# Collective Almost Synchronization in Complex Networks

M. S. Baptista1, Hai-Peng Ren2,1, J. C. M. Swarts3, R. Carareto4,1, H.
Nijmeijer3, C. Grebogi1 1Institute for Complex Systems and Mathematical
Biology, University of Aberdeen, SUPA, AB24 3UE Aberdeen, United Kingdom
2Department of Information and Control Engineering, Xi’an University of
technology, 5 Jinhua South Road, Xi’an, 710048, China  3Department of
Mechanical Engineering, Dynamics and Control Group, Eindhoven University of
Technology, WH 0.144, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
4Escola Politecnica, Universidade de São Paulo, Avenida Prof. Luciano
Gualberto, travessa 3, n. 158, 05508-900 São Paulo, SP, Brazil

###### Abstract

This work introduces the phenomenon of Collective Almost Synchronization
(CAS), which describes a universal way of how patterns can appear in complex
networks even for small coupling strengths. The CAS phenomenon appears due to
the existence of an approximately constant local mean field and is
characterized by having nodes with trajectories evolving around periodic
stable orbits. Common notion based on statistical knowledge would lead one to
interpret the appearance of a local constant mean field as a consequence of
the fact that the behavior of each node is not correlated to the behaviors of
the others. Contrary to this common notion, we show that various well known
weaker forms of synchronization (almost, time-lag, phase synchronization, and
generalized synchronization) appear as a result of the onset of an almost
constant local mean field. If the memory is formed in a brain by minimising
the coupling strength among neurons and maximising the number of possible
patterns, then the CAS phenomenon is a plausible explanation for it.

Spontaneous emergence of collective behavior is common in nature
bouchaud_MD2000 ; couzin_ASB2003 ; helbing_nature2000 . It is a natural
phenomenon characterized by a group of individuals that are connected in a
network by following a dynamical trajectory that is different from the
dynamics of their own. Since the work of Kuramoto kuramoto_LNP1975 , the
spontaneous emergence of collective behavior in networks of phase oscillators
with full connected nodes or with nodes connected by some special topologies
acebron_RMP2005  is analytically well understood. Kuramoto considered a fully
connected network of an infinite number of phase oscillators. If $\theta_{i}$
is the variable describing the phase of an oscillator $i$ in the network, and
$\overline{\theta}$ represents the mean field defined as
$\overline{\theta}=\frac{1}{N}\sum_{i=1}^{N}\theta_{i}$, collective behavior
appears in the network because every node becomes coupled to the mean field.
Peculiar characteristics of this collective behavior is that not only
$\theta_{i}\neq\overline{\theta}$ but also nodes evolve in a way that cannot
be described by the evolution of only one individual node, when isolated from
the network.

In contrast to collective behavior, another widely studied behavior of a
network is when all nodes behave equally, and their evolution can be described
by an individual node when isolated from the network. This state is known as
complete synchronization fujisaka_PTP1983 . If $x_{i}$ represents the state
variables of an arbitrary node $i$ of the network and $x_{j}$ of another node
$j$, and $\overline{x}$ represents the mean field of a network, complete
synchronization appears when $x_{i}=x_{j}=\overline{x}$, for all time. The
main mechanisms responsible for the onset of complete synchronization in
dynamical networks were clarified in pecora_PRL1998 ; nijmeijer_PHYSICAD2009 ;
nijmeijer_IEEE2011 . In networks whose nodes are coupled by non-linear
functions, such as those that depend on time-delays nijmeijer_IEEE2011  or
those that describe how neurons chemically connect baptista_PRE2010 , the
evolution of the synchronous nodes might be different from the evolution of an
individual node, when isolated from the network. However, when complete
synchronization is achieved in such networks, $x_{i}=x_{j}=\overline{x}$.

In natural networks as biological, social, metabolic, neural networks, etc,
barabasi_RMP2002 , the number of nodes is often large but finite; the network
is not fully connected and heterogeneous. The later means that each node has a
different dynamical description or the coupling strengths are not all equal
for every pair of nodes, and one will not find two nodes, say it $x_{i}$ and
$x_{j}$, that have equal trajectories. For such heterogeneous networks, as in
zhou_CHAOS2006 ; gardenes_chaos2011 , found in natural networks and in
experiments juergen_book , one expects to find other weaker forms of
synchronous behavior, such as practical synchronization femat_PLA1999 , phase
synchronization juergen_book , time-lag synchronization rosemblum_PRL1997 ,
and generalized synchronization rulkov_PRE1995 .

We report a phenomenon that may appear in complex networks “far away” from
coupling strengths that typically produce complete synchronization or these
weaker forms of synchronization. However, the reported phenomenon can be
characterized by the same conditions used to verify the existence of these
weaker forms of synchronization. We call it Collective Almost Synchronization
(CAS). It is a consequence of the appearance of an approximately constant
local mean field and is characterized by having nodes with trajectories
evolving around stable periodic orbits, denoted by $\mathbf{\Xi}_{p_{i}}(t)$,
and regarded as a CAS pattern. The appearance of an almost constant mean field
is associated with a regime of weak interaction (weak coupling strength) in
which nodes behave independently jirsa_CN2008 ; batista_PRE2007 . In such
conditions, even weaker forms of synchronization are ruled out to exist. But,
contrary to common notion based on basic statistical arguments, we show that
actually it is the existence of an approximately constant local mean field
that paves the way for weaker forms of synchronization (such as almost, time-
lag, phase, or generalized synchronization) to occur in complex networks.

Denote all the $d$ variables of a node $i$ by ${\mathbf{x}}_{i}$, then we
define that this node presents CAS if the following inequality

$|\mathbf{x}_{i}(t)-\mathbf{\Xi}_{p_{i}}(t-\tau_{i})|<\epsilon_{i}$ (1)

is satisfied for most of the time. The double vertical bar $|\ |$ represents
that we are taking the absolute difference between vector components appearing
inside the bars ($L1$ norm). $\epsilon_{i}$ is a small quantity, not
arbitrarily small, but reasonably smaller than the envelop of the oscillations
of the variables $\mathbf{x}_{i}(t)$. $\mathbf{\Xi}_{p_{i}}(t)$ is the
$d$-dimensional CAS pattern. It is determined by the effective coupling
strength $p_{i}$, a quantity that measures the influence on the node $i$ of
the nodes that are connected to it, and the expected value of the local mean
field at the node $i$, denoted by $\mathbf{C}_{i}$. The local mean field,
denoted by $\overline{\mathbf{x}}_{i}$, is defined only by the nodes that are
connected to the node $i$. The CAS pattern is the solution of a simplified set
of equations describing the network when
$\overline{\mathbf{x}}_{i}=\mathbf{C}_{i}$. According to Eq. (1), if a node in
the network presents the CAS pattern, its trajectory stays intermittently
close to the CAS pattern but with a time-lag between the trajectories of the
node and of the CAS pattern. This property of the CAS phenomenon shares
similarities with the way complete synchronization appears in networks of
nodes coupled under time-delay functions nijmeijer_IEEE2011 . In such
networks, nodes become completely synchronous to a solution of the network
that is different from the solution of an isolated node of the network.
Additionally, the trajectory of the nodes present a time-lag to this solution.

The CAS phenomenon inherits the three main characteristics of a collective
behavior: (a) the variables of a node $i$ ($\mathbf{x}_{i}$) differ from both
the mean field $\overline{\mathbf{x}}$ and the local mean field
$\overline{\mathbf{x}}_{i}$; (b) if the local mean fields of a group of nodes
and their effective coupling are either equal or approximately equal, that
causes all the nodes in this group to follow the same or similar behaviors;
(c) there can exist an infinitely large number of different behaviors (CAS
patterns).

If the CAS phenomenon is present in a network, other weaker forms of
synchronization can be detected. This link is fundamental when making
measurements to detect the CAS phenomenon.

In Ref. femat_PLA1999 , the phenomenon of almost synchronization is
introduced, when a master and a slave in a master-slave system of coupled
oscillators have equal phases but their amplitudes can be different. If a node
$i$ presents the CAS phenomenon [satisfying Eq. (1)] and $\tau_{i}=0$ in Eq.
(1), then the node $i$ is almost synchronous to the pattern
$\mathbf{\Xi}_{p_{i}}$.

