Milena Mihail Georgia Tech. with Stephen Young, Giorgos Amanatidis, Bradley Green
description
Transcript of Milena Mihail Georgia Tech. with Stephen Young, Giorgos Amanatidis, Bradley Green
![Page 1: Milena Mihail Georgia Tech. with Stephen Young, Giorgos Amanatidis, Bradley Green](https://reader033.fdocuments.in/reader033/viewer/2022051419/56815a76550346895dc7deaa/html5/thumbnails/1.jpg)
1
Milena MihailGeorgia Tech.
with
Stephen Young, Giorgos Amanatidis, Bradley Green
Flexible Models for Complex Networks
![Page 2: Milena Mihail Georgia Tech. with Stephen Young, Giorgos Amanatidis, Bradley Green](https://reader033.fdocuments.in/reader033/viewer/2022051419/56815a76550346895dc7deaa/html5/thumbnails/2.jpg)
The Internet is constantly growing and evolving giving rise to new models and algorithmic questions.2
degree
4 102 100
freq
uenc
y
, but
no sharp concentration:
Erdos-Renyi
Sparse graphs with large degree-variance.“Power-law” degree distributions.
Small-world, i.e. small diameter,high clustering coefficients.
![Page 3: Milena Mihail Georgia Tech. with Stephen Young, Giorgos Amanatidis, Bradley Green](https://reader033.fdocuments.in/reader033/viewer/2022051419/56815a76550346895dc7deaa/html5/thumbnails/3.jpg)
degree
4 102 100
freq
uenc
y
, but
no sharp concentration:
Erdos-Renyi
Sparse graphs with large degree-variance.“Power-law” degree distributions.
Small-world, i.e. small diameter,high clustering coefficients.
A rich theory of power-law random graphs has been developed [ Evolutionary & Configurational Models, e.g. see Rick Durrett’s ’07 book ].
However, in practice, there are discrepancies …
![Page 4: Milena Mihail Georgia Tech. with Stephen Young, Giorgos Amanatidis, Bradley Green](https://reader033.fdocuments.in/reader033/viewer/2022051419/56815a76550346895dc7deaa/html5/thumbnails/4.jpg)
4
“Flexible” models for complex networks:
exhibit a “large” increase in the properties of generated graphs
by introducing a “small” extension in the parameters of the generating model.
![Page 5: Milena Mihail Georgia Tech. with Stephen Young, Giorgos Amanatidis, Bradley Green](https://reader033.fdocuments.in/reader033/viewer/2022051419/56815a76550346895dc7deaa/html5/thumbnails/5.jpg)
Models with power law and arbitrary degree sequences with additional constraints, such as specified joint degree distributions (from random graphs, to graphs with very low entropy).
Models with semantics on nodes, and links among nodes with semantic proximity generated by very general probability distributions.
RANDOM DOT PRODUCT GRAPHS KRONECKER GRAPHS
Talk Outline 1. Structural/Syntactic Flexible Models
2. SemanticFlexible Models
Generalizations of Erdos-Gallai / Havel-Hakimi
Generalizations of Erdos-Renyi random graphs
![Page 6: Milena Mihail Georgia Tech. with Stephen Young, Giorgos Amanatidis, Bradley Green](https://reader033.fdocuments.in/reader033/viewer/2022051419/56815a76550346895dc7deaa/html5/thumbnails/6.jpg)
Models with power law and arbitrary degree sequences with additional constraints, such as specified joint degree distributions (from random graphs, to graphs with very low entropy).
Talk Outline
The networking community proposed that [Sigcomm 04, CCR 06 and Sigcomm 06], beyond the degree sequence , models for networks of routers should capturehow many nodes of degree are connected to nodes of degree .
Assortativity:
small large
![Page 7: Milena Mihail Georgia Tech. with Stephen Young, Giorgos Amanatidis, Bradley Green](https://reader033.fdocuments.in/reader033/viewer/2022051419/56815a76550346895dc7deaa/html5/thumbnails/7.jpg)
Networking Proposition [CCR 06, Sigcomm 06]:
A real highly optimized network G.A random graph with same average degree as G.
A random graph with same degree sequence as G.
A graph with same number of links between nodes of degree and as G.
![Page 8: Milena Mihail Georgia Tech. with Stephen Young, Giorgos Amanatidis, Bradley Green](https://reader033.fdocuments.in/reader033/viewer/2022051419/56815a76550346895dc7deaa/html5/thumbnails/8.jpg)
connected, mincost, random
The (well studied) Degree Sequence Realization Problem is:
Definitions
The Joint-Degree Matrix Realization Problem is:connected, mincost, random
![Page 9: Milena Mihail Georgia Tech. with Stephen Young, Giorgos Amanatidis, Bradley Green](https://reader033.fdocuments.in/reader033/viewer/2022051419/56815a76550346895dc7deaa/html5/thumbnails/9.jpg)
Theorem [Amanatidis, Green, M ‘08]: The natural necessary conditions for an instance to have a realization are also sufficient (and have a short description). The natural necessary conditions for an instance to have a connected realization are also sufficient (no known short description). There are polynomial time algorithms to construct a realization and a connected realization of ,or produce a certificate that such a realization does not exist.
