an exploration of power-law in use-relation of java software systems

Software Engineering Laboratory, Department of Computer Science, Graduate School of Information Science and Technology, Osaka University

An Exploration ofPower-law in Use-relation ofJava Software Systems

Makoto Ichii, Makoto Matsushita, Katsuro Inoue

Osaka University

2008/3/26 ASWEC 20081


Software Component Graph A software system is composed of software components.

Software component (component): building unit of a software system

Complex use-relation is formed between components Software component graph (component graph) represents

use-relation between components node: component / edge: use-relation

Various researches utilize component graphs to analyze software systems

It is important to know the nature of component graphs

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Power-law distribution

A graph is characterized by the degree distribution The graphs whose degree distribution follows the power-law

distribution attracts attention in various research domains Link structure of WWW pages Hosts on the Internet

Such graphs tend to have interesting characteristics Self similarity Fault tolerance

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Explore the component graphs to seek whether the degree distributions follow the power law

p(x) = Cx-α


Questions [1-2/4]Q. 1 Do the in- and out-degree distributions of a component

graph of a software system follow the power law?

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Q. 2 Do the in- and out-degree distributions of a component graph of multiple software systems follow the power law?

?

?


Questions [3-4/4]Q. 3 Do the in- and out-degree distributions of subgraph of a

component graph follow the power law?

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Q. 4 What aspects of components affects the in- and out-degree distribution of component graphs?

?


Definitions [1/2]Component: Java class (including interface)

Use-relation: Any of the following six relation types acquired by static analysis of the component source files.

A class or an interface extends another class or interface respectively.

A class implements an interface. A class or an interface declares a variable of a class or an

interface. A class instantiates a class object. A class calls a method of a class or an interface. A class or an interface references to a field variable of a class

or an interface.

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Definitions [2/2]Component graph: Directed simple graph

node: component edge: use-relation between components

In-(Out-)degree: The number of incoming (outgoing) edges to a node

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class B { … A.exec(); …}

class A { void exec() { … }}

A

B

class C { … A a = new A(); …} C

in-degree: 2

out-degree: 0

in-degree: 0

out-degree: 1

in-degree: 0

out-degree: 1


Observing the power-law Plot cumulative frequency on log-

log axis The data forms a straight line

if the distribution is the power law

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gradient : -αgradient : -(α-1)

p(x) = Cx-α

in-(or out-)degreeM. E. J. Newman, "Power laws, Pareto distributions and Zipf's law",

Contemporary Physics 46, 323-351 (2005)


Values shown in the experiments α: exponent

Derive from the gradient of the regression line

R*2: the determination coefficient adjusted for the degree of freedom

Fitness of a regression model for data

[0..1] Large value means good

fitness

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gradient : -(α-1)

p(x) = Cx-α

in-(or out-)degree


Experiment 1

Setup component sets Each set contains a single software system

Analyze component sets to create component graphs. Plot cumulative frequency of the degrees on log-log axis.

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Q. 1 Do the in- and out-degree distributions of a component graph of a software system follow the power law?

Description # of components

JDK Java 2 SE Software Development Kit 1.4 11,556

ECLIPSE Eclipse 3.0.1 13,941


Result of experiment 1 / JDK

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α R*2

in-degree 2.1 ±8.6×10-3 0.99

out-degree 3.1 ±8.2×10-2 0.88

► The in-degree follows the power law

► The out-degree does not follow the power law

# of Nodes 11,556

# of Edges 107,198

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Result of experiment 1 / ECLIPSE

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► The similar characteristics with JDK

The in-degree follows the power law

The out-degree does not follow the power lawα R*2

in-degree 2.2 ±1.6×10-2 0.96

out-degree 3.0 ±7.7×10-2 0.86

# of Nodes 13,941

# of Edges 140,678

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Experiment 2

Setup component sets Each set contains multiple software systems Use-relation across the systems exists


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Q. 2 Do the in- and out-degree distributions of a component graph for multiple software systems follow the power law?

Description # of components

ASF Various projects checked out from the repository of Apache Software Foundation

59,486

SPARS_DB The components stored in the database of demo.spars.info (includes ASF, JDK)

180,637


Result of experiment 2 / ASF

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► The similar characteristics with Exp. 1


The out-degree does not follow the power lawα R*2

in-degree 2.4 ±1.1×10-2 0.98

out-degree 3.4±6.4×10-2 0.94

# of Nodes 59,486

# of Edges 303,7551

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Result of experiment 2 / SPARS_DB

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α R*2

in-degree 2.0 ±1.5×10-3 1.00

out-degree 3.7 ±7.0×10-2 0.90

# of Nodes 180,637

# of Edges 1,808,982

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► The similar characteristics with Exp. 1


The out degree does not follow the power-law completely

► In-degree distribution fits to the power-law straight line almost ideally.


Experiment 3

Construct subsets of SPARS_DB Keyword: The components that contain a specified keyword in the

source code The keywords are randomly selected so that the number of resulting

components is about 1,000/10,000 Random: 1,000/10,000 random components


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Q. 3 Do the in- and out-degree distributions of subgraph of a component graph for software systems follow the power law?

