Distributed and Streaming Graph Processing Techniques

DISTRIBUTED AND STREAMING
GRAPH PROCESSING TECHNIQUES
P. N. LIAKOS
University of Athens, June 15th, 2018
For the Fat Lady

Say you wish to study . . .
UoA Panagiotis Liakos Motivation 2/63

Say you wish to study . . .
. . . and now imagine that the data is available . . .

. . . but it’s more than you can handle!
11100111000010000011010100100100000101100100110011101010111111000
10101011010101001111101111010001010100001100111011011101100001000
01011110001010010101000001101010101100100111000101101011010101010
00010010001101110111001010110001110010100001000101010001110001010
10001010001000111000000011010010110111101101110010000001111101110
01010110011111010101001010000001100001100100100010100010111100100
01100100001101010001100000111000010001011100100011100010100000110
00100110011000010001011100001001000100010110010111110011110101001
01000010001010110001110010000111010010101011100001111000000111001
11000100101110110011100000001001001110011110011000000001011100001
10111100101110101110110101000101000101110100100100111010001011110
10101010011011011000110011011001011101000010011010111110000100110
00010000001110001111001110001100001011001011001101111101001011010
01111101111010110101111010100011111101100111001010110100110110001
00100101001111111100110010010100001111101110100101101111100010000
00011110111001011110000111000101100001101111011010110111001100100
01010010001001000000000111101000011001000101001101000101000101110
01000011110010100101101110011000111011100110111010101110010111110
2
billion
users!
500 million tweets per day!
1
trillion
URLs!

Large Scale Graph Processing
Active Research Areas:
Distributed Graph Processing Graph Streams
Worker 1 Worker 2
Worker 3
127
2
9
11
12
10
14
17
18
20
Pregel-like
graph processing
system

Our Contribution
Distributed Graph Processing (Pregel paradigm)
Memory Optimization
Opinion Formation
Local Community Detection
Graph Streams
Streaming Community Detection
Sampling High Quality Content

Outline
1 Realizing Memory-Optimized Distributed Graph Processing
2 COEUS: Community detection via seed-set expansion on
graph streams
3 Scalable Link Community Detection: A Local
Dispersion-aware Approach
4 Rhea: Adaptively Sampling High Quality Content from
Social Activity Streams
5 On the Impact of Social Cost in Opinion Dynamics
UoA Panagiotis Liakos Realizing Memory-Optimized Distributed Graph Processing 6/63

Apache Giraph Memory Usage [CEK+
15]

Apache Giraph Memory Usage [CEK+
15]
• Ineﬀective memory usage
• Partitioning hardens the task of compression

Related memory optimization approaches
Apache Giraph [CEK+15]:
does not exploit the redundancy in real-world graphs
Ligra+ [SDB15]:
compression techniques on a shared-memory system
halved space usage at the cost of slower execution
Gbase[KTS+12]:
does not follow the vertex-centric model
requires decompression
GraphChi [KBG], FlashGraph [ZMB+15] & Graphene [LH17]:
maintain graph data on disks
reasonable performance with very modest requirements

Contribution
We present a number of novel techniques that:
1 oﬀer space eﬃcient-representations of out-edges,
2 allow fast mining (in-situ) of the graph elements without
the need of decompression,
3 enable the execution of graph algorithms in
memory-constrained settings, and
4 ease the task of memory management, thus allowing
faster execution.
– PL, Katia Papakonstantinopoulou, Alex Delis: Memory-Optimized Distributed Graph Processing through
Novel Compression Techniques. ACM CIKM 2016 & ACAC 2016
– PL, Katia Papakonstantinopoulou, Alex Delis: Realizing Memory-Optimized Distributed Graph Processing.
IEEE Trans. Knowl. Data Eng. 30(4), 2018

Properties of real-world graphs
Locality of reference: the majority of the edges of a
graph link vertices that are close to each other in the
order
Similarity (or copy property): vertices that are close to
each other in the order tend to have many common
out-neighbors
http://www.di.uoa.gr/
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http://www.di.uoa.gr/images/logo.gif
http://www.di.uoa.gr/about/
http://www.di.uoa.gr/staff/
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http://www.di.uoa.gr/images/logo.gif
http://www.di.uoa.gr/directions.html
http://www.di.uoa.gr/staff/

