Formation Of Small-World Network Containing Module Networks In Globally And Locally Coupled Map System With Changes In Global Connection With Time Delay Effects

Abstract

In this study, we performed comprehensive morphological investigations of spontaneously formed network structures among elements in coupled map systems involving global connections that change depending on the synchronicity of states of elements and spatially local connections. The model formed various hierarchical networks, some of which were classified as small-world networks containing multiple module networks, similar to the neural network of mammalian brains. Moreover, such complex networks were formed in wider parameter regions when the global connection to an element from the other element was strengthened by the synchronization between the present and past states of the former and latter elements, respectively. This study suggests that the time delay effects for connection changed among elements and local interactions promoted the self-organization of small-world networks containing module networks, such as neural networks; neural networks contain them as spike-timing-dependent plasticity and inter-neuron interaction through glial cells.

Full text

Formation Of Small-World Network Containing Module
Networks In Globally And Locally Coupled Map System With
Changes In Global Connection With Time Delay Effects
Taito Nakanishi1 and Akinori Awazu1,2
1Graduate School of Integrated Sciences for Life, Hiroshima University,
Higashihiroshima, Hiroshima 739-8526, Japan
2Research Center for the Mathematics on Chromatin Live Dynamics, Hiroshima
University, Higashihiroshima, Hiroshima 739-8526, Japan
Abstract
In this study, we performed comprehensive morphological
investigations of spontaneously formed network structures
among elements in coupled map systems involving global
connections that change depending on the synchronicity of
states of elements and spatially local connections. The model
formed various hierarchical networks, some of which were
classified as small-world networks containing multiple module
networks, similar to the neural network of mammalian brains.
Moreover, such complex networks were formed in wider
parameter regions when the global connection to an element
from the other element was strengthened by the synchronization
between the present and past states of the former and latter
elements, respectively. This study suggests that the time delay
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effects for connection changed among elements and local
interactions promoted the self-organization of small-world
networks containing module networks, such as neural
networks; neural networks contain them as spike-timing-
dependent plasticity and inter-neuron interaction through glial
cells.
1. Introduction
Various biological and social systems are regulated via their self-organized network
structures. Neural networks [1-17], metabolic networks [18-21], food webs [22,23], and
human communities [24-29] are typical networks that exhibit self-regulation of
connectivity through learning, cell differentiation and adaptation, evolution, and
communication.
The self-regulation behaviors of such systems have been studied using models that
comprise dynamic elements involving changes in mutual relationships among the
elements. For example, models of neural networks comprising elements that imitate a
nerve cell have been described as excitable or chaotic oscillators. Consequently, the
temporal change in coupling among these elements was assumed to follow a rule inspired
by Hebb's law or spike-timing-dependent plasticity (STDP) [1–17]. Recent studies have
suggested that one of the typical classes of such models, coupled chaotic map systems
with Hebbian-like rules, may frequently exhibit spontaneous formations of
asymmetrically connected network structures with the emergence of one-way hierarchies
and/or loops, multiple layers, and pacemakers [5-7,16,17].
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Recent social and medical studies have suggested that such self-organized networks
often form small-world networks with specific statistical properties in the edge
distributions among nodes. For example, small-world networks have been found in
various human social networks [30]. Neural networks in mammalian brains are known to
form small-world networks [31,32], which also involve multiple module networks [33-
35]. Recent mathematical studies have suggested simple procedures for painting small-
world networks [30]. However, recently proposed dynamic network system models
involving the self-regulation of their connection topologies did not provide sufficient
mechanisms for the manner in which such social and neural networks could organize
small-world networks simultaneously and robustly.
Most recent mathematical models for the studies of self-organized networks were
constructed based on globally coupled dynamical systems with temporally changeable
interaction strengths among elements. However, social networks appear to involve global,
distance-free interactions such as those through world-wide web systems and local
interactions such as communication in family, among friends and coworkers, and in
various local communities. In addition, neural networks in the mammalian brain involve
both the global synaptic interactions among neurons and local interactions among
spatially neighboring neurons through glial cells occupying spaces among neurons; the
volume fractions of glial cells are 2–10 times more than those of neurons [36-41]. Thus,
the combination of local and global interactions with temporally changeable interaction
strength may be considered to promote the self-organization of small-world networks.
