# Spectral cyclicality of networks

Nizar Riane† ‡

###### Abstract

We introduce the spectral influence and spectral cyclicality based on the
largest eigenvalue of a graph adjacency matrix, two novel concepts of
centrality capturing diffusion and interdependence from a local and a global
point of view respectively. We define a new clustering algorithm to
distinguish communities with high cyclicality and interdependence, allowing
overlapping, and we conclude our study with an application to the input-output
analysis in the case of the Moroccan economy.

† Université Mohammed V de Rabat<EMAIL_ADDRESS>
‡ Sorbonne Université

CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 4, place Jussieu 75005,
Paris, France

Keywords: Spectral influence; Spectral cyclicality; Cyclicality clustering;
Input-output analysis.

AMS Classification: 05C22 – 05C38 – 05C85 – 05C90

JEL Classification:

###### Contents

  1. 1 Introduction
  2. 2 Basics of graph theory
  3. 3 Spectral influence and cyclicality
    1. 3.1 Spectral influence: a local approach
    2. 3.2 Spectral influence: a global approach
  4. 4 Spectral influence and cyclicality clustering
    1. 4.1 Divisive cyclicality clustering
    2. 4.2 Agglomerative cyclicality clustering
    3. 4.3 Overlapping clustering
    4. 4.4 Discussion
  5. 5 Application to input-output analysis
    1. 5.1 Moroccan interindustrial cyclicality analysis
    2. 5.2 Propulsive industries and key cyclicality components
    3. 5.3 Discussion
  6. 6 Conclusion
  7. 7 Appendix
    1. 7.1 Industrial poles of the OECD harmonised national input-output tables
    2. 7.2 Spectral influence evolution by industrial poles

##  1 Introduction

The idea of centrality in network analysis can be traced back to early work in
graph theory, and many definitions of centrality have been proposed, each one
bringing different insights to the connections in a graph. For example, in
1987, Phillip Bonacich [Bon87] introduced a centrality measure of a vertex
based on the power centrality of its neighbors, as a way of addressing some of
the limitations of other centrality measures, such as degree centrality and
eigenvector centrality, where a high score means that a vertex is connected to
another vertices who themselves have high scores ([New18] ch.7). Bonacich
argued that "in bargaining situations, it is advantageous to be connected to
those who have few options; power comes from being connected to those who are
powerless. Being connected to powerful others who have many potential trading
partners reduces one’s bargaining power" ([Bon87]).

Linton Freeman ([Fre78]) believed that "the development of measures should
help to clarify a concept by specifying its components and their
interrelationships. In the case of centrality, however, the opposite effect
seems to have been achieved. In a sense, the introduction of new measures at
this stage is inappropriate. Ideally, measures should grow out of advanced
theoretical efforts; they should be defined in the context of explicit process
models", despite this opposition, new definitions of centrality didn’t stop
appearing.

We believe that the reservation regarding multiplicity of centrality
definitions comes from the confusion that the term "centrality" induces, it
did not designate anymore a unique measure of "importance" but a field of
study regrouping different definitions for different purposes.

Following this momentum, we introduce a new definition of centrality, based on
the spectral radius of the network adjacency matrix, which we call the
spectral influence. This new concept characterizes vertices importance from a
diffusion and interdependence perspective: a vertex score will be higher as
much as his role in the diffusion, through closed walks, across the network is
important.

The spectral influence is a local measure of a global one we call the spectral
cyclicality, which characterize the cyclicality over the graph. We go further
by developing a cyclicality based clustering algorithm. Following a divisive
or an agglomerative philosophies, one can isolate communities with highest
diffusion property in a graph.

A cycle is the simplest structure that brings redundant paths in network
connectivity and feedback effects in network dynamics ([FLSZ21]). Cycles in a
networks are synonym of interdependence, redundancy, diffusion, cascade
effect, contagion, amplification and attenuation. Cyclical patterns and
phenomena are the subject of investigation in various fields. For instance, in
the field of biology, cyclicality is studied in the context of epidemic spread
[PD04], while in biochemistry, it is explored as a possible explanation for
biochemical oscillations [GAZ09]. In ecology, researchers investigate the
robustness of food webs to biodiversity loss in the presence of cyclical
patterns [DWM02], while in finance, cyclicality is examined for its role in
the propagation of shocks within financial systems [HM11], and so on.

In the field of economics, numerous studies have focused on the importance of
network structures in understanding economic agent interactions. These
network-based studies emphasize significance of network structure in
determining economic agents interaction outcomes. For example, Acemoglu et al.
[ACOTS12] showed that microeconomic idiosyncratic shocks may lead to aggregate
fluctuations in the presence of intersectoral asymmetric interconnections, in
opposite to Robert Lucas ([Luc77]) argument that microeconomic shocks would
average out, and thus, would only have negligible aggregate effects, but the
authors conclude that in economies with balanced intersectoral network
structures, such as cycle, the aggregate volatility decays at a constant rate.
In an opposite direction, the authors of [GAZ09] expose a different conclusion
in the case of banking system, where it is shown that a cycle structure
increase risk of domino effect.

In order to give a different perspective, we have opted to apply our
definition of centrality to investigate interindustrial flows, taking
advantage of the amplification effect of cycles to highlight propulsive
industries, in line with François Perroux’s theory ([Per63] page 380): "the
actions by which the increase in the rate of growth of the product or the
productivity of a simple or complex unit A, causes the increase in the rate of
growth of the product or the productivity of another simple or complex unit B.
The propulsive unit, thus understood, acts theoretically, either by an effect
of productivity or innovation. Often combined, in practice, these effects can
however be distinguished conceptually.".

Our application could be further motivated by the work of Roland Lantner
[Lan72], who highlighted the interdependence implied by the interindustrial
flows: "The significance of a transaction between a supplier and a demander is
measured less by its absolute value than by the degree of vulnerability it
implies for one or the other". Interdependence of industrial poles is
accompanied by shocks diffusion between poles and the resulting amplification
effects, as it was highlighted by François Perroux ([Per69] page 68): "The
dominance effect of A can be transmitted to B, C or D, directly or indirectly.
If it is transmitted through an intermediary unit (T), the latter can amplify
or dampen (or even completely stop) the initial effect".

We apply our spectral cyclicality analysis to the Moroccan input-output tables
(IOTs) [OCD21]. We brings to light the profound mutations of the Moroccan
economy. By calculating the spectral influence of each industry we identify
the propulsive industries and we highlight the unique cyclical character of
the Moroccan interindustrial flows. We conclude by extracting the key
cyclicality components of the industries that form the core of the Moroccan
economy.

Our paper is organized as follows:

  1. _i_.

In Section 2, we recall the rudiments of graph topology and spectral theory of
non-negative matrix.

  2. _ii_.

In Section 3, we introduce the spectral influence of a vertex using spectral
radius, we explore its properties and we introduce the spectral cyclicality as
a measure of global diffusion on a graph.

  3. _iii_.

In Section 4, we introduce our diffusion-based clustering algorithm following
a divisive/agglomerative approach allowing overlapping.

  4. _iv_.

In Section 5, we apply our technique to the Moroccan input-output tables, from
$1995$ to $2018$.

##  2 Basics of graph theory

Throughout this paper, our results concern weighted directed graphs, those
results could be adapted to undirected graphs as well. We recall next some
fundamental results from graph topology and non-negative matrix theory, the
reader could consult for example [JLGA18] and [HJ13] for more details.

######  Definition 2.1 (Graph).

A graph $G$ is an ordered pair $G=(V,A)$ where:

  1. _i_.

$V=\\{s_{1},\ldots,s_{n}\\}$ is the set of vertices,

  2. _ii_.

$A\subset V^{2}$ is the set of unordered pairs of vertices, called edges.

When the pairs of the set $A$ are directed, the graph is called a directed
graph and the pairs are called arcs. The graph $G=(V,A,W)$, endowed with a
weight matrix $W=\left(w_{ij}\right)_{1\leqslant i,\,j\leqslant n}$ is called
a weighted graph.

######  Definition 2.2 (Graph topology [JLGA18]).

Let us denote by $G=(V,A)$ a directed graph. We define

  1. _i_.

a walk in $G$ as an alternating sequence of vertices and arcs,
$s_{0},a_{1},s_{1},\ldots,a_{n},s_{n}$, where, for $1\leqslant i\leqslant n$,
$a_{i}=s_{i-1}\,s_{i}$. The length of such a walk is $n$, which is also equal
to the number of arcs.

  2. _ii_.

a closed walk as a one with the same first and last vertex.

  3. _iii_.

a path as a walk where all vertices are distinct.

  4. _iv_.

a trail as a walk with no repeated arcs.

  5. _v_.

a circuit as a nontrivial closed trail.

  6. _vi_.

a cycle as a nontrivial closed path.

  7. _vii_.

a Hamiltonian cycle as a cycle that includes every vertex of $V$.

  8. _viii_.

a loop as an arc from a vertex to itself.

A strongly connected graph is a graph where there is a path (in each
direction) between each pair of vertices of the graph.

######  Definition 2.3 (Walk value and geometric mean value).

Given a weighted digraph $G=(V,A,W)$ and a walk
$\mathcal{W}_{s_{0}s_{n}}=s_{0},a_{1},s_{1},\ldots,a_{n},s_{n}$ in $G$ of
length $n$, one can define its value $v(\mathcal{W}_{s_{0}s_{n}})$ as the
product

$v(\mathcal{W}_{s_{0}s_{n}})=w_{01}\cdots w_{n-1\,n}$

and its geometric mean value as

$\sqrt[n]{v(\mathcal{W}_{s_{0}s_{n}})}=\sqrt[n]{w_{01}\cdots w_{n-1\,n}}$

######  Proposition 2.1.

Closed walk decomposition
Every closed walk can be decomposed into a union of Hamiltonian cycles.

