The dynamical structure of political corruption networksLuiz GA Alves1,∗, Haroldo V Ribeiro2, Alvaro F Martins2, Ervin K Lenzi3, Matjaž Perc4,5

1University of São Paulo, 2State University of Maringá, 3State University of Ponta Grossa, 4University of Maribor, 5Complexity Science Hub

Abstract

Corruptive behaviour in politics limits economic growth, embezzles public funds and promotes socio-economic inequality in modern democracies. We analyse well-documented political corruption scandalsin Brazil over the past 27 years, focusing on the dynamical structure of networks where two individualsare connected if they were involved in the same scandal. Our research reveals that corruption runs insmall groups that rarely comprise more than eight people, in networks that have hubs and a modularstructure that encompasses more than one corruption scandal. We observe abrupt changes in the sizeof the largest connected component and in the degree distribution, which are due to the coalescence ofdifferent modules when new scandals come to light or when governments change. We show further thatthe dynamical structure of political corruption networks can be used for successfully predicting partnersin future scandals. We discuss the important role of network science in detecting and mitigating politicalcorruption.

10−2

10−1

100

0 10 20 30 40 5010−2

10−1

100

0 10 20 30 40 50

Cas

oBa

nesp

a(1

987)

Ferro

via

Nor

te-S

ul(1

987)

CPI

daC

orru

pção

(198

8)M

áfia

daPr

evid

ênci

a(1

991)

Cas

oC

ollo

r(19

92)

Anõe

sdo

Orç

amen

to(1

993)

Paub

rasi

l(19

93)

Escâ

ndal

oda

Para

bólic

a(1

994)

Past

aro

sa(1

995)

Com

pra

devo

tos

para

are

elei

ção

(199

7)Es

când

alo

das

priv

atiz

açõe

s(1

997)

Fran

goga

te(1

997)

Prec

atór

ios

(199

7)D

ossi

êC

aym

an(1

998)

Gra

mpo

sdo

BND

ES(1

998)

Máf

iado

sfis

cais

(199

8)C

aso

Mar

ka/F

onte

Cin

dam

(199

9)D

esvi

osde

Verb

asdo

TRT-

SP(1

999)

Gar

otin

hoe

atu

rma

doC

huvi

sco

(200

0)Su

dam

(200

1)Vi

olaç

ãodo

pain

eldo

Sena

do(2

001)

Bunk

erpe

tista

(200

2)C

aso

Cel

soD

anie

l(20

02)

Cas

oLu

nus

(200

2)C

PIdo

Bane

stad

o(2

003)

Ope

raçã

oAn

acon

da(2

003)

Banp

ará

(200

4)C

aso

GTe

ch(2

004)

Cas

oW

aldo

miro

Din

iz(2

004)

Supe

rfatu

ram

ento

deob

ras

emSP

(200

4)C

orru

pção

nos

Cor

reio

s(2

005)

Dól

ares

nacu

eca

(200

5)Es

când

alo

doM

ensa

lão

(200

5)R

epúb

lica

deR

ibei

rão

(200

5)Va

lerio

duto

min

eiro

(200

5)Al

opra

dos

(200

6)Es

când

alo

dos

sang

uess

ugas

(200

6)Q

uebr

ade

sigi

lodo

case

iroFr

ance

nild

o(2

006)

Che

que

daG

ol(2

007)

Ren

anga

teC

aso

dola

ranj

alal

agoa

no(2

007)

Ren

anga

teC

aso

Môn

ica

Velo

so(2

007)

Ren

anga

teC

aso

Schi

ncar

iol(

2007

)R

enan

gate

Gol

peno

INSS

(200

7)C

artõ

esco

rpor

ativ

os(2

008)

Cas

oSa

tiagr

aha

(200

8)D

ossi

êco

ntra

FHC

eR

uth

Car

doso

(200

8)Pa

ulin

hoda

Forç

ae

oBN

DES

(200

8)At

osSe

cret

os(2

009)

Cas

oLi

naVi

eira

(200

9) Men

salã

odo

DEM

(200

9)Ba

ncoo

p(2

010)