BVEdges (based on [BV04])
2, 9, 10, 11, 12, 14, 17, 18, 20, 127
⇓
9 − 12, 2, 14, 17, 18, 20, 127
⇓
(1)2
4 bytes
γ(0)(9)2 ζ(13)ζ(11) ζ(2)ζ(0)ζ(1) ζ(106)
4 bytes
number of
intervals interval residuals
{
{
{
1 bit 7 bits 7 bits 4 bits 3 bits 4 bits 11 bits

BVEdges (based on [BV04])
2, 9, 10, 11, 12, 14, 17, 18, 20, 127
⇓
9 − 12, 2, 14, 17, 18, 20, 127
⇓
(1)2
4 bytes
γ(0)(9)2 ζ(13)ζ(11) ζ(2)ζ(0)ζ(1) ζ(106)
4 bytes
number of
intervals interval residuals
{
{
{
1 bit 7 bits 7 bits 4 bits 3 bits 4 bits 11 bits
• bit-encoding involves signiﬁcant overhead

IntervalResidualEdges
2, 9, 10, 11, 12, 14, 17, 18, 20, 127
⇓
9 − 12, 17 − 18, 2, 14, 20, 127
⇓
(2)2
4 bytes
number of
intervals
(9)2 (4)2 (17)2 (2)2
4 +1 bytes
1st
interval
4 +1 bytes
2nd
interval
(2)2 (14)2 (20)2 (127)2
4 bytes 4 bytes 4 bytes 4 bytes
residuals
{UoA Panagiotis Liakos Realizing Memory-Optimized Distributed Graph Processing 12/63

2, 9, 10, 11, 12, 14, 17, 18, 20, 127
⇓
9 − 12, 17 − 18, 2, 14, 20, 127
⇓
(2)2
4 bytes
number of
intervals
(9)2 (4)2 (17)2 (2)2
4 +1 bytes
1st
interval
4 +1 bytes
2nd
interval
(2)2 (14)2 (20)2 (127)2
4 bytes 4 bytes 4 bytes 4 bytes
residuals
{• avoids expensive encodings & bit streams
• signiﬁcant compression due to locality of reference

IndexedBitArrayEdges
2, 9, 10, 11, 12, 14, 17, 18 20, 127
⇓
{2}, {9, 10, 11, 12, 14}, {17, 18 20}, {127}
⇓
...
(0)2
(1)2
(2)2
(15)2
4 bytes 1 byte

2, 9, 10, 11, 12, 14, 17, 18 20, 127
⇓
{2}, {9, 10, 11, 12, 14}, {17, 18 20}, {127}
⇓
...
(0)2
(1)2
(2)2
(15)2
4 bytes 1 byte
• avoids expensive encodings & bit streams
• signiﬁcant compression due to locality of reference
• memory-eﬃcient retrieval of out-edges

VariableByteWeights
32, 378
⇓
{32}, {256 + 122}
⇓
0
1 byte
1st weight 2nd weight
{
{
1 00 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 0 01
1 byte 1 byte

VariableByteWeights
32, 378
⇓
{32}, {256 + 122}
⇓
0
1 byte
1st weight 2nd weight
{
{
1 00 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 0 01
1 byte 1 byte
• weights of edges exhibit
right-skewed distributions [BBPSV04, MAF]
• log128(n) + 1 bytes to represent an integer n

RedBlackTreeEdges
2, 9, 10, 11, 12, 14, 17, 18 20, 127
⇓
127
20
18
17
14
12
11
10
9
2
key: 8 bytes (long)
le�: 4 bytes (compressed oop)
right: 4 bytes (compressed oop)
color: 1 byte (boolean)
depth: 2 bytes (short)

RedBlackTreeEdges
2, 9, 10, 11, 12, 14, 17, 18 20, 127
⇓
127
20
18
17
14
12
11
10
9
2
key: 8 bytes (long)
le�: 4 bytes (compressed oop)
right: 4 bytes (compressed oop)
color: 1 byte (boolean)
depth: 2 bytes (short)
• does not waste space for empty buckets
• signiﬁcant savings using primitive data types
• cost-free iterations with regards to space

Experimental Evaluation
Space-eﬃciency
Performance:
when the available memory is not constrained
when the available memory is constrained
when adding/removing edges