Time-lag synchronization rosemblum_PRL1997  is a phenomenon that describes two
identical signals, but whose variables have a time-lag with respect to each
other, i.e. $\mathbf{x}_{i}(t)=\mathbf{x}_{j}(t-\tau)$. In practice, however,
an equality between $\mathbf{x}_{i}(t)$ and $\mathbf{x}_{j}(t-\tau)$ should
not be expected to be typically found, but rather

$\mathbf{x}_{i}(t)\cong\mathbf{x}_{j}(t-\tau),$ (2)

meaning that there is not a constant $\tau$ that can be found such that
$\mathbf{x}_{i}(t)=\mathbf{x}_{j}(t-\tau)$. Another suitable way of writing
Eq. (2) is by $|\mathbf{x}_{i}(t)-\mathbf{x}_{j}(t-\tau)|\leq\gamma$. If two
nodes $i$ and $j$ that present the CAS phenomenon, have the same CAS pattern,
and $\tau_{i}\neq\tau_{j}\neq 0$, then

$|\mathbf{x}_{i}(t)-\mathbf{x}_{j}(t-\tau_{ij})|\leq\epsilon_{ij}$ (3)

or alternatively $\mathbf{x}_{i}(t)\cong\mathbf{x}_{j}(t-\tau_{ij})$, for most
of the time, $\tau_{ij}$ representing the time-lag between $\mathbf{x}_{i}$
and $\mathbf{x}_{j}$. This means that almost time-lag synchronization occurs
for two nodes that present the CAS phenomenon and that are almost locked to
the same CAS pattern. Even though nodes that have equal or similar local mean
field (which usually happens for nodes that have equal or similar degrees)
become synchronous with the same CAS pattern (a stable periodic orbit), the
value of their trajectories at a given time might be different, since their
trajectories reach the neighborhood of their CAS patterns in different places
of the orbit. As a consequence, we expect that two nodes that exhibit the same
CAS should present between themselves a time-lag synchronous behavior. For
some small amounts of time, the difference
$|\mathbf{x}_{i}(t)-\mathbf{x}_{j}(t-\tau_{ij})|$ can be large, since
$\tau_{i}\neq\tau_{j}$ and $\epsilon_{i}\neq\epsilon_{j}$, in Eq. (1). The
closer $\overline{\mathbf{x}}_{i}$ and $\overline{\mathbf{x}}_{j}$ are to
$\mathbf{C}_{i}$, the smaller is $\epsilon_{ij}$ in Eq. (3).

Phase synchronization juergen_book  is a phenomenon where the phase
difference, denoted by $\Delta\phi_{ij}$, between the phases of two signals
(or nodes in a network), $\phi_{i}(t)$ and $\phi_{j}(t)$, remains bounded for
all time

$\Delta\phi_{ij}=\left|\phi_{i}(i)-\frac{p}{q}\phi_{j}(t)\right|\leq S.$ (4)

In Ref. juergen_book  $S=2\pi$ and $p$ and $q$ are two rational numbers. If
$p$ and $q$ are irrational numbers and $S$ is a reasonably small constant,
then phase synchronization can be referred as to irrational phase
synchronization baptista_PRE2004 . The value of $S$ is calculated in order to
encompass oscillatory systems that possess either a time varying time-scale or
a variable time-lag. Simply make the constant $S$ to represent the growth of
the phase in the faster time scale during one period of the slower time scale.
Phase synchronization between two coupled chaotic oscillators was explained as
being the result of a state where the two oscillators have all their unstable
periodic orbits phase-locked juergen_book . Nodes that present the CAS
phenomenon have unstable periodic orbits that are locked to the stable
periodic orbits described by $\mathbf{\Xi}_{i}(t)$. If $\mathbf{\Xi}_{i}(t)$
has a period $P_{i}$ and the phase of this CAS pattern changes $D\phi_{i}$
within one period, so the angular frequency is $\omega_{i}=D\phi_{i}/P_{i}$.
If $\mathbf{\Xi}_{j}(t)$ has a period $P_{j}$ and the phase of its CAS patter
changes $D\phi_{j}$ within one period, so the angular frequency is
$\omega_{j}=D\phi_{j}/P_{j}$. Then, the CAS patterns of these nodes are phase
synchronous by a ratio of $\frac{p}{q}=\omega_{i}/\omega_{j}$. Since the
trajectories of these nodes are locked to these patterns, the nodes are phase
synchronous by this same ratio, which can be rational or irrational. Assume
additionally that, as one changes the coupling strengths between the nodes,
the expected value $\mathbf{C}_{i}$ of the local mean field of a group of
nodes remains the same. As a consequence, as one changes the coupling
strengths, both the CAS pattern and the ratio
$\frac{p}{q}=\frac{p_{j}D\phi_{i}}{p_{i}D\phi_{j}}$ remain unaltered, and the
observed phase synchronization between nodes in this group is stable under
parameter alterations.

Consider a network of $N$ nodes with nodes connected diffusively (more general
networks are treated in the Supplementary Information) described by

$\dot{\mathbf{x}}_{i}=\mathbf{F}_{i}(\mathbf{x}_{i})+\sigma\sum_{j=1}^{N}{\mathbf{A}_{ij}}{\mathbf{E}}(\mathbf{x}_{j}-\mathbf{x}_{i}),$
(5)

where $\mathbf{x}_{i}\in\Re^{d}$ is a d-dimensional vector describing the
state variables of the node $i$, $\mathbf{F}_{i}$ represents the dynamical
system of the node $i$, and ${\mathbf{A}_{ij}}$ is the adjacent matrix. If
$A_{ij}=1$, then, the node $j$ is connected to the node $i$. ${\mathbf{E}}$ is
the coupling function The degree of a node can be calculated by
$k_{i}=\sum_{j=1}^{N}A_{ij}$.

The CAS phenomenon appears when the local mean field of a node $i$,
$\overline{\mathbf{x}}_{i}(t)=1/k_{i}\sum_{j}A_{ij}\mathbf{x}_{j}$, is
approximately constant and
$\overline{\mathbf{x}}_{i}(t)\approxeq\mathbf{C}_{i}$. Then, the equations for
the network can be described by

$\dot{\mathbf{x}}_{i}=\mathbf{F}_{i}(\mathbf{x}_{i})-p_{i}E(\mathbf{x}_{i})+p_{i}E(\mathbf{C}_{i})+\mathbf{\delta}_{i},$
(6)

where $p_{i}=\sigma k_{i}$ and the residual term is
$\mathbf{\delta}_{i}=p_{i}(\overline{\mathbf{x}}_{i}(t)-\mathbf{C}_{i})$. The
CAS pattern of the node $i$ (a stable periodic orbit) is calculated in the
variables that produce a finite bounded local average field. If all components
of $\mathbf{x}_{i}$ are bounded, then the CAS pattern is given by a solution
of

$\dot{\mathbf{\Xi}}_{p_{i}}=F_{i}(\mathbf{\Xi}_{p_{i}})-p_{i}E(\mathbf{\Xi}_{p_{i}})+p_{i}E(\mathbf{C}_{i}).$
(7)

which is just the same set of equations (6) without the residual term. So, if
$\overline{\mathbf{x}}_{i}(t)=\mathbf{C}_{i}$, the residual term
$\mathbf{\delta}_{i}=0$, and if Eq. (7) has no positive Lyapunov exponents
($\mathbf{\Xi}_{p_{i}}$ is a stable periodic orbit), then the node $x_{i}$
describes a stable periodic orbit. If
$\overline{\mathbf{x}}_{i}(t)-\mathbf{C}_{i}$ is larger than zero but
$\mathbf{\Xi}_{p_{i}}$ is a stable periodic orbit, then the node $x_{i}$
describes a perturbed version of $\mathbf{\Xi}_{p_{i}}$. The closer
$\overline{\mathbf{x}}_{i}$ is to $\mathbf{C}_{i}$, the larger the time that
Eq. (1) is satisfied at a given time. The more stable the periodic orbit is
[the larger the largest negative Lyapunov exponents of Eq. (7)], the longer
Eq. (1) is satisfied at a given time.

If the network has unbounded state variables (as it is the case of Kuramoto
networks kuramoto_LNP1975 ), the CAS pattern is the periodic orbit of period
$T_{i}$ defined in the velocity space such that
$\dot{\mathbf{\Xi}}_{p_{i}}(t)=\dot{\mathbf{\Xi}}_{p_{i}}(t+T_{i})$.

Notice that whereas Eqs. (5) and (6) represent a $Nd$-dimensional system, Eq.
(7) has only dimension $d$.

The existence of this approximately constant local mean field is a consequence
of the Central Limit Theorem, applied to variables with correlation (for more
details, see Supplementary Information). The expected value of the local mean
field can be calculated by

$\mathbf{C}_{i}=_{\lim
t\rightarrow\infty}\frac{1}{t}\int\overline{\mathbf{x}}_{i}(t)dt,$ (8)

where in practice we consider $t$ to be large, but finite. The larger the
degree of a node, the higher is the probability for the local mean field to be
close to an expected value and smaller its variance. If the probability to
find a certain value for the local mean field of the node $i$ does not depend
on the higher order moments of $\overline{\mathbf{x}}_{i}(t)$, then this
probability tends to be Gaussian for sufficiently large $k_{i}$. As a
consequence, the variance $\mu^{2}$ of the local mean field is proportional to
$k_{i}^{-1}$.

There are two criteria for the node $i$ to present the CAS phenomenon:

Criterion 1:

The Central Limit Theorem can be applied, i.e., $\mu^{2}_{i}\propto
k_{i}^{-1}$. Therefore, the larger the degree of a node, the smaller the
variation of the local mean field $\overline{\mathbf{x}}_{i}(t)$ about its
expected value $\mathbf{C}_{i}$.

Criterion 2:

The CAS pattern $\mathbf{\Xi}_{i}(t)$ describes a stable periodic orbit. The
node trajectory can be considered to be a perturbed version of its CAS
pattern. The more stable the faster trajectories of nodes come to the
neighborhood of the periodic orbits (CAS patterns), and the longer they stay
around them.

Whenever the Central Limit Theorem applies, the random variables involved are
independent. But, the Central Limit Theorem can also be applied to variables
with correlation. If nodes that present the CAS phenomenon are locked to the
same CAS pattern, their trajectories still arrive to the CAS pattern at
different “random” times, allowing for the Central Limit Theorem to be
applied. But the time-lag between two nodes ($\tau_{ij}$) is approximately
constant, since the CAS pattern has a well defined period, and the
trajectories of these nodes are locked into it. The local mean field measured
in a node $i$ remains unaltered as one changes the coupling strength either
when the network has an infinite number of nodes (e.g. Kuramoto networks) or
the nodes have a symmetric natural measured (See Secs. C, D, and E of
Supplementary Information). However, as we show in the following example, the
local mean field remains unaltered even when the network has only a finite
number of nodes and it has a natural measure with no special symmetrical
properties.