###### Proof.

Suppose the closed walk is of length $m$ and contains at least one redundant
vertex, otherwise it is Hamiltonian. The sequence between two redundant
vertices is a closed walk of some length $k<m$ on which the same argument
holds recursively, dropping all those closed walks we end with a Hamiltonian
cycle. ∎

######  Definition 2.4 (Dominant cycle).

Let $G=(V,A,W)$ be a weighted digraph and let denotes by
$\mathcal{C}_{j}^{n_{j}}$ a Hamiltonian cycle of length $n_{j}$ starting from
$s_{j}$. We call dominant cycle the Hamiltonian cycle
$\mathcal{C}_{\star}^{n_{\star}}$ with maximum geometric mean value.

######  Theorem 2.2 (Perron-Frobenius [HJ13] ch.8).

Let us consider a non-negative matrix $W$, with spectral radius $\rho(W)$.
Then:

  1. i.

$\rho(W)$ is an eigenvalue of $W$, and there exists a non-negative, non-zero
vector $\mathbf{v}$ such that:

$W\mathbf{v}=\rho(W)\mathbf{v}$

  2. ii.

$\displaystyle\min_{1\leqslant i\leqslant
n}\sum_{j=1}^{n}w_{ij}\leqslant\rho(W)\leqslant\max_{1\leqslant i\leqslant
n}\sum_{j=1}^{n}w_{ij}$ and $\displaystyle\min_{1\leqslant j\leqslant
n}\sum_{i=1}^{n}w_{ij}\leqslant\rho(W)\leqslant\max_{1\leqslant j\leqslant
n}\sum_{i=1}^{n}w_{ij}$.

Moreover, if $W$ is irreducible (which corresponds to the situation of a
strongly connected graph), then

  1. i.

the eigenvalue $\rho(W)>0$ is an algebraically simple.

  2. ii.

there is a unique positive right eigenvector $\mathbf{v}$ such that
$W\mathbf{v}=\rho(W)\mathbf{v}$ and
$\displaystyle\sum_{i=1}^{n}\mathbf{v}_{i}=1$.

  3. iii.

there is a unique positive left eigenvector $\mathbf{w}$ such that
$\mathbf{w}^{T}W=\rho(W)\mathbf{w}^{T}$ and and $\mathbf{w}\cdot\mathbf{v}=1$.

######  Corollary 2.3 (Perron-Frobenius [HJ13] ch.8).

If $\mathbf{A}$ is a non-negative irreducible matrix and $\mathbf{B}$ is any
principal square submatrix of $\mathbf{A}$, then
$\rho(\mathbf{B})<\rho(\mathbf{A})$.

######  Theorem 2.4 ([HJ13] p.539 ).

Let $\mathbf{A}$ be a non-negative matrix and Tr the trace operator. Then
$\rho(\mathbf{A})=\displaystyle\underset{m\rightarrow+\infty}{\lim\sup}\left(\text{Tr}\left(\mathbf{A}^{m}\right)\right)^{\frac{1}{m}}$.
The limit superior is replaced with the limit if the matrix is irreducible and
primitive (positive at some matrix power $k\geqslant 1$).

##  3 Spectral influence and cyclicality

In the sequel, $G=(V,A,W)$ denotes a weighted digraph with adjacency matrix
$W$.

###  3.1 Spectral influence: a local approach

######  Notation.

  * •

Given a subset $S\subset V$, $W(S)$ designates the adjacency matrix $W$
resulting from the suppression of rows and columns corresponding to the
elements of $S$.

  * •

Given a matrix $\mathbf{A}$, write $\mathbf{A}\geqslant 0$ if
$\mathbf{A}_{ij}\geqslant 0$, and $\mathbf{A}>0$ if $\mathbf{A}_{ij}>0$, for
all $i,j$.

We introduce next the central concept of this work:

######  Definition 3.1 (Spectral influence).

Given a weighted graph $G=(V,A,W)$, we define the spectral influence of a
subset $S$ of $V$, as

$\displaystyle\mathcal{S}(S)$
$\displaystyle=\frac{\rho(W)-\rho(W(S))}{\rho(W)}$

On has $0\leqslant\mathcal{S}(S)\leqslant 1$.

In the particular case of a strongly connected graph, the matrix $W$ is
irreducible, one can deduce from Perron–Frobenius corollary 2.3 that
$0<\mathcal{S}(S)$ since $\rho(W(S))<\rho(W)$.

We have the following fundamental result:

######  Theorem 3.1 (Spectral radius, closed walks and dominant cycle).

The spectral radius of the matrix $W$ is asymptotically a nondecreasing
function of the graph closed walks value, in particular, it is an increasing
function of the dominant cycle value.

###### Proof.

It follows directly from theorem 2.4 that

$\rho(W)=\displaystyle\underset{m\rightarrow+\infty}{\lim\sup}\left(\text{Tr}\left(W^{m}\right)\right)^{\frac{1}{m}}=\displaystyle\underset{m\rightarrow+\infty}{\lim\sup}\left(\sum_{j=1}^{n}\left(W^{m}\right)_{jj}\right)^{\frac{1}{m}}$

Using closed walk decomposition 2.1, let designate by $\mathcal{W}^{m}_{jj}$
an $m$ closed walk starting from $j$, $\mathcal{C}^{n_{j}}_{j}$ a $n_{j}$
Hamiltonian cycle starting from $j$ and $\mathcal{C}^{n_{\star}}_{\star}$ the
dominant cycle, one has for some $m$ sufficiently large

$\displaystyle\left(\sum_{j=1}^{n}\left(W^{m}\right)_{jj}\right)^{\frac{1}{m}}$
$\displaystyle=\left(\sum_{j=1}^{n}\sum_{\mathcal{W}^{m}_{jj}}v\left(\mathcal{W}^{m}_{jj}\right)\right)^{\frac{1}{m}}$
$\displaystyle=\displaystyle\left(\sum_{j=1}^{n}\sum_{\mathcal{W}^{m}_{jj}}\underset{\mathcal{C}^{n_{i}}_{i}\subset\mathcal{W}^{m}_{jj}}{\prod}v\left(\mathcal{C}_{i}^{n_{i}}\right)\right)^{\frac{1}{m}}$
$\displaystyle=\displaystyle\sqrt[n_{\star}]{v\left(\mathcal{C}^{n_{\star}}_{\star}\right)}\left(\sum_{j=1}^{n}\sum_{\mathcal{W}^{m}_{jj}}\underset{\mathcal{C}^{n_{i}}_{i}\subset\mathcal{W}^{m}_{jj}}{\prod}\underbrace{\left(\frac{v\left(\mathcal{C}_{i}^{n_{i}}\right)}{{v\left(\mathcal{C}^{n_{\star}}_{\star}\right)}^{\frac{n_{1}}{n_{\star}}}}\right)}_{\leqslant
1}\right)^{\frac{1}{m}}$
$\displaystyle=\mathcal{O}\left(\sqrt[n_{\star}]{v\left(\mathcal{C}^{n_{\star}}_{\star}\right)}\right)$

Since

$\displaystyle\kappa(W,\star)=\left(\sum_{j=1}^{n}\sum_{\mathcal{W}^{m}_{jj}}\underset{\mathcal{C}^{n_{i}}_{i}\in\mathcal{W}^{m}_{jj}}{\Pi}\left(\frac{v\left(\mathcal{C}_{i}^{n_{i}}\right)}{{v\left(\mathcal{C}^{n_{\star}}_{\star}\right)}^{\frac{n_{1}}{n_{\star}}}}\right)\right)^{\frac{1}{m}}$
$\displaystyle\leqslant\left(n\sum_{\mathcal{W}^{m}_{jj}}\underset{\mathcal{C}^{n_{i}}_{i}\in\mathcal{W}^{m}_{jj}}{\Pi}\left(\frac{v\left(\mathcal{C}_{i}^{n_{i}}\right)}{{v\left(\mathcal{C}^{n_{\star}}_{\star}\right)}^{\frac{n_{1}}{n_{\star}}}}\right)\right)^{\frac{1}{m}}$
$\displaystyle\leqslant\left(n\,n^{m-1}\right)^{\frac{1}{m}}$
$\displaystyle=n$

where the upper bound is the number of closed walks in a complete graph. ∎

######  Remark 3.1.

  1. 1.

The first direct consequence is that the spectral influence is maximal for
nodes whose deletion reduces the closed walks supremum value in $G$
asymptotically the most, which is equivalent to the deletion of Hamiltonian
cycles with the highest geometric mean value as a consequence of theorem 3.1:
suppose the graph has $l$ cycles decreasingly ordered by their geometric mean
values
$\displaystyle\sqrt[n_{1}]{v\left(\mathcal{C}^{n_{1}}_{i_{1}}\right)}\geqslant\ldots\geqslant\sqrt[n_{k}]{v\left(\mathcal{C}^{n_{k}}_{i_{k}}\right)}\geqslant\ldots\geqslant\sqrt[n_{l}]{v\left(\mathcal{C}^{n_{l}}_{i_{l}}\right)}$,
suppose the suppression of a vertex $s$ induce the suppression of the first
$k-1$ Hamiltonian cycles in the order, the spectral influence is thus

$\displaystyle\mathcal{S}(s)$
$\displaystyle=1-\mathcal{O}\left(\frac{\sqrt[n_{k}]{v\left(\mathcal{C}^{n_{k}}_{i_{k}}\right)}}{\sqrt[n_{1}]{v\left(\mathcal{C}^{n_{1}}_{i_{1}}\right)}}\right)$
(1)

  2. 2.

