Accurate and Fast Computation of Approximate Graph Edit Distance

based on Graph Relabeling

Sousuke Takami and Akihiro Inokuchi

School of Science and Technology, Kwansei Gakuin University, 2-1 Gakuen, Sanda, Hyogo, Japan

Keywords:

Graph Edit Distance, Graph Relabeling, Graph Classiﬁcation.

Abstract:

The graph edit distance, a well-known metric for determining the similarity between two graphs, is commonly

used for analyzing large sets of structured data, such as those used in chemoinformatics, document analysis,

and malware detection. As computing the exact graph edit distance is computationally expensive, and may

be intractable for large-scale datasets, various approximation techniques have been developed. In this paper,

we present a method based on graph relabeling that is both faster and more accurate than the conventional

approach. We use unfolded subtrees to denote the potential relabeling of local structures around a given vertex.

These subtree representations are concatenated as a vector, and the distance between different vectors is used

to characterize the distance between the corresponding graphs. This avoids the need for multiple calculations

of the exact graph edit distance between local structures. Simulation experiments on two real-world chemical

datasets are reported. Compared with the conventional technique, the proposed method gives a more accurate

approximation of the graph edit distance and is signiﬁcantly faster on both datasets. This suggests the proposed

method could be applicable in the analysis of larger and more complex graph-like datasets.

1 INTRODUCTION

Graphs are one of the most natural means of repre-

senting structured data. For instance, a chemical com-

pound can be represented as a graph in which each

vertex corresponds to an atom, each edge corresponds

to a bond between two atoms, and the label of each

vertex corresponds to the atom type. With recent im-

provements in system throughput, the need to analyze

large numbers of graphs has arisen, and the topic of

graph mining has received considerable interest be-

cause the knowledge present in structured data can

be applied to various real-world datasets. For exam-

ple, in cheminformatics, certain properties of chem-

ical compounds (e.g., mutagenicity or toxicity) can

be identiﬁed by analyzing their structural informa-

tion, and in bioinformatics, the prediction of protein–

protein interactions is beneﬁcial for drug discovery.

When analyzing datasets of graphs, one of the

most critical measures is the dissimilarity (or simi-

larity) among the graphs. A representative measure

of the dissimilarity is the graph edit distance. The

graph edit distance d(g

) between graphs g

and

is deﬁned as the minimum length of the sequence

of edit operations needed to transform g

into g

where one edit operation includes the insertion, dele-

tion, or the substitution of a vertex and edge in the

graphs. The method based on the A

algorithm is a

well-known technique for computing the exact graph

edit distance (Hart et al., 1968). However, this method

cannot be applied to large graphs, because the prob-

lem of obtaining the exact graph edit distance between

two graphs is known to be NP-complete.

To overcome this difﬁculty, a method that uses

the minimum matching problem of a complete bipar-

tite graph (V

,E, w) has been proposed to com-

pute the approximate graph edit distance between the

graphs (Riesen, 2015). In a bipartite graph, V

and

correspond to the vertices of g

and g

, respec-

tively, and w is a function assigning values of the dis-

similarities between local structures around the ver-

tices to edges in the bipartite. Given the complete

bipartite graph, the matching problem returns a map-

ping from the vertices in g

to vertices in g

that is

solvable in O(b

), where b = max{|V

|,|V

|}. Com-

puting the approximate graph edit distance using this

method is computationally simpler than computing

the exact graph edit distance. In this method, the

local structures are important in obtaining as accu-

rate an approximate distance as possible. The lo-

cal structure around each vertex v is represented by

star structures (Zeng et al., 2009), random walks

Takami, S. and Inokuchi, A.

Accurate and Fast Computation of Approximate Graph Edit Distance based on Graph Relabeling.

DOI: 10.5220/0006540000170026

In Proceedings of the 7th International Conference on Pattern Recognition Applications and Methods (ICPRAM 2018), pages 17-26

ISBN: 978-989-758-276-9

from v (Ga

ere et al., 2014), and subgraphs induced

by vertices reachable within h steps from v, called

limited-size subgraphs (Carletti et al., 2015). If com-

plex and/or large structures such as limited-size sub-

graphs are used as local structures, the computation

time required to assign values to edges in the bipar-

tite graph can be signiﬁcant. In contrast, if simple

structures such as stars or walks are used, the ac-

curacy of the approximated graph edit distance de-

creases. A recently proposed method (Carletti et al.,

2015) measures the dissimilarity between local struc-

tures around v

in g

and v

in g

with small limited-

size subgraphs. The dissimilarity in this approach

is the exact graph edit distance. Thus, although this

method gives a relatively accurate approximate graph

edit distance between g

and g

, it requires consid-

erable computation time to measure multiple exact

graph edit distances for the small subgraphs.