Space Eﬃciency Comparison
ByteArray- BVEdges IntervalRe- IndexedBit-
graph Edges sidualEdges ArrayEdges
uk-2007-05@100000 22.61 MB 6.41 MB (0.96 MB) 7.92 MB 8.91 MB
uk-2007-05@1000000 279.16 MB 67.36 MB (10.54 MB) 82.7 MB 97.79 MB
ljournal-2008 866.36 MB 386.73 MB (117.68 MB) 497.52 MB 648.52 MB
indochina-2004 1,511.67 MB 442.34 MB (48.03 MB) 646.03 MB 554.23 MB
hollywood-2011 1,381.91 MB 287.53 MB (145.85 MB) 613.52 MB 676.88 MB
uk-2002 2,733.6 MB 1,092.82 MB (116.39 MB) 1,224.07 MB 1,255.67 MB
arabic-2005 4,820.09 MB 1,428.97 MB (187.58 MB) 1,674.75 MB 1,849.83 MB
uk-2005 7,401.88 MB 2,383.54 MB (279.45 MB) 2,728.74 MB 2,928.81 MB
twitter-2010 11,189.88 MB 4,628.48 MB (2,600.07 MB) 7,127.76 MB 8,888.50 MB
sk-2005 14,829.64 MB 4,889.85 MB (607.92 MB) 5,657.79 MB 6,354.17 MB
BVEdges < 40% ByteArrayEdges
impressive savings with all techniques

Performance / small-scale graphs
0
5
10
15
20
25
30
8 workers 4 workers 2 workers
Executiontime(inminutes)
ByteArrayEdges
BVEdges
BVEdges is inferior speedwise
IntervalResidualEdges and IndexedBitArrayEdges
already show signs of improvement

Performance / large-scale graphs
0
1
2
3
4
5
0 5 10 15 20 25 30
Executiontime(inminutes)
Supersteps of PageRank execution
ByteArrayEdges
BVEdges
ByteArrayEdges’s performance ﬂuctuates
due to excessive garbage collection.

Performance / large-scale graphs
0
50
100
150
200
uk-2005 (5 workers) uk-2005 (4 workers)FAILED
Executiontime(inminutes) ByteArrayEdges
BVEdges
IntervalResidualEdges is faster than ByteArrayEdges
IndexedBitArrayEdges outperforms all

Performance / algorithms involving mutations
0
2
4
6
8
10
12
10 20 30 40 50 60 70 80 90 100
Executiontime(min)
Maximum Mutations Allowed
HashMapEdges
RedBlackTreeEdges
RedBlackEdges requires less than half of the space that
HashMapEdges needs
the performance of HashMapEdges deteriorates signiﬁcantly
as the number of allowed mutations grows

Outline
graph streams
UoA Panagiotis Liakos COEUS: Community detection via seed-set expansion on graph streams 22/63

Climate change conversation on Twitter
carbonbrief.org

carbonbrief.org
real-world
networks
are massive!

carbonbrief.org
real-world
networks
are massive!
change rapidly!

carbonbrief.org
real-world
networks
are massive!
change rapidly!
exhibit commu-
nity structure!

Motivation
We want to extract the community structure of nodes
in a network that changes rapidly.
Many useful applications:
we can provide more informative & engaging social
network feeds
we can enhance the eﬃciency of recommender systems
Size of graph data appears to be ever-increasing:
Facebook has more than 2 billion registered users
Google indexes more than 1 trillion unique URLs

Our context
5
2
8
3
6
4
7
1
9 8
2 3
...
Communities initialized
with seed-sets
Graph stream

Related Work - Static Graph
Non-overlapping Algorithms:
[GN02, NG04, BGLL08, CNM04, PL05, RB11]
Edge Betweeness
Modularity maximization
Random-walks
Overlapping Algorithms: [PDFV05, ABL10, EL09]
Clique Percolation
Hierarchical Link Clustering
More Scalable Overlapping Algorithms:
[CRGP12, YL13, GS12, WGD13]
Egonets
Matrix Factorization
Seed-set Expansion
Local Algorithms: [KG14, LHBH15, HSB+15]
Focus is shifted to local structure
Seed-set Expansion

Related Work - Graph Stream / Dynamic Graph
Yun et al. [YLP14]:
rows of the adjacency matrix of the graph are revealed
sequentially
Zakrzewska and Bader [ZB15]:
dynamic graphs
seed set expansion
incrementally adjust to dynamic changes
Hollocou et al. [HMBL17]:
if we pick uniformly at random an edge of the graph, this
edge is more likely to link nodes of the same community,
than nodes from distinct communities
if we process edges in a random order we expect many
intra-community edges to arrive before the
inter-community edges

