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A Modeling Environment for Dynamic and Adaptive Network Models Implemented in MATLAB

Conference Paper

A Modeling Environment for Dynamic and Adaptive Network Models Implemented in MATLAB

Abstract and Figures

In this paper a software environment to support Network-Oriented Modeling is presented. The environment has been implemented in Matlab. This code covers the principles of modeling by temporal-causal networks. The software environment has built-in options for network adaptation principles such as the Hebbian Learning principle from Neuroscience and the adaptation principle for bonding based on homophily from Social Science. The implementation is illustrated for an adaptive temporal-causal network model for decision making under acute stress.
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A Modeling Environment for Dynamic and Adaptive
Network Models Implemented in Matlab
S. Sahand Mohammadi Ziabari, Jan Treur
Behavioural Informatics Group, Vrije Universiteit Amsterdam
sahandmohammadiziabari@gmail.com j.treur@vu.nl
Abstract. In this paper a software environment to support Network-Oriented Modeling
is presented. The environment has been implemented in Matlab. This code covers the
principles of temporal-causal network models. The software environment has built-in
options for network adaptation principles such as the Hebbian Learning principle from
Neuroscience and the adaptation principle for bonding based on homophily from Social
Science. The implementation is illustrated for an adaptive temporal-causal network
model under acute stress for decision making.
Keywords: Network-Oriented Modeling, temporal-causal network, adaptive, software
environment, Hebbian learning, bonding by homophily, Matlab.
1 Introduction
In this paper a dedicated software environment to support Network-Oriented Modeling is
presented. The Network-Oriented Modeling approach addressed uses temporal-causal net-
work models. This means that any scientific field in which causal relations are used to explain
hypotheses, findings and theories can be used in Network-Oriented Modeling [19]. Such do-
mains vary from mental processes in individuals to social processes. For example, the inter-
actions among individuals can be modelled as a network taking into account a network adap-
tion principle like bonding based on homophily principle [8, 10]. Individual mental processes
can be modeled as interaction between mental states taking into account a network adaption
principle based on Hebbian learning [18]. The latter represents the notion of plasticity de-
scribed in Neuroscience which means that the communications within the brain are often
adaptive and change over time.
There are two different representations of a temporal-causal network model named a con-
ceptual representation (labeled graph or matrix representation) and a numerical representa-
tion (representation by difference or differential equations). Using the software environment
presented here, a conceptual representation can be used as a basis. By the software it is auto-
matically translated into a numerical representation which can be used for numerical simula-
tion, mathematical analysis, validation by comparing to empirical data or properties, tuning
of parameters to characteristics of domain, person or social context.
The example model presented in [1] has been used as illustration. This model incorporates
adaptation principles based on Hebbian learning and on suppression of connections due to
acute stress. There are other implementations for other types of network-oriented modeling,
some of which can be found in [4-6, 21, 22].
The sections of paper are as follows. In Section 2 the Network-Oriented Modeling ap-
proach based on temporal-causal networks is briefly described. In Section 3 an overview of
the Matlab software is introduced and in Section 4, more details of the Matlab code has been
described. In Section 5 the simulation results of the example model are discussed. Finally,
Section 6 is the discussion section.
2
2 The Network-Oriented Modeling Approach Addressed
This software environment covers the principles of Network-Oriented Modeling based on
temporal-causal networks discussed in the book [2]. The Network-Oriented Modeling format
used is based on a dynamic and adaptive variant of modeling, reasoning and simulation in a
causal way which is a topic with a long history in Artificial Intelligence [17]. In this respect,
any scientific field of study in which causal relations are applied can be addressed on the
basis of this Network-Oriented Modeling approach. Among the wide variety of application
areas there are two types of applications that in a sense are dominant: describing individual
mental processes specifically and describing how individuals interact with each other [19].
Table 1 shows the overview of some combination functions. The following three notions are
central in the Network-Oriented Modeling approach, and define a temporal-causal network
model, and therefore are part of a conceptual representation of a temporal-causal network
model:
Connections strength
w
X,Y
The connection strength between a state X to a state Y is called weight value wX,Y which
is normally between 0 and 1.
Aggregation of impacts of states cY(..)
Each state needs a combination function cY(..) to aggregate the impacts of other states on
state Y.
Speed of change of a state ηY
There is a speed factor ηY shows how fast a state changes over a period of time based on
the impact.