In this paper, we tackle the problem of measuring

the approximate graph edit distance between graphs

and propose an accurate and fast method. In the pro-

posed method, each of the complex local structures is

represented by a vector, which enables this method to

be applied to complex and large local structures and

ensures fast computation. The proposed method re-

quires O(b

h(|Σ| + b) time to compute the approxi-

mate graph edit distance, where Σ is the set of vertex

labels in the graphs. In addition, the method requires

O(|Σ|b +b

) memory.

The remainder of this paper is organized as fol-

lows. Section 2 formalizes the problem of measur-

ing the graph edit distance and explains the existing

method based on the minimum matching problem of

a complete bipartite graph. In Section 3, we propose

an accurate and fast method for measuring the graph

edit distance by representing local structures as vec-

tors. In Section 4, we verify the computational efﬁ-

ciency of the proposed method and compare it with

the conventional method in terms of accuracy using

real-world datasets. Finally, we conclude the paper in

Section 5.

2 PRELIMINARIES

This paper tackles the problem of computing the ap-

proximate graph edit distance between graphs. First,

we deﬁne the terminology used to solve the problem.

An undirected graph is represented as g = (V,E, Σ, `),

where V is a set of vertices, E ⊆ V × V is a set of

edges, Σ = {σ

,σ

,··· , σ

} is a set of vertex labels,

and ` : V → Σ is a function that assigns a label to each

vertex in the graph. Additionally, the set of vertices

in graph g is represented as V (g). Although we as-

ࢇ

࢈

ࢉ

ࢇ

࢈

ࢉ

ࢇ ࢈

ࢉ

ࢇ

࢈

ࢊ

ࢇ

࢈

ࢊ

ࢇ

࢈

ࢊ

ࢉ

ࢋ

૚

ࢋ

૛

ࢋ

૞

ࢋ

૝

ࢋ

૜

ࢍ

૚

ࢍ

૛

Figure 1: Sequence of edit operations for transforming g

into g

sume that only the vertices in the graphs have labels,

the methods in this paper can be applied to graphs

where both the vertices and edges have labels (Hido

and Kashima, 2009). The vertices adjacent to vertex v

are represented as N(v) = {u | (v, u) ∈ E}. The aver-

age number of adjacent vertices is represented as d. A

sequence of vertices from v to u is called a path, and

its step refers to the number of edges on that path. A

path is said to be simple if and only if it does not have

repeating vertices. The paths discussed in this paper

are not always simple.

The graph edit distance is one of the most repre-

sentative metrics to measure the dissimilarity between

graphs. The graph edit distance d(g

) between

graphs g

and g

is deﬁned as the minimum length

of the sequence of edit operations needed to trans-

form g

into g

, where one edit operation includes the

insertion or deletion of a vertex/edge and the substi-

tution of a vertex label. Although the edit distance

was originally proposed for measuring the dissimilar-

ity between two strings, the metric was extended to

graphs by introducing graph edit operations.

Figure 1 shows a certain sequence of edit oper-

ations that consists of one deletion of vertex (e

one insertion of edge (e

), one deletion of edge (e

and two substitutions of labels (e

). Computing

the edit distance between g

and g

is equivalent to

searching for the minimum length of the sequence of

edit operations needed to transform g

into g

. The

method based on the A

algorithm is a well-known

technique for computing the exact graph edit dis-

tance (Hart et al., 1968). However, this method can-

not be applied to large graphs, because the problem

of obtaining the exact graph edit distance between

two graphs is known to be NP-complete. To address

this drawback, various methods for computing the ap-

proximate graph edit distance have been proposed.

The graph edit distance is an important measure

for analyzing graphs, and is applicable to a wide

range of practical applications such as ﬁngerprint au-

thentication (Choi and Kim, 2010), malware detec-

tion (Kinable and Kostakis, 2011), chemoinformat-

ics (Kashima et al., 2003), bioinformatics, and doc-

ICPRAM 2018 - 7th International Conference on Pattern Recognition Applications and Methods

ument analysis (Wang et al., 2014). We explain three

typical fundamental problems that use the analysis

of graphs. First, given a set of graphs, we can ap-

ply methods for clustering the graphs to detect some

latent groups within them (Robles-Kelly and Han-

cock, 2003). Second, given a set of graphs and a

query graph g

, we can apply a similarity search to

the graphs (Yan et al., 2005). Third, given a set of

graphs with class labels, we can apply a Support Vec-

tor Machine (SVM) and kernel functions to forecast

the class of a graph whose label is unknown (Kashima

et al., 2003). The graph edit distance is converted to

similarity using the Gaussian kernel, which is deﬁned

k(g

) = exp



−

d(g

)

2σ



. (1)

Most machine learning and data mining algorithms

that are designed to analyze d-dimensional vectors

can be applied to graphs using this kernel.

3 APPROXIMATE GRAPH EDIT

DISTANCE

This section surveys a framework for computing the

approximate graph edit distance between two graphs

using the linear sum assignment problem (LSAP).

