start outside of the tubes they do not seem to follow any clear pattern but, if they start at the tubes, they all fall on the fixed points C_±. Once there, they start spiralling out into the attractor plane. Is then reasonable to think that all fixed points with stable and unstable directions can be caught by similar structures of the clustering in the network? We can not assure anything yet and we continue to check different behaviours of the network in order to give some clarity to the matter.

3.3.2 Double-Sized Networks

When looking at the three dimensional shapes for the undirected clustering in Fig.9, one can notice that, apart from the tubes, there are some nodes with non-vanishing clustering at the walls of the lattice. At first, we thought this could be some kind of noise from the discretization or even some finite effects.

Then following particles starting from those positions at the walls where the clustering is not null, we saw that for an integration time they go around the network and end up at C_±, crossing through the tubes in the process. This was the first hint to realize that the shapes may be stranger that what we first thought. Then, to actually be able to see the bigger shapes, we computed the double-sized networks having the same size for the volume blocks but increasing the number of them.

Fig.10 left side is the representation of the undirected clustering for the double-sized network at the peak integration time,τ = 0.65. At that time, most of the nodes have non-vanishing clustering. Even if only the nodes with at least 0.6 clustering are plotted, practically all the nodes in the network are still present. In a way, we are loosing the two dimensional shapes but, does this mean that for this integration time there is no stable manifold? Or on the other hand, all the space pertains to it?

Checking at Fig.10 right side, we see nodes with high clustering resemble shapes that do not look like tubes any more. Instead, they seem to approach to some other two-dimensional manifold. If we follow particles starting from this manifold, we see that they end atC_± after an integration time τ. Checking trajectories for particles close to the manifold, we see that those particles fall into the manifold in times of the order ofτ.

With all these considerations taken into account, the idea of the undirected clustering being the stable manifolds gets harder to fit. Actually, if one checks the figures in Ref.[9] where the stable manifolds are

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Figure 10: Undirected clustering for each node in our double-sized flow network plus the LR symmetric UPO (black). Left; network constructed for an integration time of 0.65 units. In this representation only the nodes with clustering higher than 0.6 are taken. Right;network constructed for an integration time of 0.9 units. In this representation only the nodes with clustering higher than 0.3 are taken and the trajectories of two particles starting at non-null clustering are also plotted (green and blue).

plotted, one can notice that the structures seem rather different although it is hard to actually compare them. These manifolds are infinite two-dimensional structures with many folds over themselves.

3.3.3 Backwards Networks

As we explained in Sec.2.1.4, we can apply a transformation to our adjacency matrix to get the backwards adjacency matrix. This transformation is presented in Eq.4 and it is exactly what we used to obtain the flow networks we show here. In the end, this methodology has helped us to rule out the possibility of the undirected clustering structures being the stable manifolds.

We can start by assuming that the undirected clustering structures represent the stable manifolds as hypothesized in Ref.[4]. Then, plotting the undirected clustering is equal to plot the stable manifolds or some shape resembling them. In addition, when running the time backwards and obtaining the flow network from it, if we do the plots, we expect to obtain the unstable manifolds. That is exactly what we do in Fig.11.

Figure 11: Undirected clustering for each node in our backwards double-sized flow network plus the LR symmetric UPO (black). Left; network constructed for an integration time of 0.65 units. In this representation only the nodes with clustering higher than 0.6 are taken. Right;same as the left one but the network is constructed with an integration time of 0.9 units of time. In this representation only the nodes with clustering higher than 0.3 are taken.

Surprisingly, when we plotted the undirected clustering for the backwards networks, Fig.11, the

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something we can prove. To do so, we are going to use the definitions of the different directed clustering.

The fact that the backwards undirected clustering evens the original undirected clustering finally discards our initial hypothesis. The idea of the undirected clustering of the flow network unveiling the stable manifolds is no longer possible as by definition, the stable manifolds for a system transform to unstable manifolds when inverting the time direction. In opposition to an unstable manifold which transforms to a stable manifold when inverting time.