Table 1. Overview of some combination functions c(V1, …, Vk)
Name
Description
Formula c(V1, …, Vk) =
sum (…)
Sum
V1!! Vk
ssum
l
(…)
Scaled sum
function
"#$%&$%"'
l with l > 0
min (…)
max (…)
Minimal Value
Maximal Value
Min(V1
(
(
Vk)
Max(V1
(
(
Vk)
slogistic
st
(…)
Simple
logistic sum function
#
#$)*s+"#$%,,%%$"'%*%t-% with s, t ³ 0
alogistic
s
,
t
(…)
Advanced
logistic sum function
[
#
#$)*
s
+"#$
%
,,
%%
$".
%
*
%t
-
% -
#
#$)
st% ] (1+e-st)
with s, t ³ 0
A conceptual representation of a temporal-causal network model can be transformed in a
systematic or automated manner into a numerical representation of the model as follows [2,
19]:
Y(t) represents the value of Y at time point t in the model which is in the interval [0, 1].
impactX,Y(t) = wX,Y X(t) shows the influence of a state X connected to a state Y at time
point t where wX,Y represents the weight of the connection.
The aggregated impact of some states Xi on Y at t is calculated using a combination
function cY(..):
aggimpactY(t) = cY(impactX1,Y(t), …, impactXk,Y(t))
= cY(wX1,YX1(t), …, wXk,YXk(t))
3
The impact of aggimpactY(t) on Y is applied over time gently, based on speed factor hY:
Y(t+Dt) = Y(t) + hY [aggimpactY(t) - Y(t)] Dt
or dY(t)/dt = hY [aggimpactY(t) - Y(t)]
Therefore, the difference and differential equation for Y are achieved:
Y(t+Dt) = Y(t) + hY [cY(wX1,YX1(t), …, wXk,YXk(t)) - Y(t)] Dt
dY(t)/dt = hY [cY(wX1,YX1(t), …, wXk,YXk(t)) - Y(t)]
Adaptation principles covered
The following adaptation principles are covered.
Hebbian learning
For Hebbian learning of a connection from state Xi to state Xj the following model is used
w(t+Dt) = w(t) + hw [cw(Xi(t), Xj(t), w(t)) - w(t)] Dt
with
cw(V1, V2, W) = hebbµ(V1, V2, W) = V1V2 (1W) + µ W
where µ is the persistence factor with 1 as full persistence.
State-connection modulation
For the adaptation principle for state-connection modulation with control state cs the follow-
ing model is used:
w(t+Dt) = w(t) + hw [cw(cs2(t), w(t)) - w(t)] Dt
with cw(V, W) = scma(V, W) = W + a V W (1 W)
where a is the adjustment parameter for w from cs. In combination these two adaptive com-
bination functions can be used as a weighted average with 0£ q £1as follows:
cw(V1, V2, V, W) = q hebbµ(V1, V2, W) + (1-q) scma(V, W)
w(t+Dt) = w(t) + hw [cw(Xi(t), Xj(t), cs(t), w(t)) - w(t)] Dt
All these difference equations can be used for simulation.
This state-connection adaptation principle can also by applied in a social context. The
hypothesis is based on that whenever a more intensive interplay between two persons occurs,
the connection will become solid; e.g., [12].