As an example to illustration how the CAS phenomenon appears in a complex
network, we consider a scaling-free network formed by, say, $N=1000$
Hindmarsh-Rose neurons, with neurons coupled electrically. The network is
described by

$\displaystyle\dot{x}_{i}$ $\displaystyle=$ $\displaystyle
y_{i}+3x_{i}^{2}-x_{i}^{3}-z_{i}+I+\sigma\sum_{j=1}^{N}A_{ij}(x_{j}-x_{i})$
$\displaystyle\dot{y}_{i}$ $\displaystyle=$ $\displaystyle 1-5x_{i}^{2}-y_{i}$
(9) $\displaystyle\dot{z}_{i}$ $\displaystyle=$ $\displaystyle-
rz_{i}+4r(x_{i}+1.618)$

where $I$=3.25 and $r$=0.005. The first coordinate of the equations that
describe the CAS pattern is given by

$\dot{\Xi}_{{x}_{i}}=\Xi_{{y}_{i}}+3{\Xi}_{{x}_{i}}^{2}-{\Xi}_{{x}_{i}}^{3}-{\Xi}_{{z}_{i}}+I_{i}-p_{i}{\Xi}_{{x}_{i}}+p_{i}C_{i}.$
(10)

Figure 1: [Color online] (a) Expected value of the local mean field of the
node $i$ against the node degree $k_{i}$. The error bar indicates the variance
($\mu^{2}_{i}$) of $\overline{x}_{i}$. (b) Black points indicate the value of
$C_{i}$ and $p_{i}$ for Eq. (10) to present a stable periodic orbit (no
positive Lyapunov exponents). The maximal values of the periodic orbits
obtained from Eq. (10) is shown in the bifurcation diagram in (c) considering
$C_{i}=-0.82$ and $\sigma=0.001$. (d) The CAS pattern for a neuron $i$ with
degree $k_{i}$=25 (with $\sigma=0.001$ and $C=-0.82$). In the inset, the same
CAS pattern of the neuron $i$ and some sampled points of the trajectory for
the neuron $i$ and another neuron $j$ with degree $k_{j}=25$. (e) The
difference between the first coordinates of the trajectories of neurons $i$
and $j$, with a time-lag of $\tau_{ij}=34.2$. (f) Phase difference between the
phases of the trajectories for neurons $i$ and $j$.

The others are given by
$\dot{\Xi}_{{y}_{i}}=1-\Xi_{{x}_{i}}^{2}-\Xi_{{y}_{i}}$,
$\dot{\Xi}_{{z}_{i}}=-r\Xi_{{z}_{i}}+4r(\Xi_{{x}_{i}}+1.618)$. In this
network, we have numerically verified that criterion 1 is satisfied for
neurons that have degrees $k\geq 10$ if $\sigma\leq\sigma^{*}$, with
$\sigma^{*}\cong 0.001$. In Fig. 1(a), we show the expected value $C_{i}$ of
the local mean field of the first coordinate ${x}_{i}$ of a neuron $i$ with
respect to the neuron degree (indicated in the horizontal axis), for
$\sigma=0.001$. The error bar indicates the variance of $C_{i}$ which fits to
$\propto k_{i}^{-1.0071}$. In (b), we show a parameter space to demonstrate
that the CAS phenomenon is a robust and stable phenomenon. Numerical
integration of Eqs. (9) for $p_{i}\in[0.001,1]$ produces $C_{i}\in[-0.9,0.7]$.
We integrate Eq. (10) by using $C_{i}\in[-0.9,0.7]$ and $p_{i}\in[0,0.2]$, to
show that the CAS pattern is stable for most of the values. So, variations in
$C_{i}$ of a network caused by changes in a parameter do not modify the
stability of the CAS pattern calculated by Eq. (10). For $\sigma=0.001$, Eqs.
(9) yields many nodes for which $\overline{x}_{i}\cong-0.82$. So, to calculate
the CAS pattern for these nodes, we use $C_{i}=-0.82$ and $\sigma=0.001$ in
Eqs. (10). The CAS pattern obtained, as we vary $p_{i}$, is shown in the
bifurcation diagram in (c), by plotting the local maximal points of the CAS
patterns. Criterion 2 is satisfied for most of the range of values of $p_{i}$
that produces a stable periodic CAS pattern. A neuron that has a degree
$k_{i}$ is locked to the CAS pattern calculated by integrating Eqs. (10) using
$k_{i}\sigma=p_{i}$ and the measured expected value for the local mean field,
$C_{i}$. In (d), we show the periodic orbit corresponding to a CAS pattern
associated to a neuron $i$ with degree $k_{i}=25$ (for $\sigma$=0.001) and in
the inset the sampled points of the trajectories of this same neuron $i$ and
of another neuron $j$ that has not only equal degree ($k_{j}$=25), but it
feels also a local mean field of $C_{j}\cong-0.82$. In (e), we show that these
two neurons have a typical time-lag synchronous behavior. In (f), we observe
$p/q=1$ phase synchronization between these two neurons for a long time,
considering that the phase difference remains bounded by $S=6\times 2\pi$ as
defined in Eq. (4), where the number 6 is the number of spikings within one
period of the slower time-scale. In order to verify Eq. (4) for all time, we
need to choose a ratio that is approximately equal to 1 ($p/q\cong 1$), but
not exactly 1 to account for slight differences in the local mean field of
these two neurons. Since $C_{i}$ depends on $\sigma$ for networks that have
neurons possessing a finite degree, we do not expect to observe a stable phase
synchronization in this network. Small changes in $\sigma$ may cause small
changes in the ratio $p/q$. Notice however that Eq. (4) might be satisfied for
a very long time, for $p/q=1$. If neurons are locked to different CAS patterns
(and therefore have different local mean field), Eqs. (1) and (4) are both
satisfied, but phase synchronization will not be 1:1, but with a ratio of
$p/q$ (see Sec. E in Supplementary Information for an example).

If neurons in this scaling-free network become completely synchronous, it is
necessary that $\sigma(N)\geq 2\sigma^{CS}(N=2)/|\lambda_{2}|$ (Ref.
pecora_PRL1998 ). $\sigma^{CS}(N=2)\cong 0.5$ represents the value of the
coupling strength when two bidirectionally coupled neurons become completely
synchronous. $\lambda_{2}=-2.06$ is the largest non-positive eigenvalue of the
Laplacian matrix defined as $A_{ij}-\mbox{diag}{(k_{i})}$. So,
$\sigma^{CS}(N)\geq 1/2.06\cong 0.5$. The CAS phenomenon appears when
$\sigma^{CAS}(N=1000)\leq 0.001$, a coupling strength 500 times smaller than
the one which produces complete synchronization. Similar conclusions would be
obtained when one considers networks of different sizes, with nodes having the
same dynamical descriptions and same connecting topology.

Concluding, in this work we introduce the phenomenon of Collective Almost
Synchronization (CAS), a phenomenon that is characterized by having nodes
possessing approximately constant local mean fields. The appearance of an
approximately constant mean field is a consequence of a regime of weak
interaction between the nodes responsible to place the node trajectory around
stable periodic orbits. A network has the CAS phenomenon if the Central Limit
Theorem can be applied, and it exists an approximately constant mean field. In
other words, the CAS is invariant to changes in the value of the expected
value of the local mean field, that might appear due to parameter alterations
(e.g. coupling strength). If the expected value of the local field changes,
but the Central limit Theorem can still be applied, nodes of the network will
present the CAS phenomenon and the observed weak forms of synchronization
among the nodes might (or not) be preserved. As examples of how common this
phenomenon could be, we have asserted its appearance in a large networks of
chaotic maps (see supplementary information), Hindmarsh-Rose neurons, and
Kuramoto oscillators (see supplementary information). In the Supplementary
Information, we also discuss that the CAS phenomenon is a possible source of
coherent motion in systems that are models for the appearance of collective
motion in social, economical, and animal behaviour.

##  I Supplementary Information

###  I.1 CAS and generalized synchronization

Generalized synchronization rulkov_PRE1995 ; abarbanel_PRE1996  is a common
behavior in complex networks hung_PRE2008 ; guan_chaos2009 ; hu_chaos2010 ,
and should be expected to be found typically. This phenomenon is defined as
$x_{i}=\Phi(y_{i})$, where $\Phi$ is considered to be a continuous function.
As explained in Refs. rulkov_PRE1995 ; abarbanel_PRE1996 , generalized
synchronization appears due to the existence of a low-dimensional synchronous
manifold, often a very complicated and unknown manifold.

Recent works zhou_CHAOS2006 ; ballerini_PNAS2008 ; pereira_PRE2010 ;
gardenes_chaos2011  have reported that nodes in the network that are highly
connected become synchronous. As shown in ref. guan_chaos2009 , that is a
manifestation of generalized synchronization rulkov_PRE1995 ;
abarbanel_PRE1996  in complex networks. For a fixed coupling strength among
the nodes with heterogeneous degree distributions and for the usual
diffusively coupling configuration one should expect that the set of hub nodes
(highly connected nodes) provides a skeleton about which synchronization is
developed. Reference hramov_PRE2005  demonstrates how ubiquitous generalized
synchronization is in complex networks. It is shown that a necessary condition
for its appearance in oscillators coupled in a driven-response (master-slave)
configuration is that the modified dynamics of the response system presents a
stable periodic behavior. The modified dynamics is a set of equations
constructed by considering only the variables of the response system. In a
complex network, a modified dynamics of a node is just a system of equations
that contains only variables of that node.