The problem of computing the exact graph edit dis-

tance between graphs g

= (V

,Σ,`

) and g

,Σ,`

) is formalized as follows (Riesen and

Bunke, 2009; Riesen, 2015): First, the set of vertices

is extended to

= V

∪

b−a empty vertices

z }| {

{ ε

,ε

,··· , ε

b−a

}

where |V

| = a, |V

| = b, and we assume that |V

| ≤

|. The graph edit distance computation is even-

tually performed on graphs g

= (V

,Σ,`

) and

= (V

,Σ,`

). Additionally, a cost matrix for

editing g

to g

is deﬁned as

C =

1 2 ··· b













1 c

··· c

2 c

··· c

a c

··· c

1 c

ε1

ε2

··· c

εb

b −a c

ε1

ε2

··· c

εb

, (2)

where

• c

i j

denotes the cost of replacing a label of vertex

∈ V

with a label of vertex v

∈ V

and

• c

εi

denotes the cost of inserting a vertex v

into g

to edit g

to g

Additionally, we denote the cost of editing an edge

) in g

into an edge (v

) in g

by c((v

) →

)). In this paper, we assume that only vertices

in graphs have labels, and so c((v

) → (v

)) is

the cost of inserting or deleting an edge. When g

is edited to g

, the mapping between the two sets of

vertices is denoted by a bijective function ϕ : V

→

. Then, the total cost of editing g

to g

via ϕ is

dist(g

,ϕ) =

∑

i=1

iϕ(i)

b−1

∑

i=1

∑

j=i+1



) →



ϕ(i)

ϕ( j)



. (3)

Therefore, the exact edit distance between g

and g

d(g

) = min

ϕ∈Φ

dist(g

,ϕ), (4)

where Φ is a set of all possible permutations of

integers 1, 2, · · · ,b. Equation (4) is a type of

quadratic assignment problem that is known to be NP-

complete (Koopmans and Beckmann, 1957), although

it reduces to the LSAP if Eq. (3) does not contain its

second term.

Riesen et al. proposed an efﬁcient algorithmic

framework that enables us to obtain the approximate

graph edit distance by omitting the second term of

Eq. (3). The framework ﬁrst solves

ϕ = arg min

ϕ∈Φ

∑

i=1

iϕ(i)

(5)

and then obtains the approximate edit distance be-

tween g

and g

by substituting

ϕ into Eq. (3). Equa-

tion (5) implies that each vertex v

in g

should, as

far as possible, be mapped to a vertex in g

that

has the same label as v

. Solving Eq. (5) is equiva-

lent to the minimum matching problem of a bipartite

graph whose vertices are V

and V

, and whose edge

weights are c

i j

in Eq. (2). Therefore, this problem

is tractable in O(b

). However, because the structural

information of the two graphs is ignored and only ver-

tex labels are taken into account in the optimization

problem shown in Eq. (3), we do not always obtain an

adequate mapping from V

to V

. To overcome this

difﬁculty, the cost matrix in Eq. (2) are redeﬁned to

take account of the structural information as the ma-

trix C

∗

with elements

∗

i j

= c(local(v

) → local(v

)),

Accurate and Fast Computation of Approximate Graph Edit Distance based on Graph Relabeling



















Kʻ

Kʻ







Figure 2: Limited-size subgraphs induced by vertices

within h steps from vertex v

Algorithm 1: Approximate Edit Distance.

Data: graphs g

and g

, and h

Result: approximate graph edit distance

1 while |V (g

)| < |V (g

)| do

2 V (g

) ← V (g

) ∪{ε};

3 for (v

) ∈ V (g

) ×V (g

) do

4 c

∗

i j

← d(g

);

ϕ ← LSAP(C

∗

);

d ← dist(g

ϕ);

7 return

where local(v

) is the local structure around a vertex

, and c(local(v

) → local(v

)) is the cost of edit-

ing local(v

) to local(v

). Recently, various methods

within this framework have been developed by repre-

senting local structures as stars (Zeng et al., 2009),

walks (Ga

ere et al., 2014), or limited-size sub-

graphs (Carletti et al., 2015). This enables a more

accurate mapping between sets of vertices than that

of Eq. (5).

Carletti et al. (2015) introduced local(v

), a

limited-size subgraph induced by vertices reachable

within h steps from vertex v

, as shown in Fig. 2. The

subgraph local(v

) is denoted by g

, and c

∗

i j

is the

exact edit distance between g

and g

, that is, c

∗

i j

d(g

). d(g

) is tractable for sufﬁciently small

h, although applying the problem to large graphs is

intractable, because the problem of computing the

graph edit distance is NP-complete. Additionally, be-

cause the structural information contained in limited-

size subgraphs is greater than that of walks and stars,

the graph edit distance based on limited-size sub-

graphs provides an accurate approximate graph edit

distance.