Contribution
We propose COEUS:
A novel community detection algorithm that operates on a graph stream,
using space sublinear to the number of edges.
We also suggest:
A PageRank-like A Novel Clustering Technique
Edge Quality Variation for Community Size Determination
We are extremely competitive with non-streaming approaches and
our execution time and space requirements are astonishingly low.
– PL, Alexandros Ntoulas, Alex Delis: COEUS: Community detection via seed-set expansion on graph streams.
IEEE BigData 2017 & HDMS 2018

COEUS*
*
the axis of heaven around which the constellations revolved

COEUS*
*
the axis of heaven around which the constellations revolved
Community detection
O
via seed-set Expansion
U
on graph Streams

Community Participation Value
No universal definition of what a community is!
We define community participation of node u in community C:
cp(u) =
|{(u, v) ∈ E : v ∈ C}|
|{(u, v) ∈ E}|
,
the fraction of its adjacent nodes in the graph
that are part of the community.
Our evaluation does not consider a particular quality function.
Effectiveness is measured using ground-truth communities.

Space Complexity
We maintain for every community c:
the set of nodes that constitute community c,
the degree of each node u ∈ V, and
the community degree of each node u ∈ c.
Number of communities might be large!
COUNT-MIN SKETCH:
Sublinear space data structure providing strong accuracy guarantees.

Our method in a glance
Initialize the communities
using the seed-sets
Process the edge stream and
populate the communities
Prune the communities
Termination: COEUS handles
both ﬁnite & inﬁnite streams and
can be stopped at will.
Algorithm 1: COEUS
input : A set of community seed-sets K , and a graph stream S
output : A set of communities C
begin
foreach K ∈ K do
C {};
foreach k ∈ K do
C[k] = 1;
C .put(C);
while ∃(u, v) ∈ S do
degreeV [u]+ = 1;
degreeV [v]+ = 1;
foreach C ∈ C do
if u ∈ C then
degreeC [v]+ = 1;
if v ∈ C then
degreeC [u]+ = 1;
if u ∈ C then
C.put(v);
if v ∈ C then
C.put(u);
processedElements+ = 1;
if processedElements mod W == 0 then
C prune(C, s, degreeV , degreeC );

Reckoning in edge quality
w.r.t. each community

PageRank-like Edge Quality variation
Updating the community degrees:
We do not consider the level of involvement of
the adjacent nodes in the community.
All nodes included in a community provide increments of 1
to all of their adjacent nodes.
Reckoning in edge quality:
We improve over a simple community degree measure
by considering the edge quality of nodes w.r.t. each community.

COEUScp
The increment for each node
grows with its involvement
in the community.
If this value is high, then
the probability that an adja-
cent node is a member of the
community is also high.
Algorithm 2: COEUScp
input : A set of community seed-sets K , and a graph stream S
output : A set of communities C
begin
foreach K ∈ K do
C {};
foreach k ∈ K do
C[k] = 1;
C .put(C);
while ∃(u, v) ∈ S do
degreeV [u]+ = 1;
degreeV [v]+ = 1;
foreach C ∈ C do
if u ∈ C then
degreeC [v]+ =
degreeC [u]
degreeV [u]
;
if v ∈ C then
degreeC [u]+ =
degreeC [v]
degreeV [v]
;
if u ∈ C then
C.put(v);
if v ∈ C then
C.put(u);
processedElements+ = 1;
if processedElements mod W == 0 then
C prune(C, s, degreeV , degreeC );
COEUS main-
tains its focus in
each community

Determining
the size of each community

Community size
Nodes are associated with community participation values
The size of the community may be smaller
than the one COEUS examines
We need to derive automatically the size of a community!

cp values for a random COEUS community
0
0.01
0.02
0.03
0.04
25 50 75 100
cp
Rank
community nodes
tail nodes

0
0.01
0.02
0.03
0.04
25 50 75 100
cp
Rank
community nodes
tail nodes
clearly visible tail!

0
0.01
0.02
0.03
0.04
25 50 75 100
cp
Rank
community nodes
tail nodes
clearly visible tail!
constant threshold
value won’t work!