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In this study, we developed dynamic systems comprising spatially distributed
elements connected globally and locally. All elements were connected globally with
temporal changes in connection strengths among all elements, and neighboring elements
were connected locally with constant strength. Subsequently, through comprehensive
simulations, the abovementioned conjectures were examined for the self-organization and
robust formation of small-world networks with multiple module networks.
2. Model and Methods
2.1 Model
As a simple dynamical system with local and global interactions exhibiting element
state-dependent changes in global interaction strength, the following coupled map system
with N elements was considered:
!!"#
$" #$
%
& ' (%&'(% ' ()%&*(%
)
!!
$*'!"#$!
+
%
!!
$,# * !!
$"#
)
* ()%&*(%
+
,!
$-!!
-
.,#
-/0 -
(1)
,!"#
$- "
1
#")234%
&5&'%6
7
8%
&(
9:
#";)2
<
4%
&
=
&'%6
>
8%
&(
)*+
(,-
(2),
where
!!
$
%
. / !!
$/ &
) and
,!
$-
$. / ,!
$- / &-
are the state of the
0
-th element
(
0 " .1 &12 34 ' &
) and the connection strength from the
5
-th to
0
-th element at time
6
,
respectively, and
#
$
!
-
"7!$& ' !-
(logistic map) is assumed. We assumed that the
elements were spatially arranged in a circle in the order
0 " .1 &12 18 ' &
. Further, it was
assumed that the
.
-th element and the
$8 ' &
)-th element were adjacent to each other
and provided
0 * & " .
for
0 " $4 ' &-
-th element and for
0 " .
-th element. The first,
second, and third terms in function
#
in Eq. (1) indicates the influence of each element
itself, that of two spatially neighboring elements, and that of globally coupled elements.
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In this model,
,$$ " .
(no self-connection) was assumed, and the condition of
+
,!
$- " &
.
-
was always satisfied for
0
. The parameters
7
,
(%&'(%
, and
()%&*(%
indicate
the parameters of the logistic map, strength of local interaction between two spatially
neighboring elements, and strength of the influences of global interaction, respectively.
In the case of
,!
$- " &
, the present model is equal to the coupled map model with
local couplings and uniform global couplings proposed by Ouchi and Kaneko, showing
various complex spatio-temporal pattern dynamics [42]. When
9$
:
!!
$
;
$?!- " <=>?
%
!!
$'
!-!3
) with
(%&'(% " .
and that of
9$
:
!!
$
;
$?!- " <=>?
%
!!
$' !-!,#3
) with
(%&'(% " .
,
the present model is equal to that proposed by Ito and Ohira [5]. Here, in the former case,
the connection from
5
-th to
0
-th element changes depending on the synchronicity
between the states of
0
-th and
5
-th elements. Whereas, in the latter case, the changes
depend on the synchronicity between the present state of
0
-th element and the one-step
past state of
5
-th element. These models showed a rich variety of self-organizations of
directionally connected networks with multiple layers, hierarchies, and/or loops, and a
pacemaker community in an exhaustive study of these models [16, 17].
In the present model, the set
,!
$-
represents the directional network structures of
the elements at time
6
. Herein, the connection from the
5
-th to the
0
-th element is
considered strong (weak) when
,!
$-
is large (small). In general,
,!
$-
changes in
6
.
Therefore, as employed in previous studies [4-7,9,16,17], we first classified the
connection profiles of the typical networks obtained via the proposed model according to
the following definition: the connection from
5
-th to
0
-th elements exists if the time
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average of
,!
$-
, @
,$-
A, satisfied @
,$-
A
"#
!.&%,!&%&
+
,!
$-
!.&%
!/!&%& B#
.,#
, where
#
.,#
indicates the average of
,!
$-
over the entire set of
0
and
5
(
0 C 5
). This study referred
to the network characterized by the set of @
,$-
A as the effective network.
