A Multi-agent Approach for Graph Classiﬁcation

Luca Baldini

and Antonello Rizzi

1 b

Department of Information Engineering, Electronics and Telecommunications, University of Rome “La Sapienza",

Via Eudossiana 18, 00184 Rome, Italy

Keywords:

Multi-agent Systems, Graph Embedding, Supervised Learning, Structural Pattern Recognition.

Abstract:

In this paper, we propose and discuss a prototypical framework for graph classiﬁcation. The proposed algo-

rithm (Graph E-ABC) exploits a multi-agent design, where swarm of agents (orchestrated via evolutionary

optimization) are in charge of ﬁnding meaningful substructures from the training data. The resulting set of

substructures compose the pivotal entities for a graph embedding procedure that allows to move the pattern

recognition problem from the graph domain towards the Euclidean space. In order to improve the learning

capabilities, the pivotal substructures undergo an independent optimization procedure. The performances of

Graph E-ABC are addressed via a sensitivity analysis over its critical parameters and compared against current

approaches for graph classiﬁcation. Results on ﬁve open access datasets of fully labelled graphs show interest-

ing performances in terms of accuracy, counterbalanced by a relatively high number of pivotal substructures.

1 INTRODUCTION

Multi-agent systems (Panait and Luke, 2005; Stone

and Veloso, 2000; Rizk et al., 2018; Dorri et al., 2018)

emerged in the last years as powerful approaches in

order to solve complex pattern recognition problems.

This is due to the innate nature of multi-agent sys-

tems, where independent atomic computational units

(i.e., the agents) cooperate in order to solve a complex

problem by a divide-and-conquer approach.

The ﬂexibility of multi-agent systems leaded to

different research works where such paradigm has

been employed for building supervised and unsuper-

vised learning systems. For example, in (Alamgir and

Von Luxburg, 2010) a multi-agent approach has been

used for local graph clustering in which each agent

performs a random walk on a graph with the main

constraint that such agents are "tied" together by a

rope, forcing them to be close to each other. In (Car-

valho et al., 2016) a set of self-organising agents by

means of ant colony optimisation has been applied to

anomaly detection and network control. In (Chaimon-

tree et al., 2010) each agent runs a different cluster-

ing algorithm in order to return the best one for the

dataset at hand. In (Chaimontree et al., 2011) agents

negotiate one another rather than being governed by a

master/wrapper process (e.g. evolutive algorithm) in

https://orcid.org/0000-0003-4391-2598

https://orcid.org/0000-0001-8244-0015

order to improve the clustering solution. In (

Inkaya

et al., 2015) ant colony optimisation has been used in

order to organise agents, where each ant "walks" on

the dataset, building connections amongst points. In

(Ogston et al., 2003) each agent consists in a set of

data points and agents link to each other, thus lead-

ing to clustering. In (Pan and Chen, 2012) a genetic

algorithm has been used where the agents’ genetic

code is connection-based: each agent is a clustering

result whose genetic code builds a (sub)graph and, ﬁ-

nally, such subgraphs can be interpreted as clusters.

In (Park and Oh, 2006) the multi-agent approach col-

lapses into two agents: a ﬁrst agent runs a cascade of

principal component analysis, self organizing maps

and k-means in order to cluster data and a second

agent validates such results: the two agents interac-

tively communicate with each other.

As supervised problems are concerned, in

(Bianchi et al., 2015) a multi-agent algorithm has

been proposed in which agents perform a Markovian

random walk on a weighted graph representation of

the input dataset. Each agent builds its own graph

connection matrix amongst data points, weighting the

edges according to the selected distance measure pa-

rameters, and performs a random walk on such graph

in order to discover clusters. This algorithm has been

employed in (Bianchi et al., 2016) for an unsuper-

vised identiﬁcation of frequent behaviours of mobile

network subscribers starting from a set of call data

334

Baldini, L. and Rizzi, A.

A Multi-agent Approach for Graph Classiﬁcation.

DOI: 10.5220/0010677300003063

In Proceedings of the 13th International Joint Conference on Computational Intelligence (IJCCI 2021), pages 334-343

ISBN: 978-989-758-534-0; ISSN: 2184-3236

records. In (Pan and Jiao, 2011), the authors propose

the Granular Agent Evolutionary Classiﬁcation algo-

rithm, where agents are in charge of clustering pat-

terns with similar attributes. Further, different agents

lie in different class-aware subsystems and each sub-

system is characterized by a set of classiﬁcation rules

extracted by the information collected by the granu-

lar agents. Due to the advent of large corpora, multi-

agent systems have been applied for text and doc-

ument classiﬁcation (Mostafa et al., 2005; Ahmad

et al., 2012).

