A Random Walker Can Optimize the Exploration without the Large

Capacity Memory

Tomoko Sakiyama

Department of Information Systems Science, Faculty of Science and Engineering, Soka University, Tokyo 192-8577, Japan

Keywords: Random Walk, Super-diffusion, Memory, Resource Distribution.

Abstract: A random walker explores an unknown field and sometimes changes its movement property using new spatial

information obtained by it during its exploration. An important matter is the relation between the movement

property of a random walker and the use for acquired information. I recently developed a random walk model

in which a walker coordinated its directional rule based on its experiences and found that this model presented

an optimal random walk, which demonstrated a so-called Lévy walk with μ = 2.00. Here, I investigate the

foraging efficiency for that model and verify whether a large memory capacity is required or not in order to

maintain the foraging efficiency. My findings reveal that the proposed model can apply to biological processes

where a random walker does not have a high memory capacity.

1 INTRODUCTION

Animals demonstrate random search in the absence of

prior knowledge in order to get some information,

such like spatial information (Kareiva and Shigesada,

1983; Viswanathan et al. 2001; Bartumeus et al. 2005,

2008; Bartumeus and Levin, 2008). Many random

search models such like the Lévy walk or the

Brownian walk model are effective for random

exploration and have been very well studied

(Bartumeus et al. 2005, 2008; Bartumeus and Levin,

2008). A Lévy walk, which exhibits a scale-free

distribution, is defined as a process where an agent

takes steps of length l at each time and the probability

density function of those steps decays asymptotically

as a power law:

P(l) ~ l

−μ

, where 1 < μ ≦3

Several studies of animal foraging strategies

have reported that Lévy walks are efficient where

resource is sparse and randomly distributed

(Bartumeus et al. 2005; Humphries and Sims,2014).

On the contrary, the advantage of Lévy walks will

disappear in high-density environments where

resource is abundant (Bartumeus et al. 2005;

Humphries and Sims,2014). The Lévy and

Brownian walks show similar exploration

https://orcid.org/0000-0002-2687-7228

efficiencies if extremely abundant resources are

available for random walkers.

The search ability for food resources is a matter of

life and death for random walkers. To this end, the

search ability of random walk models has been

extensively investigated (Sakiyama and Gunji, 2013).

Recently, I developed a random walk model named

as the self-reference model (Sakiyama, 2020). A

walker in that model avoids a certain direction using

the past information. At the same time however, the

walker modulates its directional rule if it experiences

some directional inconsistencies in the recent series

of its movements. The self-reference model exhibited

a so-called power-law tailed movement with optimal

μ value (μ ≈ 2.0) (Bartumeus et al. 2005). In this

paper, I check the paramenter effects by examining

the performance of resource search ability of this

model. Here, a random walker obeying that model

explores a two-dimensional field where food

resources are distributed. I investigate the parameter

effects in respect with the exploration ability of the

walker and discuss the unnecessity of a large memory

capacity.

Sakiyama, T.

A Random Walker Can Optimize the Exploration without the Large Capacity Memory.

DOI: 10.5220/0010369902090212

In Proceedings of the 14th International Joint Conference on Biomedical Engineering Systems and Technologies (BIOSTEC 2021) - Volume 3: BIOINFORMATICS, pages 209-212

ISBN: 978-989-758-490-9

209

2 MATERIALS & METHODS

2.1 The Self-reference Walks

Each trial is run for a maximum of 1,000 time steps.

Field size is set to 1,000 × 1,000. Periodic boundary

is assumed. I set the simulation stage for each trial and

set the agent at the origin; (x (0), y (0)) = (0, 0). In this

algorithm, the agent moves in two-dimensional

square lattices. On each time step, the agent selects

one direction among four discrete directions and

updates its position like follows;

(x (t+1), y (t+1)) = (x (t)+1, y (t)) with Prob (+x),

(x (t+1), y (t+1)) = (x (t)-1, y (t)) with Prob (-x),

(x (t+1), y (t+1)) = (x (t), y (t)+1) with Prob (+y),

(x (t+1), y (t+1)) = (x (t), y (t)-1) with Prob (-y),

Prob (+x) + Prob (-x) + Prob (+y) + Prob (-y) = 1.00.

At the beginning of each trial, the agent equally

selects each direction.