2.2 Simulation method
We performed simulations of the model for each set of
7
,
(%&'(%
, and
()%&*(%
for
five different initial conditions, and focused on the most frequently obtained network
structures with
6$!$ "&.@
and
6A$! " D E &.@
as the typical network structure. As the
initial condition of each simulation,
!0
$
was chosen randomly from the values satisfying
. F !0
$F &
, and
,0
$- "#
.,#
(
0 C 5
) was provided, assuming that all elements were
uniformly connected to all the elements, except for itself. We confirmed that the influence
of the initial conditions was negligible when
6$!$ "&.@
was chosen. This is supported
by the relaxation behaviors of the autocorrelation functions of the connection strength
among the elements described as
G
$
H
-
"
B2
C
2
!"D
6
,
B
C
E62
C
2
!
6
,
B
C
E6E
B
2C
2
!
6
,
B
C
E
6/
E (4)
where
I
$
6
-
" $
:
,!
$-
;
$FG--
is an
4
$
4 ' &
- dimensional vector. Here,
F I B
was assumed to be @
I
A
"
J
#
.,# 121 #
.,#
K because the average of
,!
$-
was estimated to
be
#
.,#
if
,!
$-
changed ergodically in
6
.
2.3 Estimations of small world propensity and modularity of “undirected networks”
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Neural networks in mammalian brains are typical network systems with element
state-dependent connection changes. Undirected networks described from neural
networks in mammalian brains can form small-world networks [31,32]. Additionally,
these neural networks contain multiple module networks where neurons are divided into
modules, and intra-module connections among neurons are dense, whereas inter-module
connections are sparse [33-35].
Thus, to evaluate the structural features of the formed effective networks, we
measured the small-world propensity (SWP), modularity, and number of modules of each
network to compare the statistical similarities between the networks obtained from the
present model and the neural networks in the mammalian brain as follows. Modularity is
defined as a value that increases with the differences between the connection number
among elements in the same modules and those among elements in different modules [33].
To evaluate these statistical values for each network, we first defined the undirected
connectivity between
0
-th and
5
-th elements as
L$- " L-$ " MNO
P@
,$-
A
1
@
,-$
AQ, and
connections satisfying
L$- B#
.,#
as “undirected connections”. In addition, we defined
the “undirected effective network” as that comprising connections with
L$- B#
.,#
for
each network.
Second, the SWP, modularity, and number of modules of each undirected effective
network obtained by each parameter set were evaluated based on the following recently
proposed methods. The SWP was calculated according to the method proposed by Watts
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and Strogatz [30,31], which was employed to evaluate the neural network topology of
mammalian brains [31,32]. Whereas, the modularity and number of modules in each
network were estimated using the method proposed by Newman et al [33,43], which was
also employed to extract the module structures of the neural networks [33-35]. The
network was regarded as a small-world network when the SWP
B .RS
, and it was
expected to contain certain modules when modularity
B .
. In recent studies on neural
networks in mammalian brains, the SWP was estimated
.RST3U3.RV
[31,32], and positive
modularity was obtained [33-35].
3. Results
3.1 Influence of local connections and time delay effects on global connection changes
in formed effective network structures
To reveal the influence of the local connections and the time delay effects on
connection changes in the formed network structures, we simulated the model with I)
9$
:
!!
$
;
$?!- " <=>?
%
!!
$' !-!3
) and
(%&'(% " .
, II)
9$
:
!!
$
;
$?!- " <=>?
%
!!
$' !-!,#3
)
and
(%&'(% " .
, III)
9$
:
!!
$
;
$?!- " <=>?
%
!!
$' !-!3
) and
(%&'(% B .
, and IV)
9$
:
!!
$
;
$?!- " <=>?
%
!!
$' !-!,#3
) and
(%&'(% B .
. These models were referred to as I)
model SS: considered only the effects of simple synchronization among elements, II)
model SD: considered the effects of synchronization and time delay among elements, III)
model SSL: considered the effects of simple synchronization and local connections
among elements, and IV) model SDL: considered the effects of synchronization, time
delay, and local connections among elements.