The MAS paradigm has also been successfully ap-

plied in the ﬁeld of Computer Vision. In (Chen et al.,

2019), the authors designed an agent based system

for Scene Graph generation for visual genome under-

standing where objects are agents that have to max-

imize the quality of the generated scene graph. In

(González-Briones et al., 2018), the authors proposed

an image recognition system for real time applications

where different agents collaborate for extracting rel-

evant features from the acquired images in order to

determine the age and gender of individuals in an of-

ﬁce building. The MAS paradigm has found applica-

tion also in the ﬁeld of bioinformatics (Corrêa et al.,

2020) for protein structure prediction by distributing

each relevant task among different agents in a decen-

tralized way. The distributed aspect that characterize

multi-agent systems has led various authors to inves-

tigate their application in the ﬁeld of Big Data. In

(Lombardo et al., 2019), the authors proposed a gen-

eral purpose actor-based system for distributed data

mining addressing the problem of social data analysis

such as topic detection, troll detection and hoax detec-

tion. Conversely in (Ding et al., 2018), the authors in-

troduce a multi-agent co-evolutionary adaptive strat-

egy for attribute reduction for big medical datasets

concerning infant brain Magnetic Resonance Imag-

ing. Finally, in (Modi and Shen, 2001), the authors

propose a distributed multi-agent framework for clas-

siﬁcation, where the authors assume that each agent

can see only a subset of features as the data is con-

ceived to be decentralized.

In this paper, we exploit Evolutive Agent Based

Clustering/Classiﬁcation (E-ABC, for short) for solv-

ing classiﬁcation problems. The peculiarities of E-

ABC can be summarized as follows: a) each agent

sees only a small random portion of the dataset; b)

different swarms of agents work on class-stratiﬁed

data; c) agents evolve via evolutionary optimization;

d) the synthesis of classiﬁcation models is orches-

trated by an independent evolutionary-like optimiza-

tion which allows the cooperation amongst different

swarms. The rationale behind the latter point is in

line with the divide-and-conquer approach which is

typical of multi-agent systems. In fact, the output of

the swarms serves as input for the population of the

evolutionary-like optimization, and the output of the

latter reﬁnes the behaviour of the swarms, with the ﬁ-

nal (common) goal of synthesizing a performing clas-

siﬁcation system.

E-ABC has been originally proposed in (Martino

et al., 2019a) as a clustering algorithm based on the

multi-agent paradigm conceived to work in Euclidean

spaces. In (Giampieri et al., 2018), it has been ex-

tended towards supervised problems by letting the re-

sulting clusters to compose a decision cluster classi-

ﬁer (Di Noia et al., 2020) and applied to the classiﬁca-

tion of faulty states in a real-world smart grid. In (Gi-

ampieri et al., 2020), E-ABC has been upgraded with

a multimodal optimization approach (Wong, 2015;

Preuss, 2015) in order to foster the exploration of dif-

ferent subspaces, hence foster its local metric learn-

ing capabilities. Conversely to all previous works,

in which E-ABC has been designed and employed

to process data described by vectors lying in a Eu-

clidean space, in this paper we propose an extension

of E-ABC tailored to work in the graph domain. The

transition from a structured domain such as the graph

one to a metric space such as the Euclidean one is

motivated by a well-established embedding approach

already explored in works such as (Martino and Rizzi,

2021) and (Baldini et al., 2021). As will be thor-

oughly explained later, agents are in charge of search-

ing for meaningful subgraphs amongst the training

data and these subgraphs will act as the pivotal sub-

structures for moving the classiﬁcation problem from

the graph domain towards the Euclidean space, where

any classiﬁer can freely be used.

The remainder of the paper is structured as fol-

lows: in Section 2 we describe the proposed algo-

rithm, in Section 3 we show the computational results,

in terms of sensitivity analysis to critical parameters

and comparison against similar approaches for graph

classiﬁcation. Finally, in Section 4 we conclude the

paper, remarking future directions.