A directional move that consists of a series of the

different move, such as +y, -x or +x, +y and so on, is

counted as

if (x (t+1) - x (t)) = (x (t) - x (t-1)) = ±1

or (y (t+1) - y (t)) = (y (t) - y (t-1)) = ±1,

Exp (t+1) = Exp (t),

otherwise,

Exp (t+1) = Exp (t) + 1

For example, “Exp (t+1)” can be “Exp (t) + 1” when

the agent moves in +x direction at time t-1 and is

going to move in -x direction at time t.

If Exp (t) exceeds a threshold number, th, the four

directional probabilities are changed as follows and

the agent obeys these new rules from the next time

step;

if x (t+1) - x (t) = +1,

Prob (+x) = φ,

Prob (-x) = Prob (+y) = Prob (-y) = (1-φ)/3

if x (t+1) - x (t) = -1,

Prob (-x) = φ,

Prob (+x) = Prob (+y) = Prob (-y) = (1-φ)/3

if y (t+1) - y (t) = +1,

Prob (+y) = φ,

Prob (+x) = Prob (-x) = Prob (-y) = (1-φ)/3

if y (t+1) - y (t) = -1,

Prob (-

y) = φ,

Prob (+x) = Prob (-x) = Prob (+y) = (1-φ)/3

Here, φ indicates a random number that satisfies

ratio is the element of a set [0.25, 1.00]. Here, the

maximum random number was set to 1.00 in order to

produce a straight movement toward a certain

direction. Note that Exp (t) is reset to 0 at that time.

The agent obeys a biased directional rule in order

to avoid moving in a certain direction. By doing so,

the agent can avoid visited positions to some extent

and effectively explore. At the same time however,

the agent modifies its rule when the agent experiences

several series of the different directional move such

like +x, -x or -y, -x and so on.

Only at first, i.e., at time t=0, where the agent

calculates (x (1), y (1)) by obeying a Brownian-like

walk, the four directional probabilities are modified

as follows independently of Exp (t):

if x (1) - x (0) = +1,

Prob (+x) = φ,

Prob (-x) = Prob (+y) = Prob (-y) = (1-φ)/3

if x (1) - x (0) = -1,

Prob (-x) = φ,

Prob (+x) = Prob (+y) = Prob (-y) = (1-φ)/3

if y (1) - y (0) = +1,

Prob (+y) = φ,

Prob (+x) = Prob (-x) = Prob (-y) = (1-φ)/3

if y (1) - y (0) = -1,

Prob (-y) =

φ,

Prob (+x) = Prob (-x) = Prob (+y) = (1-φ)/3

In our simulations, th is set to 5 as a default value.

3 RESULTS

Here, food resources are randomly distributed on the

field and the resource density is set to 0.001. The

agent can consume food items if those items are

located within 5.0 radii. Food depletion, which means

that food items disappear once the agent consumes

those items, does not occur. Therefore, the agent can

consume each food item whenever it detects that item

within 5.00 radii. Later however, I will check the

effect of the resource density and the food depletion.

Food depletion is an important factor for the search

ability and the movement strategy of random walkers.

This is because the random walker with sub-diffusive

movements does not have a trouble with consuming

food resources if food depletion does not occur since

it can find and consume resources again and again.

On the contrary, the random walker may need to

change its strategy if food depletion occurs due to the

fact that no items can be found by the walker once it

consumes those items. Therefore, the effects of food

depletion will reveal the performance of my model.

Paradigms-Methods-Approaches 2021 - Workshop on Novel Computational Paradigms, Methods and Approaches in Bioinformatics

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Figure 1: An example of the agent’s trajectories where the

resource depletion does not occur. The parameter th = 5.

First, I focus on movement properties of the

model. Figure 1 shows an example of an agent

trajectory obtained from 1 trial. According to this

figure, the agent seems to sometimes produce straight

movements. In fact, the mean squared displacement

(msd) between the start point and end points reveals

that the agent demonstrates a super diffusive

movement (Figure 2A). Here, each end point was

obtained every 100 time steps and each msd obtained

from 100 trials was plotted. In the random walk

analysis, the relation between the mean squared

displacement <R

> and the step is often calculated

since this property presents the diffusive property of

the walker. It is well known that his property follows

the following relation (Viswanathan et al. 1999):

> ~ t

Parameter H is determined depending on the

model (H>1/2 for a Lévy walk (super-diffusion),

H=1/2 for a Brownian walk (normal diffusion) and

H<1/2 for sub-diffusive movements). The fit for

parameter H according to Figure 2 was H ~ 0.91,

indicating that super-diffusion was achieved (R-

squared=0.99).