COEUScp
Sort the nodes
with regard to their
community participation
value
Calculate the average dis-
tance between two consecu-
tive nodes
Remove nodes until the dis-
tance becomes larger than
the average
Algorithm 3: DROPTAIL
input : A community C and the cp values ∀u ∈ C
output : The community C after irrelevant nodes are removed
begin
ˆC reverseSort(C);
totalDifference 0;
previous 0;
foreach c ∈ ˆC do
if previous > 0 then
totalDifference cp(c) − previous;
previous cp(c);
averageDifference
totalDifference
ˆC.size()−1
;
previous 0;
foreach c ∈ ˆC do
if previous > 0 then
difference cp(c) − previous;
previous cp(c);
if difference < averageDifference then
ˆC.remove(c);
else
break;

Experimental Evaluation

Dataset
Graphs Type Nodes Edges Av. Degree Av. Community Size
DBLP Co-authorship 317, 080 1, 049, 866 3.31 22.45
Amazon Co-purchasing 334, 863 925, 872 2.76 13.49
Youtube Social 1, 134, 890 2, 987, 624 2.63 14.59
LiveJournal Social 3, 997, 962 34, 681, 189 8.67 27.80
Orkut Social 3, 072, 441 117, 185, 083 38.14 215.72
Friendster Social 65, 608, 366 1, 806, 067, 135 27.53 46.81
Networks exceeding 1.8 billion links
Accompanying ground-truth communities allow for
the evaluation of accuracy

Impact of reckoning in edge quality
0
0.2
0.4
0.6
0.8
1
Am
azon
D
BLP
Youtube
LiveJournalO
rkut
Friendster
F1-score
CoEuS1
CoEuScp

Impact of reckoning in edge quality
0
0.2
0.4
0.6
0.8
1
Am
azon
D
BLP
Youtube
LiveJournalO
rkut
Friendster
F1-score
CoEuS1
CoEuScp
our variation
heavily impacts
the eﬀective-
ness of COEUS

Eﬀectiveness of dropTail algorithm
0
0.2
0.4
0.6
0.8
1
Am
azon
D
BLP
Youtube
LiveJournalO
rkut
Friendster
F1-score
CoEuScp
CoEuScp-auto
LEMON

0
0.2
0.4
0.6
0.8
1
Am
azon
D
BLP
Youtube
LiveJournalO
rkut
Friendster
F1-score
CoEuScp
CoEuScp-auto
LEMON
CoEuS handles
a graph stream

0
0.2
0.4
0.6
0.8
1
Am
azon
D
BLP
Youtube
LiveJournalO
rkut
Friendster
F1-score
CoEuScp
CoEuScp-auto
LEMON
CoEuS is
extremely competi-
tive w.r.t. accuracy!

Execution Time Comparison
Graphs COEUS LEMON
Amazon 0.0458 sec 3.1197 sec
DBLP 0.0575 sec 7.2756 sec
Youtube 0.176 sec 11.3834 sec
LiveJournal 1.573 sec 28.14 sec
Orkut 7.5171 sec −
Friendster 158.6547 sec −
COEUS is considerably faster than previous approaches
Not indicative of COEUS speed in a streaming setting
COEUS is able to derive the communities on demand!

Space Requirements Comparison
Graphs COEUS LEMON
Amazon 21.36MB 155.74MB
DBLP 21.36MB 156.49MB
Youtube 21.36MB 457.62MB
LiveJournal 21.36MB 2, 652.99MB
Orkut 21.36MB −
Friendster 21.36MB −
COEUS uses two COUNT-MIN sketches to hold a graph
its requirements depend only on the desired approximation quality
LEMON maintains the adjacency lists of a graph
thus, it requires signiﬁcantly more space

Outline
graph streams
UoA Panagiotis Liakos Scalable Link Community Detection: A Local Dispersion-aware Approach 46/63

Community Detection
Can we extract the community structure
of a node in a network?

Contribution
We focus on the neighbors of a single node in the network
to achieve eﬃciency and scalability
We build on:
Hierarchical Link Clustering Dispersion-based measures
We produce a more accurate and intuitive community
structure around a node for numerous real-world networks
– PL, Alexandros Ntoulas, Alex Delis: Scalable link community detection: A local dispersion-aware approach.
IEEE BigData 2016 & ACAC 2016 & HDMS 2017
– PL, Alexandros Ntoulas, Alex Delis: Uncovering Local Hierarchical Overlapping Communities at Scale.
Extended version undergoing review

Outline
graph streams
UoA Panagiotis Liakos Rhea: Adaptively Sampling High Quality Content from Social Activity Streams 49/63

500 million tweets
sent each day!