Algorithm 1 shows the pseudo-code for com-

puting an approximate graph edit distance between

graphs g

and g

. In Lines 1–2, the numbers of ver-

tices in g

and g

are equalized. For every pair of

vertices in V (g

) ×V(g

), the exact edit distance be-

tween limited-size subgraphs g

and g

is measured

and set as the (i, j)-th element in the cost matrix C

∗

LSAP in Line 5 returns the optimum mapping accord-

ing to the optimal bipartite graph matching. Finally,

in Line 7, Algorithm 1 returns the approximate graph

edit distance between graphs g

and g

This algorithm has a drawback in terms of com-

putational efﬁciency, because it computes the exact

edit distance multiple times. To overcome this, we

propose a novel method for computing the approxi-

mate graph edit distance more efﬁciently by compar-

ing structural information in g

and g

with a compu-

tation time that is proportional to |V (g

)| and |V (g

)|.

4 APPROXIMATE GRAPH EDIT

DISTANCE BASED ON

RELABELING GRAPHS

Given a graph g

(h)

= (V, E,Σ,`

(h)

), all labels of

vertices in g

(h)

are updated to obtain another

graph g

(h+1)

= (V,E,Σ

(h+1)

). We call this op-

eration “relabeling,” and deﬁne it as `

(h+1)

(v) =

r(v, N(v), `

(h)

). Representative methods based on re-

labeling include the Weisfeiler–Lehman Subtree Ker-

nel (WLSK) (Shervashidze et al., 2011), Neighbor-

hood Hash Kernel (NHK) (Hido and Kashima, 2009),

and Hadamard Code Kernel (HCK) (Kataoka and

Inokuchi, 2016). The vertex labels of WLSK are rep-

resented as strings and the relabeling of vertex v is de-

ﬁned as a string concatenation of the labels of N(v).

In NHK, the vertex labels are represented as ﬁxed-

length bit strings and relabeling v is deﬁned in terms

of logical operations such as XOR on the labels of

N(v). The labels of HCK are based on the Hadamard

code, which is used in spread spectrum-based com-

munication technologies, and relabeling v is deﬁned

as a summation on the labels of N(v).

Figure 3 shows an example of the framework

based on graph relabeling. Let g

(0)

be the original

graph whose vertices have labels a, b, and c. Each

of the labels is relabeled to obtain g

(1)

. Although

the actual calculation depends on the method of re-

labeling (e.g., NHK, WLSK, or HCK), the relabeling

of v is commonly applied using v, N(v), and `

(0)

(v).

In the center of Figure 3, `

(0)

) = b is relabeled

into d using adjacent vertices v

and v

. Therefore,

(1)

) = d represents the characteristics of st(v

,1),

where st(v, 1) is a tree of height 1 whose root and

leaves are v and N(v), respectively. The labels of v

and v

in g

(1)

are identical because st(v

,1) = st(v

,1)

in g

(0)

Given a graph g and its vertex v, an unfolded sub-

tree st(v, h) of height h is deﬁned recursively from the

root of the tree:

ICPRAM 2018 - 7th International Conference on Pattern Recognition Applications and Methods

Figure 3: Example of relabeling (g

(0)

→ g

(1)



















Kʻ

Kʻ











  



  



  

Figure 4: Derivation of an unfolded subtree of height 2.

• the root of st(v, h) is the vertex v in g,

• each node in st(v,h), except for leaves, has chil-

dren that are N(u) of the vertex u corresponding

to the node, and

• the height of st(v, h) is h.

Figure 4 shows an example of an unfolded subtree

st(v

,2) derived from the graph shown in Figure 2. A

vertex in the graph may appear in the unfolded subtree

(such as v

, v

, and v

), because multiple paths reach

to the vertices from the root; this causes the size of

the tree to grow exponentially as h increases.

The Label Aggregate Kernel (LAK) (Kataoka and

Inokuchi, 2016) is another method based on this

framework. Next, we present a concrete deﬁnition of

the relabeling operation in LAK. In LAK, `

(0)

(v) is a

vector in |Σ|-dimensional space. If a vertex in a graph

has a label σ

from the set Σ = {σ

,σ

,··· , σ

|Σ|

}, the

i-th element in the vector is 1 and the other elements

are 0. In LAK, `

(h)

(v) is deﬁned as

(h)

(v) = `

(h−1)

(v) +

∑

u∈N(v)

(h−1)

(u).

The i-th element in `

(h)

(v) equals the frequency of oc-

currence of σ

in st(v, h). Label `

(h)

(v), obtained by

iteratively relabeling h times, has a distribution of la-

bels that is reachable within h steps from v. Therefore,

Figure 5: Example of relabeling in LAK.

(h)

(v) represents the characteristics of st(v, h). In ad-

dition, although the size of st(v, h) increases exponen-

tially as h increases, the size of `

(h)

(v) remains |Σ|.

Therefore, retaining `

(h)

(v) in memory is tractable,

although retaining st(v, h) is intractable.

