Transcript Graphs - Minas Gjoka

Construction of Simple Graphs with a Target
Joint Degree Matrix and Beyond
Minas Gjoka, Balint Tillman, Athina Markopoulou
University of California, Irvine
Graphs
DNS
Autonomous
Systems
Social Networks
Protein interactions
World Wide Web
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
Facebook datasets
Social Networks
 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”
2a
1
3
2b
4
dK-Series
 dK-series framework [Mahadevan et al, Sigcomm ’06]
• 0K specifies the average node degree k  2 E
V
2a
1
0K 
3
24
2
4
2b
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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
D(k )  aV 1
k
2a
1
3
2b
k
D(k)
1
1
2
2
3
1
4
1K
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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
• 2K specifies the joint node degree matrix (JDM)
JDM (k , l )  aV
k
(k,l)
2a
1
3

1
1
2
bVl {{a ,b )E }
1
1
2
2
2b
3
3
1
2
2
2K
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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
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
2a
1
3
(k,l,m)
#Wedges
(k,l,m)
#Triangles
1,3,2
2
2,2,3
1
2
2b
4
3K
2
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dK-Series
 dK-series framework [Mahadevan et al, Sigcomm ’06]
•
•
OSNs •
“2K+” •
•
•
0K specifies the average node degree
1K specifies the node degree sequence
2K specifies the joint node degree matrix (JDM)
3K specifies the number of induced subgraphs of 3 nodes
…
nK specifies the entire graph
 Nice properties
• Inclusion
• Convergence
• Tradeoff : accuracy vs. complexity
9
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 [Molloy’95]
• 1K+ [Bansal ’09, Newman’09, Serrano & Boguna’05, …]
 2K Construction
• Configuration model for 2K multigraphs [Mahadevan’06]
• Balance Degree Invariant: simple graphs [Amanatidis’08], [Stanton’ 12]
 2K+ Construction
• 2K preserving, 3K targeting using edge rewiring: [Mahadevan’ 06]
• 2.5K heuristic: JDM+degree dependent clustering coefficient: [Gjoka’13]
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2K Construction
Configuration Model
Free stub
2a
2b
4a
3a
2b
3b
l
k 2 3 4
2 2 2 2
3 2 2 2
4 2 2
target JDM
l
k
2
3
4
2
3
4
0
0
0
0
0
0
0
0
current JDM
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2K Construction
Configuration Model
Used stub
Free stub
2a
2b
4a
3a
2b
Construction stuck!
2/8 (25%) of the edges
cannot be added
3b
l
k 2 3 4
2 2 2 2
3 2 2 2
4 2 2
target JDM
l
k
2
Edges added
(2a,3a) (3b,4a) (2b,3a) 3
(2b,4a) (3a,3b) (2a,2b) 4
2
3
4
2
2
1
2
2
1
1
1
current JDM
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2K Construction
Balanced Degree Invariant
JDM(3, 4) = 1
4) = 2
(3,