An important contribution to understand why generalized synchronization is a
ubiquitous property in complex network is given by the numerical work of Ref.
guan_chaos2009  and the theoretical work of Ref. hu_chaos2010 . In Refs.
guan_chaos2009 ; hu_chaos2010  the ideas of Ref. hramov_PRE2005  are extended
to complex networks. In particular, the work of Ref. hu_chaos2010  shows that
generalized synchronization occurs whenever there is at least one node whose
modified dynamics is periodic. All the nodes that have a stable and periodic
modified dynamics become synchronous in the generalized sense with the nodes
that have a chaotic modified dynamics. The general theorem presented in Ref.
hu_chaos2010  is a powerful tool for the understanding of weak forms of
synchronization or desynchronous behaviors in complex networks. However,
identifying the occurrence of generalized synchronization does not give much
information about the behavior of the network, since the function that relates
the trajectory among the nodes that are generalized synchronous is usually
unknown. The CAS phenomenon allows one to calculate, at least in an
approximate sense, the equations of motion that describes the pattern to which
the nodes are locked to. More specifically, we can derive the set of equations
governing, in an approximate sense, the time evolution of the nodes, not
covered by the theorem in Ref. hu_chaos2010 .

Finally, if there is a node whose modified dynamics describes a stable
periodic behavior and its CAS pattern is also a stable periodic stable
behavior, then the CAS phenomenon appears when the network presents
generalized synchronization.

###  I.2 CAS and other synchronous and weak-synchronous phenomena

Consider a network of $N$ nodes described by

$\dot{\mathbf{x}}_{i}=\mathbf{F}_{i}(\mathbf{x}_{i})+\sigma\sum_{j=1}^{N}{\mathbf{A}_{ij}}{\mathbf{E}}[\mathcal{H}(\mathbf{x}_{j}-\mathbf{x}_{i})]+\mathbf{\zeta}_{i}(t),$
(11)

where $\mathbf{x}_{i}\in\Re^{d}$ is a d-dimensional vector describing the
state variables of the node $i$, $\mathbf{F}_{i}$ is a $d$-dimensional vector
function representing the dynamical system of the node $i$,
${\mathbf{A}_{ij}}$ is the adjacent connection matrix, ${\mathbf{E}}$ is the
coupling function as defined in pecora_PRL1998 , $\mathcal{H}$ is an arbitrary
differentiable transformation, and $\mathbf{\zeta}_{i}(t)$ is an arbitrary
random fluctuation. Assume in the following that $\mathbf{\zeta}_{i}(t)=0$.

Assume that the nodes in the network (11) have equal dynamical descriptions,
i.e., $\mathbf{F}_{i}=\mathbf{F}$, that the network is fully connected, so
every node has a degree $k_{i}=N-1$, and that
$\mathcal{H}(\mathbf{x}_{j}-\mathbf{x}_{i})=(\mathbf{x}_{j}-\mathbf{x}_{i})$.
We can rewrite it in terms of the average field
$\overline{\mathbf{x}}(t)=\frac{1}{N}\sum_{i=1}^{N}\mathbf{x}_{i}(t)$:

$\dot{\mathbf{x}}_{i}=\mathbf{F}_{i}(\mathbf{x}_{i})-p_{i}\mathbf{E}(\mathbf{x}_{i}-\overline{\mathbf{x}}),$
(12)

where $p_{i}=\sigma k_{i}$. Therefore every node becomes “decoupled” from the
network in the sense that their interaction is all mediated by the average
field. Collective behavior is dictated by the behavior of the average field
and the individual dynamics of the node. The linear stability of the network
(12) was used in Ref. zhou_CHAOS2006  as an approximation to justify how
desynchronous behavior about the average field can appear in complex networks.
Notice that this assumption can only be rigorously fulfilled if the network is
fully connected and, therefore, it is natural to understand why the
desynchronous phenomena reported in Ref. zhou_CHAOS2006  happens for nodes
that are highly connected. One can interpret the desynchronous behavior
observed in Ref. zhou_CHAOS2006  as an almost synchronization between a node
and the mean field $\overline{\mathbf{x}}$.

The differences between complete synchronization and synchronization in the
collective sense can be explained through the following example. An
interesting solution of Eq. (12) can be obtained when
$\overline{\mathbf{x}}=\mathbf{x}_{i}(t)$, $\mathbf{x}_{i}(t)$ varying in
time. In this case, the average field is along the synchronization manifold.
The network being completely synchronous, all nodes having equal trajectories,
and $\mathbf{F}_{i}(\mathbf{x}_{i}(t))=\mathbf{x}_{i}(t)$. For such a special
network, collective behavior and complete synchronization are the same. On the
other hand, collective behavior typically appears when the coupling term
$\sigma E(\mathbf{x}_{i}-\overline{\mathbf{x}})$ is different from zero for
most of the time and $\mathbf{F}_{i}(\mathbf{x}_{i})\neq\mathbf{x}_{i}$, but
there is a majority of nodes with similar behavior. In this sense, the
desynchronous behaviors reported in Ref. zhou_CHAOS2006  can be considered as
a collective phenomena that happens to parameters close to the ones that
yields complete synchronization.

To understanding when the CAS phenomenon occurs, consider the solution of Eq.
(12) in the thermodynamics limit $N\rightarrow\infty$ when
$\overline{\mathbf{x}}$ is a constant in time, $\overline{\mathbf{x}}=C$. For
such a situation, the evolution of a node can be described by the same
following d-dimensional system of ODEs

$\dot{\mathbf{x}}=\mathbf{F}(\mathbf{x})-p\mathbf{E}(\mathbf{x}-\mathbf{C}),$
(13)

where $p=\sigma(N-1)$. If complete synchronization takes place, then
$\mathbf{F}_{i}(\mathbf{C})=0$, meaning that there can only exist complete
synchronization if all the nodes lock into the same stable steady state
equilibrium point, likely to happen if $\mathbf{F}_{i}$ is the same for all
the nodes.

Another possible network configuration that leads to
$\overline{\mathbf{x}}=\mathbf{C}$ happens when each node is only weakly
coupled (“independent”) with the others such that the Central Limit Theorem
could be applied. If the network has only a finite number of nodes and
$\overline{\mathbf{x}}(t)$ is not exactly constant in time, but
$\overline{\mathbf{x}}(t)\approxeq\mathbf{C}$, the nodes still behave in the
same predictable way if the dynamics described by
$\dot{\mathbf{x}}=\mathbf{F}(\mathbf{x})-p\mathbf{E}(\mathbf{x})+p\mathbf{E}(\mathbf{C})$
is a sufficiently stable periodic orbit. This is how the CAS phenomenon
appears in fully connected networks. All nodes become locked to the stable
periodic orbit described by
$\dot{\mathbf{x}}=\mathbf{F}(\mathbf{x})-p\mathbf{E}(\mathbf{x})+p\mathbf{E}(\mathbf{C})$.

Now, we break the symmetry of the network, allowing the nodes to be connected
arbitrarily to their neighbors. We still consider diffusive linear couplings,
$\mathcal{H}(\mathbf{x}_{j}-\mathbf{x}_{i})=(\mathbf{x}_{j}-\mathbf{x}_{i})$.
The equations of such a network can be written as

$\dot{\mathbf{x}}_{i}=\mathbf{F}_{i}(\mathbf{x}_{i})-p_{i}\mathbf{E}(\mathbf{x}_{i})+p_{i}\mathbf{E}(\overline{\mathbf{x}}_{i}(t)),$
(14)

where $k_{i}$ is the degree of node $i$ with $k_{l}\leq k_{m}$, if $l<m$, and
$\overline{\mathbf{x}}_{i}(t)$ is the local mean field defined as

$\overline{\mathbf{x}}_{i}(t)=\frac{1}{k_{i}}\sum_{j=1}^{N}A_{ij}\mathbf{x}_{j}(t).$
(15)

Our main assumption is that the local mean field of a variable that is
bounded, either $\overline{\mathbf{x}}_{i}(t)$ or
$\overline{\dot{\mathbf{x}}}_{i}(t)$, exhibits small oscillations about an
expected constant value $\mathbf{C}$. In other words, one can define a time
average $\mathbf{C}$ by either

$\mathbf{C}_{i}=\frac{1}{t}\int_{0}^{t}\overline{\mathbf{x}}_{i}(t)dt,$ (16)

or

$\mathbf{C}_{i}=\frac{1}{t}\int_{0}^{t}\overline{\dot{\mathbf{x}}}_{i}(t)dt.$
(17)

Notice that ${\mathbf{x}}_{i}\in\Re^{d}$ (or
$\dot{\mathbf{x}}_{i}\in\Re^{d}$), and so does $\mathbf{C}\in\Re^{d}$. The CAS
phenomenon appears for a node that has at least one component of the local
mean field ($\overline{\mathbf{x}}_{i}$ or $\overline{\dot{\mathbf{x}}}_{i}$)
that is approximately constant. The appearance of this almost constant value
is a consequence of the Central Limit Theorem. For networks whose nodes are
described by only bounded variables, when calculating the local mean field we
only take into consideration the component receiving the couplings from other
nodes. For networks of Kuramoto oscillators that have one variable (the phase
$\theta$) that is not bounded, a constant local mean field appears in the
component that describes the instantaneous frequency ($\dot{\theta}_{i}$).