We show an example of relabeling in LAK in Fig-

ure 5, assuming that |Σ| = 3 and relabeling is applied

only once. Consider the graph g

(0)

, whose vertices

have labels (1, 0, 0), (0, 1, 0), and (0, 0, 1). We next

relabel the graphs to obtain g

(1)

. The label of ver-

tex v in g

(1)

represents the distribution of labels con-

tained in st(v, 1). For instance, the label of v

in g

(1)

) = (2, 1, 1), which indicates that there are two

vertices labeled (1,0,0), one vertex labeled (0,1,0),

and one vertex labeled (0, 0, 1). This distribution is

equivalent to that of the labels contained in st(v

,1).

By relabeling the graphs h times, each

vertex v has h + 1 |Σ|-dimensional vectors

(0)

(v),`

(1)

(v),··· , `

(h)

(v). We concatenate these

vectors to obtain an (h + 1) × |Σ|-dimensional vector

(v) = (

(0)

(v)

z }| {

(0)

,·· · ,`

(0)

|Σ|

(1)

(v)

z }| {

(1)

,·· · ,`

(1)

|Σ|

,·· · ,

(h)

(v)

z }| {

(h)

,·· · ,`

(h)

|Σ|

(6)

Given two graphs g

and g

, if g

is isomorphic

to g

, where v

∈ V (g

) and v

∈ V (g

), then `

) is

the same as `

Based on this observation, we propose a novel

method for computing the difference between local

structures around v

and v

. Given two concatenated

vectors `

) and `

), the distance between `

)

and `

) is deﬁned as

∗

i j



) −`

)



(7)

∑

t=0



(t)

) −`

(t)

)



. (8)

Because Eq. (7) is the square of the Euclidean dis-

tance between `

) and `

), the four axioms

about this metric hold (non-negativity, identity of

indiscernibles, symmetry, and triangle inequality).

However, we need to discuss the identity of indis-

Accurate and Fast Computation of Approximate Graph Edit Distance based on Graph Relabeling

0),( =

ggd

gg =

0)()(

=−

jLiL

vv ll

)()(

jLiL

vv ll =

DOZD\V

DOPRVWDOOJUDSKV

Figure 6: Identity of indiscernibles.

Algorithm 2: Proposed Method.

Data: graphs g

and g

, and H

Result: approximate graph edit distance

1 while |V (g

)| < |V (g

)| do

2 V (g

) ← V (g

) ∪{ε};

3 C

∗

← 0;

4 for h ∈ [0, H] do

5 for (v

) ∈ V (g

) ×V (g

) do

6 c

∗

i j

← c

∗

i j

+ γ



(h)

) −`

(h)

)



;

7 g

← (V (g

),E(g

),Z

|Σ|

(h+1)

);

8 g

← (V (g

),E(g

),Z

|Σ|

(h+1)

);

ϕ ← LSAP(C

∗

);

← dist(g

ϕ);

11 return min

h∈[0,H]

;

cernibles in more detail. As shown in Fig. 6, the

identity of indiscernibles for the Euclidean distance

implies that `

) = `

) if and only if k`

) −

= 0. The above framework for relabel-

ing graphs is based on the 1-dimensional Weisfeiler–

Lehman algorithm, which checks the isomorphism

between two graphs. This algorithm is known to be a

valid isomorphism test for almost all graphs (see (Cai

et al., 1992) for examples of graphs that cannot be

distinguished by the algorithm) (Shervashidze et al.,

2011). Therefore, if g

= g

, k`

) − `

= 0

always holds. In contrast, its converse holds for al-

most all graphs.

To control the effect of steps from a central vertex

of the local structure, we add to Eq. (6) a coefﬁcient

that changes exponentially as h increases:

(v, γ) = (

(0)

(v)

z }| {

(0)

,··· , `

(0)

|Σ|

,··· ,

(h)

(v)

z }| {

(h)

,··· , γ

(h)

|Σ|

). (9)

Algorithm 2 shows the pseudo-code of the pro-

posed method for computing the approximate graph

edit distance between graphs g

and g

. In Lines 1–

2, the numbers of vertices in g

and g

are equalized.

This is repeated at most b times. For each pair of

vertices in V (g

) ×V (g

), the Euclidean distance be-

tween two vectors `

(h)

) and `

(h)

) is measured

and set as the (i, j)-th element in a square matrix C

∗

in Line 6. In Lines 7–8, using the set of nonnega-

tive integers Z, g

(h)

and g

(h)

are relabeled to obtain

(h+1)

and g

(h+1)

, respectively. In Line 9, LSAP re-

turns the mapping

ϕ from V (g

) to V (g

) according

to the optimal bipartite graph matching. The pro-

cesses in Lines 5–10 are repeated H + 1 times. Fi-

nally, the algorithm returns the minimum approximate

graph edit distance among

. This algorithm runs in

O(H(|Σ|b

+|Σ|bd +b

)) time, because the computa-

tional complexities of Lines 6, 7, 9, and 10 are O(|Σ|),

O(|Σ|bd), O(b

), and O(b

), respectively. Because

d is bounded by b, the computational complexity of

Algorithm 2 becomes O(b

H(|Σ| + b). If we com-

pute Eq. (7) in a straightforward manner, we require

O(|Σ|H) memory for each vertex of graphs g

and g

However, when we compute Lines 6, 7, and 8 in Algo-

rithm 2, we do not require `

(τ)

(v) for τ < h. Therefore,

Algorithm 2 requires O(|Σ|b +bd) memory.