In the case of an unweighted graphs, the sum
$\sum_{j=1}^{n}\left(W^{m}\right)_{jj}$ is just the number of $m$ closed walks
and the spectral influence of a vertex $s$ is a function of the number of
closed walks passing through $s$.

  3. 3.

If the graph is not strongly connected, then the matrix $W$ is reducible and
it is similar to bloc triangular matrix
$\displaystyle\left(\begin{matrix}W_{11}&W_{12}&\ldots&W_{1k}\\\
0&W_{22}&\ldots&W_{2k}\\\ \vdots&\ddots&\ddots&\vdots\\\ 0&\ldots&0&W_{kk}\\\
\end{matrix}\right)$ ([HJ13] page 63). The spectral radius is
$\rho(W)=\displaystyle\max\\{\rho(W_{11}),\ldots,\rho(W_{nn})\\}$ and the same
reasoning holds per bloc, while the spectral influence is null for non
dominant blocs.

  4. 4.

Like other mathematical means, the geometric mean is sensible to extreme
values.

  5. 5.

One can enumerate the different configurations of spectral influence: denote
by $\mathcal{C}^{n_{i_{1}}}_{i_{1}}$ the dominant cycle and
$\mathcal{C}^{n_{i_{s}}}_{i_{s}}$ the dominant cycle after the suppression of
a vertex $s$, there are four situations

    1. (a)

the vertex $s$ does not belongs to the dominant cycle and

      1. i.

$s$ does not belongs to a Hamiltonian cycle crossing the dominant cycle:
$\kappa(W(s),i_{1})=\kappa(W,i_{1})$ and $\mathcal{S}(s)=0$.

      2. ii.

$s$ belongs to a Hamiltonian cycle crossing the dominant cycle:
$\kappa(W(s),i_{1})<\kappa(W,i_{1})$ and $\mathcal{S}(s)>0$ (irreducibility
corollary 2.3).

    2. (b)

the vertex $s$ belongs to the dominant cycle and

      1. i.

the vertex $s$ does not belongs to a Hamiltonian cycle crossing
$\mathcal{C}^{n_{i_{s}}}_{i_{s}}$: $\kappa(W(s),i_{2})=\kappa(W,i_{2})$ and
$\mathcal{S}(s)>0$.

      2. ii.

the vertex $s$ belongs to a Hamiltonian cycle crossing
$\mathcal{C}^{n_{i_{s}}}_{i_{s}}$: $\kappa(W(s),i_{2})<\kappa(W,i_{2})$ and
$\mathcal{S}(s)$ is greater than situation (b).i.

  6. 6.

The spectral influence is a relative local measure of interdependence and
diffusion in networks, it measure the relative cyclicality involving the
vertex in question.

###  3.2 Spectral influence: a global approach

We introduced above the concept of spectral influence as a measure of local
influence, and it was demonstrated that this quantity is reliant on closed
walks that involve the vertex under consideration, which in turn require the
existence of Hamiltonian cycles (theorem 3.1). The total sum of the elementary
influences will be as large as the number of vertices involved in Hamiltonian
cycles, to achieve the maximum when all the vertices are contained in one
dominant cycle (redundancy effect). Conversely, the total sum will decreases
as much as the flow is split into small Hamiltonian cycles and loops.

Given a subset $S\subseteq V$ of cardinal $k$, one can write

$\displaystyle 0\leqslant\sum_{s\in S}\left(\rho(W)-\rho(W(s))\right)\leqslant
k\,\rho(W)$ (2)

In the case of strongly connected graphs: the left hand bound is never reached
by Perron–Frobenius corollary 2.3. Conversely, the right hand bound is strict
except the case where the graph is a Hamiltonian cycle (the suppression of a
vertex destruct every closed walk). The redundancy effect is the fact that a
dominant cycle of length $l\geqslant 2$ contains $l$ vertices whose
suppression reduce the spectral radius and increase the spectral influence, it
has the following consequence:

######  Corollary 3.2 (The whole is not the sum of the parts).

Let $G=(V,A,W)$ be a weighted graph and $S$ be a subset of $V$ of cardinal
$k\geqslant 2$. The whole is not the sum of the parts in the sens that, the
equality

$\sum_{s\in S}\left(\rho(W)-\rho(W(s))\right)=\rho(W)-\rho(W(S))$

does not hold in general. In particular, $\mathcal{S}$ is not a mathematical
measure on $V$.

###### Proof.

Figure 1 gives a counter-example. Consequently, the spectral influence
$\mathcal{S}$ does not respect countable additivity of mathematical measures.
∎

Figure 1: The whole is not the sum of the parts:
$\sum_{s=1}^{2}\left(\rho(W)-\rho(W(s))\right)=0.42<\rho(W)-\rho(W(S))=1.74$
while
$\sum_{s=4}^{5}\left(\rho(W)-\rho(W(s))\right)=0.08>\rho(W)-\rho(W(S))=0.06$.

######  Definition 3.2 (Spectral cyclicality).

Given a weighted graph $G=(V,A,W)$, we define the spectral cyclicality of $G$
as

$\displaystyle\mathbf{S}$
$\displaystyle=\displaystyle\sum_{s=1}^{n}\mathcal{S}(s)$

The spectral cyclicality characterize interdependence and diffusion in the
graph. According to equation 2, $0\leqslant\mathbf{S}\leqslant n$ and three
extreme configurations could occur:

  1. 1.

The situation of perfect cyclicality

     * •

Hamiltonian cycle: when the connection structure is reduced to a Hamiltonian
cycle and the elementary spectral influence is $1$ (the destruction of a
vertex disconnect the graph). The spectral cyclicality is $n$.

  2. 2.

Situations of weak cyclicality

     * •

Fair division: each vertex shares uniformly the weight $w_{i}$ with all the
vertices. The spectral radius of the matrix equals the mean flow
$\displaystyle\frac{\sum_{i=1}^{n}w_{i}}{n}$ and the spectral influence is

$\mathcal{S}(s)=\frac{\displaystyle\frac{\sum_{i=1}^{n}w_{i}}{n}-\displaystyle\frac{\sum_{i\neq
s}w_{i}}{n}}{\displaystyle\frac{\sum_{i=1}^{n}w_{i}}{n}}=\frac{w_{s}}{\sum_{i=1}^{n}w_{i}}$

The spectral cyclicality is $1$.

     * •

Triangular structure: each vertex $s_{i}$ serves the next vertices $s_{j}$,
$i\leqslant j$, until the last one which is self-connected (loop). The
spectrum of the adjacency matrix contains one positive non null eigenvalue,
the spectral influence equals one for the looped vertex and null elsewhere and
the spectral cyclicality is $1$.

  3. 3.

No cycles

     * •

Chain: each vertex $s_{i}$ is connected to the next vertex $s_{i+1}$
constituting a chain.

     * •

Loops: every pole is looped and there is no diffusion. The spectral influence
is null except eventually the pole with the dominant loop of value
$w_{{i^{\star}_{1}}{i^{\star}_{1}}}$, the spectral cyclicality is
$\displaystyle\frac{w_{{i^{\star}_{1}}{i^{\star}_{1}}}-w_{{i^{\star}_{2}}{i^{\star}_{2}}}}{w_{{i^{\star}_{1}}{i^{\star}_{1}}}}$,
where $w_{{i_{s}}{i_{s}}}$ is the second maximum loop value.

Following remark 3.1, one can deduce that the spectral cyclicality is greater
as much as the graph dominant cycle is longer and the graph is strongly
connected, and the maximum is reached when the graph is a Hamiltonian cycle.
Conversely, the spectral cyclicality is reduced by the introduction of non-
dominant cycles with high geometric mean values, disjoint from the dominant
one, reducing the vertices spectral influence by reducing dominant cycles gap
and redundancy effect.

We close this section by plotting in figure 2 a bestiary of unweighted
digraphs with the corresponding spectral influences and cyclicality.

(a) $\mathbf{S}=3$.

(b) $\mathbf{S}=2.25$.

(c) $\mathbf{S}=1.64$.

(d) $\mathbf{S}=1$.

(e) $\mathbf{S}=0.76$.

(f) $\mathbf{S}=0$.

Figure 2: The spectral cyclicality for different disposition of a three
vertices unweighted connected digraph: a) perfect diffusion on a cycle, b)
diffusion slowed down by a circuit, c) diffusion slowed down by a loop, d)
slowest diffusion in a complete graph, e) partial diffusion, f) one way
diffusion.

##  4 Spectral influence and cyclicality clustering

The spectral cyclicality is a measurement of cyclicality and diffusion on
graphs. In the sequel, we define a clustering algorithm based on this
measurement to regroup vertices with strong cyclicality. In order to implement
our algorithm, we need to adopt the following convention about isolated
vertices:

######  Assumption 4.1.

If the digraph $G=(V,A,W)$ is reduced to a unique vertex $s$, then the
spectral cyclicality is $\mathbf{S}=1$.