2 GRAPH E-ABC

Graph E-ABC is a graph classiﬁcation system based

on multi-agent evolutive strategy. The system is in-

tended to be multi-agent by means of a collabora-

tive approach established between the individuals be-

longing to different swarms Σ = {S

,...,S

}, where

S is the number of classes in the classiﬁcation prob-

lem. In particular, each swarm S

∈ Σ is a team of

agents that individually performs a simple data min-

ing task exploiting graphs belonging to the i

class

A Multi-agent Approach for Graph Classiﬁcation

335

(i.e., each swarm is devoted to extract meaningful in-

formation by observing class-stratiﬁed data). The re-

sults of this process are later examined and collected

together in order to be exploited for a ﬁnal collab-

orative scope, i.e. the synthesis of candidate alpha-

bets set A

,...,A

, which enable to generate differ-

ent embedding spaces F

,...,F

exploiting the sym-

bolic histograms approach. In the resulting embed-

ding spaces, will ﬁnally be possible to set up com-

mon classiﬁcation systems, being the generic embed-

ding space F

an Euclidean space. In order to improve

the quality of the alphabets, we designed a genetic

algorithm-inspired evolutive strategy which combines

and mutates the promising alphabets with custom ge-

netic operators via an independent set of agents Z.

2.1 Agent Behaviour

The ﬁrst action performed by an agent a belonging to

a swarm S

is to gather a set of class-stratiﬁed sub-

graphs D

sampled from graphs in D

belonging to

class i, where the number of subgraph W = |D

| is

a user-deﬁned parameter. The sampling process can

be designed accordingly to different graph traversal

strategies which deﬁne the topology of the resulting

subgraphs (Baldini. et al., 2019; Baldini et al., 2020;

Baldini et al., 2021).

Once D

is ready, a can start the data min-

ing process. The information extraction is per-

formed according to the Basic Sequential Algorith-

mic Scheme (BSAS) clustering algorithm (Theodor-

idis and Koutroumbas, 2008) working directly in

the graph domain. BSAS relies on three key fac-

tors: a suitable dissimilarity measure d : G × G → R

between patterns; a resolution parameter θ ∈ [0,1]

that deﬁnes the threshold on d for including patterns

into a speciﬁc cluster; the maximum number of al-

lowed clusters Q. The adopted dissimilarity mea-

sure is a heuristic based on a Graph Edit Distance

named weighted node Best Match First (nBMF) (Bal-

dini et al., 2021). In the nBMF procedure, w =

node

ins

node

del

node

sub

edge

ins

edge

del

edge

sub

] ∈ {0,1}

, is a

set of nodes and edges insertion, deletion, substitu-

tion weights for the weighted nBMF and γ

γ is the

set of parameters for the dissimilarity measures be-

tween nodes and edges (if applicable). Full details

on nBMF, along with detailed pseudocodes, can be

found in (Baldini. et al., 2019; Martino and Rizzi,

2021). Clearly, the choice of w and γ

γ is critical since

wrong values could undermine the ability of d to

correctly capture the proximity between semantically

close graphs. The agent runs BSAS according to d

and Q, generating Π = {P

,...,P

}, i.e. a set of par-

titions, each of which is obtained for a given θ

i=1

Each cluster C ∈ P

undergoes the evaluation

phase which determines the reliability r (C) of the

cluster according to two internal properties, namely

its compactness f

and its cardinality f

|C| − 1

∑

g∈C

d (g,g

∗

) (1)

|C|

(2)

where g

∗

is the cluster representative of C, i.e. the

MinSOD of the cluster (Martino et al., 2019b). The

reliability r(C) reads as follows:

r(C) = η · f

+ (1− η) · f

(3)

where η is a trade-off parameter that weights the im-

portance of compactness against cardinality. Accord-

ing to a threshold τ, g

∗

can be promoted to be a sym-

bol s or simply discarded as not relevant information.

The evaluation phase is repeated for every cluster in

every partition P

∈ Π, leading to a set of symbols

B = {s

,...,s

} discovered by agent a, where M de-

pends on the number of symbols survived in the eval-

uation stage.

2.2 Synthesis of Classiﬁcation Models

Once every agent in every swarm in Σ has completed

its data mining process, a class speciﬁc set of sym-

bols is returned. The sets of symbols B synthesized

by each agent in a speciﬁc swarm S

can be merged in

a class-speciﬁc (i.e., swarm-speciﬁc) bucket of sym-

bols H

i=1

. In other words, H

contains the collective

information gathered by agents working on class i.