Cas

oEr

enic

e(2

010)

Os

novo

sal

opra

dos

(201

0)Es

când

alo

emC

idad

es(2

011)

Escâ

ndal

ona

Agric

ultu

ra(2

011)

Escâ

ndal

ono

Espo

rte(2

011)

Escâ

ndal

ono

sTr

ansp

orte

s(2

011)

Escâ

ndal

ono

Trab

alho

(201

1)Es

când

alo

noTu

rism

o(2

011)

Cas

oC

acho

eira

(201

2)Es

când

alo

naPe

sca

(201

2)O

pera

ção

Porto

Segu

ro(2

012)

Máf

iado

ISS

(201

3)O

pera

ção

Lava

Jato

(201

4)Pe

trolã

o(2

014)

0

10

20

30

40

−0.20.00.20.40.60.81.0

0 2 4 6 8 10

−0.20.00.20.40.60.81.0

0 2 4 6 8 100

15

30

45

60

75

1986 1992 1998 2004 2010 20160

15

30

45

60

75

1986 1992 1998 2004 2010 2016

Num

ber o

f peo

ple

Cum

ulat

ive

dist

ribut

ion

Number of people Year Lag (years)

Num

ber o

f peo

ple

Auto

corre

latio

n

Corruption case

DCB

A

7.51 ± 0.03

1.2 ± 0.4

Figure 1: Demography and evolving behaviour of corruption scandals in Brazil. A) The number ofpeople involved in each corruption scandal in chronological order (from 1987 to 2014). B) Cumulativeprobability distribution (on a log-linear scale) of the number of people involved in each corruption scandal(red circles). C) Time series of the number of people involved in corruption scandals by year (red circles).The alternating gray shades indicate the term of each general election that took place in Brazil between1987 and 2017. D) Autocorrelation function of the time series of the yearly number of people involved inscandals (red circles).

Figure 2: Complex network representation of people involved in corruption scandals. Complex networkof people involved in all corruption cases in our dataset (from 1987 to 2014). Each vertex represents aperson and the edges among them occur when two individuals appear (at least once) in the same corruptionscandal. Node sizes are proportional to their degrees and the colour code refers to the modular structure ofthe network. There are 27 significant modules, and 14 of them are within the giant component (indicatedby the red dashed loop).

Figure 3: Characterization of nodes based on the within-module degree (Z) and participation coefficient(P ). Each dot in the Z-P plane corresponds to a person in the network and the different shaded regionsindicate the different roles according to the network cartography proposed by Guimerà and Amaral. Themajority of nodes (97.5%) are classified as ultraperipheral (R1) or peripheral (R2), and the remaining arenon-hub connectors (R3, three nodes), provincial hubs (R5, three nodes), and connector hubs (R6, twonodes).

0

4

8

12

16

20

1986 1992 1998 2004 2010 20160

4

8

12

16

20

1986 1992 1998 2004 2010 2016

10−3

10−2

10−1

100

0 20 40 60 80 10010−3

10−2

10−1

100

0 20 40 60 80 100

1991

2003

2014

17.6

7.2

2.4Cum

ulat

ive

dist

ribut

ion

10−3

10−2

10−1

100

0 1 2 3 4 510

−3

10−2

10−1

100

0 1 2 3 4 5

1988

1992

1996

2000

2004

2008

2012

Year

1

Cum

ulat

ive

dist

ribut

ionA B

C

Vertex degree Rescaled vertex degree

Cha

ract

eris

tic d

egre

e

Year

Dilm

a

Lula

FHC

Itam

arC

ollo

r

Sarn

ey

PTPTPSDB

PMDB

PMDB

PRN

Pres

iden

tPa

rty

2.4 ± 0.1

5.9 ± 0.8

17.9 ± 0.9

Figure 4: The vertex degree distribution is exponential, invariant over time, and the characteristic degreeexhibits abrupt changes over the years.