3.3.4 Invariance under time inversion

We already gave strong reasons to discard the hypothesis postulated in Ref.[4]. Here though, we try and go deeper in why this can not be right. To do so, we analyse it mathematically and more formally than just checking visually some fancy graphs.

In Sec.2.2 we presented the mathematical descriptions to obtain the different clustering coefficients for an unweighted directed network. In this part, we are interested in analysing if Eq.5 changes after applying Eq.4 to our adjacency matrixA.

Then, as we previously said, Eq.4 only transposes the matrix and applies a renormalization. For our purpose, we can omit the renormalization part because we work with an unweighted network. Remember that for unweighted networks all the weights are the unity or zero. Checking Eq.5 we see that the numerator is formed by the sum of the adjacency matrix and its transposed. Applying Eq.4 to it, (Aback+A^T_back)³_ii = (A+A^T)³_ii leaving the numerator unchanged. Then for the denominator the same occurs,k_i,back^tot =k_i^out+k_iⁱⁿ=k_i^tot. Obviously, the directional degree,k^↔_i does not change either as it is a symmetric link.

This result does actually make sense if one understands that when computing the undirected clustering, one loses the directionality of the edges in network terms (undirected networks have symmetric adjacency matrices). On the other hand, analysing it as a dynamical system, this represents a system that has no time direction. In other words, the system is invariant under a time inversion.

Finally, studying the definitions of the directed clustering, we checked that not only the undirected clustering is invariant to time inversions. The cyclic and middle clustering by their own and the in and out clustering as a group are also invariant to time inversions (note that we are working with unweighted networks). We can express this ideas formally such that;

C_i^cyc(Aback) =C_i^cyc(A) C_i^mid(Aback) =C_i^mid(A)

C_iⁱⁿ(Aback) =C_i^out(A) C_i^out(Aback) =C_iⁱⁿ(A)







=⇒ Ci(Aback) =Ci(A) (14)

As a recap, we have finally shown that the initial hypothesis postulated in Ref.[4] is not possible.

That is so because we have shown that the undirected clustering is invariant under a time inversion transformation. This implies that the undirected clustering structures can not characterise any stable or unstable manifold as they must swap under a time inversion transformation.

3.4 Directed clustering

Now that we know that the undirected clustering is not representing the stable manifolds for the attractor, we would like to understand why such structures appear. Always looking to relate them with some other structure borrowed from dynamical systems theory.

To do so, we are going to study the different types of directed clustering by their own as we have seen that studying the undirected as a whole is not enough.

3.4.1 Cyclic clustering

The first directed clustering we compute is the cyclic clustering. If we do so is because it has already been studied in Ref.[4]. Then, comparing results with previous works is useful to test our networks.

We start by plotting the average cyclic clustering for different integration times. Ref. [4] demonstrated that this figure displays peaks at times of one third of the orbit periods.

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Figure 12: Mean cyclic clustering for the Lorenz flow network as function of the integration timeτ.

We know that the Lorenz Attractor has an infinite set of UPOs, which are usually really hard to compute. A small perturbation in the particle following one of those orbits may kick the particle far from it. In our case though, we do not need that much precision as our network nodes have a certain volume.

Then if the particle follows a trajectory close enough to the periodic orbit, our clustering measure can catch this trajectory as a closed one.

As example, the first peaks of Fig.12 are forτ1= 0.52,τ2= 0.77 andτ3= 1.03. Thenτ1represents the first set of UPOs, LR and RL, with the full period beingT1≈1.5587≈3τ1= 1.56. Similar for LLR, RRL with T2 ≈2.3059 ≈3τ2 and LLRR, RRLL with T3 ≈3.0843≈3τ3. Higher period UPOs would give the peaks at righter places in the figure. The valleys of the graphs are one third of times for which there are no closed or nearly closed trajectories.

Figure 13: Cyclic clustering for each node in our flow network. Left; network constructed for an integration time of 0.52 units. The respective UPO LR have been also plotted (black). Right;network constructed for an integration time of 0.77 units. The respective UPOs LRR and RRL have been also plotted (black).