The procedure moves to the generation of candi-

date alphabets leveraging the buckets H

in order to

enable the graph embedding stage. This process is ad-

dressed by a separated population Z of K individuals,

whose actions can be summarized as follows:

1. The agent z ∈ Z evaluates the maximum number

of symbols per class t = T /S that can be extracted

according to a user-deﬁned bound T

2. The agent explores the buckets H

and extracts

uniformly at random at most t symbols

3. When all the classes are explored, the selected

symbols are collected into the multi-class alpha-

bet of symbols A

The procedure continues for all z ∈ Z, leading to the

synthesis of A

,...,A

According to the symbolic histogram approach

(Del Vescovo and Rizzi, 2007a; Del Vescovo and

Rizzi, 2007b), the alphabets can now be exploited for

building the vectorial representation of both training

NCTA 2021 - 13th International Conference on Neural Computation Theory and Applications

336

and validation sets (D

and D

). In short, the sym-

bolic histogram for a given graph G can be evaluated

via the following two-steps procedure:

1. the graph G is decomposed in k atomic units, i.e.

exp

= {g

,...,g

}

2. the symbolic histogram h

is deﬁned as:

= [occ(s

exp

),...,occ(s

exp

)] (4)

where the function occ : A × G → N performs the

process of counting the occurrences of a symbol

∈ A in G

exp

occ(s

exp

) =

∑

g∈G

exp

Γ(s

,g) (5)

where

Γ(s

,g) =

(

1 if d(s

,g) ≤ ζ

0 otherwise

(6)

and d(·,·) is the nBMF dissimilarity measure in-

troduced in Section 2.1.

Building the symbolic histograms of all graphs in D

and D

leads to the deﬁnition of F

and F

, respec-

tively an |D

|×|A

| and an |D

|×|A

| matrix, whose

rows are the symbolic histograms obtained with the

alphabet, for i = 1,...,K. A set of classiﬁcation

models c

,...,c

can be build accordingly to speciﬁc

embedding spaces:

1. Train a classiﬁer c

in the respective embedding

space F

spanned by the symbolic histograms ma-

trix F

2. Test the classiﬁer by predicting the embedded val-

idation set F

3. Evaluate the performance measure ω

as the accu-

racy of c

in correctly classify F

The resulting classiﬁers, c

,...,c

can be retained to-

gether with their performance measure ω

,...,ω

2.3 Driving Evolutions

In Sections 2.1 and 2.2, we described the goal of the

main actors: the swarms Σ must be successful in ﬁnd-

ing relevant information for building informative al-

phabet sets; conversely, the population Z must be able

to select and collect effective symbols together in or-

der to build meaningful embedding spaces whose va-

lidity can be assessed by the classiﬁer performances

ω. Hence, the evolutive process must take into ac-

count the different roles played by Σ and Z in order to

converge to suitable solutions. For sake of clearness,

before diving into the design of the evolutionary strat-

egy, we give a formal description of the fundamental

quantities involved in the optimization phase.

2.3.1 Symbol Quality

The quality of a single symbol Q

shall reﬂect the per-

formances ω of the alphabets in which the symbol s

under analysis appears in. A lookup table is built in

order to indicate the update value t that will be as-

signed to s via the following two steps:

1. Since the classiﬁcation accuracy assumes values

in range [0, 1], the [0,1] range is uniformly dis-

cretized into a ﬁnite number of bins

2. each accuracy bin is mapped with a reward value,

also uniformly discretized into the same number

of bins, where the admissible range [t

min

max

] is

user-conﬁgurable.

The quality update strategy works as follow:

1. Select the candidate alphabet A

2. Select the symbol s ∈ A

3. Find the accuracy bin in which ω lies and gather

the corresponding reward value t

4. Reward the symbols by applying the following

update rule:

(new)

= Q

(old)

+t (7)

5. Repeat from step 2 for all symbols in the selected

alphabet A.

The procedure repeats from step 1 for all the alpha-

bets under evaluation. In this approach, we interpret

the performance ω as a critic about the effectiveness

of the embedding space F built accordingly to the al-

phabet A. In this way, the quality Q

can be seen as

a measure for determining if the symbol s is useful

for building a valuable alphabet which can attain high

level of performance.

2.3.2 Agent Quality

The agent quality measure Q

is deﬁned accord-

ing to the qualities Q

,...,Q

and the reliabilities

r(C

),...,r(C

), where C

,...,C

are the clusters

related to s

,...,s

symbols the agent a has found

(see Section 2.1). The overall procedure for assigning

can be break down in the following steps:

1. Select the agent a from S

∈ Σ

2. According to the agent set of symbols B, evaluate

the mean qualities

|B|

∑

s∈B

3. According to the agent set of symbols B evaluate

the mean reliability r

|B|

∑

s∈B

is normalized in [0,1] according to the symbols

qualities observed in H

A Multi-agent Approach for Graph Classiﬁcation

337

4. Set agents quality Q

= ρ ·Q

+ (1− ρ) · r

(8)

5. Repeat from step 1 for all agents a ∈ S

with i =

1,...,S.