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Figure 1 shows the phase diagrams of typical effective networks obtained using the
abovementioned four models with
4 " W.
as a function of
7
and
()%&*(%
. Here, the
case of
(%&'(% " .R&
is shown for model SSL, and the cases of
(%&'(% " .R&
,
(%&'(% "
.R.T
, and
(%&'(% " .RD
are shown for model SDL. Each symbol in Fig. 1 represents the
mean of the typical network structures observed for a given parameter set. The phase
diagrams of the SS models shown in Fig. 1(a) exhibited the same results as those reported
in a previous study [17]. Additionally, although the symbols used to indicate each formed
network were different, the phase diagrams of the SD models shown in Fig. 1(c) exhibited
the same results as those reported in another previous study [16] for
()%&*(% X .R&
. The
results of
()%&*(% " .R.T
are shown in Fig. 1(c). Furthermore, the behaviors of the SSL
and SDL models for
()%&*(% B .RW
were obtained as similar to those observed in the
parameter regions marked U, G, and P, as described later.
The temporal change in
Y!!
$Z
exhibited periodic or chaotic motion depending on
the parameter set, as shown in Fig. 1. In the parameter regions marked U and G, the
following simple trivial networks were obtained, where the elements were effectively
connected symmetrically and uniformly (partially uniformly) with each other, and the
parameter regions marked U, @
,$-
A
[#
.,#
and @
,-$
A
[#
.,#
were obtained. In the
parameter regions marked as G, the elements were divided into groups such that @
,$-
A
[
#
.0,#
and @
,-$
A
[#
.0,#
were obtained in each group, where
4H
indicates the number
of elements in each group. In the subsequent arguments, we do not focus on simple, trivial
cases.
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For other parameter regions, various networks with different structural
characteristics were obtained as follows. The typical example structures of the effective
networks in the case of
4 " W.
are shown in Fig. 2.
P: Pair-driven network, where certain elements form pairs that are connected
symmetrically with each other, whereas the other elements form one-way directionally
connected networks without any loops.
L: Loop-driven network, where a few elements are located in the upper stream of
the network and form a one-way connected loop, whereas the other elements form one-
way directionally connected networks without any loops.
T: Trio-driven networks, where certain elements form trios, which are connected
almost symmetrically to each other, whereas the other elements form one-way
directionally connected networks without any loops.
S: Sym-community-driven networks where some elements form a community,
whereas the other elements form one-way directionally connected networks without any
loops. The elements in the community are connected almost symmetrically, and each
element in this community is connected to certain non-community-forming elements.
These networks are large hierarchical networks containing one such community in their
uppermost stream.
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B: Bipartite layer-driven networks, where certain elements form paired layers
upstream of the network. Each element in one upstream layer is connected to and from
other elements that belong to another upstream layer. Further, certain elements in these
layers are connected to non-layer-forming elements.
M: Multilayer networks, where the elements are divided into more than three layers,
and each element is connected to and from other elements that belong to other layers.
Here, certain layers are located in the upper stream of the network and form a one-way
connected loop, whereas the other elements form one-way directionally connected
networks without any loops at the lower stream of the network.
The detailed features of the abovementioned networks, P, L, T, and S, have already
been reported in a previous study [17], and those of networks B and M have also been
reported [16]. Conversely, the following network structures have not been reported in
previous studies:
D: Double-strand network, where
,!
$-
changes chaotically in
6
, and @
,$-
A with
@
,$-
A
B#
.,#
exhibits @
,$F$I#
A
"
@
,$I#F$
A
F
@
,$F$I+
A
"
@
,$I+F$
A.
A: Asym-community-driven network, where certain elements formed a community
upstream of the network, similar to a sym-community-driven network. However, the
elements in the community were connected almost asymmetrically to each other, unlike
a sym-community-driven network.
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Comparisons among the phase diagrams of the four models showed that the local
connections appeared to expand the parameter region to form a pair-driven network
(network P) and a sym-community-driven network (network S). In particular, the SSL
model exhibited network P in most cases, whereas both P and S networks emerged in the
SDL model. In addition, the parameter region wherein networks P or S emerged was wider
with an increase in
(%&'(%
. Moreover, the SS model could also exhibit network S; however,
the SDL model showed network S in a wider parameter region than SS model.