Bonding based on homophily
Bonding based on homophily shows that the more look like the states of two connected states,
the stronger their connection will become: ‘the more you are alike, the more you like (each
other)’ [2]; see, for example, [7, 8, 9]. When also the states are assumed dynamic, this prin-
ciple can be combined with contagion of states into a circular causal relation [11]: Stat e «
Link. See also, for example [13-16]. The homophily principle can be as represented numeri-
cally by a combination function cA,B (V1 , V2 , W) as follows :
wA,B(t+Dt) = wA,B(t) + hA,B [cA,B(XA(t), XB(t), wA,B(t)) - wA,B(t)] Dt
dY(t)/dt = hA,B [cA,B(XA(t), XB(t), wA,B) - wA,B]
Three variants of models for the homophily axiom are the linear, quadratic and logistic vari-
ants:
Linear
c(V1 , V2 , W) = slhomo(V1 , V2 , W) = W + W(1- W)( t - | V1 V2| )
Quadratic
c(V1 , V2 , W) = sqhomo(V1 , V2 , W) = W + W(1- W)( t2 – (V1V2)2 )
Logistic
c(V1 , V2 , W) = sloghomo(V1 , V2 , W)
4
= W + W (1- W)(0.5 – 1/(1 +
/*
s
+0"1*%230*
t
%-
))
Based on these options that can be chosen the following numerical differential and difference
equations are generated
dwC,D /dt = hC,D wC,D (1- wC,D)( tC,D - |XCXD| )
wC,D(t+Dt) = wC,D + hC,D wC,D (t) (tC,D - |XCXD|)Dt
dwC,D /dt = hC,D wC,D (1- w C,D)(t2 C,D - |XC XD|)2)
wC,D (t+Dt) = wC,D + hC,D wC,D (t)(1 - wC,D)(t2C,D – (XA(t)XB(t))2) Dt
dwC,D /dt = hC,D wC,D (1- wC,D)(0.51/(1 +
/*
s
+024*%250*%
t
4(5-
))
wC,D (t+Dt) = wC,D + hA,B wC,D (t)(1 - wC,D)(0.5 – 1/(1 +
/*
s
+024*%250*%
t
4(5-
)) Dt
Here XC, XD are the states of person C and D, wC,D is the connection weight from person C
to person D, hC,D the update speed factor for the connection from person C to person D, and
tC,D the threshold or tipping point for connection adaption
3 Modeling a Temporal-Causal Network in Matlab
The advantage of using Matlab for simulation is that (as the abbreviation of that says) ‘Matrix
laboratory’ can easily work with matrices. The format is defined in Table 2, if there is a
connection between X1 and X2 as states, the connection weight assigned to the matrix repre-
sentation (w) of states between aforementioned states. Due to simplicity of Matlab and also
providing many functions, the Matlab software became one of the often used software envi-
ronments for engineers and computer science developers. It has been used in different aspects
of sciences from image processing, due to providing many toolboxes, and also machine learn-
ing, analyse and simulate behavioral dynamics of agents in cognitive science, social science,
and in artificial Intelligence.
The initialization of the matrices for doing actions, calculating based on the combination
functions (Identity, Advance logistic, Advance advanced logistic, scaled sum, …), Matlab
representation of functions based on notations of states, relations and all principles are shown
in the Table in the appendix.
In Figure 1 the functional view of the Matlab code is shown. The process starts with the
inputs named number of states (nodes), connection weights, speed factors, initial values. In
the next step all parameters are allocated in matrices with 1´ N dimensions, where N is the
number of the states in the model.
Table 2. Notions and Matlab representations
Notions
Matlab representations
N
W
Sp_f
STDX
First row of Matr ix ‘O
id=O(1,:);
Second row of MatrixO
sum function
3rd and fourth rows of Matrix ‘O
Scaled sum,
Scaling factor
5
5th and 6th rows of Matrix ‘O
normalised sum, normalizing factor
7th row of Matrix ‘O
adnorsum
8th, 10th and 11th rows of Matrix ‘O
slogistic(...)
9th and 10th and 11th rows of Matri x ‘O
Alogistic, steepness, threshold
12th and 13th and 14th rows of MatrixO
adaptive advanced logistic, steepness,
steepness
eta
hebb, mu
slhomo, htau
alhomo, htau
sqhomo, htau
aqhomo, htau
Adcon, amp
time=0:dt:398;
L=length(time);
dt
plot(time(1:230),STDX(1:230,i),'lineWi
dth',3)
RMS = sqrt (nansum ((Output -
emp_data) .^2 )) / (col * row)
The next phase is allocating the primitive values of the states in the first row of the matrix
(STDX) and then specifying the time period for having simulation. If there is a Hebbian
learning in the model, then parameters of Hebbian learning, h, hebb, and µ are allocated in
three different matrices, similarly for the Homophily principle, simple homophily (slhom),
advanced homophily (alhom) simple quadratic homophily (sqhom) and advanced quadratic
homophily (qhom), threshold (hthau), and finally for state-connection modulation (scma) and
the number of the states which have modulation ability. Figure 1 illustrates functional view
of written Matlab code.