We already know that the times match with the orbit periods but as shown in Fig.13, plotting the directed clustering per node, we also get some interesting structures. The structures that appear are none other than the UPOs. Rigorously, they are just a coarse-grained version of them as our construction blocks have a predetermined size (discretization length). These results are in perfect concordance with what was concluded in Ref.[4].

When checking the explicit values of the clustering (not only the cyclic one), it is not easy to give a meaning to them. In fact, as we commented in previous sections, the size of the discretization volume blocks is arbitrary. Then, as the value for the clustering depends on it, we can not expect to conclude

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one in Ref.[4]. The peaks and valleys appear for the same integration times but the values for the peaks are quite different. In our computation we get higher values for all the figure as if we had rescaled it.

Notice that we used exactly the same volume blocks and shapes as they did. What we did different is that Ref.[4] started with a bigger lattice hence a higher number of blocks. Those extra blocks where placed at further positions from the attractor, positions which we did not account. What happens is that those blocks are far from the attractor and they do not have any closed orbit. Not having closed orbits implies having null cyclic clustering which produces a lower average cyclic clustering.

As we pointed in Sec.2.2.2, the cyclic clustering is invariant under a time inversion transformation.

Then, applying this transformation to the whole network does not change the values or shapes we obtain. We also explained that there are actually two possibilities to form a cyclic cluster and that both are topologically identical. In fact, one can check that both forms are just the time inverted version of each other.

Figure 14: Representation of the cyclic directed clustering per node and the LLRR UPO in black solid line. Left;network constructed for an integration time of 0.96 units of time. Right;network constructed for an integration time of 1.03 units of time.

Fig.14 is a representation of the cyclic clustering for two different times. Right side plots the clustering for an integration time ofτ= 1.03 which is one third of the period for the LLRR UPO,T ≈3.08. Then left side plots the clustering for an integration time of τ = 0.96 which does not map with any UPO period. Recalling Fig.12, we can see that there is no peak for that integration time. More precisely, this timeτ = 0.96 is between the two peaks of the UPOs LLR/RRL and LLRR, with integration times τ = 0.77 andτ = 1.03. For τ = 0.96, particles following trajectories close to LLR/RRL do more than one cycle and particles following trajectories close to LLRR do not have time to do one whole cycle.

Next, we need an explanation to why there are non-null values for the left figure. Before explaining it, note that the values for the left figure of Fig.14 are much smaller than the right ones, implying that less particles are closing cyclic triangles. The fact that, for an integration timeτ = 0.96 we have cyclic clustering, comes as a consequence of the discretization. There is no closed orbit with a period T = 3·0.96 = 2.88 but, as our nodes have a finite size, they are catching some trajectories as closed cycles. Those are the trajectories that after a time τ = 2.88, the final position is really close from the starting one. This produces a noisy cloud of points around the attractor as only the particles close to the attractor may fulfil those conditions (particles far from it just fall into the attractor).

As a recap for this part, we have confirmed that computing the cyclic clustering of a flow network reveals the periodic orbits of the dynamical system as well as the periods they have. This is an important step as it gives an utility to compute the clustering of a flow network.

3.4.2 In clustering

In the following pages, we are studying the remaining types of directed clustering introduced in Sec.2.2.2.

We have not found any record of these clusterings being studied for dynamical systems or flow networks before.

Recall from previous comments that the in and out clustering form a group. A group in the sense that they are invariant under an inversion of time transformation. They do not fulfil it by themselves though. In fact, they transform from one to the other if a time inversion transformation is applied. As

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an example, if we compute the in clustering for one of our flow networks and the out clustering for the respective backwards flow network, we get exactly the same result.

This is easy to understand looking at the figures of the definitions in Sec.2.2.2. Check that inverting the time in Fig.4(changing the direction of the arrows), the node being the in-node becomes the out- node. Then, we swap from havingk_iⁱⁿ= 2 andk^out_i = 0 to the opposite hence swapping from having in clustering to out clustering. Same idea applies for the out-nodes or out clustering but in the opposite direction hence changing to in-nodes or in clustering.