JDM (3, 4)
target JDM
JDM(3, 4) <

k =3
3a
3b
k =3
3a
3b
l
4a
4b
l
4a
4b
k =3
3a
3b
l
4a
4b
=4
Construction
Used stub
Free stub
constrained!
=4
=4
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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
JDMtarget
 Input: Joint Degree Matrix
• JDMtarget must be graphical
 Goal:
• Construct a simple graph with
exactly JDMtarget
1
2
3
4
1
1
1
2
1
1
3
1
1
4
1
1
4
4
2
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2K Construction
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
…
JDM/JDMtarget
1
…
add edge between (x, y)
3
4
1
0/1 0/1
2
0/1 0/1
3
0/1 0/1
4
0/1 0/1 0/4 0/2
0/4
2a
…
…
2
3a
4a
4b
3b
1a
1b
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2K Construction
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
…
JDM/JDMtarget
1
…
add edge between (x, y)
JDM(k, l)++
3
4
1
0/1 1/1
2
0/1 0/1
3
0/1 0/1
4
1/1 0/1 0/4 0/2
0/4
2a
…
…
2
3a
4a
4b
3b
1a
1b
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2K Construction
Initialize:
JDM/JDMtarget
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
1
3
4
1
0/1 1/1
2
0/1 0/1
3
0/1 0/1
4
1/1 0/1 0/4 0/2
0/4
2a
neighbor switch for x
if y does not have free stubs
2
3a
4a
4b
3b
1a
1b
neighbor switch for y
add edge between (x, y)
JDM(k, l)++
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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
k=3
l=4
x
y
Add edge between x and y
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Case 2
x has free stubs but y does not
JDM(k, l) < JDMtarget(k, l)
node x has degree k
node y has degree l
k=3
l=4
x
b
y
t
Add edge between x and y
Neighbor switch
between y and b
using t
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Case 3
neither x nor y have free stubs
JDM(k, l) < JDMtarget(k, l)
node x has degree k
node y has degree l
k=3
l=4
Neighbor switch
between x and
b2 using t2
t2
x
b2
b1
y
t1
Add edge between x and y
Neighbor switch
between y and
b1 using t1
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Properties of 2K Algorithm
 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
22
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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Flexibility of 2K Algorithm
 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
target
 Previously known: space of graphs with JDM
is connected; but


slow MCMC mixing
Property 1: clustering
Property 2: attribute correlation
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Extension 1: Target JDM + Clustering
0 triangles
1 triangles
2
3
2
2
3
2
3
2 triangles
2
2
2
3
3
2
3
2
l 2
k
3
4
4
4
2
2
3
2
2
2
JDM
Intuition: by controlling the order we add edges we can control
clustering.
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Extension 1: Target JDM + Clustering
0 triangles
2 triangles
2a
3a
2d
2c
l 2
k
3
4
4
4
2
2
3b
3
2a
3a
3b
JDM
2b
2c
25
2a
50
2c
2d
75
2d
2a
2b
3a
2b
0
3b
nodes randomly on a circle,
consider node pairs’ distance
2b
3a
2c
2d
3b
[INFOCOM 2013]: add edges in
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increasing distance high clustering
“Sortedness” of node pairs’ list controls clustering
• Example: JDMtarget of Facebook Caltech Network
• Consider many orders of node pairs  create graphs with JDMtarget
 compute avg clustering c.
2a
2b
3a
2c
2d
3b
[INFOCOM 2015]: control order of
node pairs  control clustering
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Extension 1: Target JDM + 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 Node Closeness
Average Node Shortest Path Length
Average Clustering Coefficient
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Real world examples
target JDM+avg clustering
2K+MCMC did not finish after several days
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Extension 2: Node Attributes
l
k 1 2
2
1
2 2 6
JDM
2 2
2 4
JAM
l
k 1 2
2
1
2 2 6
JDM
Joint Attribute Matrix
(or Attribute Mixing Matrix)
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Extension 2: Node Attributes Mixing
l
k 1 2
2
1
2 2 6
JDM
2 2
2 4
JAM
l
k 1 2
2
1
2 2 6
JDM
4
6
JAM
Joint Attribute Matrix
(or Attribute Mixing Matrix)
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Extension 2: Degree+Attribute Mixing
l
k 1 2
2
1
2 2 6
2 2
2 4
JDM
1
2
1
1
2
1
2
1
1
l
k 1 2
2
1
2 2 6
JAM
2
JDM
1
1
1
2
4
2
4
6
1
JAM
2
2
2
2
6
Joint Degree and Attribute Matrix (JDAM)
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Extension 2: target JDAM
2K Algorithm also works for target JDAM
1
2
1
1
2
1
2
1
1
2
1
1
1
1
2
4
2
2
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2
2
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Joint Degree and Attribute Matrix (JDAM)
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Real world examples
graphs with node attributes
Average Node Closeness
Average Node Shortest Path Length
Average Clustering Coefficient
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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)
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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)
http://odysseas.calit2.uci.edu/osn/
37
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)
2a
2b
3a
2c
2d
3b
Questions?
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