6
Inputs
Number of state s
Connection weights
Speed factors,
initial values
Mixing
Is there any
Hebbian
Learning
principle
Multiplying
Is there any
Homophily Multiplying
principle
Is there any Multiplying
state-
connection
modulation
Add Add Ad d
Figure 1. Functional view of the Matlab process
Allocating each row
of combi nati on func -
tion and also all other
parameters with ma-
trices 1*N dimension
in order to save time
in Matlab
Allocating ini-
tial values of
states in the first
row of Matrix
STDX (1,N)
Specifying period
of time for simu-
lation (L)
Making a ma-
trix with
(time, number
of states)
STDX(L,
N)
Allocating etha,
hebb, mu with
matrices as the pa-
rameters of
Hebbian learn ing
Allocating slhom,
alhom, sqhom,
qhom, hthau as a
threshold of ho-
mophily in Matlab
Allocating state-
connection am-
plification (sca),
scaampl , and the
name of the state
for modulat ion
Allocating condy matrix with
the weights of states in col-
umn-wise way. (for 10 states,
100 columns, condy(1,100))
Showing relationship a mong
states
Making condy(i,k) if there is a homoph-
ily, suppression, Hebbian learning
Allocating le gends for showing each state
Allocating time for condy to make a graph if there is
any state-connection suppression
Making final STDX for simulating graph
Allocating le gends for showing each conn ection
Simulating the graph of each connection if there is state-connec-
tion modulation
Simulating the graph of each states
7
Then in the next step, if there is any above-mentioned principle in the model they will
multiply by the STDX matrix formed already with time. Meanwhile, the matrix called condy
with the weights of states formed in a column-wise order and then make a new row based on
any principle existed and then add the influence of them in the matrix. Finally, all matrices
with STDX and condy (if there was any state-connection modulation) results in generating
the simulation.
The Human Interaction flowchart is depicted in Figure 2. As it can be seen, a first step is
initialization as providing inputs, for instance number of states, weights among states, speed
factors, and combination functions. In the second step it is needed to be decide in which
system the user want to work. In this phase, there are two systems, Multi-Agent system and
Single-Agent system. The former offers Simple linear, Advanced Linear, Simple Quadratic,
Advanced Quadratic Homophily principles and for latter there are Hebbian learning and
state-connection suppression and finally the plotting of simulation occurs.
Figure 2. Human Interaction flowchart
And for more specification, the flowchart of the code is presented in Figure 3
Initialization
(Inputs)
States – Weights - Speed factors -
Combination Functions
Multi-Agent
System
Single-Agent
System
Simple linear Homophily
Advanced Linear Homop hily
Simple Quadratic Homophily
Advanced Quadratic Homophily
Hebbian Learning Principle
State-Connection Modulat ion Principl e
Plotting
8
Figure 3. Structural view of the code
To enable easy use, one can also use a user interface between Excel and Matlab. As Matlab
works with Matrices it might be easier to use an Excel interface of matrices and just read the
matrices from the Excel file and then do the execution in Matlab. Such a matrix expresses
the parameters, combination functions, Identity function, sum function, scaled sum with scale
factor, Normalized with normalizing factor, adaptive normalized sum, simple logistic, ad-
vanced logistic, advanced logistic and adaptive advanced logistic function with steepness and
threshold and other principles. As can be seen from Figure 4 and 5 the Matlab code reads
the matrices based on the matrix representation in Excel; for instance in these figures it would
be from column C1 to L32 to read matrix for all weights of states, speed factors, deltaT, maxt,
combination functions and finally initial values from the first sheet and if there is any princi-
ple combined with the model using the second sheet to read from Excel for Hebbain Learning
principle and Homophily for their principles. Figs 4 and 5 show this option in Excel sheets.
Variables Initializations in matrices
(STDX (X))
Number of
States (N)
Weights of
Connections
(W)
Combination
Functions
(O)
Simple
Logistic
(C12)
Advanced
Logistic
(C13)
Adaptive
Normalized
Sum (C7)
Normalized
Sum (C5)
Sum (C2)
Identity (C1)
Adaptive
Advanced
logistic
(C14)
Scaled Sum
(C3)
Scaling Factor
Normalizing
Factor
Steepness
Threshold
Steepness
Threshold
Steepness
Threshold
Homophily Principles
(Condy (i,k))
Speed Factors
(Sp_f)
9
Figure 4. Excel interface to read for Matlab programming (Parameters, Speed factor, combination
function and initial values)
Figure 5. Excel interface to read for Matlab programming (Hebbian, Homophily and suppression
principles)
10
Parameter tuning using the Sum of Squared residuals and Root Mean Square
For comparison between empirical data and simulation results and optimisation of parame-
ters Matlab components are available; the sum of squared residuals (SSR) have been imple-
mented to calculate the difference.
