MULTI-MODAL ANALYSIS OF COMPLEX NETWORK

Point Stimulus Response Depending on its Location in the Network

Takeshi Ozeki, Teruhiko Kudo

Faculty of Science and Technology, Sophia University, 7-1 Kioicho, Chiyodaku, Tokyo,102-8554, Japan

Yosuke Takeda

Facultyof Economics, Sophia University, 7-1 Kioicho, Chiyodaku, Tokyo, 102-8554, Japan

Keywords: Adjacency matrix, Multi-modal Analysis, Markov transition, Euclidian Norm.

Abstract: We report a new method of diagnosis of a node in a network by “Point Stimulus Response”. The “Point

Stimulus Response” corresponds to the impulse response of the network, that is, the state temporal variation

in the Markov transition with the delta-function of initial state. We can evaluate the reaction of the system

against a point stimulus such as a point failure. In this report, for the first, we summarize our mathematical

platform for analysing complex network system using the adjacency matrix as the transition matrix in

Markov transition approximation. On this basis, we formulate the point stimulus response. The location

dependence of the point stimulus response is demonstrated in Tokyo Metropolitan Railway Network System.

For a concrete example, the total amount of suffered passengers and time response of recovery from a point

failure will be discussed depending on the location of point failure in the network system. It can be said that

a way to find a point for effective stimulus response is one of key approaches for knowledge discovery.

However, the real indication or meaning of the point stimulus is in the stage of speculation.

1 INTRODUCTION

Knowledge Discovery is an interdisciplinary area

focusing upon methodologies for identifying valid,

novel, potentially useful and meaningful patterns

from data, often based on underlying large data sets.

Our mathematical platform is aiming extraction and

analysis of knowledge from the mutual interaction

patterns, obtained by such network log data (Onnela,

2008). The mutual interaction pattern is described as

the adjacency matrix in the Markov process

approximation (Ozeki, 2010).

Brin and Page reported, in their first paper on

“Google”(Page, 1990), that it was a great surprise

the PageRank is obtained purely mechanically from

the pattern of mutual page links. That is the surprise

of discovery that the pattern is entangled with the

real world. The “Google” approximates a Web

surfer as a random walker in the Markov process and

combines the dominant eigenvector with the list of

coincidence as the PageRank .

The “Google”, however, uses only the dominant

eigenmode because the eigenvectors of higher-order

modes are not positive valued so that the probability

finding the Web surfer at a page cannot be defined

for the higher-order modes (Langville, 2006).

Here, we have proposed a mathematical platform

for analysing the network pattern in multi-modal

scheme (Ozeki, 2009). Each mode corresponds to a

substructure of the pattern. Various pattern

dependant behaviours can be analysed for

knowledge discovery.

In this paper, we would like to report a new

method for the diagnosis of various objectives, such

as security and activation, of a network system by a

“Point Stimulus Response”.

The “Point Stimulus Response” corresponds to

the impulse response of the network system, that is,

the state variation in the Markov transition with the

delta-function of initial state. We can evaluate the

system activity against the point stimulus.

It can be said that a way to find a point of

effective stimulus response or “tsubo” is one of the

key approaches of “Knowledge Discovery”.

In Japan, “Shiatsu” is a popular therapy by

pressing “shiatsu point” to enhance the body’s

226

Ozeki T., Kudo T. and Takeda Y..

MULTI-MODAL ANALYSIS OF COMPLEX NETWORK - Point Stimulus Response Depending on its Location in the Network.

DOI: 10.5220/0003090302260231

In Proceedings of the International Conference on Knowledge Discovery and Information Retrieval (KDIR-2010), pages 226-231

ISBN: 978-989-8425-28-7

 2010 SCITEPRESS (Science and Technology Publications, Lda.)

natural healing ability and prevent the progression of

disease. Shiatsu points are called “tsubo”, in

Japanese and their locations and effects are based on

understanding of modern anatomy and physiology.

The concept of “tsubo” is our stimulus point of the

network system.

The point stimulus has been used as a way of

reactivation of an old city (Horiike, 2002).

This report is believed as the first theoretical

approval of locating stimulus points of a network

system.

2 MATHEMATICAL PLATFORM

In this session, we would like to summarize our

mathematical platform for network system analysis,

briefly.

2.1 Adjacency Matrix

The adjacency matrix

of a network can be used

as a Markov-transition matrix to simulate the

evolution of states:

qAq )

()

(

⋅=

where

(

the probability amplitude vector of the state at the n

transition step. The probability amplitude is

normalized with respect to the Euclidean norm after

each transition step by application of

|( ) | 1

−

∑

,where

q )(

is the

i component of

( . The probability

p )(

of finding a random

walker at the node “

i “ is given by

|)(|)(

nini

qp =

The eigen-equation is

)()( m

φλφ

⋅=⋅

where

the eigenvalue of mode “

” and

)(m

is its

eigenvector; the eigenvectors form a complete

orthogonal basis under the assumption of a

symmetric adjacency matrix. This fact is the reason

of using the adjacency matrix as the transition

matrix in the Markov transition.