0.0

0.2

0.4

0.6

0.8

1.0

1986 1992 1998 2004 2010 20160.0

0.2

0.4

0.6

0.8

1.0

1986 1992 1998 2004 2010 2016

−0.2

−0.1

0.0

0.1

0.2

0.3

0.4

1986 1992 1998 2004 2010 2016

−0.2

−0.1

0.0

0.1

0.2

0.3

0.4

1986 1992 1998 2004 2010 2016

Siz

e o

f th

e la

rge

st

clu

ste

r

Year

Dilm

a

Lu

la

FH

C

Ita

ma

r

Sa

rne

y

PT

PTPS

DB

PM

DB

PM

DB

PR

N

Year

Dilm

a

Lu

la

FH

C

Ita

ma

r

Sa

rne

y

PT

PTPS

DB

PM

DB

PM

DB

PR

N

Pre

sid

en

tP

arty

Gro

wth

ra

te o

f th

e la

rge

st

clu

ste

r

C

BA

Máfia da

previdência

Caso Collor

Ferrovia

Norte-SulCPI da corrupção

Grampo do BNDES

Dossiê Cayman

Máfia dos fiscais

Dossiê Cayman Máfia dos fiscais

CPI Banestado

Grampo do

BNDES

Máfia da

previdência

Caso Marka

Fonte Cindam

Ferrovia

Norte-Sul

CPI da corrupção

Desvios de verba

do TRT

Precatórios

Frangogate

Paubrasil Compra de votos Caso Banespa

Anões do orçamento

Escândalo da

parabólica

Pasta rosa

Frangogate

Paubrasil

Superfaturamento

de obras em SP

Precatórios

Escândalo da parabólica

República de

Ribeirão

Valerioduto mineiro

Corrupção nos correiosDólares na

cueca

Escândalo do

Mensalão

Caso

Celso DanielOperação

anaconda

Caso

Waldomiro Diniz

Caso GTech

Banpará Sudam

Pasta rosa

Violação do

painel do Senado

Bunker petista

Anões do

orçamento

Caso Banespa

Garotinho e a turma

do chuvisco

Caso

Lunus

Compra de votos

para reeleição

Escâncalo das

privatizações

Escâncalo das

privatizações

New nodes

Year: 1998Year: 1997

Year: 2005Year: 2004

Máfia da Previdência

CPI da Corrupção

Ferrovia Norte-Sul

Caso Banespa

Máfia da Previdência

CPI da Corrupção

Ferrovia Norte-Sul

Caso Banespa

Caso Collor

Year: 1992

Year: 1991

Co

llor

Co

llor

Figure 5: Changes in the size of the largest component of the corruption network over time are causedby a coalescence of network modules.

Fra

ctio

n o

f co

rre

ct

pre

dic

tion

s

Predictor

Sim

Ra

nk

Ro

ote

d P

ag

eR

an

k

Re

so

urc

e a

lloca

tion

Ja

cca

rd

Co

sin

e

Asso

cia

tion

str

en

gth

De

gre

e p

rod

uct

Ad

mic

\Ad

ar,

co

mm

on

ne

igh

bo

urs

,

HR

V,

an

d K

atz

Ra

nd

om

0.00

0.05

0.10

0.15

0.20

0.25

Figure 6: Predicting missing links between people in the corruption network may be useful for investigat-ing and mitigating political corruption. We tested the predictive power of eleven methods for predictingmissing links in the corruption networks. These methods are based on local similarity measures (degreeproduct, association strength, cosine, Jaccard, resource allocation, Adamic-Adar, and common neighbours),global (path- and random walk-based) similarity measures (rooted PageRank and SimRank), and on thehierarchical structure of networks (hierarchical random graph – HRV). To access the accuracy of thesemethods, we applied each algorithm to snapshots of the corruption network in a given year (excluding2014), ranked the top-10 predictions, and verified whether these predictions appear in future snapshots ofthe network. The bar plot shows the fraction of correct predictions for each method. We also included thepredictions of a random model where missing links are predicted by chance (error bars are 95% bootstrapconfidence intervals).

Reference

• HV Ribeiro, LGA Alves, AF Martins, EK Lenzi, M Perc The dynamical structure of politicalcorruption networks, Journal of Complex Networks CNY002, 1-15 (2018). ∗lgaalves@usp.br