The idea behind the assumption 4.1 is to decide whether two vertices should be
split into two isolated vertices or not, the spiting decision will be taken if
the spectral cyclicality is not greater than $1$. Let’s analyze the
implication of this assumption: Given a graph with two vertices and the
corresponding non-negative weight matrix $W=\left(\begin{matrix}a&b\\\
c&d\end{matrix}\right)$, the spectrum of the matrix $W$ is

$\\{\lambda_{1},\lambda_{2}\\}=\displaystyle\left\\{\frac{1}{2}\left(a+d+\sqrt{(a-d)^{2}+4bc}\right),\frac{1}{2}\left(a+d-\sqrt{(a-d)^{2}+4bc}\right)\right\\}$

The spectral cyclicality is therefore

$\displaystyle\mathbf{S}$
$\displaystyle=\displaystyle\frac{\frac{1}{2}\left(a+d+\sqrt{(a-d)^{2}+4bc}\right)-d}{\frac{1}{2}\left(a+d+\sqrt{(a-d)^{2}+4bc}\right)}+\frac{\frac{1}{2}\left(a+d+\sqrt{(a-d)^{2}+4bc}\right)-a}{\frac{1}{2}\left(a+d+\sqrt{(a-d)^{2}+4bc}\right)}$
$\displaystyle=\displaystyle\frac{2}{\displaystyle
1+\frac{a+d}{\sqrt{(a-d)^{2}+4bc}}}$

$\mathbf{S}$ reaches its maximum $2$ in the case of a cycle ($a=d=0$). If
either $c$ or $b$ is null, then $\mathbf{S}<1$ and our algorithm split the
vertices since there is no diffusion on the graph. In the general case, the
splitting condition is $\frac{a+d}{\sqrt{(a-d)^{2}+4bc}}\geqslant 1$ which is
equivalent to $ad\geqslant cd$, and this occurs when the product of loop
values is greater than the cycle value.

######  Notation.

We will designate by

  * •

$[1,n]$ the set of integers from $1$ to $n$.

  * •

$\tilde{W}(S)$ the resulting matrix obtained by deleting all rows and columns
of $W$, except for those that correspond to the elements of $S$.

  * •

$\overline{W}(S)$ the matrix resulting from $W$ by setting $w_{ij}=0$ if
$i\neq j$ and $\\{i,j\\}\subset S$.

###  4.1 Divisive cyclicality clustering

We explain hereafter how the divisive cyclicality clustering algorithm works:
given a weighted digraph $G=(V,A,W)$, the algorithm eliminates the vertex
whose removal increases the spectral cyclicality the most, if no increase is
possible, the algorithm eliminates the vertices whose removal does not
decrease spectral cyclicality. The same operation is applied to deleted
vertices until all vertices are classified.

Algorithm 1: | Divisive cyclicality clustering
---|---
Input: | a weight matrix $W\in M_{n}$ and a vertices set $V$
Initialization: | set $S=\\{\\}$
While: | $\displaystyle\max_{i\in V}\mathbf{S}(W(i))\geqslant\mathbf{S}(W)$
| update $W=W(i^{\star})$ , $S=S\cup\\{i^{\star}\\}$, where
$i^{\star}=\displaystyle\arg\max_{i\in V}\mathbf{S}(W(i))$
Update: | set $V=S$.
Output: | a partition $\\{V_{1},\ldots,V_{k}\\}$.

###  4.2 Agglomerative cyclicality clustering

Given a digraph $G=(V,A,W)$, the agglomerative cyclicality clustering
algorithm groups vertices based on their spectral cyclicality, with the
highest values being grouped first. If no increase in cyclicality is possible,
the algorithm repeats the process with the remaining vertices until all
vertices are classified.

Algorithm 2: | Agglomerative cyclicality clustering
---|---
Input: | a weight matrix $W\in M_{n}$ and a vertices set $V$
Initialization: | set $S=\displaystyle\arg\max_{(j,k)\in V^{2}}\mathbf{S}(\tilde{W}(\\{j,k\\}))$
While: | $\displaystyle\max_{i\in V}\mathbf{S}(\tilde{W}(S\cup\\{i\\}))>\mathbf{S}(\tilde{W}(S))$
| update $S=S\cup\\{i^{\star}\\}$, where
$i^{\star}=\displaystyle\arg\max_{i\in V}\mathbf{S}(\tilde{W}(S\cup\\{i\\}))$
Update: | set $V=V\setminus S$.
Output: | a partition $\\{V_{1},\ldots,V_{k}\\}$.

###  4.3 Overlapping clustering

Let’s denote by $S_{1}(W)$ the first partition obtained by applying the
divisive (resp. agglomerative) cyclicality algorithm. Overlapping clustering
can be achieved by preserving the vertices and deleting the partition intra-
connections at each stage:

Algorithm 3: | Overlapping divisive/agglomerative cyclicality clustering
---|---
Input: | a weight matrix $W\in M_{n}$ and a vertices set $V$
Initialization: | set $O=S_{1}(W)$
While: | $\displaystyle\mathbf{S}(\overline{W}(O))>1$
| update $W=\overline{W}(O)$ , $O=S_{1}(W)$
Output: | an overlapping partition $\\{O_{1},\ldots,O_{k}\\}$.

###  4.4 Discussion

The schemes of divisive and agglomerative clustering are opposite in nature.
Divisive clustering starts with a single cluster that contains all the
vertices, and then recursively splits it into smaller clusters, whereas
agglomerative clustering starts with each vertex as a separate cluster and
then merges them into larger clusters.

Overall, both divisive and agglomerative cyclicality clustering share the same
philosophy of grouping vertices with high cyclicality, even though they
achieve this goal through different methods. Let’s analyze how the algorithm
practically works in the case of a divisive scheme:

Following remark 3.1.5, the algorithm starts deleting vertices not strongly
connected to the dominant cycle, then it starts deleting competitor cycles
reducing the cyclicality. In the case of unweighted graphs, the algorithm
chooses the longest cycle as the dominant one. It important to notice that the
algorithm outcome may not be a perfect cycle, but a strong cyclicality
component, depending on the structure of the network. When overlapping is
allowed, the algorithm cuts off all the arcs between the element of a
partition keeping the loops, and preventing the same cycles to appears in
other partitions.

One limitation of the cyclicality clustering algorithm is the exaequo
situation where the algorithm should pick a vertex randomly, moreover, the
clustering algorithm depends on the division/agglomeration path, i.e.,
vertices suppression/concatenation order, and some vertices could be
eliminated/regrouped earlier producing a component not cyclically optimal.

##  5 Application to input-output analysis

The application of spectral cyclicality analysis to input-output analysis can
help to better understand interindustrial flows and connections between
industrial poles. On the one hand, Some poles plays the role of propulsive
industries driving other poles to increase their production and creating
transformation cycles. On the other hand, such mechanism is synonym with
interdependence, influence, shock diffusion and amplification effects.

In opposition to input-output analysis based on Leontieff or Ghosh
normalization, our technique does not require a direction of normalization
(supply or demand) and it is applied directly to the absolute flow, moreover,
it does not require any coefficients stability hypothesis ([Lan72]).

In the sequel, we analyze the Moroccan interindustrial flows of the OECD
Input-Output Tables (IOTs) [OCD21], from $1995$ to $2018$. The data traces
flows of intermediate goods and services of $45$ industrial poles in million
US Dollars (see appendix 7.1).

###  5.1 Moroccan interindustrial cyclicality analysis

We report in figure 3 the evolution of spectral cyclicality of interindustrial
exchanges, the curve reflects a clear upward trend in cyclicality.

Figure 3: Evolution of Moroccan spectral cyclicality from $1995$ to $2018$.

We perform a benchmark for a pool of $30$ countries, each country is
designated by its three letters ISO code222International Organization for
Standardization https://www.iso.org/iso-3166-country-codes.html., we can
notice that Morocco manifests a distinguished cyclical character:

| ARG | AUS | BEL | BRA | CAN | CHN | COL | CZE | DEU | DNK | FRA | GBR
---|---|---|---|---|---|---|---|---|---|---|---|---
$\mathbf{S}$ | $0.687$ | $0.599$ | $0.363$ | $0.663$ | $0.783$ | $0.297$ | $0.538$ | $0.480$ | $0.379$ | $0.872$ | $0.566$ | $0.377$

| HUN | ITA | JPN | KAZ | MAR | MEX | MYS | POR | RUS | SAU | SPA | THA
---|---|---|---|---|---|---|---|---|---|---|---|---
$\mathbf{S}$ | $0.445$ | $0.729$ | $0.426$ | $0.387$ | $1.133$ | $0.349$ | $0.957$ | $0.293$ | $0.776$ | $1.024$ | $0.750$ | $0.398$

| TUN | TUR | SGP | USA | VNM | ZAF
---|---|---|---|---|---|---
$\mathbf{S}$ | $0.217$ | $0.200$ | $0.331$ | $0.615$ | $0.589$ | $0.617$

###  5.2 Propulsive industries and key cyclicality components

We perform next a spectral cyclicality analysis of the Moroccan
interindustrial flows from $1995$ to $2018$. The next two tables report, for
each industry, the spectral influence $\mathcal{S}$ and the corresponding
cyclical cluster in the case of nonoverlapping divisive (DC) and agglomerative
(AC) clustering, the results illustrate the profound changes in the
interindustrial structure.

