Improvement of Bias and Rising Threshold Algorithm based on Local

Information Sharing

Nhuhai Phung, Masao Kubo and Hiroshi Sato

Department of Computer Science, National Defense Academy, Yokosuka, Kanagawa, Japan

Keywords:

Agreement Algorithm, Local Information Sharing, Trial and Error.

Abstract:

This paper improves BRT algorithm based on Local Information Sharing (BRT-Lis) to apply to the practical

problems which locations related environments. We consider how to adjust the local parameters so that agreed

behavior could be obtained optimally. We also examine how the Local Information Sharing inﬂuences the

agreement with a simple position changing model.

1 INTRODUCTION

In a swarm, macro candidate behaviors are created by

micro interactions among individuals, and the behav-

ior of individuals is inﬂuenced by their macro behav-

ior. The relationship between the micro interactions

and the macro states is usually nonlinear. Thus, it is

not easy to analytically obtain micro behaviors that

lead to an appropriate macro state. Therefore, the

principle and method leading to a bottom-up mech-

anism by the highly organized behavior found in so-

cial organisms is still attracting the attention of re-

searchers. Therefore, the principle and method lead-

ing to a bottom-up mechanism by the highly orga-

nized behavior found in social organisms is still at-

tracting the attention of researchers. Research on

foraging behavior by ant and bee colonies has been

undertaken for a long time, and excellent models

have been proposed to assign individuals to feeding

and resting roles. Tofts et al. (1992) introduced

foraging-for -work (FFW) model in which individu-

als seek work and engage in task performance when

they encounter a stimulus. Page et al. (1991) de-

veloped a model of task allocation in bees based on

the threshold principle, where individuals are repre-

sented as boolean automata embedded within a net-

work. However, they did not try to relate quantita-

tively their results to any speciﬁc experimental ob-

servation. Gordon et al. (1992) developed a model

based on a connectionist model, which can be seen as

a more complex and a more experiment-driven ver-

sion of Page et al.’s boolean network. Bonabeau et

al. (1998) introduced a simple mathematical model

of the regulation of division of labor in social insects

based on a ﬁxed response thresholds. They showed

that this simple model explains the experimental ob-

servation of Wilson (1984), it was possible to ex-

tend the model to a more complicated situation, ex-

plore its characteristics, and studied under conditions

that could explain temporary multifaceted phenom-

ena. Castello analyzed the simulation results using

the adaptive respond threshold model (ARTM), and

conducted experiments with real robots. And, in this

study, SARTM, a simpliﬁed version of ARTM, was

proposed in order to improve the adaptation and emer-

gent capabilities of robotic swarms in which the re-

sponse threshold is calculated dynamically.

On the other hand, there are cases where it is

not possible to adjust the division of labor, so it is

not possible to lead a swarm to attain the targeted

macroscopic state. Thus, general models about how

a swarm discovers, memorizes, and learns new mi-

cro interactions to enable the attainment of the de-

sired macro state have not been proposed. This results

in a bottleneck in the construction of swarm systems

(Kubo et. el’s reseach (2015)).

In previous work (Phungnhu et al., 2017), we have

developed an agreement algorithm using a trial and

error method at the macro level (BRT model, Bias and

Rising Threshold model) based on the hypothesis that

the opinion of a swarm is always made by agreements

between the individuals while the agreement content

changes on an hourly basis. In the conventional di-

vision of labor models, for example, ﬁxed response

thresholds model Bonabeau et al.’s research (1998),

the swarm’s tasks are ﬁxed. They considered only one

or some limited purposes of the swarm. In contrast,

in the BRT algorithm, the swarm can create various

Phung, N., Kubo, M. and Sato, H.

Improvement of Bias and Rising Threshold Algorithm based on Local Information Sharing.

DOI: 10.5220/0006751603390344

In Proceedings of the 10th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2018) - Volume 1, pages 339-344

ISBN: 978-989-758-275-2

339

responses against environmental changes and the like

by switching its behaviors until discovering an suit-

able agreement behavior.

The BRT model is a model that expands the 2-

choice consensus-building model proposed by Na-

matame bias model (Namatame, 2001) that can be

used even in cases where more than two choices are

available. In the BRT model, an agent determines its

attitude by its own preference, in which the thresh-

old is rising with time, and by looking at the ratio of

the population. However, this ratio is the ratio of the

number of agents who are choosing the same behav-

ior to the total number. In other words, as shown in

Figure 1 (A), the information sharing must be global.

However, in reality, the global information sharing is

impossible. Moreover, even if the information on the

whole swarm can not be obtained, macro behaviors

will be able created by local interaction among neigh-

bors (Figure 1 (B)).