3.2 Characterizations of formed effective networks by small world propensity and
modularity of undirected effective networks
To evaluate the structural features of effective networks, the SWP and modularity
were estimated for undirected effective networks obtained using each parameter in each
model. Figure 3 shows (a) SWP as a function of
7
and
()%&*(%
, and (b) modularity as a
function of
7
and
()%&*(%
, where
(%&'(% " .R&
for models SSL and SDL. SWP was
measured when networks P, L, T, S, B, M, A, or D were formed. Additionally, modularity
was estimated only for the parameter region where the undirected effective network was
regarded as a small-world network with an SWP larger than 0.6. In these figures, the parts
of networks P and S, as shown in Fig. 4, exhibited a value of SWP larger than 0.6, and a
value of modularity larger than 0. Additionally, the SDL model exhibited such effective
networks in a wider range of parameter regions than that in the other models.
3.3 Sym-community-driven networks exhibited similar small world propensity and
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modularity of undirected networks to neural network of mammalian brain.
Certain pair-driven networks (network P) and sym-community-driven networks
(network S) tend to exhibit high values of SWP and modularity. It was noted that both
networks commonly involved a hierarchical structure, with an upper community of
elements connected symmetrically and lower elements with few connections from them.
However, in contrast to network P, network S contained intermediate elements containing
connections from certain upper elements and those to certain lower elements. In addition,
although the upper community in network P always comprised only two elements, the
number of elements constructing the upper community in network S was generally greater
than 2.
To evaluate the robustness of the abovementioned features, the system-size-
dependent behaviors of the model SDL with the parameters of networks P or C were
obtained. Figure 5 shows the averages and standard deviations of (a) SWP, (b) modularity,
and (c) number of modules as a function of the number of elements
4
in the cases where
networks P and S were formed; typical shapes of (d) network P and (e) network S were
formed in the case of
4 " &D.
. The averages and standard deviations were calculated
using simulations from 10 different initial conditions for each
4
. This figure showed the
SWP was almost independent of
4
. However, the modularity and number of modules
increased with
4
when the network S was formed. Whereas, the modularity decreased
with
4
when the network P was formed. These facts suggest that the SDL model with
appropriate parameters can form a larger small-world network with a multitude of
modules according to the increase in the number of elements.
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4. Discussion
In this study, we comprehensively focused on the morphologies of spontaneously
formed network structures using globally and spatially (local) coupled map systems
involving global connection changes dependent on the synchronicity of states of elements.
Various types of hierarchical directional networks were obtained through the simulations
of the proposed model, and some of them were classified as small-world networks
containing multiple module networks, as observed in the neural network of mammalian
brains. In addition, we found such small world networks were formed in wide range of
parameter regions when the temporal change in each global connection strength between
two elements was influenced by their present and past states.
The comparison with the results of the present arguments and the behaviors of neural
networks in mammalian brains indicated the following common features:
I) The synaptic connections among neurons in the brain exhibited spike-timing-
dependent-plasticity [44], indicating that the change in each global connection from one
element to another in both systems was strengthened by the synchronization of the past
and present states of the former and latter elements, respectively. II) The spatially
neighboring neurons mutually influenced each other through the glial cells that occupied
spaces among neurons [36-41], indicating that both systems involved interactions among
spatially neighboring elements. III) Both systems can form a small-world network
containing modular networks [31-35].
Conversely, the basic dynamics of each element were significantly different between
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preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in
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the proposed model and neural networks in the brain when each element was isolated,
where the element in the proposed model exhibited chaotic dynamics, whereas that in
neural networks showed excitable dynamics. Thus, in the formation of small-world
networks containing multiple modular networks, the effects of the time delay involved in
the change in the connection strength to an element from other elements and the local
interactions between spatially neighboring elements appeared to contribute more
significantly than the detailed features of each element.
In this study, the question “Why and the manner in which the local interactions and
the effects of the time-delay involved in the change in the global connection strengths
promote the small world networks?” was not answered. To answer this question,
additional theoretical studies should be conducted from the perspective of dynamical
systems with large degrees of freedom in the future.
The present results are expected to provide novel insights into the self-organization
of small-world networks by not only neural networks but also various other biological
and social networks. To reveal the detailed processes and the mechanism of the formation
of such biological networks, the construction of models or modifications of recently
proposed models comprising more realistic model elements with interaction rules based
on the present arguments should be conducted, which is also one of the future issues.