Industry | $\mathbf{1995}$ | $\mathbf{2000}$ | $\mathbf{2005}$
---|---|---|---
| $\mathcal{S}$ | DC | AC | $\mathcal{S}$ | DC | AC | $\mathcal{S}$ | DC | AC
D01T02 | $0.16774$ | $1$ | $\iota$ | $0.301$ | $1$ | $\iota$ | $0.33535$ | $1$ | $\iota$
D03 | $0.0001$ | $\iota$ | $\iota$ | $0.00008$ | $4$ | $6$ | $0.00016$ | $3$ | $5$
D05T06 | $4*10^{-7}$ | $1$ | $6$ | $4*10^{-7}$ | $1$ | $4$ | $6*10^{-7}$ | $\iota$ | $5$
D07T08 | $0.00025$ | $1$ | $\iota$ | $0.00002$ | $2$ | $2$ | $6*10^{-6}$ | $3$ | $4$
D09 | $0.00005$ | $1$ | $6$ | $0.00001$ | $1$ | $4$ | $0.00001$ | $1$ | $\iota$
D10T12 | $0.16811$ | $1$ | $\iota$ | $0.30329$ | $1$ | $\iota$ | $0.38024$ | $1$ | $\iota$
D13T15 | $0.00542$ | $\iota$ | $\iota$ | $0.00179$ | $\iota$ | $\iota$ | $0.00095$ | $\iota$ | $\iota$
D16 | $0.00016$ | $\iota$ | $\iota$ | $0.0001$ | $\iota$ | $\iota$ | $0.00007$ | $3$ | $\iota$
D17T18 | $0.0002$ | $1$ | $\iota$ | $0.00015$ | $\iota$ | $\iota$ | $0.00011$ | $\iota$ | $\iota$
D19 | $0.00138$ | $1$ | $1$ | $0.00123$ | $1$ | $1$ | $0.00136$ | $1$ | $1$
D20 | $0.00581$ | $1$ | $\iota$ | $0.00315$ | $1$ | $\iota$ | $0.00266$ | $3$ | $\iota$
D21 | $0.00062$ | $1$ | $\iota$ | $0.00043$ | $1$ | $\iota$ | $0.00037$ | $3$ | $\iota$
D22 | $0.00072$ | $\iota$ | $\iota$ | $0.00033$ | $\iota$ | $6$ | $0.00037$ | $1$ | $5$
D23 | $0.0004$ | $2$ | $\iota$ | $0.00014$ | $\iota$ | $\iota$ | $0.00011$ | $3$ | $\iota$
D24 | $0.00018$ | $1$ | $4$ | $0.00007$ | $3$ | $4$ | $0.00009$ | $3$ | $4$
D25 | $0.00038$ | $\iota$ | $4$ | $0.00021$ | $3$ | $4$ | $0.00015$ | $3$ | $4$
D26 | $0.00005$ | $1$ | $\iota$ | $0.00001$ | $\iota$ | $\iota$ | $0.00001$ | $1$ | $\iota$
D27 | $0.0004$ | $\iota$ | $\iota$ | $0.00016$ | $\iota$ | $\iota$ | $0.00012$ | $3$ | $\iota$
D28 | $0.00007$ | $1$ | $\iota$ | $0.00003$ | $1$ | $\iota$ | $0.00003$ | $1$ | $\iota$
D29 | $0.00005$ | $\iota$ | $\iota$ | $0.00002$ | $\iota$ | $\iota$ | $0.00002$ | $3$ | $\iota$
D30 | $2*10^{-6}$ | $1$ | $\iota$ | $2*10^{-6}$ | $\iota$ | $\iota$ | $1*10^{-6}$ | $3$ | $\iota$
D31T33 | $0.00033$ | $1$ | $\iota$ | $0.00013$ | $4$ | $\iota$ | $0.00008$ | $3$ | $5$
D35 | $0.00077$ | $1$ | $1$ | $0.00029$ | $1$ | $\iota$ | $0.0002$ | $1$ | $\iota$
D36T39 | $0.00001$ | $1$ | $5$ | $0.00001$ | $1$ | $2$ | $0.00001$ | $1$ | $\iota$
D41T43 | $0.00115$ | $2$ | $2$ | $0.00036$ | $2$ | $2$ | $0.00028$ | $2$ | $2$
D45T47 | $0.01764$ | $1$ | $\iota$ | $0.01603$ | $1$ | $6$ | $0.01614$ | $1$ | $5$
D49 | $0.00051$ | $1$ | $2$ | $0.0004$ | $1$ | $1$ | $0.0005$ | $1$ | $1$
D50 | $0.00003$ | $1$ | $\iota$ | $0.00001$ | $1$ | $6$ | $0.00002$ | $1$ | $2$
D51 | $0.00005$ | $1$ | $\iota$ | $0.00005$ | $1$ | $\iota$ | $0.00006$ | $1$ | $\iota$
D52 | $0.0001$ | $1$ | $\iota$ | $0.00007$ | $1$ | $\iota$ | $0.00009$ | $1$ | $\iota$
D53 | $0.00002$ | $1$ | $\iota$ | $7*10^{-6}$ | $4$ | $6$ | $8*10^{-6}$ | $1$ | $5$
D55T56 | $0.00279$ | $1$ | $3$ | $0.00172$ | $1$ | $3$ | $0.0013$ | $1$ | $3$
D58T60 | $0.0001$ | $1$ | $\iota$ | $0.00007$ | $1$ | $\iota$ | $0.00006$ | $1$ | $\iota$
D61 | $0.00015$ | $1$ | $\iota$ | $0.00006$ | $1$ | $6$ | $0.00006$ | $1$ | $\iota$
D62T63 | $0.00001$ | $1$ | $\iota$ | $7*10^{-6}$ | $4$ | $\iota$ | $5*10^{-6}$ | $3$ | $\iota$
D64T66 | $0.00019$ | $1$ | $\iota$ | $0.00041$ | $1$ | $\iota$ | $0.00039$ | $1$ | $\iota$
D68 | $0.00015$ | $1$ | $2$ | $0.00008$ | $4$ | $5$ | $0.00006$ | $2$ | $2$
D69T75 | $0.00011$ | $1$ | $\iota$ | $0.0001$ | $\iota$ | $\iota$ | $0.0001$ | $3$ | $\iota$
D77T82 | $0.00005$ | $1$ | $\iota$ | $0.00003$ | $1$ | $\iota$ | $0.00003$ | $1$ | $\iota$
D84 | $0.00038$ | $1$ | $3$ | $0.00041$ | $1$ | $3$ | $0.00022$ | $1$ | $3$
D85 | $0.00023$ | $1$ | $\iota$ | $0.00023$ | $1$ | $\iota$ | $0.00012$ | $1$ | $\iota$
D86T88 | $0.00005$ | $1$ | $\iota$ | $0.00006$ | $1$ | $\iota$ | $0.00003$ | $1$ | $2$
D90T93 | $2*10^{-6}$ | $1$ | $\iota$ | $1*10^{-6}$ | $1$ | $3$ | $8*10^{-7}$ | $1$ | $\iota$
D94T96 | $0.00001$ | $1$ | $5$ | $0.00001$ | $1$ | $5$ | $7*10^{-6}$ | $1$ | $3$
D97T98 | $0$ | $\iota$ | $\iota$ | $0$ | $\iota$ | $\iota$ | $0$ | $\iota$ | $\iota$