The notable difference between Algorithms 1 and

2 is that the former requires the exact graph edit dis-

tance to be computed, which is known to be an NP-

complete problem. Although the computation is con-

ducted for small graphs, Algorithm 1 runs the compu-

tation b

times, which entails a signiﬁcant computa-

tional cost. In contrast, the proposed method does not

require the exact graph edit distance to be computed,

replacing this with computations of the Euclidean dis-

tance. The complexity of this operation is indepen-

dent of the size of the graphs, which enables us to use

LSAP multiple times in our proposed method. The

ﬁnal output of the proposed method is selected from

for 0 ≤ h ≤ H, with our method returning the most

accurate graph edit distance.

In the proposed method, we used LAK to charac-

terize the local structure around each vertex. If we

use WLSK, HCK, or NHK, the difference between g

and g

is represented as only a binary value, rather

than in a quantitative form. The binary value repre-

sents whether g

and g

are isomorphic or not. This

is why we use LAK in our proposed method.

5 EXPERIMENTAL EVALUATION

The proposed method was implemented in Java. All

experiments were conducted on an Intel Xeon E5-

2609 2.50 GHz computer with 32 GB memory run-

ning Microsoft Windows 7. We used two real-world

datasets. The ﬁrst dataset, MUTAG (Debnath et al.,

1991), contains information on 188 chemical com-

pounds and their class labels. The class labels are

binary values that indicate the mutagenicity of chem-

ICPRAM 2018 - 7th International Conference on Pattern Recognition Applications and Methods

Figure 7: Conversion of a graph.

Table 1: Summary of evaluation datasets.

MUTAG ENZYMES

Number of graphs |D| 188 600

Maximum graph size 84 126

Average graph size 53.9 32.6

Number of labels |Σ| 12 3

Number of classes 2 6

(class distribution) (126,63) (100,100,100,)

100,100,100)

Avg. degree of vertices 2.1 3.8

ical compounds. Each chemical compound is rep-

resented as an undirected graph where each vertex,

edge, vertex label, and edge label corresponds to an

atom, chemical bond, atom type, and bond type, re-

spectively. Because we assume that only the vertices

in the graphs have labels, the chemical graphs are con-

verted using a previous method (Hido and Kashima,

2009), that is, an edge labeled with ` that is adjacent

to vertices v and u in a chemical graph is replaced

with a vertex labeled with ` that is adjacent to v and u

with unlabeled edges, as shown in Figure 7. The sec-

ond dataset, ENZYMES, contains information on 600

proteins and their class labels. These are one of six la-

bels denoting the six EC top-level classes (Schomburg

et al., 2004). Table 1 summarizes the datasets.

5.1 Computational Efﬁciency

Figures 8 and 9 show the average computation time

required to compute the approximate graph edit dis-

tance for each pair of graphs in the MUTAG and EN-

ZYMES datasets for various H, respectively. These

ﬁgures do not contain results of the exact graph edit

distance, because we could not compute the exact

graph edit distance between most of the graphs with

more vertices than 30. The horizontal axes are log-

arithmic. To plot the computation times of both

methods for H = 0, note that the horizontal axes re-

fer to H + 1 rather than H. The result of the pro-

posed method for H = 0 is almost the same as that

of the conventional method for H = 0, because they

use the same local structures around vertices. As H

increases, the computation time of the conventional

method aforementioned in Algorithm 1 (Carletti et al.,

2015) increases dramatically. We could compute up

to H = 2 within a total computation time of 3 hours,

ϭ͘Ϭ

ϭϬ͘Ϭ

ϭϬϬ͘Ϭ

ϭϬϬϬ͘Ϭ

ϭ ϭϬ ϭϬϬ ϭϬϬϬ

ĐŽŵƉƵƚĂƚŝŽŶƚŝŵĞ΀ŵƐĞĐ΁

,нϭ

WƌŽƉŽƐĞĚ

ŽŶǀĞŶƚŝŽŶĂů

Figure 8: Average computation time for various H (MU-

TAG).