In Eq. (8), ρ weights the importance between the

symbols qualities and its reliabilities. In particular, in

the early generations of the evolution, we give more

importance to the right hand term in order to initially

synthesize well-formed clusters. Next, the quality of

an agent shall be better described as its ability in ﬁnd-

ing informative symbols according to Q

, since the

ﬁnal scope of the whole procedure is devoted to syn-

thesize meaningful alphabets employed for the clas-

siﬁcation problem. Indeed, this information is con-

tained in symbols qualities Q

whose values are actu-

ally backtracked in the agent quality Q

2.3.3 Evolutive Orchestration

The optimization of the agents in the swarms Σ is

driven by a genetic evolution. As stated in Section

2.1, each swarm S must be able to synthesize a candi-

date set of symbols H that will be later employed in

the formation of pivotal alphabets A. For this reason,

the genetic code a

code

of each agent reads as follow:

code

= [Q w γ

γ τ] (9)

which summarize the crucial parameters for the in-

formation extraction described in Section 2.1. Agents

in the same swarm follow a classic (µ + λ) selection

scheme (Beyer and Schwefel, 2002), where λ off-

spring is generated according to common genetic op-

erators applied to the parents population µ, i.e. muta-

tion, crossover and random spawn of new individuals.

The optimization aims at maximizing the agent ﬁtness

function deﬁned as the quality Q

given in Eq. (8).

The classiﬁcation models optimization is an

evolutionary-like procedure speciﬁcally designed ac-

cording to the unconventional nature of the individ-

ual’s genetic code z

code

. Indeed, z

code

is a direct rep-

resentation of a speciﬁc alphabet A, whose cardinal-

ity is not ﬁxed a priori, rather depends on a uniformly

distributed at random variable bounded in [S,T ]. The

genetic code z

code

reads as follow:

code

= A = [s

,...,s

] (10)

where J = |A|. The ﬁtness function f

reﬂects the

critic obtained by the classiﬁer c trained in the em-

bedding space built according to A, i.e. the accuracy

ω the classiﬁer c attained on the validation set. Given

the set of elite individuals Z

∗

, i.e. top-Z individuals

evaluated so far, the offspring

Z is generated accord-

ing to custom operators deﬁned as follows:

Crossover: two individuals z

and z

are selected uni-

formly at random from Z ∪ Z

∗

. A cut point is de-

termined according to the shortest code between

the two and a new individual is created according

to a one-point crossover operator.

Union: two individuals z

and z

are selected uni-

formly at random from Z ∪ Z

∗

and eventually a

new individual is deﬁned as the union set z

∪ z

Mutation: a single individual z

is uniformly at ran-

dom extracted from Z ∪ Z

∗

. With a given proba-

bility α

mut

, each symbol s ∈ z

has the chance to be

swapped with a symbol belonging to a randomly

extracted individual z

∈ Z ∪ Z

∗

The recombination process is repeated until

Z is pop-

ulated with L different individuals. The whole evo-

lutive orchestration can be schematically described as

follow:

1. Run the agent swarms Σ

2. Collect the agent symbols into H

,...,H

class-

speciﬁc buckets

3. Generate Z

4. Generate

Z by recombination of Z and Z

∗

5. Evaluate Z and

6. Reward the symbols in Z ∪

Z according to the

classiﬁcation performance

7. Evaluate the agents ﬁtnesses for the swarms in Σ

8. Evolve the agent swarms Σ

9. Replace Z

∗

by selecting the top-Z individuals

amongst Z ∪

Z ∪ Z

∗

The latter procedure highlights how the two different

swarms Σ and Z cooperate together. To summarize,

the agents in the swarms observe new sampled data

at each iteration in order to detect useful information

and improve their ability thanks to genetic optimiza-

tion that tries to maximize the quality of the agents’

output, i.e. their symbols. On the contrary, the pop-

ulation Z explores possible combinations of agents’

outputs for testing prospective embedding spaces and

simultaneously provides a supervised critic about the

quality of the agents’ output. The recombination of

Z with Z

∗

can be considered as an exploitation phase

where the most effective individuals observed so far

are mixed with just-explored ones in order to possibly

generate improved alphabets. Finally, Z

∗

is intended

to be created according to the selection pressure held

by the limited environment space, that is, only the best

Z individuals survive and will be part of the next gen-

eration.