Industry | $\mathbf{2010}$ | $\mathbf{2015}$ | $\mathbf{2018}$
---|---|---|---
| $\mathcal{S}$ | DC | AC | $\mathcal{S}$ | DC | AC | $\mathcal{S}$ | DC | AC
D01T02 | $0.54341$ | $1$ | $4$ | $0.54843$ | $1$ | $4$ | $0.53322$ | $1$ | $4$
D03 | $0.00008$ | $2$ | $\iota$ | $0.00011$ | $\iota$ | $\iota$ | $0.00007$ | $\iota$ | $\iota$
D05T06 | $2*10^{-7}$ | $2$ | $5$ | $4*10^{-8}$ | $2$ | $5$ | $1*10^{-8}$ | $4$ | $2$
D07T08 | $0.00019$ | $2$ | $1$ | $0.00008$ | $4$ | $1$ | $0.00008$ | $5$ | $1$
D09 | $7*10^{-6}$ | $1$ | $4$ | $4*10^{-6}$ | $1$ | $4$ | $3*10^{-6}$ | $1$ | $4$
D10T12 | $0.52652$ | $1$ | $4$ | $0.58081$ | $1$ | $4$ | $0.58233$ | $1$ | $4$
D13T15 | $0.00026$ | $\iota$ | $\iota$ | $0.00018$ | $\iota$ | $\iota$ | $0.00015$ | $\iota$ | $\iota$
D16 | $0.00005$ | $\iota$ | $\iota$ | $0.00003$ | $\iota$ | $\iota$ | $0.00003$ | $\iota$ | $\iota$
D17T18 | $0.00016$ | $1$ | $4$ | $0.00014$ | $1$ | $4$ | $0.00013$ | $1$ | $4$
D19 | $0.00016$ | $1$ | $2$ | $0.00005$ | $1$ | $2$ | $7*10^{-7}$ | $4$ | $2$
D20 | $0.00298$ | $\iota$ | $\iota$ | $0.00229$ | $4$ | $\iota$ | $0.00293$ | $5$ | $\iota$
D21 | $0.00032$ | $\iota$ | $\iota$ | $0.00028$ | $\iota$ | $\iota$ | $0.00022$ | $\iota$ | $\iota$
D22 | $0.00028$ | $2$ | $5$ | $0.00019$ | $2$ | $5$ | $0.00018$ | $3$ | $5$
D23 | $0.00013$ | $\iota$ | $\iota$ | $0.00007$ | $\iota$ | $\iota$ | $0.00005$ | $2$ | $\iota$
D24 | $0.00004$ | $\iota$ | $\iota$ | $0.00001$ | $\iota$ | $\iota$ | $0.00001$ | $\iota$ | $\iota$
D25 | $0.0001$ | $\iota$ | $\iota$ | $0.00005$ | $\iota$ | $\iota$ | $0.00006$ | $\iota$ | $\iota$
D26 | $5*10^{-6}$ | $\iota$ | $4$ | $3*10^{-6}$ | $\iota$ | $\iota$ | $3*10^{-6}$ | $\iota$ | $\iota$
D27 | $0.0001$ | $\iota$ | $\iota$ | $0.00004$ | $\iota$ | $\iota$ | $0.00004$ | $\iota$ | $\iota$
D28 | $0.00002$ | $1$ | $4$ | $6*10^{-6}$ | $1$ | $4$ | $8*10^{-6}$ | $1$ | $4$
D29 | $0.00003$ | $\iota$ | $\iota$ | $0.00007$ | $\iota$ | $\iota$ | $0.00009$ | $\iota$ | $\iota$
D30 | $2*10^{-6}$ | $\iota$ | $\iota$ | $2*10^{-6}$ | $\iota$ | $\iota$ | $1*10^{-6}$ | $\iota$ | $\iota$
D31T33 | $0.00004$ | $\iota$ | $\iota$ | $0.00002$ | $\iota$ | $\iota$ | $0.00002$ | $\iota$ | $\iota$
D35 | $0.00014$ | $1$ | $4$ | $0.00012$ | $\iota$ | $\iota$ | $0.00015$ | $2$ | $\iota$
D36T39 | $3*10^{-6}$ | $1$ | $4$ | $3*10^{-6}$ | $1$ | $4$ | $4*10^{-6}$ | $1$ | $4$
D41T43 | $0.00036$ | $1$ | $1$ | $0.0002$ | $2$ | $1$ | $0.0002$ | $2$ | $1$
D45T47 | $0.01347$ | $2$ | $5$ | $0.01056$ | $2$ | $5$ | $0.01057$ | $3$ | $5$
D49 | $0.00035$ | $2$ | $2$ | $0.00029$ | $\iota$ | $2$ | $0.00033$ | $3$ | $5$
D50 | $0.00003$ | $1$ | $4$ | $0.00003$ | $\iota$ | $\iota$ | $0.00005$ | $\iota$ | $\iota$
D51 | $0.00003$ | $2$ | $5$ | $0.00004$ | $2$ | $5$ | $0.00004$ | $1$ | $4$
D52 | $0.00008$ | $\iota$ | $\iota$ | $0.00007$ | $\iota$ | $\iota$ | $0.00007$ | $\iota$ | $\iota$
D53 | $4*10^{-6}$ | $2$ | $5$ | $2*10^{-6}$ | $2$ | $5$ | $1*10^{-6}$ | $3$ | $5$
D55T56 | $0.00059$ | $\iota$ | $3$ | $0.0005$ | $\iota$ | $3$ | $0.00067$ | $\iota$ | $3$
D58T60 | $0.00003$ | $\iota$ | $\iota$ | $0.00003$ | $\iota$ | $\iota$ | $0.00003$ | $\iota$ | $\iota$
D61 | $0.00004$ | $\iota$ | $5$ | $0.00002$ | $\iota$ | $5$ | $0.00002$ | $3$ | $5$
D62T63 | $2*10^{-6}$ | $\iota$ | $6$ | $3*10^{-6}$ | $3$ | $\iota$ | $3*10^{-6}$ | $\iota$ | $\iota$
D64T66 | $0.00034$ | $\iota$ | $6$ | $0.00031$ | $3$ | $\iota$ | $0.00035$ | $\iota$ | $6$
D68 | $0.00002$ | $2$ | $6$ | $0.00002$ | $3$ | $5$ | $0.00002$ | $2$ | $6$
D69T75 | $0.00007$ | $\iota$ | $\iota$ | $0.00006$ | $\iota$ | $\iota$ | $0.00007$ | $\iota$ | $\iota$
D77T82 | $0.00001$ | $\iota$ | $6$ | $0.00002$ | $3$ | $\iota$ | $0.00002$ | $\iota$ | $\iota$
D84 | $0.00029$ | $\iota$ | $3$ | $0.00028$ | $\iota$ | $3$ | $0.00031$ | $2$ | $3$
D85 | $0.00013$ | $1$ | $4$ | $0.0001$ | $1$ | $4$ | $0.00011$ | $1$ | $4$
D86T88 | $0.00005$ | $1$ | $4$ | $0.00003$ | $1$ | $4$ | $0.00005$ | $1$ | $4$
D90T93 | $1*10^{-6}$ | $2$ | $5$ | $1*10^{-6}$ | $\iota$ | $\iota$ | $2*10^{-6}$ | $\iota$ | $\iota$
D94T96 | $4*10^{-6}$ | $\iota$ | $5$ | $4*10^{-6}$ | $\iota$ | $\iota$ | $4*10^{-6}$ | $3$ | $5$
D97T98 | $0$ | $\iota$ | $\iota$ | $0$ | $\iota$ | $\iota$ | $0$ | $\iota$ | $\iota$

According to our algorithm, the interindustrial Moroccan flows in $2018$ could
be regrouped and ordered from a cyclicality point of view in one the following
two ways:

Divisive clustering:

  1. 1.

Component I:

    1. (a)

D19: Coke and refined petroleum products,

    2. (b)

D05T06: Mining and quarrying, energy producing products.

  2. 2.

Component II:

    1. (a)

D23: Other non-metallic mineral products,

    2. (b)

D41T43: Construction,

    3. (c)

D35: Electricity, gas, steam and air conditioning supply,

    4. (d)

D84: Public administration and defense, compulsory social security,

    5. (e)

D68: Real estate activities.

  3. 3.

Component III:

    1. (a)

D10T12: Food products, beverages and tobacco,

    2. (b)

D01T02: Agriculture, hunting, forestry,

    3. (c)

D17T18: Paper products and printing,

    4. (d)

D85: Education,

    5. (e)

D86T88: Human health and social work activities,

    6. (f)

D51: Air transport,

    7. (g)

D28: Machinery and equipment, nec,

    8. (h)

D36T39: Water supply, sewerage, waste management and remediation activities,

    9. (i)

D09: Mining support service activities.

  4. 4.

Component IV:

    1. (a)

D45T47: Wholesale and retail trade, repair of motor vehicles,

    2. (b)

D49: Land transport and transport via pipelines,

    3. (c)

D22: Rubber and plastics products,

    4. (d)

D61: Telecommunications,

    5. (e)

D53: Postal and courier activities,

    6. (f)

D94T96: Other service activities.

  5. 5.

Component V:

    1. (a)

D20: Chemical and chemical products,

    2. (b)

D07T08: Mining and quarrying, non-energy producing products.

Agglomerative clustering:

  1. 1.

Component I:

    1. (a)

D41T43: Construction,

    2. (b)

D07T08: Mining and quarrying, non-energy producing products.

  2. 2.

Component II:

    1. (a)

D19: Coke and refined petroleum products,

    2. (b)

D05T06: Mining and quarrying, energy producing products.

  3. 3.

Component III:

    1. (a)

D84: Public administration and defense, compulsory social security,

    2. (b)

D55T56: Accommodation and food service activities.

  4. 4.

Component IV:

    1. (a)

D10T12: Food products, beverages and tobacco,

    2. (b)

D01T02: Agriculture, hunting, forestry,

    3. (c)

D17T18: Paper products and printing,

    4. (d)

D85: Education,

    5. (e)

D86T88: Human health and social work activities,

    6. (f)

D51: Air transport,

    7. (g)

D28: Machinery and equipment, nec,

    8. (h)

D36T39: Water supply, sewerage, waste management and remediation activities,

    9. (i)

D09: Mining support service activities.

  5. 5.

Component V:

    1. (a)

D45T47: Wholesale and retail trade, repair of motor vehicles,

    2. (b)

D49: Land transport and transport via pipelines,

    3. (c)

D22: Rubber and plastics products,

    4. (d)

D61: Telecommunications,

    5. (e)

D53: Postal and courier activities,

    6. (f)

D94T96: Other service activities.

  6. 6.

Component VI:

    1. (a)

D64T66: Financial and insurance activities,

    2. (b)

D68: Real estate activities.

By allowing overlapping, we deduce a different clustering:

Overlapping divisive clustering:

  1. 1.

Component I:

    1. (a)

D19: Coke and refined petroleum products,

    2. (b)

D05T06: Mining and quarrying, energy producing products.

  2. 2.

Component II:

    1. (a)

D23: Other non-metallic mineral products,

    2. (b)

D41T43: Construction,

    3. (c)

D35: Electricity, gas, steam and air conditioning supply,

    4. (d)

D84: Public administration and defense, compulsory social security,

    5. (e)

D68: Real estate activities,

    6. (f)

D36T39: Water supply, sewerage, waste management and remediation activities.

  3. 3.

Component III:

    1. (a)

D10T12: Food products, beverages and tobacco,

    2. (b)

D01T02: Agriculture, hunting, forestry,

    3. (c)

D17T18: Paper products and printing,

    4. (d)

D85: Education,

    5. (e)

D86T88: Human health and social work activities,

    6. (f)

D51: Air transport,

    7. (g)

D28: Machinery and equipment, nec,

    8. (h)

D36T39: Water supply, sewerage, waste management and remediation activities,

    9. (i)

D09: Mining support service activities.

  1. 4.

Component IV:

    1. (a)

D45T47: Wholesale and retail trade, repair of motor vehicles,

    2. (b)

D49: Land transport and transport via pipelines,

    3. (c)

D22: Rubber and plastics products,

    4. (d)

D61: Telecommunications,

    5. (e)

D53: Postal and courier activities,

    6. (f)

D94T96: Other service activities,

    7. (g)

D03: Fishing and aquaculture.

  2. 5.