Description # of componentsKWD1K The components that contain “labels” 1,002KWD10K The components that contain “getstring” 8,938RND1K Randomly-selected 1,000 components 1,000

RND10K Randomly-selected 10,000 components 10,000


Result of experiment 3 / KWD1K

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out-degree

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α R*2

in-degree 2.2 ±3.3×10-2 0.98

out-degree 3.7 ±2.0×10-1 0.93

# of Nodes 1,002

# of Edges 1,564

► The similar characteristics with SPARS_DB


The out-degree does not follow the power law


Result of experiment 3 / KWD10K

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out-degree

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α R*2

in-degree 2.1 ±9.3×10-3 0.99

out-degree 3.4 ±2.7×10-1 0.93

# of Nodes 8,938

# of Edges 24,317

► The similar characteristics with SPARS_DB


The out-degree does not follow the power law


Result of experiment 3 / RND1K

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α R*2

in-degree 2.3 ±1.8×10-1 0.93

out-degree N/A N/A

# of Nodes 1,000

# of Edges 52

► The original characteristics is almost lost


Result of experiment 3 / RND10K

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out-degree

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α R*2

in-degree 1.9 ±2.1×10-2 0.98

out-degree 4.3 ±3.3×10-1 0.91

# of Nodes 10,000

# of Edges 6,184

► The similar characteristics with SPARS_DB, however

# of edges is small


Experiment 4

List top-ten components in the in- and out-degree Calculate correlation between degrees and metric values.

Spearman's rank correlation coefficient Target: SPARS_DB

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Q. 4 What aspects of components affects the in- and out-degree distribution of component graphs?

Metric Description

LOC Non-comment source lines of code

WMC1 A variation of weighted methods per class (WMC)

Weight of a method: constant value (1)WMC2 A variation of WMC

Weight of a method: Cyclomatic complexityLCOM A variation of lack of cohesion of methods: LCOM5


Result of experiment 4 / In-degree Top-ten components

The components that have fundamental/general role

Correlation with metrics In-degree have low

correlation with the metrics

The in-degree relates to the role

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Name LOC

In-degree

Out-degree

1 java.lang.String 675 116,239 21

2 java.lang.Object 35 98,261 4

3 java.lang.Class 605 29,682 41

4 java.lang.Exception 15 21,046 2

5 java.lang.Throwable 136 19,519 12

6 java.lang.System 170 19,175 27

7 java.util.Iterator 5 15,522 1

8 java.util.List 27 14,462 4

9 java.util.ArrayList 200 13,656 19

10 java.lang.Integer 285 12,736 9

Out-degree

LOC WMC1 WMC1 LCOM

In-degree

0.00 0.07 0.24 0.08 0.12


Result of experiment 4 / Out-degree Top-ten components

Simply large/complex classes Correlation with metrics

High correlation with LOC and WMC

The out-degree relates to the size/complexity of a component

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In-degree

LOC WMC1 WMC1 LCOM

Out-degree

0.00 0.82 0.64 0.75 0.39

Name LOC

In-degree

Out-degree

1 org.apache...FunctionEval 364 1 354

2 org.jgraph.GPGraphpad 2,196 130 255

3 com.jgraph.GPGraphpad 2,200 131 253

4 org.jgraph.GPGraphpad 542 209 252

5

org.eclipse...

ASTConverter 4,520 3 223

6 org.eclipse...JavaEditor 1,368 115 220

7

net.sourceforge...

GanttProject 3,055 98 216

8

it.businesslogic...

MainFrame 7,177 46 204

9

org...

InstConstraintVisitor 1,626 3 197

10

org...

ASTInstructionCompiler 2,449 1 189


Answers: summary of experiments [1/4]Q. 1 Do the in- and out-degree distributions of a component

graph of a software system follow the power law?The in-degree follows the power lawThe out-degree does not follow the power law

Mixture of the power-law distribution and the lognormal distribution

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Answers: summary of experiments [2/4]Q. 2 Do the in- and out-degree distributions of a component

graph for multiple software systems follow the power law?The in-degree follows the power lawThe out-degree does not follow the power law

The similar results with that of single software systems

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Answers: summary of experiments [3/4]Q. 3 Do the in- and out-degree distributions of subgraph of a

component graph for software systems follow the power law?Depends on how the subgraph is created.

Keyword-based subgraph has similar characteristics with the superset Related components likely share words

Random-selection-based subgraph with small number of nodes has different characteristics

• Few edges exist.

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out-degree

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Answers: summary of experiments [4/4]Q. 4 What aspects of components affects the in- and out-degree

distribution of component graphs?In-degree relates to the roles of components

Most of the components are used at the specific part Components with fundamental/general role are used from

everywhere The more the size of component set grows, the more the

value of in-degree becomes large.Out-degree relates to size/complexity of components

Many components have reasonable size/complexity Some components may have relatively large size/complexity

Extremely large components are unreasonable

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Summary Component graphs are investigated to seek whether the in-

and out-degree distribution follows the power-law As the results, following characteristics are revealed.

The in-degree distribution follows the power-law The in-degree of a component relates to the role of the component

The out-degree distribution does not follows the power-law The out-degree of a component relates to the size/complexity of

the component Some sort of subgraph of a component graph have the same

characteristics of degree distribution with the graph.

Future works Explore the other types of component graph

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+

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Discussion Generative models of a power-law graph

If a node is added to a graph, the nodes with large degree tend to get the edge to the new node. “rich get richer”

Meanings for component graphs If a new component is added to (developed for) a software

system, the new component uses the component that is already used by many components

The members of frequently-used components hardly change even if the software development proceeds If the member changes, it means that the fundamental structure

(design, architecture) of the software is changed

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an exploration of power-law in use-relation of java software systems

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