Thus, in this paper, we point out improving the

original BRT algorithm in order to make it more real-

istic and practical based on Local Information Shar-

ing. By using a simple position changing model, we

also ﬁnd out that the suitable macro behavior can be

discovered more quickly than in the original BRT al-

gorithm when the number of agents is very small.

This paper is organized as follows. Section 2 de-

scribes the original BRT algorithm, Section 3 intro-

duces BRT-Lis, and then Section 4 shows the exper-

imental results and discussions. Conclusions and fu-

ture work are described in Section 5.

Figure 1: The image of (A) global information sharing in

BRT and (B) local information sharing in BRT-Lis.

2 ORIGINAL BRT MODEL

The BRT model is a model that expanded from Na-

matame model. First of all, we will introduce this

model.

2.1 Namatame Bias Model

Namatame got inspiration from the research in crit-

ical mass of Thomas Crombie Schelling. As shown

Figure 2: The role of social skin in individual decision-

making (Namatame, 2001).

Figure 3: Bias design method. (A) Probability density func-

tion. (B) Cumulative distribution function and equilibrium

point of collective consensus decision (E1 and E3 are stable

points, E2 is an unstable point).

in Figure 2, he concluded that individual decision-

making does not only depend on personal philosophy

and personal preferences (bias θ

), but it also depends

on the atmosphere of the whole group p(t). He pro-

posed a decision-making framework based on the in-

troduction of individual differences (bias value). An

agent determines its attitude by its own preference and

by looking at the ratio of the population in favor and

opposite to it. As shown in Eq.(1), the agents disagree

if the proportion of the consensus faction is less than

the threshold and agrees if it exceeds the threshold.



p(t) ≥ θ

: Agree

p(t) < θ

: Opposite

(1)

Namatame investigated various distributions and

clarifed that we distribute the bias value θ

in a bell

shape as shown on the left side of Figure 3, one unsta-

ble point and two stable points are generated, and all

members smoothly move their opinions to ultimately

agree or disagree, as shown on the right side of Figure

3. The number of iterations required to attain conver-

gence depends on the bell shape of the distribution

and not on the number of agents, so it is expected that

prompt convergence can be achieved even with a large

number of agents.

2.2 The BRT Model

Namatame bias model can only deal with two options

that agree or disagree with an opinion. In the BRT

model, we have proposed an algorithm that enables

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340

agreement to be achieved promptly even when there

are multiple opinions (M ≥ 2).

Now, we assumed that there are N

agents P

,··· ,P

(1 ≤ i ≤ N). A =

,··· ,a

} is the set of agent candi-

date behavior with M ≥ 2 (1 ≤ j ≤ M). A

(t) ∈ A

is the behavior of agent P

at time t. The agent P

has a bias θ

(0 < θ

< 1). n(a

) is the number of

agents selecting behavior a

. a

goal

is the desirable

candidate behavior (a

goal

∈ A). The agent does not

know a

goal

, in advance, only when all have agreed,

and it is understood that a

goal

is an agreed behavior.

At this time, the agent P

decides on behavior

(t +1) at the time t as follows: If

n(A

(t))/N ≥ θ

+ τ ·c ·(t −t

i,last

(t))

(2)

is satisﬁed, A

(t + 1) = A

(t). Otherwise, a behavior

other than A

(t) is stochastically chosen and becomes

(t + 1) ∈A\A

(t). Here, τ is a constant representing

the increment in the number of proponents.

In addition, t

i,last

(t) is the time at which the agent

last changed its behavior, and ((t −t

i,last

(t))) is the

time over which the same action continues to be se-

lected.

i,last

(t +1) =



t +1 A

(t +1) 6= A

(t)

i,last

(t) otherwise

(3)

c(t) is a function that is equal to 1 when the desir-

able macro purpose is not achieved as follows:

c =



0 ∀i,A

(t) = a

goal

1 otherwise

(4)

If the ratio of agents who select the same behavior

as theirs is lower than θ

+τ·c·(t −t

i,last

(t)), the agent

randomly selects a new behavior from other candidate

behaviors.

3 BRT-LIS

3.1 Proposed Method: BRT-Lis

In BRT model, an agent knows all the behaviors of

other agents. Therefore, it can know the number

of agents who choose the same behavior (n(A

(t))).

Here, we assume that the agent gets only the informa-

tion of neighbors that is within a ﬁxed radius r. We

call NB

(t) is the set of neighbors of agent P

at time

step t. N

(t) is size of this set (the number of neigh-

bors).

Here, we also assume that there are

N agents P

,··· ,P

(1 ≤ i ≤ N).