Component V:

    1. (a)

D45T47: Wholesale and retail trade, repair of motor vehicles,

    2. (b)

D84: Public administration and defense, compulsory social security,

    3. (c)

D35: Electricity, gas, steam and air conditioning supply,

    4. (d)

D51: Air transport,

    5. (e)

D77T82: Administrative and support services,

    6. (f)

D68: Real estate activities,

    7. (g)

D31T33: Manufacturing nec, repair and installation of machinery and equipment,

    8. (h)

D36T39: Water supply; sewerage, waste management and remediation activities,

    9. (i)

D30: Other transport equipment,

    10. (j)

D62T63: IT and other information services,

    11. (k)

D19: Coke and refined petroleum products.

Overlapping agglomerative clustering:

  1. 1.

Component I:

    1. (a)

D41T43: Construction,

    2. (b)

D07T08: Mining and quarrying, non-energy producing products.

  2. 2.

Component II:

    1. (a)

D19: Coke and refined petroleum products,

    2. (b)

D05T06: Mining and quarrying, energy producing products.

  3. 3.

Component III:

    1. (a)

D84: Public administration and defense, compulsory social security,

    2. (b)

D55T56: Accommodation and food service activities.

  4. 4.

Component IV:

    1. (a)

D23: Other non-metallic mineral products,

    2. (b)

D41T43: Construction,

    3. (c)

D35: Electricity, gas, steam and air conditioning supply,

    4. (d)

D07T08: Mining and quarrying, non-energy producing products,

    5. (e)

D68: Real estate activities,

    6. (f)

D36T39: Water supply; sewerage, waste management and remediation activities.

  5. 5.

Component V:

    1. (a)

D84: Public administration and defense, compulsory social security,

    2. (b)

D41T43: Construction.

  6. 6.

Component VI:

    1. (a)

D45T47: Wholesale and retail trade; repair of motor vehicles,

    2. (b)

D41T43: Construction,

    3. (c)

D84: Public administration and defense, compulsory social security,

    4. (d)

D22: Rubber and plastics products,

    5. (e)

D55T56: Accommodation and food service activities,

    6. (f)

D19: Coke and refined petroleum products,

    7. (g)

D05T06: Mining and quarrying, energy producing products.

  7. 7.

Component VII:

    1. (a)

D10T12: Food products, beverages and tobacco,

    2. (b)

D01T02: Agriculture, hunting, forestry,

    3. (c)

D17T18: Paper products and printing,

    4. (d)

D85: Education,

    5. (e)

D41T43: Construction,

    6. (f)

D86T88: Human health and social work activities,

    7. (g)

D51: Air transport,

    8. (h)

D28: Machinery and equipment, nec,

    9. (i)

D36T39: Water supply, sewerage, waste management and remediation activities,

    10. (j)

D09: Mining support service activities.

  8. 8.

Component VIII:

    1. (a)

D49: Land transport and transport via pipelines,

    2. (b)

D41T43: Construction.

  1. 9.

Component IX:

    1. (a)

D20: Chemical and chemical products,

    2. (b)

D07T08: Mining and quarrying, non-energy producing products.

    3. (c)

D22: Rubber and plastics products,

    4. (d)

D41T43: Construction,

    5. (e)

D31T33: Manufacturing nec, repair and installation of machinery and equipment,

    6. (f)

D55T56: Accommodation and food service activities,

    7. (g)

D19: Coke and refined petroleum products.

  2. 10.

Component X:

    1. (a)

D35: Electricity, gas, steam and air conditioning supply,

    2. (b)

D84: Public administration and defense, compulsory social security,

    3. (c)

D94T96: Other service activities.

  3. 11.

Component XI:

    1. (a)

D49: Land transport and transport via pipelines,

    2. (b)

D19: Coke and refined petroleum products,

    3. (c)

D05T06: Mining and quarrying, energy producing products.

  4. 12.

Component XII:

    1. (a)

D45T47: Wholesale and retail trade; repair of motor vehicles,

    2. (b)

D49: Land transport and transport via pipelines,

    3. (c)

D61: Telecommunications,

    4. (d)

D28: Machinery and equipment, nec,

    5. (e)

D53: Postal and courier activities,

    6. (f)

D94T96: Other service activities,

    7. (g)

D90T93: Arts, entertainment and recreation.

  5. 13.

Component XIII:

    1. (a)

D64T66: Financial and insurance activities,

    2. (b)

D68: Real estate activities,

    3. (c)

D41T43: Construction,

    4. (d)

D07T08: Mining and quarrying, non-energy producing products,

    5. (e)

D36T39: Water supply; sewerage, waste management and remediation activities,

    6. (f)

D19: Coke and refined petroleum products,

    7. (g)

D05T06: Mining and quarrying, energy producing products.

  6. 14.

Component XIV:

    1. (a)

D84: Public administration and defense, compulsory social security,

    2. (b)

D61: Telecommunications,

    3. (c)

D68: Real estate activities,

    4. (d)

D36T39: Water supply; sewerage, waste management and remediation activities.

  7. 15.

Component XV:

    1. (a)

D35: Electricity, gas, steam and air conditioning supply,

    2. (b)

D05T06: Mining and quarrying, energy producing products.

To visualize the principal (non atomic) cyclicality components, we produce a
graph proportional representation of arcs (with respect to the weights) and
vertices (with respect to spectral influence in the component):

(a) $\mathbf{S}=1.35676$.

(b) $\mathbf{S}=1.27431$.

(c) $\mathbf{S}=1.16459$.

(d) $\mathbf{S}=1.11335$.

(e) $\mathbf{S}=1.03187$.

Figure 4: The five divisive clusters of $2018$ Moroccan input-output table.

(a) $\mathbf{S}=1.6033$.

(b) $\mathbf{S}=1.35676$.

(c) $\mathbf{S}=1.34825$.

(d) $\mathbf{S}=1.16459$.

(e) $\mathbf{S}=1.11335$.

(f) $\mathbf{S}=1.00877$.

Figure 5: The six agglomerative clusters of $2018$ Moroccan input-output
table.

(a) $\mathbf{S}=1.35676$.

(b) $\mathbf{S}=1.27467$.

(c) $\mathbf{S}=1.16459$.

(d) $\mathbf{S}=1.15275$.

(e) $\mathbf{S}=1.02185$.

Figure 6: The five overlapping divisive clusters of $2018$ Moroccan input-
output table.

(a) $\mathbf{S}=1.6033$.

(b) $\mathbf{S}=1.35676$.

(c) $\mathbf{S}=1.34825$.

(d) $\mathbf{S}=1.2768$.

(e) $\mathbf{S}=1.21107$.

(f) $\mathbf{S}=1.19601$.

Figure 7: The fifteen overlapping agglomerative clusters of $2018$ Moroccan
input-output table.

(a) $\mathbf{S}=1.16461$.

(b) $\mathbf{S}=1.12018$.

(c) $\mathbf{S}=1.11361$.

(d) $\mathbf{S}=1.10433$.

(e) $\mathbf{S}=1.06336$.

(f) $\mathbf{S}=1.05842$.

Figure 8: The fifteen overlapping agglomerative clusters of $2018$ Moroccan
input-output table-bis.

(a) $\mathbf{S}=1.02491$.

(b) $\mathbf{S}=1.01399$.

(c) $\mathbf{S}=1.00407$.

Figure 9: The fifteen overlapping agglomerative clusters of $2018$ Moroccan
input-output table-ter.

A direct comparison of the nonoverlapping and overlapping situations reveals
the following:

  * •

Divisive algorithm: The two algorithm produce almost the same first four
components (in the overlapping situation, component number two has a sixth
pole D36T39: Water supply, sewerage, waste management and remediation
activities, and component number four has a seventh pole D03: Fishing and
aquaculture). The difference is recorded for the component number five where
it is a two-poles component of mining and quarrying for the nonoverlapping
algorithm and an eleven poles component of public expenditures.

  * •

Agglomerative algorithm: The difference is much greater and additional
clusters arise when employing overlapping clustering. The first three clusters
are identical, the overlapping cluster number seven has an extra pole (D41T43:
Construction) compared to nonoverlapping cluster number four. The
nonoverlapping cluster number five correspond to the overlapping cluster
number twelve and the nonoverlapping cluster number six is included in
overlapping cluster number thirteen.

###  5.3 Discussion

In $2018$, Moroccan interindustrial flows structure is characterized by high
cyclicality. Compared to the countries panel, Morocco along with Saudi Arabia,
are the only countries with a spectral cyclicality greater than one, making
them the countries with the highest interindustrial flows cyclicality.

We can distinguish three high cyclicality disjoint key components in the
Moroccan interindustrial structure, we find convenient to do an analogy with
cyclical clusters and planet systems in terms of influence. One could describe
those components according to the following themes:

  1. 1.

Mining and quarrying represented in figure 4.(a), formed by a planet: D19-Coke
and refined petroleum products; and a moon: D05T06-Mining and quarrying,
energy producing products.

  2. 2.

Construction represented in figure 4.(b), characterized as planetary system of
decreasing size (influence): D23-Other non-metallic mineral products;
D41T43-Construction; D35-Electricity, gas, steam and air conditioning supply;
D84-Public administration and defense, compulsory social security; D68-Real
estate activities.

  3. 3.

Agri-food and related activities represented in figure 4.(c), formed by two
planet-hosting stars: D10T12-Food products, beverages and tobacco; and
D01T02-Agriculture, hunting, forestry; and seven planets: D17T18-Paper
products and printing; D85-Education; D86T88-Human health and social work
activities; D51-Air transport; D28-Machinery and equipment, nec; D36T39-Water
supply, sewerage, waste management and remediation activities; D09-Mining
support service activities.