Ϭ͘ϭ

ϭ͘Ϭ

ϭϬ͘Ϭ

ϭϬϬ͘Ϭ

ϭ ϭϬ ϭϬϬ

ĐŽŵƉƵƚĂƚŝŽŶƚŝŵĞ΀ŵƐĞĐ΁

,нϭ

WƌŽƉŽƐĞĚ

ŽŶǀĞŶƚŝŽŶĂů

Figure 9: Average computation time for various H (EN-

ZYMES).

with more than 95% of the computation time in the

conventional method taken up by computing the exact

graph edit distance between local structures. There-

fore, the conventional method can only take account

of small local structures around vertices to compute

the approximate graph edit distance. In contrast, the

computation time of the proposed method is propor-

tional to H, as discussed in the ﬁnal part of Section 4.

As the proposed method does not require the exact

graph edit distance to be computed, we obtained re-

sults for up to H = 200 within a total computation

time of 3 hours, which indicates that the proposed

method takes account of large local structures around

vertices. Although the proposed method computes

LSAP multiple times, the computational complexity

of LSAP is O(b

), which is sufﬁciently tractable for

the size of the graphs in the experimental datasets.

From these results, we can conﬁrm that the proposed

method is much faster than the conventional method

for computing approximate graph edit distances.

5.2 Accuracy of the Graph Edit

Distance

Figures 10 and 11 show the average approximate

graph edit distance measured by the proposed method

Accurate and Fast Computation of Approximate Graph Edit Distance based on Graph Relabeling

for each pair of graphs in the MUTAG and EN-

ZYMES datasets, respectively. The approximate

graph edit distance measured by the proposed method

is not less than the exact graph edit distance, be-

cause the exact graph edit distance is deﬁned as the

“minimum length” of the sequence of edit operations

needed to transform g

into g

and the approximate

graph edit distance measured by the proposed method

is computed by Eq. (3). As H increases, the aver-

age approximate graph edit distance decreases in both

datasets, because large local structures around the ver-

tices are used to map the vertices in one graph to

vertices in another graph. In addition, by tuning γ,

we can obtain more accurate edit distances than with

γ = 1. In Figure 12, each point (

) shows the

approximate graph edit distances

and

measured

by the conventional and proposed methods, respec-

tively, for a pair of graphs in the MUTAG dataset.

There are many points above the red line, which sug-

gests that most of the approximate graph edit dis-

tances measured by the proposed method are more

accurate than those given by the conventional method,

because

. For the MUTAG dataset, the average

approximate graph edit distances of the conventional

and proposed methods are 78.5 and 82.4, respectively.

Figure 13 for ENZYMES indicates the same tendency

as Figure 12. For the ENZYMES dataset, the average

approximate graph edit distances of the conventional

and proposed methods are 115.8 and 129.0, respec-

tively. One of the reasons why the proposed method

computes accurate approximate graph edit distances

is that it very efﬁciently takes account of large local

structures around vertices, which enables the mini-

mum approximate graph edit distance to be selected

from among

,··· ,

In general, there is a trade-off between fast com-

putation and accuracy. However, as shown in the pre-

vious subsection and here, the proposed method is

much faster and more accurate than the conventional

method.

5.3 Application of Graph Edit Distance

To validate the applicability of the graph edit dis-

tance, we compared the accuracy of the predictions

given by the conventional and proposed methods. The

graph classiﬁcation problem is deﬁned as follows.

Given a set of n training examples D = {(g

)}

(i = 1, 2, ··· , n), where each example is a pair consist-

ing of a labeled graph g

and the class y

∈ {+1, −1}

to which it belongs, the objective is to learn a function

f that correctly predicts the classes of the test exam-

ples. In this experiment, graphs are classiﬁed by an

SVM using a graph kernel. The graph kernel used in

ϳϳ

ϳϴ

ϳϵ

ϴϬ

ϴϭ

ϴϮ

ϴϯ

ϴϰ

Ϭ Ϭ͘ϱ ϭ ϭ͘ϱ

ƉƉƌŽǆŝŵĂƚĞ ŐƌĂƉŚĞĚŝƚĚŝƐƚĂŶĐĞ

,сϬ

,сϭ

,сϮ

,сϯ

Figure 10: Average approximate graph edit distance for var-

ious H and γ (MUTAG).

ϭϭϬ

ϭϭϭ

ϭϭϮ

ϭϭϯ

ϭϭϰ

ϭϭϱ

ϭϭϲ

ϭϭϳ

ϭϭϴ

ϭϭϵ

Ϭ Ϭ͘ϱ ϭ ϭ͘ϱ Ϯ

ƉƉƌŽǆŝŵĂƚĞ ŐƌĂƉŚĞĚŝƚĚŝƐƚĂŶĐĞ

,сϬ ,сϭ ,сϮ ,сϯ

Figure 11: Average approximate graph edit distance for var-

ious H and γ (ENZYMES).

this experiment is deﬁned as

k(g

) = exp

−

d(g

)

2σ

, (10)

where

d(g

) is the approximate graph edit dis-

tance between g

and g

measured by the conven-

tional and proposed methods

. To learn from the

kernel matrices generated by the above graph kernel,

we used the LIBSVM package with 10-fold cross-

validation (Chang and Lin, 2001).