NCTA 2021 - 13th International Conference on Neural Computation Theory and Applications

338

2.4 Test Phase

After the evolution converges and/or reach the maxi-

mum number of generations N

stop

, the ﬁnal elite pop-

ulation Z

∗

can be exploited for evaluate the solutions

obtained on a graph test set D

. Recalling that each

∗

code

∈ Z

∗

is an alphabet of symbols A, we can build

the symbolic histogram matrix F

of test set and train

a classiﬁer c according to F

, where as usual F

and

are respectively an |D

| × |A| and an |D

| × |A|

matrix. By repeating the embedding procedure for all

the solutions in Z

∗

, c

,...,c

classiﬁers are placed in

ensemble in order to possibly exploit simultaneously

all the information carried by the different embedding

space spanned by the symbolic histograms matrices.

The ﬁnal performance of the system is obtained by

equipping the ensemble with a winner-takes-all pol-

icy rule. That is, each classiﬁer in ensemble emits the

label for the symbolic histogram belonging to the test

set under analysis. Afterwards, the most voted label

is retained as the ﬁnal prediction.

3 TEST AND RESULTS

3.1 Datasets and Algorithmic Setup

In order to test Graph E-ABC, we consider the follow-

ing ﬁve open-access datasets from the IAM Reposi-

tory (Riesen and Bunke, 2008): AIDS, GREC, Letter-

L, Letter-M and Letter-H.

As the dissimilarities between nodes’ and edges’

attributes are concerned, all of them are customized

according to the nodes and edges attributes for each

dataset. GREC is the only dataset for which the

dissimilarity measures between nodes and edges are

parametric themselves: such values populate γ

γ which

shall be optimized, as described in Section 2.3.3. Full

details on the datasets, alongside formal deﬁnitions of

their dissimilarity measures between nodes and edges

can be found in (Martino and Rizzi, 2021) and (Bal-

dini et al., 2021).

The algorithm parameters are set as follows. For the

orchestration of agents in A:

• The sets of sampled subgraphs D

is built accord-

ingly to a random walk extraction

• The BSAS resolution θ assumes linearly spaced

values in range [0,1] with step size 0.1

• µ = 5 and λ = 15, the parents and offspring popu-

lation sizes (respectively)

• η = 0.5, weight between compactness f

and car-

dinality f

• The maximum number of generation N

stop

= 20.

For the orchestration of the model evolution:

• K = 10, the number of individuals in Z

• Z = 10, the number of best individuals in Z

∗

• L = 20, the number of recombined individuals in

• α

mut

= 0.15, the swapping probability in Section

2.2.

Other parameters include:

• t

min

= −10 and t

max

= 10, respectively the maxi-

mum penalty and the maximum reward values for

updating the symbol quality Q

• ρ assumes equally spaced values in the range [0,1]

with step size

stop

• The number of bins for the lookup table in Section

2.3.1 equals the number of classes, so it changes

in a dataset-dependent fashion

• ζ

= 1.1 · f

(C), where C is the cluster who gen-

erated the symbol s

The remaining two parameters (T and W ) are subject

to a sensitivity analysis, as detailed in the following.

3.2 Sensitivity Analysis

Amongst the parameters that characterize Graph E-

ABC, to the best of our judgement, two of them are

critical and need a careful tuning: the number of

subgraphs that agents need to extract before running

BSAS (W ) and the maximum number of class-related

subgraphs to be included in the alphabet (T ). In order

to address how these two parameters affect the per-

formances of the overall system, we perform a sensi-

tivity analysis via grid search. In particular, we chose

a set of candidate values for both T and W , namely

T = W = [10,50,100,200], and for each hT,W i-pair,

we run Graph E-ABC and record the average accu-

racy on the test set and the average number of sym-

bols across 10 different runs, in order to account the

stochastic nature of the algorithm.

In Figure 1 we show the sensitivity analysis of

Graph E-ABC with respect to the grid over T × W .

From the sensitivity analysis emerges that T is the

most critical parameter affecting the performances: in

particular, for AIDS and Letter-L, performances tend

to deteriorate as T → 0, whereas for harder classiﬁca-

tion problems such as Letter-M and Letter-M, perfor-

mances tend to deteriorate for larger T . GREC is the

only dataset for which performances seem to be rather

stable, regardless of T . As W is concerned, Letter-M

is the only dataset which shows an increasing trend in

performances as W grows, whereas for the remaining

four datasets a clear trend does not emerge.

A Multi-agent Approach for Graph Classiﬁcation

339

100

200

100

200

Accuracy Test Set [%]

(a) AIDS.