  4. 4.

Logistics and transports in figure 4.(d), with a central star:
D45T47-Wholesale and retail trade, repair of motor vehicles; two planets:
D49-Land transport and transport via pipelines; D22-Rubber and plastics
products; and three moons: D61-Telecommunications; D53-Postal and courier
activities; D94T96-Other service activities.

If we study the intersecting clusters, we can identify additional interesting
productive components. Among them, we can enumerate the homogeneous ones:

  1. 1.

Energy products represented in figure 7.(f), represented by a sun: D45T47:
Wholesale and retail trade, repair of motor vehicles, three planets: D41T43:
Construction, D84: Public administration and defense, compulsory social
security, D22: Rubber and plastics products, and three satellites: D55T56:
Accommodation and food service activities, D19: Coke and refined petroleum
products, D05T06: Mining and quarrying, energy producing products.

  2. 2.

Manufacture of chemical products represented in figure 8.(c), represented by a
sun: D20: Chemical and chemical products, two planets: D07T08: Mining and
quarrying, non-energy producing products, D22: Rubber and plastics products,
and four satellites: D41T43: Construction, D31T33: Manufacturing nec, repair
and installation of machinery and equipment, D55T56: Accommodation and food
service activities, D19: Coke and refined petroleum products.

If we study the historical evolution from $1995$ to $2018$, one can conclude,
according to figure 3, that Moroccan interindustrial flows structure is
clearly tending towards a structure with strong cyclicality characterized with
strong diffusion and interdependence. The Moroccan passes from one big cluster
with weak cyclicality regrouping almost all sectors to three key components
with strong cyclicality. One can give a detailed lecture:

  1. 1.

The top "propulsive industries", according to the spectral influence, are in
order and persistently the poles: D10T12-Food products, beverages and tobacco;
D01T02-Agriculture, hunting, forestry; D45T47-Wholesale and retail trade,
repair of motor vehicles; D20-Chemical and chemical products. While the
weakest industries, in the sense of the spectral influence, are in an
increasing order: D97T98-Activities of households as employers,
undifferentiated goods and services-producing activities of households for own
use; D05T06-Mining and quarrying, energy producing products; D30-Other
transport equipment.

  2. 2.

According to spectral influences evolution graphs (see the appendix 7.2) one
can notice the structural mutation of the Moroccan economy through a process
of concentration and complementary of the interindustrial flows. One can
notice poles with increasing spectral influence: D01T02-Agriculture, hunting,
forestry; D10T12-Food products, beverages and tobacco; D29-Motor vehicles,
trailers and semi-trailers; D50-Water transport. This increasing influences
absorb the influences of others poles like: D13T15-Textiles, textile products,
leather and footwear; D16-Wood and products of wood and cork;
D21-Pharmaceuticals, medicinal chemical and botanical products; D22-Rubber and
plastics products; D23-Other non-metallic mineral products; D24-Basic metals;
D25-Fabricated metal products; D26-Computer, electronic and optical equipment;
D27-Electrical equipment; D28-Machinery and equipment, nec;
D31T33-Manufacturing nec; repair and installation of machinery and equipment;
D35-Electricity, gas, steam and air conditioning supply; D41T43-Construction;
D45T47-Wholesale and retail trade; repair of motor vehicles; D53-Postal and
courier activities; D55T56-Accommodation and food service activities;
D61-Telecommunications; D68-Real estate activities.

##  6 Conclusion

In summary, the new concept of spectral influence is a local measurement of
centrality of vertex in a graph from a diffusion perspective, this local
measurement induce a global measurement characterizing cyclicality of the
whole graph. One can use this global measure in the definition of clustering
algorithms to isolated high cyclicality components.

The application of this new technique to input-output analysis sheds new light
to interindustrial organization by highlighting interindustrial transformation
unites involving many industries and detecting propulsive industries which are
the driving force of the economy. In opposite to the classical Leontieff/Ghosh
analysis, supposing a flow relative orientation (demand/supply) and supposing
a structural stabitity of the coefficients, the spectral cyclicality analyzes
the absolute flows and trace the bilateral influences at time $t$.

Cyclicality in graph reflects many interesting features that could be very
helpful in understanding social networks, like interdependence (dominance,
resilience,…), diffusion (ideas, diseases, economic shocks…), feedback effects
and amplification (productivity, transformation, …) and many potential uses of
this new technique are conceivable.

##  7 Appendix

###  7.1 Industrial poles of the OECD harmonised national input-output tables

D01T02: Agriculture, hunting, forestry |  D03: Fishing and aquaculture |  D05T06: Mining and quarrying, energy producing products
---|---|---
D07T08: Mining and quarrying, non-energy producing products |  D09: Mining support service activities |  D10T12: Food products, beverages and tobacco
D13T15: Textiles, textile products, leather and footwear |  D16: Wood and products of wood and cork |  D17T18: Paper products and printing
D19: Coke and refined petroleum products |  D20: Chemical and chemical products |  D21: Pharmaceuticals, medicinal chemical and botanical products
D22: Rubber and plastics products |  D23: Other non-metallic mineral products |  D24: Basic metals
D25: Fabricated metal products |  D26: Computer, electronic and optical equipment |  D27: Electrical equipment
D28: Machinery and equipment, nec |  D29: Motor vehicles, trailers and semi-trailers |  D30: Other transport equipment
D31T33: Manufacturing nec; repair and installation of machinery and equipment |  D35: Electricity, gas, steam and air conditioning supply |  D36T39: Water supply; sewerage, waste management and remediation activities
D41T43: Construction |  D45T47: Wholesale and retail trade; repair of motor vehicles |  D49: Land transport and transport via pipelines
D50: Water transport |  D51: Air transport |  D52: Warehousing and support activities for transportation
D53: Postal and courier activities |  D55T56: Accommodation and food service activities |  D58T60: Publishing, audiovisual and broadcasting activities
D61: Telecommunications |  D62T63: IT and other information services |  D64T66: Financial and insurance activities
D68: Real estate activities |  D69T75: Professional, scientific and technical activities |  D77T82: Administrative and support services
D84: Public administration and defense; compulsory social security |  D85: Education |  D86T88: Human health and social work activities
D90T93: Arts, entertainment and recreation |  D94T96: Other service activities |  D97T98: Activities of households as employers; undifferentiated goods- and services-producing activities of households for own use

###  7.2 Spectral influence evolution by industrial poles

D01T02.

D03.

D05T06.

D07T08.

D09.

D10T12.

D13T15.

D16.

D17T18.

D19.

D20.

D21.

D22.

D23.

D24.

D25.

D26.

D27.

D28.

D29.

D30.

D31T33.

D35.

D36T39.

Figure 10: Evolution of the spectral radius of Moroccan industrial poles.

D41T43.

D45T47.

D49.

D50.

D51.

D52.

D53.

D55T56.

D58T60.

D61.

D62T63.

D64T66.

D68.

D69T75.

D77T82.

D84.

D85.

D86T88.

D90T93.

D94T96.

D97T98.

Figure 11: Evolution of the spectral radius of Moroccan industrial poles-bis.

## References

  * [ACOTS12] Daron Acemoglu, Vasco M. Carvalho, Asuman Ozdaglar, and Alireza Tahbaz-Salehi.  The network origins of aggregate fluctuations.  Econometrica, 80(5):1977–2016, 2012.
  * [Bon87] Phillip Bonacich.  Power and centrality: A family of measures.  American Journal of Sociology, 92(5):1170–1182, 1987.
  * [DWM02] Jennifer A. Dunne, Richard J. Williams, and Neo D. Martinez.  Network structure and biodiversity loss in food webs: robustness increases with connectance.  Ecology Letters, 5(4):558–567, 2002.
  * [FLSZ21] Tianlong Fan, Linyuan Lü, Dinghua Shi, and Tao Zhou.  Characterizing cycle structure in complex networks.  Communications Physics, 4, 2021.
  * [Fre78] Linton C. Freeman.  Centrality in social networks conceptual clarification.  Social Networks, 1(3):215–239, 1978.
  * [GAZ09] B.N. Goldstein, A.M. Aksirov, and D.T. Zakrjevskaya.  A new kinetic model for biochemical oscillations: Graph-theoretical analysis.  Biophysical Chemistry, 145(2):111–115, 2009.
  * [HJ13] Roger A. Horn and Charles R. Johnson.  Matrix Analysis.  Cambridge University Press, 2013.
  * [HM11] Andrew G. Haldane and Robert M. May.  Systemic risk in banking ecosystems.  Nature, 469:351–355, 2011.
  * [JLGA18] J. Yellen J. L. Gross and M. Anderson.  Graph theory and its applications.  Chapman and Hall/CRC, 2018.
  * [Lan72] R. Lantner.  L’analyse de la dominance économique.  Revue d’économie politique, 82(2):216–283, 1972.
  * [Luc77] Robert E Lucas.  Understanding business cycles.  Carnegie-Rochester Conference Series on Public Policy, 5:7–29, 1977\.
  * [New18] Mark Newman.  Networks.  Oxford University Press, 07 2018.
  * [OCD21] OCDE.  Input-output tables (iots), 2021.
  * [PD04] Thomas Petermann and Paolo De Los Rios.  Cluster approximations for epidemic processes: a systematic description of correlations beyond the pair level.  Journal of Theoretical Biology, 229(1):1–11, 2004.
  * [Per63] François Perroux.  Les industries motrices et la croissance d’une économie nationale.  L’Actualité économique, 39(3-4):377–441, 1963.
  * [Per69] Fraonçois Perroux.  L’Économie du XXe siècle.  Presses universitaires de France, 1969.