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%
% loading empirical data
load('Data1.mat', 'Data1');
[row, col]=size(Data1);
% Calculating Root Mean Square
RMS = sqrt (nansum ((Output - emp_data) .^2 )) / (col * row)
4 Illustration for an Example Network Model
It is shown how the model presented in [1] can be executed in Matlab. The conceptual rep-
resentation of temporal causal network of the model used in [1] is illustrated in Figure 5 and
the explanation of states has been shown in Table 3. As can been seen from Figure 5 here
both the Hebbian learning principle and state-connection modulation was used. The Hebbian
learning principle used between states srss and (psa1 and psa2) and also State-connection sup-
pression between state cs2 and the connections with Hebbian Learning principle. The number
of states considered to be 10. Figure 7 shows simulation result of weights of states and Figure
8 shows state-connection suppression.
In Fig. 6 shows temporal-causal network model. An overview of explanation of the states
is illustrated in Table 3.
Table 3. States explanations in the model [1]
X1
Sensory representation of stimulus s
X2
Sensory representation of context c
X3
Sensory representation of action effect e1
X4
Sensory representation of action effect e2
X5
Feeling state for extreme emotion ee
X6
Preparation state for action a1
X7
Preparation state for action a2
X8
Preparation state for response of extre me emotion ee
X9
Control state for timing of suppression of connections
X10
Control state for suppression of connections
11
Figure 6. Adaptive temporal-causal network model’s conceptual representation [1]
% SRSs SRSc SRSe1 SRSe2 FSee PSa1 PSa2 PSee CS1 CS2
% X1 X2 X3 X4 X5 X6 X7 X8 X9 X10
W=[ 0 0 0 0 0 0.9 0.4 0 0 0 %X1 SRSs
0 1 -0.1 0.3 0 0 0 1 0 0 %X2 SRSc
0 0 0 0 0 0.7 0 0 0 0 %X3 SRSe1
0 0 0 0 0 0 0.7 0 0 0 %X4 SRSe2
0 0 0 0 0 0 0 1 0 0 %X5 FSee
0 0 0.7 0 0 0 -0.2 0 0 0 %X6 PSa1
0 0 0 0.7 0 -0.2 0 0 0 0 %X7 PSa2
0 0 0 0 1 0 0 0 0 0 %X8 PSee
0 0 0 0 0 0 0 0 0 0 %X9 CS1
0 0 0 0 0 0 0 0 0 -0.9 %X10 CS2
];
Sp_f=[ 0 0.05 0.5 0.5 0.5 0.5 0.5 0.4 0.02 0.6];
O=[ 0 0 0 0 1 0 0 0 1 0 % identity function id(.)
0 0 0 0 0 0 0 0 0 0 % sum function sum (...)
0 0 1 1 0 1 1 1 0 1 % Scaled sum sum (...)
0 0 0.7 1 0 2 2 2 0 1 % Scaling factor
0 0 0 0 0 0 0 0 0 0 % normalised sun norsum(...)
12
0 0 0 0 0 0 0 0 0 0 % normalizing factor
0 0 0 0 0 0 0 0 0 0 % adnorsum
0 0 0 0 0 0 0 0 0 0 % slogistic(...)
0 0 0 0 0 0 0 0 0 0 % alogistic(...)
0 0 0 0 0 0 0 0 0 0 % steepness
0 0 0 0 0 0 0 0 0 0 % threshold
0 1 0 0 0 0 0 0 0 0 % adaptive advanced logistic (...)
0 18 0 0 0 0 0 0 0 0 % steepness
0 0.2 0 0 0 0 0 0 0 0 % threshold factor
% Suppression amo ng connections in Hebbian learning
adcon(13,6)=0.15;
adcon(13,7)=0.15;
adcon(14,6)=0.5;
adcon(14,7)=0.5;
adcon(24,6)=-0.7;
adcon(24,7)=-0.7;
% Hebbian learnin g among stat es 1,6
eta(1,6)=0.5;
eta(1,7)=0.8;
hebb(1,6)=0.85;
hebb(1,7)=0.85;
mu(1,6)=0.8;
mu(1,7)=0.8;
% Assign time for plotting
dt=0.25;
time=0:dt:398;
L=length(time);
STDX=zeros(L,N);
% Assign the init ialization f or states 1 and 2 (can be fo r any stat es)
STDX(1,1)=1;
STDX(1,2)=0.1;
Figure 7. Simulation outcome of presented model: states
13
Figure 8. Simulation outcome for suppression and Hebbian learning for w1 (connection X1-X6) and
w2 (connection X1-X7)
Figure 9. shows a difference between simulation result and empirical data provided.