When reasoning about what we expect to obtain when plotting the in clustering, we got to the argument that the shapes drawn by the in clustering must include the attractor and the stable fixed points. The idea is that the attractor and stable fixed points are coordinates in the space where the particles from close positions end up. Then, there is a good chance that the attractor and the stable fixed points have non-null in clustering.

Figure 15: Representation of in directed clustering per node and the LR UPO in black solid line. Left;

network constructed for an integration time of 0.90 units of time. Right; network constructed for an integration time of 0.65 units of time.

Checking the in clustering for different integration times (two in Fig.15), we see that it graphs the typical butterfly shape of the Lorenz Attractor. That was expected as all the particles starting away from the attractor eventually fall into it. The hard part comes when studying the differences between integration times, left and right side of Fig.15 are an example. In these cases, we had to plot different particle trajectories to try and give some insight to these behaviours. From doing so, we found several distinct behaviours.

For integration times near the peak of the undirected clustering at t= 0.65 (Fig.6), there are many particles that fall intoC_±. Once there, for the next integration time, they cycle around the fixed points without leaving really far (see the brighter colors at the center of both figures in Fig.15). That was something we have already noticed before but, what happens to other times?

Checking the trajectories of particles with integration times of lower average clustering, we see that for really small times the particles can not reach the fixed points C_±. In this case, the particles do not compute any clustering as they can not close triangles. With bigger integration times than the peak, the particles had already fallen into the attractor and some of them changed from one wing of the attractor to the other one. This may be of some relevance when understanding why there are the lower peaks in Fig.9.

3.4.3 Out clustering

Before commenting it, remember that the out clustering is the inverse of the in clustering if we do a time inversion transformation. Then as we did with the in clustering, we reason its behaviour looking at the trajectories that the particles follow on the dynamical system. We know that the particles outside the attractor fall to it. What we also know is that they must diverge at the start as we have a chaotic system. Then, when they finally fall into the attractor, they start cycling the attractor and some of them end at close positions. If they do, this produces a non-null clustering for the clustering types in, out and middle.

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conclusion that we should get a cloud of points around the attractor, bigger and bigger as we increase the integration time. This is close to what happens but it is not all, the cloud of points getting bigger and bigger is just a direct consequence of the particles having a finite velocity to arrive to the attractor.

If a particle can not reach the attractor in time, it can not close any cluster hence its out clustering is null. In fact, the whole undirected clustering for the nodes fulfilling that condition vanishes, Fig.9.

Then, all the directed clusterings do too.

Figure 16: Representation of out directed clustering per node and the LR UPO in black solid line. Top- Left;network constructed for an integration time of 0.40 units of time. Top-Right;network constructed for an integration time of 0.65 units of time. Bottom;network constructed for an integration time of 0.9 units of time.

Checking at the figures in Fig.16, we see that for the lower integration time τ = 0.4, we can not see the cloud of points we were expecting. In fact, only the tubes we talked in previous sections are bright, the respective nodes have non-null out clustering. Checking the trajectories we see that the particles starting at the tubes are the fastest approaching to the fixed points. The particles starting there, have already fallen intoC_± for an integration time and they cycle around it going to the attractor as commented before. The particles starting at the lower red region under the attractor are the particles that fall directly to the attractor. These ones have it harder to close orbits as the orbit of the attractor is bigger than the ones cycling aroundC±, this generates the lower values of clustering we see. For the peak time of the average undirected clusteringτ = 0.65, all the nodes in the network are coloured and quite brighten except the ones including the attractor. This time is the critical time for our lattice dimensions as is the time when all the particles have been able to fall into the attractor. Note also that we can compare it with the right figure of Fig.15 to realize that they are nearly complementary (both have equal integration time). They do not necessarily fulfil this condition as one node can have in and out clustering at the same time but it is interesting to realize that it happens for this critical time. Finally, for bigger times the cloud starts becoming smaller again as we have a finite number of particles and our attractor has alreadyeaten all of them. If we had filled all the space with particles, we would expect Fig.8 to have even bigger peaks as particles from farther positions fall into the attractor and cycle around it.