It should be noted that using a Markov process

normalized by the Euclidean norm makes it possible

to describe the network states in a multi-modal way.

Previously, in such systems as the Google search

engine (Langville, 2006) using a stochastic transition

matrix normalized by the 1-norm, higher order

modes cannot define the probability of finding a

random walker because the components of

eigenvectors are not always positive.

2.2 Non-linear Markov Transition

To examine multi-modal dynamics of the network,

we define a Markov transition with weak non-

linearity; a non-linear Markov process can be

formulated as follows: the transition coefficient from

node ”

j “ to node ”i” is affected by the probability

amplitude

q )(

of node ”k” linked to node ”i”. Such

a non-linear Markov transition is given by

1, ,,

() () ()()

in ij jn ji ki jn kn

jjk

qAq AAqq

=⋅+⋅⋅ ⋅

∑

(1)

, where

is a measure of the strength of the non-

linearity. Since the Markov property states that the

probability distribution for the system at the next

step depends only on the current state of the system,

the non-linear state transition given by equation (1)

indeed defines a Markov process. It is possible to

define higher-order non-linear interactions in a

similar way (Ozeki,2009). Since we have a complete

basis of orthogonal eigenvectors, the mode

amplitudes

()

() ()

mn in

−

=⋅

∑

can describe the

mode evolution of the system (Haken,1987).

2.3 Node, Mode and Network

Entropies

The entropy may be efficient measure of network

optimazation. We define three kinds of entropies

based on the Shanonn entropy (Shanonn,1948) using

the probability finding a random walker at each node.

The node entropy

is defined by

()ln(())

iii

ϕϕ

=−

∑

that is the sum of the

Shanonn entropy -

)ln(()(

φφ

of node

over all

of mode m.The mode entropy

is defined by

∑

−=

ME ))ln(()(

φφ

that is the sum of Shanonn

entropy of the mode

over all of node i. The

MULTI-MODAL ANALYSIS OF COMPLEX NETWORK - Point Stimulus Response Depending on its Location in the

Network

227

network entropy

is defined by

GE NE NE==

∑∑

3 POINT STIMULUS

3.1 Formulation of “Point Stimulus

Response”

The point stimulus response is the impulse response

in the electronic circuit system: that is, the temporal

response stimulated by a delta-function provides the

network system characteristics. The point stimulus

response is defined by the temporal response in the

non-linear Markov transition for the positive point

stimulus;

),( piPPS

, where node

""p

is a

location of stimulus. We found that the inverse or

negative delta function is more effective in some

network with particular symmetric nature. In a kind

of network having skew degeneracy (Ozeki,2010), a

negative point stimulus,

),( piNPS

−=

is effective

to stimulated the mode competition among the skew

degenerate modes. In the following, for the first, the

positive point stimulus response is discussed by a

concrete network example and in later the negative

point stimulus response is discussed. The

stimulation of the mode competition between the

modes close to quasi-skew degeneracy is interesting

related to the potential activity or development of

nodes.

3.2 Diagnosis of Tokyo Railway System

Fig.1 denotes the complexity of a central part of

Tokyo Railway System including subways. The

adjacency matrix is assumed to be symmetric and

the total number of stations (nodes) is truncated to

736 (Rail Map of Tokyo Area, 2004). A distorted

hexagonal in Fig.1 is “Yamanote Circular Line”

which includes several well-known stations such as

Tokyo, Akihabara, Ikebukuro, Shinjuku, Shibuya

and etc. Before the detail analysis of point stimulus

response, it seems better to summarize the mode

structure of the network. The list of eigenmode

naming and eigenvalue is shown the top of Fig.2.

The probability amplitude distributions of the

important four modes are shown in Fig.2. The

dominant mode with the largest positive eigenvalue

is named mode #2 of which probability amplitude is

positive. The mode # 0 has the largest negative

eigenvalue and its mode amplitude is similar with

that of mode #3, that is the mode with the second

largest positive eigenvalue. These mode relations are

important to understand the mode competition.

It is

our surprise that the probability distribution of the

Figure1: Tokyo Metropolitan Railway Network System.

Figure 2: Eigenmode naming, eigenvalue and eigenvectors.

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228

Figure 4: Location Dependence of Positive Point Stimulus Response.

Figure 3: Probability distribution of mode #0.

mode having the largest negative eigenvalue shown

in Fig.3 extract the world largest three stations from

viewpoint of number of passengers without any

passenger statistics. In Google-like matrix, the

dominant mode provides only the degree vectors.