Figure 9. Simulation result for empirical data (X1) and simulation result (X2).
And the result is 0.01309.
5 Discussion
The implementation of a dedicated Matlab-based software environment for Network-Ori-
ented Modeling has been described. The modeling approach covered can be found in [2]; see
also [19]. This implementation has been used in dynamic and adaptive network-oriented
modelling. The environment was illustrated for the example model described in [1] for which
previously only an Excel-based model was available.
An important advantage of the software environment is that modeling can take place at
the level of conceptual representations expressed as labeled graphs or matrices. Therefore, it
is suitable in a multidisciplinary context where different disciplines play a role, also disci-
plines where technical knowledge from computer science or AI is minimal. The more tech-
nical numerical representations and the actual execution are taken care of by the software
environment and therefore for users no programming skills are needed.
14
Acknowledgement
We would like to thank our colleague Fakhra Jabeen, Ph.D. candidate at Vrije Universiteit
Amsterdam, for her assistance with making possible to have an Excel interface with current
Matlab code.
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lection from influence. Sociological methodology 40, 329393 (2010).
16. Mundt, M.P., Mercken, L., Zakletskaia, L.I.: Peer selection and influence effects on adolescent
alchohol use: a stochastic actor-based model. BMC Pediatrics 12, 115 (2012).
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studying an expert physician, in ed. By F.R.G. Karlsruhe, proceedings English International joint
conference on Artificial Intelligence, IJCAI’83 (William Kaufman, Los Altos, CA, 1983).
18. Hebb, D.: The organization of behavior, Wiley, 1949.
15
19. Treur, J.: The Ins and Outs of Network-Oriented Modeling: from biological networks and mental
networks to social networks and beyond. Transactions on Computational Collective Intelligence.
Paper for keynote lecture at ICCCI’18, 2018.
20. Ziabari, S.S.M, Treur, J.: An adaptive cognitive temporal-causal network model of a mindfulness
therapy based on music. Proceedings of the 10th International Conference on Intelligent Human
Computer Interaction (IHCI2018), Springer, India (2018).
21. Ziabari, S.S.M.: Integrative cognitive and affective modeling of deep Brain stimulation. Proceed-
ings of the 32nd International conference on industrial, engineering and other applications of ap-
plied intelligent systems (IEA/AIE 2019), submitted.
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mitted.
Appendix: Inputs Description of the Matlab Code
% Initializing th e matrices w ith zeroes in or der to get less time in operating the main codes
id=zeros(1,N);
sum=zeros(1,N);
ssum=zeros(1,N);
lambda=zeros(1,N) ;
norsum=zeros(1,N) ;
norlambda=zeros(1 ,N);
adnorsum=zeros(1, N);
slog=zeros(1,N);
alog=zeros(1,N);
s=zeros(1,N);
t=zeros(1,N);
adalog=zeros(1,N) ;
adas=zeros(1,N);
adat=zeros(1,N);
id=O(1,:);
sum=O(2,:);
ssum=O(3,:);
lambda=O(4,:);
norsum=O(5,:);
norlambda=O(6,:);
adnorsum=O(7,:);
slog=O(8,:);
alog=O(9,:);
s=O(10,:);
t=O(11,:);
adalog=O(12,:);
adas=O(13,:);
adat=O(14,:);
eta=zeros(N);
hebb=zeros(N);
16
mu=zeros(N);
slhomo=zeros(N);
alhomo=zeros(N);
sqhomo=zeros(N);
aqhomo=zeros(N);
htau=zeros(N);
amp=zeros(N);
adcon=zeros(14+N, N^2);
clc
clear
close all
format long
N=10; % Number of states
% Weights of states (Connection among states)
% X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 . . .
W=[ 0 0 0 0 0 0 0 0 0 0 %X1
0 0 0 0 0 0 0 0 0 0 %X2
0 0 0 0 0 0 0 0 0 0 %X3
0 0 0 0 0 0 0 0 0 0 %X4
0 0 0 0 0 0 0 0 0 0 %X5
0 0 0 0 0 0 0 0 0 0 %X6
0 0 0 0 0 0 0 0 0 0 %X7
0 0 0 0 0 0 0 0 0 0 %X8
0 0 0 0 0 0 0 0 0 0 %X9
0 0 0 0 0 0 0 0 0 0 %X10
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
];
% Speed factor connections
Sp_f=[ 0 0 0 0 0 0 0 0 0 0 . . .];
% Combination functions
O=[ 0 0 0 0 0 0 0 0 0 0 . . . % identity function id(.)