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3.4.4 Middle clustering

The middle clustering has been the hardest to understand as it has a stronger dependence on the integration time. In addition, it is not as clear as the other ones how to check possible particle trajectories.

To do so, we had to use the knowledge we obtained from studying the in and out clustering. Then, for an integration time and knowing both clusterings, we kind of know where the particles closing triangles start and end. We use that knowledge to contrast all the graphs and conclude which path is followed by the particles to close triangles (not closed cycles!).

Recall that inverting the direction of time does not affect to the values or forms of the middle clustering. The inversion only swaps which of their neighbours is the out-node and which is the in-node.

This is another way of detecting which nodes are the ones contributing to the middle clustering, we just have to check that the nodes for which the clustering is equal forA andAback. We should discard the cyclic ones too but their contribution in our network is really small in comparison.

Figure 17: Representation of middle directed clustering per node and LR UPO in black solid line. Left;

network constructed for an integration time of 0.90 units of time. Right; network constructed for an integration time of 0.65 units of time.

In Fig.17 we plotted two different times for the middle clustering. At left side one, we changed from the usual perspective as we hoped to get a better visualization of the tubes we already spoke of. If we check with care, we can see that the middle clustering does not vanish around the attractor. Its value is really low (red) but not null. If we look at the center of the attractor we also see nodes with brighter reds or even yellows. Those nodes are part of the emerging tubes that go through the fixed pointsC_±. For this integration timeτ = 0.9, we can say that the middle clustering is doing both jobs, displaying as the in and the out clustering.

For the right figure, with its integration time at the peak of the undirected clusteringτ= 0.65, we can see that the middle clustering fills the attractor as well as the space between the attractor and the fixed pointsC_±. Checking the particles for this times, we see that here the particles are cycling around the fixed points and then closing the clusters. This produces non vanishing middle clustering for a similar but smaller domain than the in clustering.

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4 Discussion and Conclusion

In this section we sum up all the relevant points we extracted from our results. Then, we discuss them to explain the different behaviour we obtained.

First results we want to discuss are the ones we obtained for the cyclic clustering. At Ref.[4] the authors concluded that, the average cyclic clustering of a flow network provides a kind of “spectroscopy”

by which the presence of orbits of different periods are revealed. Then, the times in which the average cyclic clustering displays a peak correspond to times of one third of the orbit periods. As an example, if we plot the average cyclic clustering of a dynamical system and we obtain a peak for an integration time τ1, we know that this system has an orbit with period 3τ1. We checked these conclusions in our flow network at Fig.12 where we can clearly see the different peaks for the clustering. We also checked that the times of those peaks correspond to UPOs of the attractor not only by computing their periods but also by plotting some of them, Fig.13 and Fig.14. In those figures we also saw how the plots of the cyclic clustering per node resemble the actual orbits in a coarse-grained way.

Next, comparing our results against the ones obtained in Ref.[4], we analysed the discrepancies we have for the values of the average cyclic clustering. Doing so, we concluded that the actual values of the clustering are not to be considered when extracting conclusions. This is so because they strongly depend on the discretization of the flow network which is in principle, arbitrary. This specially aggravates if we unweight the flow network. Then, to extract conclusions from the flow network clustering one has to compare the results to extract relative values instead of absolute ones. As an example, if we want to check the relevance of the cyclic clustering versus the middle clustering for two integration timesτ1 and τ2, we should compare the relative valueCcyc(τ1)/Cmid(τ1) versusCcyc(τ2)/Cmid(τ2).

