The (not so) Critical Nodes of Criminal Networks
Donatella Firmani1 , Giuseppe F. Italiano1 , Luigi Laura2
1 University
of Rome “Tor Vergata”
2 Sapienza
University of Rome
CrimeNet 2014 - Barcellona, España - November 10th , 2014
Motivations
“Criminal networks are not simply social networks operating in
criminal contexts. The covert settings that surround them call for
specific interactions and relational features within and beyond the
network.”
Carlo Morselli, Inside Criminal Networks. Springer, 2009.
Peculiar features of Criminal Networks
Sparrow, in a seminal paper [23], listed four peculiarities:
I
limited size;
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information incompleteness (i.e., criminal network data is
inevitably incomplete);
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undefined borders, i.e. it is not easy to discover all the
connections of a node;
I
dynamics, that is, many of the useful networks questions
depends heavily on the temporal dimension.
In this scenario, SNA tools are limited, due to this intrinsic
differences!
Who are the key players in a network?
I
One of the most basic questions associated to the analysis of
a network is who are the key players, or who are the central
nodes in the network?
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Node-centrality measures, such as degree, betweenness and
closeness centrality, can successfully identify most key players
in social networks (see the survey of Borgatti, [8]).
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On the contrary, in criminal networks, the most important
actors do not necessarily display high centrality scores (Baker
and Faulkner, [4]).
Our contribution
We show that that criminal networks can suffer high disconnection
when few nodes are deleted. However, while in social networks
nodes whose removal disconnects most nodes from the network
belong to their cores, in criminal networks this is not the case.
This suggests that:
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nodes that pulls together the network deliberately operate
from network peripheries, thus being protected from detection;
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the key players in the network may not be the ones whose
removal affects large portions of the network.
Preliminaries: Graph connectivity
D
A
F
B
J
G
C
E
H
M
K
I
N
L
An example graph with two connected components and four
articulation points (B, G, H, K).
Preliminaries: Graph Cores
An example graph, its 2-core (light and dark grey nodes), and its
3-core (dark grey nodes). Nodes in the 3-core are a subset of the
nodes in the 2-core of the graph.
Preliminaries: Critical Nodes
I
The concept of critical nodes has been introduced by Ausiello
et al. in [2], building upon the concepts of articulation points
and core of a connected network.
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Precisely, the critical nodes are defined as the articulation
points belonging to the network core.
Using different samples of the Autonomous Systems network,
Ausiello et al. show that:
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I
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the removal of few critical nodes can affect large portions of
the network, thus they are central in a very strong sense [2];
the critical nodes have orders of magnitude higher centrality
scores than other nodes [3].
Relations between Articulation Points and Critical Nodes
If we focus on the 2-core, we can state the following
Lemma
Let G be an undirected connected graph. The critical nodes of G
(i.e. the articulation points of G 2 ) are articulation points of G .
[The proof is in the paper]
Relations between Articulation Points and Critical Nodes
If we focus on the 3-core, we can say that the previous relation
does not hold, as shown in this figure
The red node is not an articulation point of the graph, but it is
after the removal of the blue node (that is the only node removed
when we consider the 3-core of the graph).
The network datasets
Name
Drug-Traffic
Terrorists
Karate
Science
Facebook
Lindenstrasse
Marvel
Airlines
PowerGrid
ASes
Type
Criminal Network
Criminal Network
Social Network
Social Network
Social Network
Fictitious SN
Fictitious SN
Other
Other
Other
Nodes (n)
2749
4275
34
1589
4039
233
10822
235
4941
22963
Edges (m)
13578
7874
78
2742
88234
325
314054
1297
6594
48436
Source
Mainas [15]
Mainas [15]
GEPHI [13]
Pajek [5]
SNAP [22]
Pajek [5]
GEPHI [13]
GEPHI [13]
GEPHI [13]
GEPHI [13]
Experimental results
We measured, for each dataset:
1. the number of articulation points, in short APs;
2. the number of critical nodes, in short CNs;
3. the number of critical nodes in the 3-core, in short CN3 s;
4. the number of CNs and CN3 s that belong to the top K
articulation points, sorted by the impact, where K is
respectively the number of CNs or CN3 s.