0 0 0 0 0 0 0 0 0 0 . . . % sum function sum (...)
0 0 0 0 0 0 0 0 0 0 . . . % Scaled sum sum (...)
0 0 0 0 0 0 0 0 0 0 . . . % Scaling factor
0 0 0 0 0 0 0 0 0 0 . . . % normalized sum norsum(...)
0 0 0 0 0 0 0 0 0 0 . . . % normalizing factor
0 0 0 0 0 0 0 0 0 0 . . . % adaptive normalized sum adnorsum
(...)
0 0 0 0 0 0 0 0 0 0 . . . % simple logistic slogistic
0 0 0 0 0 0 0 0 0 0 . . . % advanced logistic alogistic
17
0 0 0 0 0 0 0 0 0 0 . . . % steepness
0 0 0 0 0 0 0 0 0 0 . . . % threshold
0 0 0 0 0 0 0 0 0 0 . . . % adaptive advanced logistic
0 0 0 0 0 0 0 0 0 0 . . . % steepness
0 0 0 0 0 0 0 0 0 0 . . . % threshold factor
k=0;
for i=1:N
for j=1:N
k=k+1;
condy(1,k)=W(i,j);
end
end
for i=2:L
for j=1:N
k=0;
for ii=1:N
for jj=1:N
k=k+1;
CC=0;
for iii=15:14+N
jjj=iii-14;
CC=CC+adcon(iii,k)*STDX(i-1,jjj);
End
% identity functi on id
C2=1-condy(i-1,k);
C1=con dy(i-1,k);
% Scaled sum sum
C3=htau( ii,jj)-abs(STDX(i-1,ii)-STDX(i-1,jj));
% Scalin g factor
C4=abs(C 3);
% normal ized sum norsum
C5=htau( ii,jj)^2-abs (STDX(i-1,ii)-STDX(i -1,jj))^2;
C6=abs(C 5);
% Homophily princ iples
condy(i,k)= condy(i-1,k)+eta(ii,jj)*(hebb(ii,jj)*(...
STDX(i-1,ii)*STDX(i-1,jj)*C2+mu(ii,jj)*C1)+...
adcon(13,k)*(C1+adcon(14,k)*(CC)*(C2)*...
C1)+slhomo(ii,jj)*(C1+amp(ii,jj)*C1*C2*...
C3)+al homo(ii,jj)* (C1+amp(ii,j j)*C2*((C4+C3)/2 )+C1*...
((C4-C3)/2))+sqhomo(ii,jj)*(C1+amp(ii,jj)*C1*C2*C5)+...
aqhomo (ii,jj)*(C1* amp(ii,jj)*C 2*((C6+C5)/2)+C1 *((C6-C5)/ 2))-...
C1)*dt ;
end
C7=0;
18
% adapt ive normalized s um adnorsum
for jj=1:N
C7=C7+condy(i-1,(jj-1)*N+j)*STDX(i-1,jj);
end
wxsum(i-1,j)=C7;
C11=0;
for jj=1:N
C11=C11+condy(i-1,(jj-1)*N+j);
end
landa(i-1,j)=C11;
if ssum(j)>0
C8=ssum(j)*C7/lambda(j);
else
C8=0;
end
if norsum(j)> 0
C9=norsum(j)*C7/norlambda;
else
C9=0;
end
if adnorsum(j )>0
C10=adnorsum(j)*C7/C11;
else
C10=0;
End
% s imple logist ic slogistic
C12=slog(j)*(1/(1+exp(-s(j)*(C7-t(j)))));
% advanced log istic alogis tic
C13=alog(j)*(((1/(1+exp(-s(j)*(C7-t(j)))))-.. .
(1/(1+exp(s(j)*t(j)))))*(1+exp(-s(j)*t(j))));
% adaptive adv anced logistic
C14=adalog(j)*(((1/(1+exp(-adas(j)*(C7-adat(j)*C1 1))))-...