Then for the rest of the clustering types, the undirected and the three directed, the results have been harder to analyse. We started with the undirected clustering, proving incorrect the hypothesis postulated in Ref.[4]. The hypothesis was already hard to fit when we output Fig.10. Figure where the spatial structures formed by the undirected clustering transformed from the two tubular shapes to a bigger two dimensional surface. This surface could only be seen when we expanded our discretization space. The displayed surface in Fig.10 did not easily match with the stable manifolds of the Lorenz attractor shown in Ref.[9]. We did not stop with a visual confirmation and helped by Fig.11, we saw that the undirected clustering of the backwards adjacency matrix and of the original adjacency matrix are identical. These directly implied that the structures drawn by the undirected clustering can not be stable or unstable manifolds as those must interchange for both adjacency matrices.

A fact we realized while studying the undirected clustering is that it has more peaks and valleys after the big first peak. Looking at the trajectories that the particles follow, we saw that the clustering grows until it arrives to the typical time for which one particle does a whole cycle to the wing attractor, Fig.8 second and third peak. What happens is that after doing a cycle, some particles change to the opposite wing of the attractor. If that happens, it becomes harder for them to close triangles. We characterize this fact as specific of our dynamical system.

Once arrived to this point, we already achieved the initial purpose of this work. We did not stop here and continued to investigate the different types of directed clustering. In the process, we realized how the directed clustering changed under a time inversion transformation to the flow network. The cyclic and middle clusterings are not affected at all if we swap from the original network to the backwards one. This is so because their relative structure in the triangles forming clusters does not change if we invert the time direction, Fig.2 to Fig.5. On the other hand, the in and out clustering are affected under a time inversion transformation to the flow network. They transform from one to the other as we showed in Sec.3.3.4 that their definitions are complementary under this transformation. Note that all these conclusions only work for unweighted networks. Weighted networks also maintain their spatial structures but they redistribute their values. This may not be that relevant as we already talked about why we should not regard the actual values of the clustering.

As final thoughts for this section, we are discussing the different behaviours of the directed clustering and how they can be related to the attractor dynamics. To do so, we have focused on the different results we presented in Sec.3.4 and in the analysis of all the trajectories we visualized. We are aware that these kind of analysis is not decisive but our purpose here is just to shed some light into the matter as this is the first approach to it.

We know that the particles in our system have a finite velocity. As a consequence, they need a certain amount of time to arrive to the attractor, time that depends on where they started. In fact, this happens for any attractor. Then, for the flow network of a dynamical system that has an unique attractor, we recall that the average out clustering grow with their integration time until a critical time. This implies

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that the average undirected clustering grows until this critical time too. Being this critical time, the time for which all the particles in the space can arrive to the attractor. Note that we are not saying that the peak in Fig.6 is bigger if we do a bigger discretization, is just that the peak moves to bigger times.

Something we realized looking at the particle trajectories is how they fall into the attractor. To do so, they have two different approaches. One is falling directly into one of the wings and getting closer to the attractor and the second one which triggers thetubeswe discussed so much about. In this second one, particles first fall into the stationary pointsC_± following trajectories close to the stable manifolds.

Once there, they cycle around the stationary points approaching to the attractor. This is an interesting behaviour but hard to generalize as it is really specific of the structure of our dynamical system, the Lorenz attractor. Then, the tubes formed by the undirected or out clustering are just particularities of our attractor.

One expected behaviour of the in clustering and which we expect to be generalizable, is the fact that it displays the actual attractor in the space. The in clustering displays the positions of the dynamical system where some particles end for an integration time. Note that this may only happen if the particles falling into the attractor have a way of closing triangles. On the other hand, we expect the out clustering to be kind of opposite. The opposite in the sense that it does not represent the attractor but points out of it. This is because the out-nodes are those from which the particles leave and do not come back. Then we expect the out clustering to characterize shapes outside the attractor although we can not confirm which are those shapes.

In general terms, the middle clustering is probably the less useful as it displays behaviours mixing the in and out clustering.

In summary, we have confirmed the methodology proposed in Ref.[4] to detect periodic orbits using the cyclic clustering. Next, we have proved that the undirected clustering does not represent the stable manifolds of the dynamical system underlying the flow network. Finally, we have showed when the different types of directed clustering are significant for the flow network as well as how they relate to the attractor and the trajectories of its dynamical system.

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