100

200

100

200

Accuracy Test Set [%]

(b) GREC.

100

200

100

200

100

Accuracy Test Set [%]

100

200

100

200

Accuracy Test Set [%]

(d) Letter-M.

100

200

100

200

Accuracy Test Set [%]

(e) Letter-H.

Figure 1: Graph E-ABC sensitivity analysis (average accuracy on the test set).

3.3 Comparison against Current

Granular Approaches for Graph

Classiﬁcation

In order to compare the performances of Graph

E-ABC, we select three suitable competitors that

(like Graph E-ABC) exploit the steps on information

granulation and graph embedding via symbolic his-

tograms: the class-aware version of GRALG, pro-

posed in (Baldini et al., 2021), and the RECTIFIER

and Dual-RECTIFIER classiﬁers proposed in (Mar-

tino and Rizzi, 2021). The major difference be-

tween Graph E-ABC and the three competitors is

that Graph E-ABC follows a ‘cooperative approach’,

where different agents exploit independent portions

of the dataset in order to search for suitable symbols

and later they join forces (via another set of agents)

for building the classiﬁcation model. GRALG, REC-

TIFIER and Dual-RECTIFIER, as instead, follow an

‘individualistic approach’, where all individuals start

from the same set of subgraphs extracted from the

training data that does not change throughout the evo-

lution and each individual independently looks for

suitable granules of information, performs the em-

bedding procedure and trains the classiﬁer in the em-

bedding space. The three competitors are driven by

a single-objective unimodal evolutionary procedure,

hence the best individual is retained for the synthe-

sis of the ﬁnal classiﬁcation system, to be validated

NCTA 2021 - 13th International Conference on Neural Computation Theory and Applications

340

on the test set. Amongst the three competitors, Dual-

RECTIFIER is the only one that (like Graph E-ABC)

exploit different optimization procedures in a class-

aware fashion, yet it is worth remarking that in Dual-

RECTIFIER such different optimization procedures

are independent one another, whereas in Graph E-

ABC there exist some sort of cooperation between

different swarms.

In Table 1 we report the results of the compari-

son amongst Graph E-ABC, GRALG, RECTIFIER

and Dual-RECTIFIER in terms of accuracy on the

test set and resulting number of symbols (i.e., size

of the embedding space). Results for GRALG, REC-

TIFIER and Dual-RECTIFIER are reported as func-

tion of visited symbols, expressed in percentage with

respect to the maximum attainable number of sub-

graphs that can be drawn from the training set (de-

tails in (Baldini et al., 2021) and (Martino and Rizzi,

2021)). For Graph E-ABC, as instead, we report (for

each dataset) the best point on the grids in Figure 1

(that is, the hT,W i-pair leading to the maximum ac-

curacy on the test set) and the average across all points

in the grid. In terms of accuracy, Graph E-ABC does

not rank amongst the most performing methods for

any of the datasets. However, the following observa-

tions arise: for easy classiﬁcation problems such as

AIDS and Letter-L, the shift of Graph E-ABC (best

point) with respect to the best performing algorithm

is negligible (≈ 1% and < 2%, respectively). The

same is not true for harder problems such as GREC

and Letter-H, where the accuracy shift with respect

to the best point is approximately 15% (GREC), 9%

(Letter-H). For Letter-M, a mid-hardness classiﬁca-

tion problem, the accuracy shift with respect to the

best performing algorithm is approximately 4%.

In terms of alphabet size, Graph E-ABC ranks

as the least suitable algorithm. This is due to the

ensemble-like nature of the synthesis of the model.

In fact, recall from Section 2.2, K different classiﬁers

are in charge of classifying the data in K different em-

bedding spaces. Hence, if we sum together the size of

each of the K embedding spaces, this inevitably leads

to a drastically higher number of symbols. However,

if we take the average number of symbols that each

model has to exploit (i.e., the sum of symbols divided

by K models), we can see that the results are rather in

line with the competitors for low subsampling rates

(i.e., more than 50%).

4 CONCLUSIONS AND FUTURE

DIRECTIONS

In this paper, we presented a promising multi-agent-

based algorithm (Graph E-ABC) for graph classiﬁca-

tion. Graph E-ABC exploits a two-fold optimization

routine in order to simultaneously improve the agents

(hence their ability in searching for meaningful sub-

graphs) and the set of subgraphs that compose the

classiﬁcation model. Computational results on open

Table 1: Comparison against current approaches for graph classiﬁcation system in terms of accuracy on the test set and, in

brackets, size of the embedding space (i.e., number of symbols). For each dataset, we highlight in bold the most performing

technique in terms of accuracy (in case of ties, the size of the embedding space acts as a tiebreaker).