Acknowledgments
We thank Junji Ito, Takahiro Chihara, Amika Ohara, and Masashi Fujii for providing
.CC-BY 4.0 International licenseperpetuity. It is made available under a
preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in
The copyright holder for thisthis version posted November 15, 2022. ; https://doi.org/10.1101/2022.11.13.516347doi: bioRxiv preprint

fruitful information. This work was supported by Grants-in-Aid for Scientific Research
(KAKENHI) [Grant Number 21K06124 (A.A.) ] from the Japan Society for Promotion
of Science. Computations were partially performed on the NIG supercomputer at ROIS
National Institute of Genetics.
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The copyright holder for thisthis version posted November 15, 2022. ; https://doi.org/10.1101/2022.11.13.516347doi: bioRxiv preprint

Fig. 1. Phase diagrams of the typical effective networks obtained by the models (a) SS,
(b) SSL with
(%&'(% " .R&
, (c) SD, and (d-f) SDL with
(%&'(% " .R&
,
(%&'(% " .R.T
, and
(%&'(% " .RD
. Each symbol indicates each formed effective network ( details in test). Rigid
box indicates the connections among elements
,!
$-
are stationary, and broken box
indicates that connections
,!
$-
changed temporally. Shadowed background indicates that
the states of all elements
!!
$
exhibit periodic changes in time, whereas white background
indicates that the states of certain or all elements exhibit chaotic motions.
!!"#$%"
0.3
0.2
0.1
"
!!"#$%"
0.3
0.2
0.1
"
P
G
G
G
G
G
3.9
T
T
G
G
G
P
3.7
L
P
G
G
P
P
U
L
G
P
P
P
3.9
P
U
P
P
P
T
P
S
P
G
P
T
4
P
S
G
G
G
R
P
P
G
G
P
U
3.7
P
G
P
P
P
P
P
P
P
P
P
P
3.8
P
P
G
P
P
P
P
P
G
P
P
P
D
P
G
G
P
G
4
D
A
G
G
G
G
3.8
S
R
G
P
P
P
3.6
G’
P
U
G
G
G
3.6
P
P
U
U
P
P
P
P
G
G
P
P
(a) Model SS (!"#&%" = 0.0)(b) Model SSL (!"#&%" = 0.1)
"
0.3
0.2
0.1
B
B
U
U
U
P
3.6
B
B
U
B
P
P
3.7
M
B
B
B
P
B
M
B
B
B
B
B
3.8
P
B
B
U
U
B
L
M
U
U
U
B
3.9
A
M
U
U
U
B
A
P
U
U
R
B
4
R
P
U
U
R
B
0.3
0.2
0.1
"
B
B
U
U
U
R
B
S
U
U
U
P
3.6
B
P
U
U
P
P
3.7
B
B
U
B
P
P
L
B
U
R
S
S
3.8
L
B
U
U
S
S
P
B
B
U
S
S
3.9
B
B
U
U
U
R
4
B
P
U
U
U
R
(c) Model SD (!"#&%" = 0.0)(d) Model SDL (!"#&%" = 0.1)
!!"#$%"
!!"#$%"
" "
0.3
0.2
0.1
P
P
U
U
U
B
43.6
B
B
U
U
U
P
B
B
U
U
P
P
3.7
P
B
U
P
P
P
L
B
U
U
S
S
3.8
L
B
R
R
R
A
L
B
B
U
U
P
3.9
B
B
U
U
U
B
P
P
U
U
U
B
0.3
0.2
0.1
3.6
B
S
U
U
U
P
B
P
U
U
P
P
3.7
P
P
U
B
P
P
B
S
U
R
S
S
3.8
A
S
U
U
U
S
B
R
U
B
U
U
3.9
P
A
U
U
U
U
P
P
U
U
U
U
4
P
P
U
U
U
U
(e) Model SDL (!"#&%" = 0.05)(f) Model SDL (!"#&%" = 0.2)
!!"#$%"
!!"#$%"

Fig. 2. Typical structures of effective networks for respective symbols in Fig. 1. Each
circle with index
0
indicates
0
-th element, and the arrow from the
0
-th circle to the
5
-th
one indicates existence of a connection from
0
-th element to
5
-th one. Index
0
also
indicates its spatial position (details in text).