In Ref. hu_chaos2010 , it was shown that for chaotic networks described by a
system of equations similar to Eq. (14), generalized synchronization can
appear if the modified dynamics described by
$\dot{\mathbf{x}}_{i}=\mathbf{F}_{i}(\mathbf{x}_{i})-\sigma
k_{i}\mathbf{E}(\mathbf{x}_{i})$ of a certain number of nodes are either
stable equilibrium points ($\dot{\mathbf{x}}_{i}$=0) or they describe stable
periodic solutions (limit cycle). Generalized synchronization appears between
the nodes that have modified dynamics describing stable periodic states and
the nodes that have modified dynamics describing chaotic states.

To understand the phenomenon of collective almost synchronization (CAS),
introduced in this work, consider that
$\mathcal{H}(\mathbf{x}_{j}-\mathbf{x}_{i})=(\mathbf{x}_{j}-\mathbf{x}_{i})$.
It is a phenomena that appears necessarily when
$\overline{\mathbf{x}}_{i}\approxeq\mathbf{C}_{i}$ or
$\overline{\dot{\mathbf{x}}}_{i}\approxeq\mathbf{C}_{i}$. The equations for
the network can then be described by

$\dot{\mathbf{x}}_{i}=F_{i}(\mathbf{x}_{i})-p_{i}\mathbf{E}(\mathbf{x}_{i})+p_{i}\mathbf{E}(\mathbf{C}_{i})+\mathbf{\delta}_{i},$
(18)

where the residual term is
$\delta_{i}=p_{i}(\overline{\mathbf{x}}_{i}-\mathbf{C}_{i})$. This term is
small most of the time but large for some intervals of time;
$\mathbf{\delta}_{i}(t)>0$ for all time, but $\mathbf{\delta}_{i}(t)<\epsilon$
for most of the times. Another requirement for the CAS phenomenon to appear is
that the CAS pattern $\mathbf{\Xi}_{i}(t)$ of a node $i$ that is described by
Eq. (18) ignoring the residual term

$\dot{\mathbf{\Xi}}_{i}=\mathbf{F}_{i}(\mathbf{\Xi}_{i})-p_{i}\mathbf{E}(\mathbf{\Xi}_{i})+p_{i}\mathbf{E}(\mathbf{C}_{i}).$
(19)

must be a stable periodic orbit. We define that a node presents collective
almost synchronization (CAS) if

$|\mathbf{x}_{i}(t)-\mathbf{\Xi}_{i}(t-\tau_{i})|<\epsilon_{i},$ (20)

for most of the time,

Notice from Eq. (19) that for $p_{i}>0$, the CAS pattern will not be described
by $\mathbf{F}(\mathbf{x}_{i})$ and therefore does not belong to the
synchronization manifold. On the other hand, $\mathbf{\Xi}_{i}$ is induced by
the local mean field as typically happens in synchronous phenomenon due to
collective behavior. This property of the CAS phenomenon shares similarities
with the way complete synchronization appears in networks of nodes coupled
under time-delay functions nijmeijer_IEEE2011 . In such networks, nodes become
completely synchronous to a solution of the network that is different from the
solution of an isolated node of the network. Additionally, the trajectory of
the nodes present a time-lag to this solution.

To understand the reason why the CAS phenomenon appears when
$\mathbf{\Xi}_{i}(t)$ is a sufficiently stable periodic orbit, we study the
variational equation of the CAS pattern (19)

$\dot{\mathbf{\xi}}_{i}=[D\mathbf{F}_{i}(\mathbf{\xi}_{i})-p_{i}\mathbf{E}]\mathbf{\xi}_{i}.$
(21)

obtained by linearizing Eq. (19) around $\mathbf{\Xi}_{i}$ by making
$\mathbf{\xi}_{i}=\mathbf{x}_{i}-\mathbf{\Xi}_{i}$. This equation produces no
positive Lyapunov exponents. As a consequence, neglecting the existence of the
time-lag between $\mathbf{x}_{i}(t)$ and $\mathbf{\Xi}(t)_{i}$, the trajectory
of the node $i$ oscillates about $\mathbf{\Xi}_{i}$, and
$\mathbf{x}_{i}-\mathbf{\Xi}_{i}\leq\epsilon_{i}$, for most of the time,
satisfying Eq. (20), where $\epsilon_{i}$ depends on $\mathbf{\delta}_{i}$. If
there are two nodes $i$ and $j$, which feel similar local mean fields,
$\mathbf{\Xi}_{i}\approxeq\mathbf{\Xi}_{j}$, then
$\mathbf{x}_{i}\approxeq\mathbf{x}_{j}$, for most of the time.

To understand why the nodes that present CAS have also between them a time-lag
type of synchronization, integrate Eq. (18), using Eq. (19), to obtain

$\mathbf{x}_{i}(t)=\int_{0}^{t}[\dot{\mathbf{\Xi}}_{i}(t)+\mathbf{\delta}_{i}(t)]dt.$
(22)

This integral is not trivial in the general case. But we have a simple
phenomenological explanation for its solution. When the CAS pattern is
sufficiently stable, the asymptotic time limit state of the variable
$\mathbf{x}_{i}(t)$ is the CAS pattern $\mathbf{\Xi}_{i}(t)$. But due to the
residual term $\mathbf{\delta}_{i}(t)$, the trajectory of $\mathbf{x}_{i}(t)$
arrives in the neighborhood of $\mathbf{\Xi}(t)$ at time $t$ with a time-lag.
As a result, nodes that are collectively almost synchronous obey Eq. (20). In
addition, two nodes that present CAS have also a time-lag between their
trajectories for the same reason. There is an extra contribution to the time-
lag between the trajectories of two nodes if their initial conditions differ.

Phase synchronization juergen_book  is a phenomena where the phase difference,
denoted by $\Delta\phi_{ij}$ between the phases of two signals (or nodes in a
network), $\phi_{i}(t)$ and $\phi_{j}(t)$, remains bounded for all time

$\Delta\phi_{ij}=\left|\phi_{i}(i)-\frac{p}{q}\phi_{j}(t)\right|\leq S,$ (23)

where $S=2\pi$, and $p$ and $q$ are two rational numbers juergen_book . For
coupled chaotic oscillators one can also find irrational phase synchronization
baptista_PRE2004 , where Eq. (23) can be satisfied for all time with $p$ and
$q$ irrational. $S$ is a reasonably small constant, that can be larger than
2$\pi$ in order to encompass oscillatory systems that either have a time
varying time-scale or whose time-lag varies in time. This bound can be simply
calculated by making $S$ to represent the growth of the phase in the faster
time scale after one period of the slower time scale.

The link between the CAS phenomenon and phase synchronization can be explained
by thinking that it is a synchronous phenomenon among the nodes that is
mediated by their CAS patterns. The phase of the periodic orbit of the CAS
pattern of the node $i$ grows as
$\tilde{\phi}_{i}(t)=\omega_{i}t+\xi_{i}(t)+\phi_{i}^{0}$ and of the node $j$
grows as $\tilde{\phi}_{j}(t)=\omega_{j}t+\xi_{j}(t)+\phi_{j}^{0}$. The
quantities $\phi_{i}^{0}$ and $\phi_{j}^{0}$ are displacements of the phase
caused by the existence of time-lag, and $\xi_{i}(t)$ and $\xi_{j}(t)$ are
small fluctuations. For $t\rightarrow\infty$ these can be neglected and we
have that

$\frac{\tilde{\phi}_{i}(t)}{\tilde{\phi}_{j}(t)}=\frac{\omega_{i}}{\omega_{j}}=\frac{p}{q},$
(24)

where $\omega_{i}=\lim_{t\rightarrow\infty}\frac{\tilde{\phi}_{i}(t)}{t}$
gives the average frequency of oscillation of the CAS pattern of node $i$, and
$p$ and $q$ are two real numbers.

The phase of the nodes can be written as a function of the phase of the
periodic orbits of the CAS pattern. So,
$\phi(t)_{i}=\tilde{\phi(t)}_{i}+\delta\phi_{i}(t)$ and
$\phi(t)_{j}=\tilde{\phi(t)}_{j}+\delta\phi_{j}(t)$, $\delta_{i}(t)$
represents a variation of the phase of the node $i$ with respect to the phase
of the CAS pattern, and depends on the way the phase is defined
pereira_PLA2007 . The phase difference $\Delta\phi_{ij}(t)$, as written in Eq.
(23), becomes equal to
$|t(q\omega_{i}-p\omega_{j})+q\delta_{i}(t)-p\delta\phi_{j}(t)|$. But, from
Eq. (24), $q\omega_{i}-p\omega_{j}=0$, and therefore,
$\Delta\phi_{ij}(t)\leq\max{(q\delta\phi_{i}(t)-p\delta\phi_{j}(t))}$. But
since the node orbit is locked to the CAS pattern, $\Delta\phi_{ij}(t)$ is
always a small quantity.