Contribution
We study the problem of extracting high-quality samples
of a social activity stream.
Related work:
White-lists of users [GSB+12, WLP+12, GZB+13, ZBG+16].
Authoritative users through network attributes
[ZAA07, JA07, ACD+08, PC11, BBC+13] (not streams).
We propose RHEA:
A high quality content sampling algorithm that forms a
network of authorities as it processes a social activity stream,
and samples only the activity of the top-K authoritative users.
– PL, Alexandros Ntoulas, Alex Delis: Rhea: Adaptively Sampling Authoritative Content from Social Activity
Streams. IEEE BigData 2017

Outline
graph streams
UoA Panagiotis Liakos On the Impact of Social Cost in Opinion Dynamics 52/63

Formation of opinions in a social context
intrinsic belief
+
friends’ expressed
opinions
expressed
opinion

Basic notions of the model
We use:
a variation of the DeGroot model due to Friedkin and Johnsen [FJ90]
and the corresponding game of [BKO11].
Each user i maintains:
An intrinsic belief si An expressed opinion zi
Remains constant Updated iteratively through averaging
The cost a user suﬀers emanates from:
Suppressing her intrinsic belief Disagreeing with her friends

Contribution
1 We analyze user activity in and verify that social interaction
results in inﬂuence on opinions among the participants.
2 We implement over Spark (GraphX) a distributed algorithm
At each time step user i updates zi to minimize her cost:
zi =
si+ j∈N(i) wijzj
1+ j∈N(i) wij
N(i): the set of nodes that i follows
wij : the strength of the inﬂuence of j on i
3 The algorithm terminates when z converges to the unique Nash
equilibrium, where the social cost is minimized
4 The resulting Nash equilibria are illustrative of how users really
behave.
– PL, Katia Papakonstantinopoulou: On the Impact of Social Cost in Opinion Dynamics. AAAI ICWSM 2016
& AGATHA 2016

Open Directions
1 Many opportunities for memory optimization in
distributed graph processing systems
2 Ground-truth communities should better portray
the functional role of a network’s nodes
3 Distributed streaming community detection
4 Authorities VS Fake news
5 Empirical analysis of the opinion formation process in
other social networks
UoA Panagiotis Liakos Open Directions 56/63

References I
[ABL10] Yong-Yeol Ahn, James P Bagrow, and Sune Lehmann.
Link communities reveal multiscale complexity in networks.
Nature, 466(7307):761–764, 2010.
[ACD+08] Eugene Agichtein, Carlos Castillo, Debora Donato, Aristides Gionis, and Gilad Mishne.
Finding high-quality content in social media.
In Proc. of the Int. Conf. on Web Search and Web Data Mining, WSDM 2008, Palo Alto, California,
USA, February 11-12, 2008, pages 183–194, 2008.
[BBC+13] Alessandro Bozzon, Marco Brambilla, Stefano Ceri, Matteo Silvestri, and Giuliano Vesci.
Choosing the right crowd: expert ﬁnding in social networks.
In Joint 2013 EDBT/ICDT Conferences, EDBT ’13 Proceedings, Genoa, Italy, March 18-22, 2013, pages
637–648, 2013.
[BBPSV04] A. Barrat, M. Barthélemy, R. Pastor-Satorras, and A. Vespignani.
The architecture of complex weighted networks.
Proc. of the National Academy of Sciences of the United States of America, 101(11):3747–3752, 2004.
[BGLL08] Vincent D Blondel, Jean-Loup Guillaume, Renaud Lambiotte, and Etienne Lefebvre.
Fast unfolding of communities in large networks.
Journal of Statistical Mechanics: Theory and Experiment, 2008(10):P10008, 2008.
[BKO11] David Bindel, Jon M. Kleinberg, and Sigal Oren.
How bad is forming your own opinion?
In FOCS, pages 57–66, 2011.
[BV04] Paolo Boldi and Sebastiano Vigna.
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thank you!
Special thanks to:
Alex Delis, Katia Papakonstantinopoulou, Alexandros Ntoulas,
Michael Sioutis, Nikos Leonardos, Katerina El Raheb & Alexis Antoniadis
UoA Panagiotis Liakos Acknowledgements 63/63

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