(1/(1+exp(adas(j)*adat(j)))))*(1+exp(-adas(j)*adat(j))) );
aggimpact(i-1,j)=id(j)*C7+sum(j)*C7+C8+C9+C10+C12+C13+C14;
STDX(i,j)=STDX(i-1,j)+Sp_f(j) *(aggimpact(i-1, j)-STDX(i-1,j))*dt;
end
end
end
% plotting Sectio n
for i=1:N
plot(time(1:230), STDX(1:230,i ),'lineWidth ',3)
hold on
end
19
for i=1:N
if i<10
ch(i,1:3)=['X',num2str(i) ,' '];
else
ch(i,1:4 )=['X',num2str(i),' '];
end
end
legend(ch)
grid on
figure(2)
for i=1:N
plot(time,STDX(:, i),'lineWidth',3)
hold on
end
for i=1:N
if i<10
ch(i,1:3)=['X',num2str(i) ,' '];
else
ch(i,1:4 )=['X',num2str(i),' '];
end
end
legend(ch)
grid on
% Plotting figure of suppression of conne ctions
figure(3)
plot(time,condy,'lineWidth',3)
hold on
legend('X1 _ 6','X1 _ 7')
grid on
Table 4. Notions of states and Matlab representations of functions
Notions of states
Matlab representations
Combination function (identity)
C1=condy(i-1,k);
Combination function (Sum)
C3=htau(ii,jj)-ab s(STDX(i-1,i i)-STDX(i-1,jj));
Combination function (Scaled sum)
C8=ssum(j)*C7/lam bda(j);
Combination function (Normalised sum)
C9=norsum(j)*C7/n orlambda
Combination function (Adaptive normal-
ized)
C7=C7+condy(i-1,( jj-1)*N+j)*STDX(i-1,jj);
Combination function (Simple Logistic)
C12=slog(j)*(1/(1 +exp(-s(j)*( C7-t(j)))));
Combination function (Advance Logistic)
C13=alog(j)*(((1/ (1+exp(-s(j)*(C7 -t(j)))))-
(1/(1+exp(s(j)*t( j)))))*(1+ex p(-s(j)*t(j))));
Combination function (Adaptive advance
logistic)
C14=adalog(j)*((( 1/(1+exp(-ad as(j)*(C7-
adat(j)*C11))))-
Hebbian Learning Principle
condy(i-1,k)+eta(ii,jj)*(hebb(ii,jj)*(STDX(i-
1,ii)*STDX(i-1,jj)*C2 +mu(ii,jj)*C 1)
20
Homophily Principles (Simple linear Ho-
mophily)
slhomo(ii,jj)*(C1+amp(ii,jj)*C1*C2*C3)
Homophily Principles (Advanced linear
Homophily)
alhmo(ii,jj)*(C1+ amp(ii,jj)*C 2*((C4+C3)/2 )+C1*
((C4-C3)/2))
Homophily Principles (Simple Quadrtic
Homophily)
sqhomo(ii,jj)*(C1 +amp(ii,jj)* C1*C2*C5)
Homophily Principles (advanced Quadrtic
Homophily)
aqhmo(ii,jj)*(C1* amp(ii,jj)*C 2*((C6+C5)/2 )+C1*
((C6-C5)/2))-C1)*dt;
State-connection suppression
adcon(13,k)*(C1+a dcon(14,k)*( CC)*(C2)* C1 )
All values for States
STDX(i,j)=STDX(i-1,j) +Sp_f(j)*(ag gimpact(i-
1,j)-STDX(i-1,j))*dt;
All values for connection weights
condy(i,k)=condy( i-
1,k)+eta(ii,jj)*( hebb(ii,jj)* (...
STDX(i-1,ii)*STDX(i-
1,jj)*C2+mu(ii,jj )*C1)+...
adcon(13,k)*(C1+ad-
con(14,k)*(CC)*(C 2)*...
C1)+slhomo(ii,jj) *(C1+amp(ii, jj)*C1*C2*.. .
C3)+alhomo(ii,jj) *(C1+amp(ii, jj)*C2*((C4+ C3)/2
)+C1*...
((C4-
C3)/2))+sqhomo(ii ,jj)*(C1+amp (ii,jj)*C1*C 2*C5)
+...
aqhomo(ii,jj)*(C1 *amp(ii,jj)* C2*((C6+C5)/ 2)+C1
*((C6-C5)/2))-...
C1)*dt;
RMS (Root Mean Square)
% loading empirical data
load('Data1.mat', 'Data1');
[row, col]=size(Data1);
% Calculating Root Mean Square
RMS = sqrt (nansum ((Output - emp_data)
.^2 )) / (col * row)
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