In practice, for networks composed by a finite number of nodes, we do not
expect that the quantities $\delta\phi_{i}(t)$ and $\delta\phi_{j}(t)$ to
remain small for all the time. The reason is that the CAS pattern can only be
approximately calculated and in general we do not know the precise real value
of the local mean field. However, our simulations show that these quantities
remain small for time intervals that comprise many periods of oscillations of
the node trajectories. For networks having an expected value of the mean field
$\mathbf{C}_{i}$ that is independent on the coupling strength $\sigma$, the
ratio $p/q$ does not change as one changes the value of $\sigma$, and then
phase synchronization is stable under a parameter variation. For the network
of Kuramoto oscillators, Eq. (23) can be verified for all time with a value of
$p/q$ that remains invariant as one changes $\sigma$.

Assume for now that the nodes have equal dynamics, so
$\mathbf{F}_{i}=\mathbf{F}$. If a node $i$ with degree $k_{i}$ has a periodic
CAS pattern that is sufficiently stable under Eq. (21), all the nodes with
degrees close to $k_{i}$ also have similar CAS patterns that are sufficiently
stable under Eq. (21). Node $i$ is locked to $\mathbf{\Xi}_{i}$ and node $j$
is locked to $\mathbf{\Xi}_{j}$. But since $\mathbf{\Xi}_{i}$ is approximately
equal to $\mathbf{\Xi}_{j}$, thus, $\mathbf{x}_{i}\cong\mathbf{x}_{j}$, for
most of the time. So, if the pattern solution is sufficiently stable, the
external noise $\mathbf{\zeta}_{i}(t)$ can be different from zero, and still
have similar trajectories for that interval of time. The same argument remains
valid if $\mathbf{F}_{i}\neq\mathbf{F}_{j}$, as long as the CAS pattern is
sufficiently stable.

In Ref. pereira_PRE2010 , synchronization was defined in terms of the node
$\mathbf{x}_{N}$ that has the largest number of connections, when
$\mathbf{x}_{i}(t)\cong\mathbf{x}_{N}$ (which is equivalent to stating that
$|\mathbf{x}_{i}(t)-\mathbf{x}_{N}|<\epsilon$), where $\mathbf{x}_{N}$ is
assumed to be very close to the synchronization manifold $\mathbf{s}$ defined
by $\dot{\mathbf{s}}=\mathbf{F}(\mathbf{s})$. This type of synchronous
behavior was shown to exist in scaling free networks whose nodes have equal
dynamics and that are linearly connected. This was called hub synchronization.

The link between the CAS phenomenon with the hub synchronization phenomenon
pereira_PRE2010 , and generalized synchronization can be explained as in the
following. It is not required for nodes that present the CAS phenomenon for
their error dynamics $\mathbf{x}_{j}-\mathbf{x}_{i}$ to be small. But for the
following comparison, assume that
$\mathbf{\vartheta}_{ij}=\mathbf{x}_{j}-\mathbf{x}_{i}$ is small so that we
can linearise Eq. (14) about another node $j$. Assume also that
$\mathbf{F}_{i}=\mathbf{F}$. The variational equations of the error dynamics
between two nodes $i$ and $j$ that have equal degrees are described by

$\dot{\mathbf{\vartheta}}_{ij}=[D\mathbf{F}(\mathbf{x}_{i})-p_{i}E]\mathbf{\vartheta}_{ij}+\mathbf{\eta}_{i}.$
(25)

In Ref. pereira_PRE2010 , hub synchronization exists if Eq. (25), neglecting
the coupling term $\mathbf{\eta}_{i}$, has no positive Lyapunov exponents.
That is another way of stating that hub synchronization between $i$ and $j$
occurs when the variational equations of the modified dynamics
$[\dot{\mathbf{x}}_{i}=\mathbf{F}(\mathbf{x}_{i})-p_{i}E(\mathbf{x}_{i})]$
presents no positive Lyapunov exponent. In other words, in order to have hub
synchronization it is necessary that the modified dynamics of both nodes be
describable by stable periodic oscillations. Hub synchronization is the result
of a weak form of generalized synchronization, defined in terms of the linear
stability of the error dynamics between two highly connected nodes. Unlike
generalized synchronization, hub synchronization offers a way to predict, in
an approximate sense, the trajectory of the synchronous nodes.

In contrast, the CAS phenomenon appears when the CAS pattern, which is
different from the solution of the modified dynamics, becomes periodic.
Another difference between the CAS and the hub synchronization phenomenon is
that whereas $\overline{\mathbf{x}}_{i}\approxeq\mathbf{C}$ in the CAS
phenomenon, $\overline{\mathbf{x}}_{i}\approxeq{\mathbf{x}}_{i}$ in the hub
synchronization, in order for $\mathbf{\eta}_{i}$ to be very small, and
$\mathbf{x}_{i}$ to be close to the synchronization manifold. So, whereas hub
synchronization can be interpreted as being a type of practical
synchronization femat_PLA1999 , CAS is a type of almost synchronization.

In the work of Refs. politi_PRE2006 ; politi_PRL2010 , it was numerically
reported a new desynchronous phenomenon in complex networks. The network has
no positive Lyapunov exponents but it presents a desynchronous non-trivial
collective behavior. A possible situation for the phenomenon to appear is when
$\mathbf{\delta}_{i}$ and $\mathbf{C}_{i}$ in Eq. (18) are either zero or
sufficiently small such that the stability of the network is completely
determined by Eq. (21), and this equation produces no positive Lyapunov
exponent. Assume now that $p_{i}$ in Eq. (19) is appropriately adjusted such
that the CAS pattern for every node $i$ is a stable periodic orbit. The
variational Eqs. (21) for all nodes have no positive Lyapunov exponents. If
additionally, $\overline{\mathbf{x}}_{i}(t)\approxeq\mathbf{C}$, then the
network in Eq. (14) possesses no positive Lyapunov exponent. Therefore,
networks that present the CAS phenomenon for all nodes might present the
desynchronous phenomenon reported in Refs. politi_PRE2006 ; politi_PRL2010 .
The CAS phenomenon becomes different from the phenomenon of
Refs.politi_PRE2006 ; politi_PRL2010  if for at least one node, Eq. (19)
produces a chaotic orbit.

To understand the occurrence of CAS in networks formed by heterogeneous nodes
connected by nonlinear functions such as networks of Kuramoto oscillators, we
rewrite the Kuramoto’s network model in terms of the local mean field,
$\overline{\theta}_{i}=\frac{1}{k_{i}}\sum_{j=1}^{N}A_{ij}\theta_{j}$. Using
the coordinate transformation

$\frac{1}{k_{i}}\sum_{j=1}^{N}A_{ij}\exp{{}^{\mathbb{j}(\theta_{j}-\theta_{i})}}=\tilde{r}_{i}\exp{{}^{\mathbb{j}(\overline{\theta}_{i}-\theta_{i})}},$
(26)

the dynamics of the node $i$ is described by

$\dot{\theta_{i}}=\omega_{i}+p_{i}\tilde{r}_{i}sin(\overline{\theta}_{i}-\theta_{i}).$
(27)

The phase ${\theta_{i}}$ is not a bounded variable and therefore we expect
that typically $\overline{\theta}_{i}$ has not a well defined average. But,
$\overline{\dot{\theta_{i}}}(t)$ is bounded and has a well defined average
value which is an approximately constant quantity ($C_{i}$) for nodes in
networks with sufficiently large number of connections and with sufficiently
small coupling strengths. When $\overline{\dot{\theta}_{i}}\cong C_{i}$, the
node $i$ has the propensity to exhibit the CAS phenomenon, and the CAS pattern
is calculated by Eq. (27) considering that $\overline{\theta_{i}}=C_{i}t$.
Notice that $\overline{\theta_{i}}=\overline{\dot{\theta}_{i}}t\cong C_{i}t$.

Phase synchronization between two nodes in the networks of Eq. (27) is stable
under parameter variations (coupling strength in this case) if these nodes
present the CAS phenomenon. There is irrational (rational) phase
synchronization if
$\frac{\overline{\dot{\theta_{i}}}}{\overline{\dot{\theta_{j}}}}$ is
irrational (rational). If nodes are sufficiently “decoupled” we expect that
$\frac{\overline{\dot{\theta_{i}}}}{\overline{\dot{\theta_{j}}}}\approxeq\omega_{i}/\omega_{j}$.
Phase synchronization will be rational whenever nodes with different natural
frequencies become locked to Arnold tongues’s, induced by the coupling
$p_{i}\tilde{r}_{i}sin(\overline{\theta}_{i}-\theta_{i})$.

There is a special solution of Eq. (27) that produces a bounded state in the
variable ${\theta_{i}}$ when the network is complete synchronous to an
equilibrium point. In such case, $\overline{\theta}_{i}$ becomes constant, and
Eq. (27) has one stable equilibrium
$\theta_{i}=\arcsin{\left(\frac{\omega_{i}}{p_{i}}\right)}$, obtained when
$p_{i}>\omega_{i}$. But, the local mean field becomes constant due to complete
synchronization and not due to the fact that the nodes are “decoupled”. These
conditions do not produce the CAS phenomenon.