3.3 Location-Dependent Positive

Point-Stimulus

We set the point stimulus on from Shinsjuku to

Tokyo, along the Yamanote-line in CCW. The point

stimulus responses of these stations calculated by the

non-linear Markov process are shown in panels of

Fig.4 with station name and code number. The

mode amplitude

nnnn

aaaa 3,2,1,0

correspond to the

mode #0,#1,#2 and #3, being shown in Fig.2. The

point stimulus responses of from Shinjuku #0 to

Ebisu #4 dominantly consist of damped oscillation

of mode #0 (red) and a quick build-up of mode #2

(green). The damped oscillation amplitudes decrease

toward Ebisu #4. On the other hand, in the point

stimulus responses of from Ohsaki#7 to Tokyo #13,

damped oscillation of the mode #1 denoted by blue,

becomes dominant, and the damped oscillation

amplitudes reach at the peak around Shinagawa #8

and Shinbashi #11.

3.4 Location-Dependent Negative

Point-Stimulus

Fig.5 denotes the negative point stimulus responses

For typical three stations: Shinjuku, Shibuya and

Tokyo. The bottom panels of Fig.5 show the

probability amplitude distribution

finding a

random walker, calculated by the superposition of



MULTI-MODAL ANALYSIS OF COMPLEX NETWORK - Point Stimulus Response Depending on its Location in the

Network

229

Figure 5: Location Dependant Point Stimulus Response.

modes using Eq.2. In the case of Shinjuku and

Shibuya, since the sustainable oscillation of mode #0

is observed, the probability distribution of finding

random walker also oscillates between the in-phase

superposition and the out-of-phase superposition,

just as shown in the bottom panels. The red line

denotes the in-phase superposition and the blue line

denotes the out-of-phase superposition. (Here, we

should note that the sign of out-of-phase

superposition is inverted for clear understanding.)

The distance between nodes included in red and blue

lines is only one link distance: For example,

Shinjuku (code #0) and Ikebukuro(code #25) in one-

link distance due to the Saikyo-line, so that the

random walker can transit between red/blue station-

groups within one step. In the case of Tokyo, the

superposition of modes of Eq.2 shows no temporal

variation after damped oscillation is vanished.

∑

⋅=

inmin

asp

)(

(2)

3.5 Categorization of Point Stimulus

Response and Response Time

It is convenient to categorize the point stimulus

response into the following two: The point stimulus

response with the sustainable oscillation is named

“the infinite response point”. The point stimulus

response with the finite response is named “ the

finite response point”. The categorization of stations

within the Yamanote circular line is shown in the

bottom panels of Shinjuku and Shibuya, in Fig.5,

that is, the stations with larger probability amplitude,

such as Shinjuku, Yoyogi, Harajuku, Shibuya,

Ikebukuro, Shinohkubo and Yotsuya, are the

infinite response nodes. These are the stations within

one-link distance of Shinjuku and can be said as

satellite stations: The others are the finite response

nodes.

It should be noted that the build-up time of the

dominant mode #2 takes longer steps to reach the

stationary state due to the mode competition with

mode #0, in the case that the positive point stimulus

is applied to from Shinjuku to Ebisu, as shown

inFig.4. The response time of nodes in the network

is mainly determined by this mode competition. For

further study, the recovery time from the point

failure of the Tokyo Railway Network will be

analysed from these viewpoints.

4 POINT FAILURE OF

NETWORK SYSTEM

The point failure of the station in the Tokyo

Metropolitan Railway Network System is one of

concrete image of the point stimulus. We can

estimate the total suffered passengers as shown in

Fig.6 using the following;

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230

∑

⋅⋅=

immi

MAstimulusS

(3)

where

stimulus

is the projection of the positive

point stimulus

PPS

on the eigenvector

. The total

number of suffered passengers denoted by red line is

rather independent of the location of point failure

compared with larger variation in the number of

passengers.

We feel that the number of suffered passenger

calculated seems rather larger than the reported

figures. Tokyo metropolitan railway system has a lot

of redundancy in it structure for reliable operation,

but we define the link topologically, that is, multiple

duplication of trucks between adjacent stations is

neglected. It is necessary to improve the accuracy of

the adjacency matrix expression.

Figure 6: Total number of Suffered Passengers.

5 CONCLUSIONS AND FUTURE

WORKS

We discuss on the multi-modal analysis method for

discovery of knowledge from pattern information.

We proposed a diagnosis tool of the point stimulus

response and demonstrated it in Tokyo Metropolitan

Railway Network system. The point stimulus is

effective to find interesting nodes to characterise the

system, such as the excitation sustainable oscillation.

It is not verified by physical data yet, but seems to

be a way of an approval of “Tsubo” in “shiatsu

therapy”.

As for future work, we would like to discuss on

the knowledge discovery based on pattern structure

embedded in data, automatically collected in the

network systems. It is believed that the adjacency

matrix obtained automatically, in such Facebook,

gives us interesting chances to analyse the social

substructures and their stability, using these new

knowledge discovery technology.

ACKNOWLEDGEMENTS

Author T.O. wishes to thank Dr.H.Kasai and

Dr.H.Horiike, architect, for their continuous

stimulus and advices.

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