Figures 14 and 15 show the classiﬁcation accu-

racy of the conventional and proposed methods, as

well as that of WLSK (Shervashidze et al., 2011), for

various H using the MUTAG and ENZYME datasets.

The maximum accuracy of the proposed method is

greater than that of the conventional method, because

the proposed method computes accurate approximate

graph edit distances, as shown in the previous section.

In addition, the maximum accuracy of the proposed

Their experimental results are represented with “Pro-

posed” and “Conventional” in Figs. 14 and 15.

ICPRAM 2018 - 7th International Conference on Pattern Recognition Applications and Methods

ϮϬ

ϰϬ

ϲϬ

ϴϬ

ϭϬϬ

ϭϮϬ

ϭϰϬ

ϭϲϬ

ϭϴϬ

ϮϬϬ

Ϭ ϮϬ ϰϬ ϲϬ ϴϬ ϭϬϬ ϭϮϬ ϭϰϬ ϭϲϬ ϭϴϬ ϮϬϬ

ƉƉƌŽ͘ŐƌĂƉŚĞĚŝƚĚŝƐƚĂŶĐĞĨŽƌƚŚĞ

ĐŽŶǀĞŶƚŝŽŶĂůŵĞƚŚŽĚ

ƉƉƌŽ͘ŐƌĂƉŚĞĚŝƚĚŝƐƚĂŶĐĞĨŽƌƚŚĞƉƌŽƉŽƐĞĚ

ŵĞƚŚŽĚ

Figure 12: Approximate graph edit distance for H = 2 (MU-

TAG).

ϱϬ

ϭϬϬ

ϭϱϬ

ϮϬϬ

ϮϱϬ

ϯϬϬ

ϯϱϬ

Ϭ ϱϬ ϭϬϬ ϭϱϬ ϮϬϬ ϮϱϬ ϯϬϬ ϯϱϬ

ƉƉƌŽ͘Ő ƌĂƉŚĞĚŝƚĚŝƐƚĂŶĐĞĨŽƌƚŚĞ

ĐŽŶǀĞŶƚŝŽŶĂůŵĞƚŚŽĚ

ƉƉƌŽ͘ ŐƌĂƉŚĞĚŝƚĚŝƐƚĂŶĐĞĨŽƌƚŚĞƉƌŽƉŽƐĞĚ

ŵĞƚŚŽĚ

Figure 13: Approximate graph edit distance for H = 2 (EN-

ZYMES).

method is comparative with that of WLSK, which is

one of the most representative graph kernels.

6 CONCLUSION

As computing the exact graph edit distance is compu-

tationally expensive, and may be intractable for large-

scale datasets, in this paper, we proposed a method

based on graph relabeling that is both faster and more

accurate than the conventional approach. We used un-

folded subtrees to denote the potential relabeling of

local structures around a given vertex. These subtree

representations are concatenated as a vector, and the

distance between different vectors is used to charac-

terize the distance between the corresponding graphs.

This avoids the need for multiple calculations of the

exact graph edit distance between local structures.

ϳϬ

ϳϱ

ϴϬ

ϴϱ

ϵϬ

ϵϱ

ϭϬϬ

Ϭ ϭ Ϯ ϯ ϰ ϱ

ůĂƐƐŝĨŝĐĂƚŝŽŶĐĐƵƌĂĐǇ΀й΁

WƌŽƉŽƐĞĚ

ŽŶǀĞŶƚŝŽŶĂů

t>^<

Figure 14: Prediction accuracy for various H (MUTAG).

ϮϬ

ϯϬ

ϰϬ

ϱϬ

ϲϬ

ϳϬ

Ϭ ϭ Ϯ ϯ ϰ ϱ

ůĂƐƐŝĨŝĐĂƚŝŽŶĐĐƵƌĂĐǇ

΀й΁

WƌŽƉŽƐĞĚ

ŽŶǀĞŶƚŝŽŶĂů

t>^<

Figure 15: Prediction accuracy for various H (ENZYMES).

Simulation experiments on two real-world chemical

datasets were reported. Compared with the conven-

tional technique, the proposed method gave a more

accurate approximation of the graph edit distance and

is signiﬁcantly faster on both datasets. This suggested

the proposed method could be applicable in the anal-

ysis of larger and more complex graph-like datasets.

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APPENDIX

The Label Aggregate Kernel (LAK) was designed to

measure the similarity between two graphs g

and g

to enable the application of a Support Vector Machine

(SVM). The kernel is deﬁned as

k(g

) =

∑

t=0

∑

∈V (g

)

∑

∈V (g

)

δ(`

(t)

),`

(t)

)),

where δ is the Kronecker delta. In contrast, in this pa-

per, LAK has been used to measure the approximate

graph edit distance corresponding to the dissimilarity

between two graphs g

and g

ICPRAM 2018 - 7th International Conference on Pattern Recognition Applications and Methods