A = {a

,··· ,a

} is the set of agent can-

didate behavior with M ≥ 2 (1 ≤ j ≤ M). A

(t) ∈ A

is the behavior of agent P

at time t. The agent P

has a bias θ

( 0 < θ

< 1). n

) is the number of

neighbors selecting behavior a

. a

goal

is the desirable

candidate behavior (a

goal

∈ A). The agent does not

know a

goal

, in advance, only when all have agreed,

and it is understood that a

goal

is an agreed behavior.

At this time, the agent P

decides on behavior

(t +1) at the time t as follows: If

(t))/N

(t) ≥ θ

+ τ ·c ·(t −t

i,last

(t))

(5)

is satisﬁed, A

(t + 1) = A

(t). Otherwise, a behavior

other than A

(t) is stochastically chosen and becomes

(t +1) ∈A\A

(t). Here, τ, c(t) and t

i,last

(t) are same

to BRT model.

In NB

(t), if the ratio of agents who select the

same behavior as theirs is lower than θ

+ τ ·c ·(t −

i,last

(t)), the agent randomly selects a new behavior

from other candidate behaviors.

3.2 Generation of the Bias Value θ

Here, we also use Gaussian distribution to generate

the bias value θ

The Gaussian distribution method:

= min(1,max(0,G(µ,σ

)))

(6)

Where: G(µ,σ

) is a random variable with a Gaussian

distribution, µ is the expected distribution, σ

is the

variance.

G(µ,σ

) =

√

2Πσ

exp(−

(x−µ)

)

2σ

)

(7)

4 THE EXPERIMENTAL

RESULTS

4.1 Experimental Setup

In this section, we present the details of the environ-

ment that are important for completely understanding

our experimental setup. In this work, we use a simple

position changing model. Picture of domain is shown

in Figure 4. Black dot represents the agent P

and the

dotted circle represents the neighborhood of P

at the

center of the circle, light black dot represents a neigh-

bor of P

, blue dot represents an agent who is not the

neighbor of P

. Red arrows represent 4 behaviors of

agent (M = 4), behavior a

is Turn Left (move to the

left side of the domain), behavior a

is Go Up, behav-

ior a

is Turn Right, behavior a

is Go Down.

The experimental settings for variables are given

in Table 1. We chose the variables that yielded the

best results in our previous research on BRT model

Improvement of Bias and Rising Threshold Algorithm based on Local Information Sharing

341

[8]. The domain is toroidal. This means that agents

that move off one edge of our domain reappear on the

opposite edge moving in the same behavior.

Figure 4: Image of the simple position changing model. The

black dot represents the agent P

and the dotted circle rep-

resents the neighborhood of P

at the center of the circle,

the light black dot represents a neighbor of P

, the blue dot

represents an agent who is not the neighbor of P

. The red

arrows represent 4 behaviors of agent (M = 4), behavior a

is Turn Left (move to the left side of the domain), behavior

is Go Up, behavior a

is Turn Right, behavior a

is Go

Down.

Table 1: Experimental setting for variables.

Variable Value

domain height 600

domain width 600

units moved by each agent per time step 1

the number of candidates behavior M 4

the expected distribution µ 1/M

the variance of Gaussian distribution σg 1/(3M)

the increment value τ 0.001

4.2 Conﬁrmation of Behavior

In the following, we describe computer experiments

that were used to show how the agreement state can

be switched as a function of the BRT-Lis. In order

to observe the behavior-switching more clearly, we

assume that there is no desirable behavior (a

goal

) in

the set of agent candicate behavior A.

For this experiment, the number of agents was set

at N = 1000. We show the candidate behaviors se-

lection over time where all members started from an

agreed behavior at the initial time when the radius r is

r = 20, 50, 100, 200.

Figure 5 shows an example of agents, which are

randomly located in the domain, at the initial time.

Figures 6 ∼ 9 show a transition example of each case

of radius values. The horizontal axis represents time

step and the vertical axis represents n(a

),i = 1, 2, 3, 4,

which is the number of agents who choose behavior

. As can be seen in Figure 6, agreement by all indi-

viduals has never been emerged with too small radius

r (r = 20). In Figure 7, we see that the number of

agents who choose behavior a

increases rapidly in

ﬁrst 1200 time steps and reachs 1000 instantaneously

several times. Thus, it can not be said that the agree-

ment behavior has be emerged. However, as can be

seen in Figure 7, when the radius r is set in high value

(r = 100), the number of agents choosing the same be-

havior is beginning to be maintained at a certain and

high value. This means that although the agreement

is unstable, it has been emerged. When the radius r

is set in very high value (r = 200), which means the

agent can share the information with about one-third

of all individuals (πr

/(wid th ∗height)), all individu-

als choose the same behavior for nearly constant time.

We found that all individuals switched behavior by

agreeing with a behavior.

Thess results show that the higher radius value we

set, the more clearly the agreement can be seen. In

the next subsection, we will examine the inﬂuence of

the radius r in more details.