P : Model SD (!!"#$! = 0.0, !%!"&$! = 0.05, ' = 3.85)
S: Model SDL (!!"#$! = 0.1, !!"#$! = 0.2,' = 3.75)
L : Model SD (!!"#$! = 0.0, !%! "&$! = 0.05,' = 3.85)
T : Model S S (!!"#$! = 0.0, !%!"&$!= 0.15, ' = 3.9)
M: Model SD (!!"#$! = 0.0, !%! "&$! = 0.1,' = 3.9)
D: Model SDL (!!"#$! = 0.1, !!"#$! = 0.2,' = 3.75)
B : Model SDL (!!"#$! = 0.1, !%! "&$! = 0.1, ' = 3.7)
A: Model SSL (!!"#$! = 0.1, !!"#$! = 0.1, ' = 4)

Fig. 3. (a) Small world propensity (SWP) and (b) modularity measured by the models
SS, SD, SSL with
(%&'(% " .R&
, and SDL with
(%&'(% " .R&
. SWP was put as 0 in (a)
when the trivial network (U or G) was formed. Modularity is calculated only when SWP
B .RS
and is set as 0 when SWP
/ .RS
in (b).
!
"!"#$%"
!
!
"!"#$%"
!
Model SS (""#&%" = 0.0)Model SSL (""#&%" = 0.1)
Model SD (""#&%" = 0.0)Model SDL (""#&%" = 0.1)
"!"#$%"
"!"#$%"
SWPSWP
SWPSWP
(a)
P
PP
P
P
P
P S S
S S S
S S
S
!
"!"#$%"
!
!
"!"#$%"
!
Model SS (""#&%" = 0.0)Model SSL (""#&%" = 0.1)
Model SD (""#&%" = 0.0)Model SDL (""#&%" = 0.1)
"!"#$%"
"!"#$%"
Modularity
(b)
Modularity
ModularityModularity
P
PP
P
P
P
P S S
S S S
S S
S

Fig. 4. Typical structural features of (a) pair-driven networks P formed by model SDL
with
7 " WRST
,
()%&*(% " .R&T
, and
(%&'(% " .R&
, and (b) hidden community-driven
networks S formed by model SDL with
7 " WR\T
,
()%&*(% " .R&T
, and
(%&'(% " .R&
,
exhibiting small world propensity
B .RS
and positive modularity in cases of
4 " W.
.
The structure of network P is obtained as stationary for sufficiently large
6
, and then, the
network at time
6
converges to effective network in (a). Conversely, the structure of
network S changes in
6
even for sufficiently large
6
, where the obtained structures are
different among the effective network (upper), the network at
6 " &......
(lower left),
(a) P : Model SDL (!!"#$! " #$%, !%!"&$! " #$%&' ( " )$*&)
+'
(
,
-
Effective network
( Network P )
(b) S: Model SDL (!!"#$! " #$%, !%!"&$! " #$%&'( " )$*&)
+'
(
,
-
, " %###### , " %%#####
Effective network
( Network S )

and that at
6 " &&.....
(lower right), as shown in (b).
Modularity
SWP
P1
P2
S1
S2
P1
P2
S1
S2
(a) (b)
. .
Number3of3modules
P1
P2
S1
S2
.
(c)
(d) P : Model SDL (!!"#$! = 0.1, !%!"&$! = 0.15, ( = 3.65, + = 120)
Effective network
( Network P )

Fig. 5. Average (curves) and standard deviation (error bar) of (a) small world propensity,
(b) modularity, and (c) number of modules as a function of
4
when the network P or S
are obtained by the SDL model. P1 and P2 indicate that the model exhibits the network P,
where parameters are
7 " .RWS
(P1) and
7 " .RWST
(P2) with
()%&*(% " .R&
and
(%&'(% " .R&T
. S1 and S2 indicate that the model exhibits the network P, where parameters
are
7 " .RW\T
(S1) and
7 " .RW]
(S2) with
()%&*(% " .R&
and
(%&'(% " .R&T
. (d)
Typical structures of networks P and (e) that of network S obtained as effective networks
in the case of
4 " &D.
, where parameters are
7 " .RWST
in (d) and
7 " .RW\T
in (e)
with
()%&*(% " .R&
and
(%&'(% " .R&T
.
(e) S: Model SDL (!!"#$! = 0.1,!%!"&$! = 0.15, ( = 3.75, + = 120)
Effective network
( Network S )