For the evaluation of the parameter effects, I

replaced the parameter th from 5 with 50. Figure 2B

represents the diffusive property in case of th = 50.

Results suggest that super-diffusive movements can

be maintained even after the parameter replacement

(Figure 2B: threshold = 50, H ~ 0.91, R-squared =

0.99).

Figure 2: Log-scale plot of mean squared displacement

(msd) and t

obtained from 100 trials for each threshold. A.

th = 5. B. th = 50.

Figure 3: % of resource consumption in respect with

resource density; 0.001 and 0.01 for each threshold value

and for the Brownian walker under the condition where the

resource depletion does not occur. *** indicates p < 1.0E-

03, ns indicates non-significant.

In fact, the resource search ability of this model

seems to be not dependent on the parameter threshold.

According to Figure 3, which showed the fraction of

the resource consumption, I found that there was no

significant difference between th =5 and 50 (Figure 3:

resource density = 0.001, th =5 (2.06) vs. th = 50

(1.95), Mann-Whitney U test, P = 0.58, NS).

Furthermore, this tendency is not changed even after

the resource density is replaced with 0.01(Figure 3:

resource density = 0.01, th =5 (1.94) vs. th = 50

(1.97), Welch Two Sample t-test, t = -0.69, df =

195.88, P = 0.49, NS) Here, I counted the number of

resources consumed by the agent on each trial and

converted it to the percentage against the total number

distributed on the field. Importantly, I found that the

A Random Walker Can Optimize the Exploration without the Large Capacity Memory

211

proposed model outperformed the Brownian walk

model when resource density was low (Figure 3:

resource density = 0.001, th =5 (2.06) vs. Brownian

(1.78), Mann-Whitney U test, P < 1.0E-04, th =50

(1.95) vs. Brownian (1.78), Mann-Whitney U test, P

< 1.0E-04, resource density = 0.01, th =5 (1.94) vs.

Brownian (1.90), Welch Two Sample t-test, t = 0.57,

df = 134.35, P = 0.57, NS, th =50 (1.97) vs. Brownian

(1.90), Welch Two Sample t-test, t = 0.96, df =

141.83, P = 0.34, NS). These results suggest that the

proposed model can search effectively in the low-

density environment and the performance is not

affected by the parameter threshold. In other words,

the agent is not necessarily to remember a large

number of “Exp”.

To investigate the influence of the food depletion

to the search ability, I also conducted the same

analysis under the condition where resource items

were depleted once the agent consumed items. Figure

4 indicates that the proposed model again

outperforms the Brownian walker model.

Interestingly, this tendency is found not only in the

low density environment but also in the (relative)

high density environment (Figure 4: resource density

= 0.001, th =5 (0.17) vs. th = 50 (0.16), Mann-

Whitney U test, P = 0.31, NS, resource density = 0.01,

th=5 (0.17) vs. th = 50 (0.17), Mann-Whitney U test,

P = 0.39, NS, resource density = 0.001, th =5 (0.17)

vs. Brownian (0.03), Mann-Whitney U test, P < 1.0E-

15, th =50 (0.16) vs. Brownian (0.03), Mann-Whitney

U test, P < 1.0E-15, resource density = 0.01, th =5

(0.17) vs. Brownian (0.03), Mann-Whitney U test, P

< 1.0E-15, th =50 (0.17) vs. Brownian (0.03), Mann-

Whitney U test, P < 1.0E-15). This is perhaps because

a Brownian walker presents normal-diffusive

movements, which may result in the inefficient search

of the food resources under the condition where

resource items are depleted.

Figure 4: % of resource consumption in respect with

resource density; 0.001 and 0.01 for each threshold value

and for the Brownian walker under the condition where the

resource depletion occurs. *** indicates p < 1.0E-03, ns

indicates non-significant.

4 CONCLUSIONS

In the developed random walker algorithm, the agent

modulates its directional rule and avoids a certain

direction. However, it modifies its directional rule

when the inconsistency of the recent series of the

directional move beyond a threshold value. As a

results, I found that the agent presented and

maintained super-diffusive movements in some

threshold values. Thanks to this, that model

outperforms the Brownian walk model when the

resource density is low or when resources are

depleted once the agent consumes those items.

Moreover, the performance of resource search ability

was not influenced by the threshold replacement.

These results suggest that the proposed model does

not require the large number of “Exp” to achieve an

effective search.

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