We take the thermodynamics limit when the network has infinite nodes with
infinite degrees. $C_{i}$ calculated using Eq. (17) does not change as one
change the coupling $\sigma$, since
$\overline{\dot{\theta}_{i}}=\lim_{k_{i},N\rightarrow\infty}\frac{1}{k_{i}}[\sum_{j=1}^{N}A_{ij}(\omega_{i}+p_{i}\tilde{r}_{i}sin(\overline{\theta_{i}}-\theta_{i}))]$=$\lim_{k_{i},N\rightarrow\infty}\frac{1}{k_{i}}[\sum_{j=1}^{N}A_{ij}\omega_{j}]+[\sum_{j=1}^{N}A_{ij}(\sigma\tilde{r}_{j}sin(\overline{\theta_{j}}-\theta_{i}))]$
=
$\frac{1}{k_{i}}[\sum_{j=1}^{N}A_{ij}\omega_{j}]+\sigma\sum_{j=1}^{N}A_{ij}(\tilde{r}_{j}sin(\overline{\theta_{j}}-\theta_{i}))$.
But, if nodes are sufficiently decoupled
$\sum_{j=1}^{N}A_{ij}(\tilde{r}_{j}sin(\overline{\theta_{j}}-\theta_{i}))$
approaches zero, and therefore, $C_{j}$ only depends on the natural
frequencies:
$\overline{\dot{\theta}_{i}}=C_{i}=\frac{1}{k_{i}}[\sum_{j=1}^{N}A_{ij}\omega_{j}]$.

Assume that there are two nodes, $i$ and $j$, and that for most of the time
$\Xi_{i}\approxeq\Xi_{j}$. Then, for most of the time it is also true that
$\Xi_{i}-\theta_{i}\approxeq\Xi_{j}-\theta_{j}$, which allow us to write that
$sin(\Psi_{j}-\theta_{j})-sin(\Psi_{i}-\theta_{i})\approxeq\cos{(\Psi_{i}-\theta_{i})}[(\Psi_{j}-\theta_{j})-(\Psi_{j}-\theta_{j})]\approxeq\cos{(\Psi_{i}-\theta_{i})}[\theta_{j}-\theta_{j}]$.
Since $\Psi_{i}\approxeq\theta_{i}$, then
$\cos{(\Psi_{i}-\theta_{i})}\approxeq 1$ and
$sin(\Psi_{j}-\theta_{j})-sin(\Psi_{i}-\theta_{i})\approxeq[\theta_{j}-\theta_{j}]$.
Defining the error dynamics between the two nodes to be
$\xi_{ij}=\theta_{j}-\theta_{i}$, we arrive that

$\dot{\xi}_{ij}\approxeq(\omega_{j}-\omega_{i})-p_{i}\xi_{ij}.$ (28)

Therefore, it implies that we expect to find two nodes having the same similar
CAS behavior when both the local mean field is close and when the difference
between their natural frequencies $(\omega_{j}-\omega_{i})$ is small.

The CAS phenomenon can also appear in a system of driven particles
vicsek_PRL1995  that is a simple but powerful model for the onset of pattern
formation in population dynamics couzin_ASB2003 , economical systems
gregoire_physicaD2003  and social systems helbing_nature2000 . In the work of
Ref. vicsek_PRL1995 , it was assumed that individual particles were moving at
a constant speed but with an orientation that depends on the local mean field
of the orientation of the individual particles within a local neighborhood and
under the effect of additional external noise. Writing an equivalent time-
continuous description of the Vicsek particle model vicsek_PRL1995 , the
equations of motion for the direction of movement of a particle $i$, can be
written as

$\dot{\mathbf{x}}_{i}=-\mathbf{x}_{i}+\overline{\mathbf{x}}_{i}+\Delta\mathbf{\theta}_{i},$
(29)

where $\overline{\mathbf{x}}_{i}$ represents the local mean field of the
orientation of the particle $i$ within a local neighborhood and
$\Delta\mathbf{\theta}_{i}$ represents a small noise term. When
$\overline{\mathbf{x}}_{i}$ is approximately constant, the CAS pattern is
described by a solution of
$\dot{\mathbf{x}}_{i}=-\mathbf{x}_{i}+\overline{\mathbf{x}}_{i}$, which will
be a stable equilibrium point as long as $\Delta\mathbf{\theta}_{i}$ is
sufficiently small. From the Central Limit Theorem,
$\overline{\mathbf{x}}_{i}$ will be approximately constant as long as the
neighborhood considered is sufficiently large or the density of particles is
sufficiently large.

###  I.3 About the expected value of the local mean field: the Central Limit
Theorem

The Theorem states that, given a set of $t$ observations, each set of
observation containing $k$ measurements
($x_{1},x_{2},x_{3},x_{4},\ldots,x_{k}$), the sum
$S_{N}=\sum_{i=1}^{k}x_{i}(N)$ (for $N=1,2,\ldots,t$), with the variables
$x_{i}(N)$ drawn from an independent random process that has a distribution
with finite variance $\mu^{2}$ and mean $\overline{x}$, converges to a Normal
distribution for sufficiently large $k$. As a consequence, the expected value
of these $t$ observations is given by the mean $\overline{x}$ (additionally,
$\overline{x}=\frac{1}{t}\sum_{N=1}^{t}S_{N}$), and the variance of the
expected value is given by $\frac{\mu^{2}}{k}$. The larger the number $k$ of
variables being summed, the larger is the probability with which one has a sum
close to the expected value. There are many situations when one can apply this
theorem for variables with some sort of correlation hilhorst_BJP2009 , as it
is the case for variables generated by deterministic chaotic systems with
strong mixing properties, for which the decay of correlation is exponentially
fast. In other words, a deterministic trajectory that is strongly chaotic
behaves as an independent random variable in the long-term. For that reason,
the Central Limit Theorem holds for the time average value $\overline{x}(t)$
produced by summing up chaotic trajectories from nodes belonging to a network
that has nodes weakly connected. Consequently, the distribution of
$\overline{x}_{i}(t)=\frac{1}{N}\sum_{j}A_{ij}x_{j}(t)$ for node $i$ should
converge to a Gaussian distribution centered at
$C_{i}=\frac{1}{t}\int_{0}^{t}\overline{x}_{i}(t)dt$ as the degree of the node
is sufficiently large. In addition, the variance $\mu^{2}_{i}$ of the local
mean field $\overline{x}(t)_{i}$ decreases proportional to $k_{i}^{-1}$, as we
have numerically verified for networks of Hindmarsh-Rose neurons
($\mu^{2}_{i}\propto k_{i}^{-1.0071}$) and networks of Kuramoto oscillators
($\mu^{2}_{i}\propto k_{i}^{-1.055}$).

If the network has no positive Lyapunov exponents, we still expect to find an
approximately constant local mean field at a node $i$, as long as the nodes
are weakly connected and its degree is sufficiently large. To understand why,
imagine that every node in the network stays close to a CAS pattern and one of
its coordinates is described by $sin(\omega_{i}t)$. Without loss of generality
we can make that every node has the same frequency $\omega_{i}=\omega$. The
time-lag property in the node trajectories, when they exhibit the CAS pattern,
results in that every node is close to $sin(\omega_{i}t)$ but they will have a
random time-lag in relation to the CAS pattern (due to the decorrelated
property between the node trajectories). So, the selected coordinate can be
described by $sin(\omega t+\phi^{0}_{i})+\delta_{i}(t)$, where $\phi^{0}_{i}$
is a random initial phase and $\delta_{i}(t)$ is a small random term
describing the distance between the node trajectory and the CAS pattern.
Neglecting the term $\delta_{i}(t)$, the distribution of the sum
$\sum_{i=1}^{k}sin(\omega t+\phi^{0}_{i})$ converges to a normal distribution
with a variance that depends on the variance of $sin(\phi^{0}_{i})$.

From previous considerations, if the degree of some of the nodes tend to
infinite, the variance of the local mean field for those nodes tends to zero
and, in this limit, the residual term $\delta_{i}$ in Eq. (18) is zero and the
local mean field of these nodes is a constant. As a consequence, the node is
perfectly locked with the CAS pattern ($\epsilon=0$ in Eq. (20)).

###  I.4 CAS in a network of coupled maps

As another example to illustrate how the CAS phenomenon appears in a complex
network, we consider a network of maps whose node dynamics is described by
$F_{i}(x_{i})=2x_{i}$ mod(1). The network composed, say, by $N=1000$ maps, is
represented by
$x_{i}^{(n+1)}=F_{i}(x_{i}^{(n)})+\sigma\sum_{j=1}^{N}A_{ij}(x_{j}^{(n)}-x_{i}^{(n)})$
mod(1), where the upper index $n$ represents the discrete iteration time, and
$A_{ij}$ is the adjacency matrix of a scaling-free network. The map has a
constant probability density. When such a map is connected in a network, the
density is no longer constant, but still symmetric and having an average value
of 0.5. As a consequence, nodes that have a sufficient amount of connections
($k\geq 10$) feel a local mean field, say, within $[0.475,0.525]$, (deviating
of 5$\%$ about $C_{i}$=0.5) and $\mu^{2}_{i}\propto k_{i}^{-1}$ (criterion 1),
as shown in Fig. 2(a). Therefore, such nodes have propensity to present the
CAS phenomenon. In (b) we show a bifurcation diagram of the CAS pattern,
$\Xi_{i}$, obtained from Eq. (19) by using $C_{i}=C=0.5$, as we vary $p_{i}$.
Nodes in this network that have propensity to present the CAS phenomenon will
present it if additionally $p_{i}\in[1,3]$; the CAS pattern is described by a
period-2 stable orbit (criterion 2). This interval can be calculated by
solving $|2-p_{i}|\leq 1$. In (c) we show the probability density function of
the trajectory of a node that present the CAS phenomenon. The density is
centered at the position of the period-2 orbit of the CAS pattern and for most
of the time Eq. (20) is satisfied. The filled circles are fittings assuming
that the probability density is given by a Gaussian distribution. Therefore,
there is a high probability that $\epsilon_{i}$ in Eq. (20) is small. In (d)
we show a plot of the trajectories of two nodes that have the same degree
which is equal to 80. We chose nodes which present no time-lag between their
trajectories and the trajectory of the pattern. If there was a time-lag, the
points in (d) would not be only aligned along the diagonal (identity) line,
but they would also appear off-diagonal.

