Construction Algorithms for Online Social...

Construction Algorithms for Online Social Networks

Minas Gjoka, Balint Tillman, Athina Markopoulou

University of California, Irvine

Graphs

Social Networks

Protein interactions World Wide Web

Autonomous Systems

DNS

2

Motivation §  Measurements/sampling OSNs

•  http://odysseas.calit.uci.edu/osn/ •  [INFOCOM 2010],[ SIGMETRICS 2011],

3x[JSAC 2011], [WOSN 2012]… •  ~3500 researchers have requested our

FB datasets

§  Generate synthetic graphs that resemble real social networks •  to use in simulations •  for anonymization

§  Q1: resemble in terms of what? §  Q2: generate how?

3

dK-Series

§  dK-series framework [Mahadevan et al, Sigcomm ’06] •  “A set of graph properties that describe and constrain random

graphs, using degree correlations, in successively finer detail”

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dK-Series

§  dK-series framework [Mahadevan et al, Sigcomm ’06] •  0K specifies the average node degree

VE

k2

=

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dK-Series

§  dK-series framework [Mahadevan et al, Sigcomm ’06] •  0K specifies the average node degree •  1K specifies the node degree sequence

o  node degree “sequence” vs “distribution”

∑∈=

kVakD 1)(

1a

2a

4a

3b

3a

1b

4b

2

1

2

2

k #

1

2

3

4

1K 6

dK-Series

§  dK-series framework [Mahadevan et al, Sigcomm ’06] •  0K specifies the average node degree •  1K specifies the node degree sequence •  2K specifies the joint node degree matrix (JDM)

∑ ∑∈ ∈ ∈=k lVa Vb EbalkJDM }),{{1),(

1a

2a

4a

3b

3a

1b

4b

(k,l)

1 1

1 1

1 1 4

1 1 4 2

1 2 3 4

1

2

3

4

2K 7

§  dK-series framework [Mahadevan et al, Sigcomm ’06] •  0K specifies the average node degree •  1K specifies the node degree distribution •  2K specifies the joint node degree matrix (JDM) •  3K specifies the number of induced subgraphs of 3 nodes

o  nodes are labeled by their degree k

dK-Series

#Wedges(k1,k2,k3) #Triangles(k1,k2,k3)

k1 k2

k3

k1 k2

k3

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dK-Series

§  dK-series framework [Mahadevan et al, Sigcomm ’06] •  0K specifies the average node degree •  1K specifies the node degree distribution •  2K specifies the joint node degree matrix (JDM) •  3K specifies the distribution of subgraphs of 3 nodes •  … •  nK specifies the entire graph

§  Nice properties •  Inclusion •  Convergence •  Tradeoff : accuracy vs. complexity

OSNs “2K+”

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Related Work §  Graph Construction Approaches:

•  Stochastic: reproduces dk-distribution in expectation. •  Configuration (“pseudograph”): reproduces dk-distribution exactly.

o  Deterministic algorithms up to d=2. MCMC for d>=2.

§  1K Construction •  Configuration: 1K multigraphs or simple graphs [Molloy’1995] •  1K+ [Bansal ’2009, Newman’2009, Serrano & Boguna’2005, …] •  What else is known: conditions for 1K to be graphical [Erdos-Gallai, Havel]; space of

graphs with degree sequence connected [Havel-Hakimi]; MCMC for sampling. §  2K Construction

•  Configuration model [Mahadevan’2006] •  Balance Degree Invariant: [Amanatidis’2008], [Stanton’ 2012],[Czabarka’2014]. •  What else is known: conditions for 2K to be graphical [Amantidis’08][Stanton’12];

space of graphs with a given JDM is connected[Stanton’12 [Czabarka’14]; MCMC convergence speed is an open problem.

§  2K+ Construction

•  2K preserving, 3K targeting using edge rewiring: [Mahadevan’ 2006] •  2.5K heuristic: JDM+degree dependent clustering coefficient: [Gjoka’13]

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Our Contributions §  New 2K Construction Algorithm

§  can produce any simple graph §  Main benefit: no constraints in constructed graphs

§  with the exact JDMtarget §  in O(|E|dmax)

§  2K+ Framework: JDMtarget+ Additional Properties §  2K + Node Attributes (exactly) §  2K + Avg Clustering (approx)

§  Main benefit: orders of magnitude faster than 2K+MCMC

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2K Construction

1 1

1 1

1 1 4

1 1 4 2

1 2 3 4

1

2

3

4

JDMtarget §  Input: Joint Degree Matrix

•  JDMtarget must be graphical

§  Goal:

•  Construct a simple graph with exactly JDMtarget

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2K Construction

0/1 0/1

0/1 0/1

0/1 0/1 0/4

0/1 0/1 0/4 0/2

1 2 3 4

1

2

3

4

JDM/JDMtarget

1a

2a

4a

3b

3a

1b

4b

Initialize: 1K: create nodes and stubs

JDM(k,l)=0 for all k,l

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2K Construction

0/1 1/1

0/1 0/1

0/1 0/1 0/4

1/1 0/1 0/4 0/2

1 2 3 4

1

2

3

4

JDM/JDMtarget

1a

2a

4a

3b

3a

1b

4b



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2K Construction

0/1 1/1

1/1 0/1

0/1 1/1 0/4

1/1 0/1 0/4 0/2

1 2 3 4

1

2

3

4

JDM/JDMtarget

1a

2a

4a

3b

3a

1b

4b



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2K Construction

1a

2a

4a

3b

3a

1b

4b

Initialize:

1K: create nodes and stubs JDM(k,l)=0 for all k,l

Pick (k, l) degree pair, in any order While JDM(k, l) < JDMtarget(k, l)