Figure 5: An example of agents at the initial time.

Figure 6: Swiching behavior when r = 20.

4.3 Searching Ability of the Desired

Behavior and Its Features

In this subsection, we show that it is possible to dis-

cover the desired behavior using computer simula-

tions.

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342

Figure 7: Swiching behavior when r = 50.

Figure 8: Swiching behavior when r = 100.

Figure 9: Swiching behavior when r = 200.

4.3.1 Inﬂuence of the Radius r

We conduct the experiment with the number of agents

N = 100. Assuming that all the agents selects

an agreed behavior at the initial time, we veriﬁed

whether it is possible to reach the desired behavior

by changing the radius r. We selected behavior a

and set it as the desired behavior (a

goal

= a

). Then,

we changed the radius r and counted the cases where

all of the agents were able to select a

, conducted

100 trials on each parameter set, and determined the

probability of discovery and average of time required

to reach the desired behavior from an agreed state.

We set time limit is 30000, and if this time limit was

reached the trial was stopped. These experimental re-

sults are shown in Figure 10 and 11.

Figure 10 shows the changing of the success ra-

tio and average of time required to reach the desired

behavior from an agreed state in the initial time. The

vertical axis on the left is the ratio at which behav-

ior a

was successfully agreed (blue line), the verti-

cal axis on the right is average of time required to do

that (red bar), and the horizontal axis is value of the

radius r. As we can see, the success ratio increased

with a decrease in r and when r was set to r = 200,

the success ratio was close to 100%. This means that

BRT-Lis model could achieve the same result as BRT

model, which is equivalent to BRT-Lis model with

r = 400. On the other hand, we can see that the higher

radius value we set, the more rapidly the desired be-

havior be discovered with a proportion.

Figure 11 shows the cumulative distribution of ra-

tios that reached the desired behavior from an agreed

state in initial time. As a result, it is found that when

the radius r was too small, the desired behavior could

not be discover any time. When the radius r is set

to the bigger value (about 50 ∼ 100), the number

of cases discovering the desired behavior increases.

When the radius r is set to even bigger value (around

200), the desired behavior is reached promptly in all

cases.

From the above, it can be concluded that BRT-Lis

model could discover the desired behavior in the same

as of BRT model with an suitable radius r.

Figure 10: The changing of the success ratio and the aver-

age of time required to discover the desired behavior from

an agreed state in the initial time. r = 20,50, 100,200.

Figure 11: The cumulative distribution of ratios that

reached the desired behavior from an agreed state in initial

time.

Improvement of Bias and Rising Threshold Algorithm based on Local Information Sharing

343

4.3.2 Inﬂuence of the Number of Agents N

Here, we make clear how the number of agents in-

ﬂuence the searching ability of the desired behavior.

We set r = 100, a

goal

= a

. Assuming that all the

agents selects an agreed behavior at the initial time.

We changed the radius r and counted the cases where

all of the agents were able to select a

. Figure 12

shows the results of performing 100 trials for a num-

ber of agents N = 4,8,16,64,256. A trial is termi-

nated at time step 30000.

The vertical axis on the left is the ratio at which

behavior a

was successfully agreed (red line), the

vertical axis on the right is average of time required

to do that (blue bar), and the horizontal axis shows

the number of agents N. As can be seen, when the

number of agents is set a very small value (about 4

∼ 16), the success ratio surprisingly decreased with

an increase in N. It is considered that the number of

agents is too small, and there is almost no neighbor.

Therefore, each agent makes decisions without being

affected by other agents. At this time, the probability

of selecting the desired behavior a

by all individu-

als could be 1/(3N). Thus, the smaller the number of

agents N is, the higher success ratio could be. How-

erver, when the number of agents is set a bigger value,

the density of agents in the domain increases and the

number of neighbors of each agents increases too.

Therefore, the probability that individual information

is indirectly conveyed to everyone increases. As can

be seen, the success ratio increases when N ≥ 16.

Figure 12: The changing of the success ratio and the aver-

age of time required to reach the desired behavior from an

agreed state in the initial time. N = 4,8, 16,64,256.

5 CONCLUSIONS

In this paper, we introduced the improved BRT al-

gorithm based on Local Information Sharing (BRT-

Lis) and considered how to adjust the local parameters

so that the desired behavior could be obtained opti-

mally. We conducted the experiments that conﬁrm the

behavior switching function of BRT-Lis with a sim-

ple position changing model. We also examined how

the Local Information Sharing inﬂuences the search-

ing ability of the desired behavior. The experimen-

tal result revealed that the following implications: (1)

BRT-Lis model can discover the desired behavior in

the same as of BRT model with an suitable radius r

and (2) the extremely low density or high density of

agents in the domain is the condition of high success

ratio of searching the desired behavior.

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