Experimental Results:
Articulation Points and Critical Nodes
G
Drug-Traffic
Terrorists
Karate
Science
Facebook
Lindenstrasse
Marvel
Airlines
PowerGrid
ASes
n
2749
4275
34
1589
4039
233
10822
235
4941
22963
m
13578
7874
78
2742
88234
325
314054
1297
6594
48436
n
1554
4085
34
379
4039
233
10822
235
4941
22963
largest CC
m
2216
6358
78
914
88234
325
314054
1297
6594
48436
APs
280
521
1
57
11
74
107
9
1229
1870
n
454
1303
33
352
3964
123
10543
201
3353
14966
G2
m
1116
3576
78
887
88159
215
313775
1263
5006
40439
APs
15
77
1
44
7
1
3
2
94
10
n
163
681
22
265
3856
12
9935
162
231
4383
G3
m
588
2440
55
736
87952
19
312565
1190
479
19678
APs
0
7
1
1
1
0
2
1
1
1
Metric: Impact of Articulation Point
We define impact of a node v is given by the number of nodes that
get disconnected from the largest connected component when v is
removed.
We use the impact to derive two ratios:
|CNs∩(Top APs)|
|CNs|
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membership ratio:
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weighted impact ratio: the sum of the impacts of the CNs (or
of the CN3 ) divided by the sum of the impacts of the top K
APs.
An example
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The Marvel network has 107 APs, 3 CN, and 2 CN3 s.
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The top APs have impacts: {39, 27, 15, 13, 12, . . .}.
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The three CNs have impact values equals to {39, 4, 3}.
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Of the two CN3 s, none belongs to the top two APs.
Thus:
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The membership ratio of the CNs is 1/3.
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The weighted membership ratio of the CNs is
39/(39 + 27 + 15).
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The membership ratio and weighted membership ratio of the
CN3 s is zero.
Impact of Articulation Points and Critical Nodes
Drug-Traffic
Terrorists
Karate
Science
Facebook
Lindenstrasse
Marvel
Airlines
PowerGrid
ASes
lCC
APs
CNs
280
521
1
57
11
74
107
9
1229
1870
15
77
1
44
7
1
3
2
94
10
G2
CNs in
TOP APs
8
40
1
44
7
1
1
1
47
10
CN3 s
0
7
1
1
1
0
2
1
1
1
G3
CN3 s in
TOP APs
−
2
1
1
1
−
0
1
0
1
Max Impact
AP CN CN3
172
83
7
60
197
13
39
11
106
333
57
83
7
60
197
13
39
11
106
333
−
83
7
60
197
−
1
11
8
333
Overall impact of Critical Nodes
Criminal Networks
Social Networks
Fictitious SNs
Other
100
80
60
40
20
0
Dr
ug
-Tr
Te
aff
ic
rro
ris
Ka
ts
rat
e
Sc
ien
ce
Fa
Lin
ce
Ma
de
bo
o
k
ns
tr
rve
as
s
e
l
Air
lin
es
Po
we
AS
rG
rid
es
Membership ratio (yellow) and the weighted impact ratio (red).
Overall impact of CN3 , the Critical Nodes of the 3-core
Criminal Networks
Social Networks
Fictitious SNs
Other
100
80
60
40
20
0
Te
rro
ris
ts
Ka
rat
e
Sc
ien
ce
Fa
ce
bo
ok
Ma
rve
l
Air
lin
es
Po
AS
we
rG
rid
es
Membership ratio (yellow) and the weighted impact ratio (red).
Conclusions
We used the notion of Critical Nodes to compare Criminal
Networks and Social Networks (and other networks). Our findings
are that:
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as for the SNs, the removal of few nodes can have a huge
impact on the overall network...
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...but, differently from SNs, these nodes do not belong to the
core of the network!