Figure 2: (a) Expected value of the local mean field of the node $i$ against
the node degree $k_{i}$. The error bar indicates the variance ($\mu^{2}_{i}$)
of $\overline{x}_{i}$. (b) A bifurcation diagram of the CAS pattern [Eq. (19)]
considering $C_{i}=0.5$. (c) Probability density function of the trajectory of
a node with degree $k_{i}$=80 (therefore, $p_{i}=\sigma k_{i}=1.3$,
$\sigma=1.3/80$). (d) A return plot considering two nodes ($i$ and $j$) with
the same degree $k_{i}=k_{j}=$80.

###  I.5 CAS in the Kuramoto network

An illustration of this phenomenon in a network composed by nodes having
heterogeneous dynamical descriptions and a nonlinear coupling function is
presented in a random network of $N$=1000 Kuramoto oscillators. We rewrite the
Kuramoto network model in terms of the local mean field,
$\overline{\theta}_{i}=\frac{1}{k_{i}}\sum_{j=1}^{N}A_{ij}\theta_{j}$. Using
the coordinate transformation
$\frac{1}{k_{i}}\sum_{j=1}^{N}A_{ij}\exp{{}^{\mathbb{j}(\theta_{j}-\theta_{i})}}=\tilde{r}_{i}\exp{{}^{\mathbb{j}(\overline{\theta}_{i}-\theta_{i})}}$,
the dynamics of node $i$ is described by

$\dot{\theta_{i}}=\omega_{i}+p_{i}\tilde{r}_{i}sin(\overline{\theta}_{i}-\theta_{i}),$
(30)

where $\omega_{i}$ is the natural frequency of the node $i$, taken from a
Gaussian distribution centered at zero and with standard deviation of 4\. If
$\tilde{r}_{i}$=1, all nodes coupled to node $i$ are completely synchronous
with it. If $\tilde{r}_{i}$=0, there is no synchronization between the nodes
that are coupled to the node $i$. Since the phase is an unbounded variable,
the CAS phenomenon should be verified by the existence of an approximate
constant local mean field in the frequency variable $\dot{\theta_{i}}$. If
$\overline{\dot{\theta}_{i}}(t)\cong C_{i}$, which means that
$\overline{\theta_{i}}=\overline{\dot{\theta}_{i}}t\cong C_{i}t$, then Eq.
(30) describes a periodic orbit (the CAS pattern), regardless the values of
$\omega_{i}$, $p_{i}$, and $\tilde{r}_{i}$, since it is an autonomous two-
dimensional system; chaos cannot exist. Therefore, criterion 2 is always
satisfied in a network of Kuramoto oscillators. We have numerically verified
that criterion 1 is satisfied for this network for
$\sigma\leq\sigma^{CAS}(N=1000)$, where $\sigma^{CAS}(N=1000)\cong 0.075$.
Complete synchronization is achieved in this network for
$\sigma\geq\sigma^{CS}=1.25$. So, the CAS phenomenon is observed for a
coupling strength that is 15 times smaller than the one that produces complete
synchronization.

For the following results, we choose $\sigma=0.001$. Since the natural
frequencies have a distribution centered at zero, it is expected that, for
nodes with higher degrees, the local mean field is close to zero (see Fig.
3(a)). In (b), we show the variance of the local mean field of the nodes with
degree $k_{i}$. The fitting produces $\mu^{2}_{i}\propto k_{i}^{-1.055}$
(criterion 1). In (c), we show the relationship between the value of
$p_{i}\tilde{r}_{i}$ and the value of the degree $k_{i}$. In order to
calculate the CAS pattern of a node with degree $k_{i}$, we need to use the
value of $p_{i}\tilde{r}_{i}$ (which is obtained from this figure) and the
measured $C_{i}$ as an input in Eq. (30). We pick two arbitrary nodes, $i$ and
$j$, with degrees $k_{i}=96$ and $k_{j}=56$, respectively, with natural
frequencies $\omega_{i}\approxeq-5.0547$ and $\omega_{j}\approxeq-5.2080$. In
(d), we show that phase synchronization is verified between these two nodes,
with $p/q=\omega_{i}/\omega_{j}$. We also show the phase difference
$\delta\phi_{j}=\theta_{j}-\Xi_{\theta_{j}}$ between the phases of the
trajectory of the node $i$ with degree $k_{j}=96$ and the phase of its CAS
pattern, for a time interval corresponding to approximately 2500/$P$ cycles,
where the period of the cycles in node $i$ is calculated by
$P=\frac{2\pi}{5.0547}$. Phase synchronization between nodes $i$ and $j$ is a
consequence of the fact that the phase difference between the nodes and their
CAS patterns is bounded.

Figure 3: Results for $\sigma=0.001$. (a) Expected value of the local mean
field $\overline{\dot{\theta}_{i}}$ of a node with degree $k_{i}$. (b) The
variance $\mu^{2}_{i}$ of the local mean field. (c) Relationship between the
value of $p_{i}\tilde{r}_{i}$ and $k_{i}$. (d) Phase difference
$\Delta\phi_{ij}=\theta_{i}-p/q\theta_{j}$ between two nodes, one with degree
$k_{i}=96$ and the other with degree $k_{j}=56$; the phase difference
$\delta\phi_{i}=\theta_{i}-\Xi_{\theta_{i}}$ between the phases of the
trajectory of the node $i$ with degree $k_{i}=96$ and the phase of its CAS
pattern.

In the thermodynamic limit, when a fully connected network has an infinite
number of nodes, $C_{i}$ does not change as one changes the coupling $\sigma$,
since it only depends on the mean field of the frequency variable
($\overline{\dot{\theta}}$). As a consequence, if there is the CAS phenomenon
and phase synchronization between two nodes with a ratio of $p/q$ for a given
value of $\sigma$, changing $\sigma$ does not change the ratio $p/q$.
Therefore phase synchronization is stable under alterations in $\sigma$. Phase
synchronization will be rational and stable whenever nodes with different
natural frequencies $\omega_{i}$ become locked to Arnold tongues jensen ;
arnold_tongue  induced by the coupling
$p_{i}\tilde{r}_{i}sin(\overline{\theta}_{i}-\theta_{i})$.

There is a special solution of Eq. (30) that produces a bounded state in the
variable ${\theta_{i}}$ when the network is complete synchronous to an
equilibrium point. In such case, $\overline{\theta}_{i}$ becomes constant, and
Eq. (30) has one stable equilibrium
$\theta_{i}=\arcsin{\left(\frac{\omega_{i}}{p_{i}}\right)}$, obtained when
$p_{i}>\omega_{i}$. But, the local mean field becomes constant due to complete
synchronisation and not due to the fact that the nodes are “decoupled”. These
conditions do not produce the CAS phenomenon.

###  I.6 Preserving the CAS pattern in different networks: a way to predict
the onset of the CAS phenomenon in larger networks

Consider two networks, $n_{1}$ and $n_{2}$, whose nodes have equal dynamical
descriptions, the network $n_{1}$ with $N_{1}$ nodes and the network $n_{2}$
with $N_{2}$ nodes ($N_{2}>N_{1}$), and two nodes, $i$ in the network $n_{1}$
and $j$ in the network $n_{2}$. Furthermore, assume that both nodes have
stable periodic CAS patterns (criteria 1 is satisfied), and assume that the
nodes have sufficiently large degrees such that the local mean field of node
$i$ is approximately equal to node $j$. Then the CAS pattern of node $i$ will
be approximately the same as the one of node $j$ if

$\sigma^{CAS}(n_{1})k_{i}(n_{1})=\sigma^{CAS}(n_{2})k_{j}(n_{2}).$ (31)

$\sigma^{CAS}(n_{1})$ and $\sigma^{CAS}(n_{2})$ represent the largest coupling
strengths for which the variance of the local mean field of a node decays with
the inverse of the degree of the node (criterion 2 is satisfied) in the
networks, respectively, and $k_{i}(n_{1})$ and $k_{j}(n_{2})$ are the degrees
of the nodes $i$ and $j$, respectively. In other words, the CAS phenomenon
occur in the network if $\sigma\leq\sigma^{CAS}$.

Therefore, if $\sigma^{CAS}(N_{1})$ is known, $\sigma^{CAS}(N_{2})$ can be
calculated from Eq. (31). In other words, if the CAS phenomenon is observed at
node $i$ for $\sigma\leq\sigma^{CAS}(N_{1})$, the CAS phenomenon will also be
observed at node $j$ for $\sigma(n_{2})\leq\sigma^{CAS}(n_{2})$, where
$\sigma^{CAS}(n_{2})$ satisfies Eq. (31).

Acknowledgment MSB acknowledges the partial financial support of the Northern
Research Partnership. HPR acknowledges the partial financial support of NSFC
Grant 60804040.

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