Pick (x, y) any pair of disconnected nodes with degrees k and l

if x does not have free stubs neighbor switch for x

if y does not have free stubs neighbor switch for y

add edge between (x, y) JDM(k, l)++

0/1 1/1

1/1 0/1

0/1 1/1 0/4

1/1 0/1 0/4 0/2

1 2 3 4

1

2

3

4

JDM/JDMtarget

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Case 1 x, y both have free stubs JDM(k, l) < JDMtarget(k, l) node x has degree k node y has degree l

x

y

Add edge between x and y

k=3

l=4

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Case 2 x has free stubs but y does not

x

y

k=3

l=4

t

Neighbor switch between y and b using t

b


JDM(k, l) < JDMtarget(k, l) node x has degree k node y has degree l

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Case 3 neither x nor y have free stubs

x b2

y

k=3

l=4

t1

Neighbor switch between y and b1 using t1

b1

Neighbor switch between x and b2 using t2

t2


JDM(k, l) < JDMtarget(k, l) node x has degree k node y has degree l

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Properties of 2K Algorithm

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§  Terminates with exact JDMtarget in O(|E|dmax) •  It adds 1 edge at a time, while staying below JDMtarget

§  It can produce ALL graphs with the JDMtarget §  Output graph depends on the order of adding edges

Space of constructed graphs Example: all 7-node graphs Generate all non-‐isomorphic

7-‐node simple graphs G1, .., G1044

All Unique Joint Degree Matrices JDM1, ... , JDM768

2K ConstrucOon Algorithm

Output

Input

768 syntheOc graphs (not all disOnct)

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Flexibility of 2K Algorithm

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§  Output graph depends on the order of considering edges to add §  It can produce ALL graphs with JDMtarget §  Family of algorithms: add one edge at a time, while staying below

JDMtarget •  any order of degree pairs (k,l) •  any order of node pairs (x,y), even before completing a degree pair •  Can start with an empty or partially built graph

§  2K+: can target additional properties fast §  Previously known: space of graphs with JDMtarget is connected; but

slow MCMC mixing §  Property 1: clustering §  Property 2: attribute correlation

Extension 1: Target JDM + Clustering

2

2

3

3

2 2

2

2

3

3

2

2

2 2

3

3

2 2

JDM

2

3

2 3k l

4 4 4 2

Intuition: by controlling the order we add edges we can control clustering.

0 triangles 1 triangles 2 triangles

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2a

2c

3b

3a

2b

2d

2a 2b

3b

3a

2d 2c

JDM

2

3

2 3k l

4 4 4 2

0 triangles 2 triangles

0

25

75

50

2b

3a

3b 2d

2a

2c

2b 3a

3b 2d

2a

2c

2b 3a

3b 2d

2a

2c


[INFOCOM 2013]: add edges in increasing distance àhigh clustering

nodes randomly on a circle, consider node pairs’ distance

Random order of node pairs à low clustering 24

“Sortedness” of node pairs’ list controls clustering

•  Example: JDMtarget of Facebook Caltech Network •  Consider many orders of node pairs à create graphs with JDMtarget

à compute avg clustering c.

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2b 3a

3b 2d

2a

2c

[INFOCOM 2015]: control order of node pairs à control clustering

2K+ Avg Clustering Input: target JDM, avg clustering coefficient c Stage 1 E’ = list of node pairs s.t. sortedness(E’)≈S(c)

FOR each candidate node pair (v,w) in E’: IF both nodes v and w have free stubs and the corresponding JDM(k, l) < JDMtarget(k, l): add edge (v,w)

Stage 2 If not all |E| edges have been added:

Add remaining edges using 2K_Simple


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Real world examples target JDM+avg clustering

Average Clustering Coefficient

Average Node Shortest Path Length

Average Node Closeness

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2K+MCMC did not finish after several days

Real world examples target JDM+avg clustering

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Extension 2: Node Attributes

JDM

1

2

1 2k l

2 2 6

JDM

1

2

1 2k l

2 2 6

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JAM

2 2 2 4

Joint Attribute Matrix (or Attribute Mixing Matrix)

Extension 2: Node Attributes Mixing

JDM

1

2

1 2

JAM

k l

2 2 6

2 2 2 4

JDM

1

2

1 2

JAM

k l

2 2 6

4 6

Joint Attribute Matrix (or Attribute Mixing Matrix)

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JDM

1

2

1 2

JAM

k l

2 2 6

2 2 2 4

JDM

1

2

1 2

JAM

k l

2 2 6

4 6

1

2

2

1 2 21 1

1 1

1 1 4

1

2

2

1 2 22

2

6

Joint Degree and Attribute Matrix (JDAM)

Extension 2: Degree+Attribute Mixing

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1

2

2

1 2 21 1

1 1

1 1 4

1

2

2

1 2 22

2

6

Joint Degree and Attribute Matrix (JDAM)

Extension 2: target JDAM

2K Algorithm also works for target JDAM

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Real world examples graphs with node attributes

Average Clustering Coefficient

Average Node Shortest Path Length

Average Node Closeness

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Real world examples small graphs with node attributes

Simulation takes ~1 day to target 2K and c = 0.24 with MCMC (using double edge swaps) 34

Construction of 2K+ Graphs §  New 2K Construction Algorithm

•  can produce any simple graph with exact JDMtarget in O(|E|dmax)

§  2K+ Framework: JDMtarget+ Additional Properties §  Extension 1: 2K (exactly) + Avg Clustering (approx) §  Extension 2: 2K (exactly) + Node Attributes (exactly)

§  Future directions §  Construction: target attributes + structure (towards 3K) §  Applications to privacy

http://odysseas.calit2.uci.edu/osn/

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Construction of 2K+ Graphs

QuesOons?

Construction Algorithms for Online Social...

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