Technique AIDS GREC Letter-L Letter-M Letter-H

Graph E-ABC (best point) 98.07 (34) 80.09 (1284) 96.89 (2260) 86.27 (4407) 79.78 (2517)

Graph E-ABC (average) 96.95 (222) 74.56 (1748) 92.20 (1460) 81.79 (3879) 74.76 (4592)

RECTIFIER (5%)

†

98.12 (6) 92.64 (87) 93.33 (23) 86.36 (52) 84.57 (90)

RECTIFIER (10%)

†

98.41 (8) 92.91 (169) 94.58 (32) 86.01 (71) 85.62 (145)

RECTIFIER (30%)

†

98.30 (9) 92.78 (415) 95.99 (79) 87.94 (173) 86.89 (300)

RECTIFIER (50%)

†

98.40 (16) 92.65 (439) 95.57 (81) 89.37 (232) 86.31 (431)

RECTIFIER (80%)

†

98.57 (16) 93.09 (741) 94.80 (157) 87.21 (294) 88.92 (668)

Dual-RECTIFIER (5%)

†

99.11 (6) 94.37 (67) 94.77 (19) 85.74 (27) 82.59 (49)

Dual-RECTIFIER (10%)

†

98.97 (9) 95.25 (77) 94.37 (22) 86.50 (42) 84.84 (86)

Dual-RECTIFIER (30%)

†

98.95 (11) 95.59 (173) 94.19 (29) 90.14 (77) 87.58 (162)

Dual-RECTIFIER (50%)

†

98.81 (13) 95.22 (264) 95.11 (35) 89.88 (104) 86.67 (177)

Dual-RECTIFIER (80%)

†

98.94 (37) 95.57 (215) 95.37 (50) 90.31 (136) 86.83 (275)

Class-Aware GRALG (10%)

‡

98.99 (9) 88.36 (288) 98.25 (146) 89.00 (214) 83.29 (287)

Class-Aware GRALG (30%)

‡

99.11 (11) 86.96 (335) 98.28 (205) 88.77 (277) 83.05 (319)

Class-Aware GRALG (50%)

‡

99.11 (12) 87.15 (322) 98.37 (218) 89.40 (311) 83.47 (313)

†

Averaged across different trade-off values in the objective function.

‡

Class-Aware Frequency Scaling heuristic for driving the granulation procedure.

A Multi-agent Approach for Graph Classiﬁcation

341

access datasets show interesting results in terms of ac-

curacy for 3 out of 5 datasets. However, these interest-

ing accuracy results are counterbalanced by a high di-

mensionality of the feature space. Despite promising,

this prototypical implementation has some drawbacks

that might affect the behaviour of Graph E-ABC. For

example, we can investigate whether there exist a

good trade-off when employing the ensemble, that is,

how the performance change as a function of K. In

fact, while a higher number of models in the ensemble

may lead to higher accuracy, it also leads to a higher

number of symbols (since, trivially, more embedding

spaces have to be tested). Further, another potential

drawback relates to how the quality of the symbols

is evaluated. In fact (recall from Section 2.3.1), the

quality of each symbol is evaluated independently to

the one of other symbols: this approach does not al-

low to capture the correlation amongst symbols, that

is, whether there exist ‘groups of symbols’ that are

responsible for a fruitful embedding space.

Certainly one of the most striking facets of Graph

E-ABC is the highly stochastic nature of the agent ﬁt-

ness functions due to the observation of continuously

different data shards sampled from the training set.

For such reason, a repeated evaluation of the same

individual can lead to a different ﬁtness value, mostly

due to the quality of the sampled data. A plain (µ +λ)

evolutionary scheme might not be suitable to efﬁ-

ciently deal with uncertain environments and future

investigation will be directed to design speciﬁc evo-

lutionary strategies able to take actions against noisy

ﬁtness functions (Bhattacharya et al., 2014; Branke

et al., 2001; Merelo et al., 2016).

An additional improvement may regard the deﬁni-

tion of the symbol quality (see Section 2.3.1). In fact,

suitable reward values may change drastically from

one dataset to another and a poor user-deﬁned choice

can undermine not only the symbol quality (see Eq.

(7)), but also the agent quality (see Eq. (8)). A suit-

able countermeasure would be using adaptive strate-

gies for populating the reward values in the lookup

table. This would also limit the huge number of free-

parameters to be deﬁned by the end-user.

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