The Joint-Degree Matrix Realization Problem is:
Open: Mincost, Random realizations of
![Page 10: Milena Mihail Georgia Tech. with Stephen Young, Giorgos Amanatidis, Bradley Green](https://reader033.fdocuments.in/reader033/viewer/2022051419/56815a76550346895dc7deaa/html5/thumbnails/10.jpg)
10
Reduction toperfect matching:
2
Given arbitrary
Is this degree sequence realizable ?
If so, construct a realization.
Degree Sequence Realization Problem:Advantages: Flexibility in:enforcing or precluding certain edges,adding costs on edges and finding mincost realizations,close to matching close to sampling/random generation.
![Page 11: Milena Mihail Georgia Tech. with Stephen Young, Giorgos Amanatidis, Bradley Green](https://reader033.fdocuments.in/reader033/viewer/2022051419/56815a76550346895dc7deaa/html5/thumbnails/11.jpg)
11
Theorem [Erdos-Gallai]: A degree sequence is realizable iff the natural necessary condition holds:
moreover, there is a connected realization iff the natural necessary condition holds:
![Page 12: Milena Mihail Georgia Tech. with Stephen Young, Giorgos Amanatidis, Bradley Green](https://reader033.fdocuments.in/reader033/viewer/2022051419/56815a76550346895dc7deaa/html5/thumbnails/12.jpg)
12
4
3
2
2
1
11
[ Havel-Hakimi ] Construction Algorithm: Greedy: any unsatisfied vertex is connected withthe vertices of highest remaining degree requirements.
032
2
0
0 0
1
0
1
0
0Connectivity, if possible, attained with 2-switches.
add
delete
add
delete
![Page 13: Milena Mihail Georgia Tech. with Stephen Young, Giorgos Amanatidis, Bradley Green](https://reader033.fdocuments.in/reader033/viewer/2022051419/56815a76550346895dc7deaa/html5/thumbnails/13.jpg)
Random generation of graph with a given degree sequence:
Theorem [Cooper, Frieze & Greenhill 04]:The Markov chain corresponding to a general 2-link switch is rapidly mixingfor degree sequences with .
![Page 14: Milena Mihail Georgia Tech. with Stephen Young, Giorgos Amanatidis, Bradley Green](https://reader033.fdocuments.in/reader033/viewer/2022051419/56815a76550346895dc7deaa/html5/thumbnails/14.jpg)
14
Random generation of connected graph with a given degree sequence:
Theorem [Feder,Guetz,M,Saberi 06]:The Markov chain corresponding to a local 2-link switch is rapidly mixingif the degree sequence enforces diameter at least 3, and for some .
![Page 15: Milena Mihail Georgia Tech. with Stephen Young, Giorgos Amanatidis, Bradley Green](https://reader033.fdocuments.in/reader033/viewer/2022051419/56815a76550346895dc7deaa/html5/thumbnails/15.jpg)
Theorem, Joint Degree Matrix Realization [Amanatidis, Green, M ‘08]:
Proof [sketch]:
![Page 16: Milena Mihail Georgia Tech. with Stephen Young, Giorgos Amanatidis, Bradley Green](https://reader033.fdocuments.in/reader033/viewer/2022051419/56815a76550346895dc7deaa/html5/thumbnails/16.jpg)
Balanced Degree Invariant:
Example Case Maintaining Balanced Degree Invariant:
deletedelete
add
add add
Note: This may NOT be asimple “augmenting” path.
![Page 17: Milena Mihail Georgia Tech. with Stephen Young, Giorgos Amanatidis, Bradley Green](https://reader033.fdocuments.in/reader033/viewer/2022051419/56815a76550346895dc7deaa/html5/thumbnails/17.jpg)
Theorem, Joint Degree Matrix Connected Realization [Amanatidis, Green, M ‘08]:
Proof [sketch]:
![Page 18: Milena Mihail Georgia Tech. with Stephen Young, Giorgos Amanatidis, Bradley Green](https://reader033.fdocuments.in/reader033/viewer/2022051419/56815a76550346895dc7deaa/html5/thumbnails/18.jpg)
Main Difficulty: Two connected components are amenable to rewiring by 2-switches, only using two vertices of the same degree.
connectedcomponent
connectedcomponent
The algorithm explores vertices of the same degree in different components,transforming the graph to bring it to a form amenable to rewiring by 2-switch, if possible .
![Page 19: Milena Mihail Georgia Tech. with Stephen Young, Giorgos Amanatidis, Bradley Green](https://reader033.fdocuments.in/reader033/viewer/2022051419/56815a76550346895dc7deaa/html5/thumbnails/19.jpg)
Certificates of non-existence of connected realizationsresult from contractions of subsets of performed by the algorithm (as it was searching for transformations amenable to 2-switch rewirings across connected components.)
0 4 0 2 1
4 0 1 0 1
0 1 0 2 2
2 0 2 0 1
1 1 2 1 0
D
1
1
1
2
1
4
2
2
4
3 2
9 available edges & 11 vertices.
There are not enough edges to connect all the vertices!
![Page 20: Milena Mihail Georgia Tech. with Stephen Young, Giorgos Amanatidis, Bradley Green](https://reader033.fdocuments.in/reader033/viewer/2022051419/56815a76550346895dc7deaa/html5/thumbnails/20.jpg)
20
Open Problems for Joint Degree Matrix Realization
Construct mincost realization. Construct random realization. Satisfy constraints between arbitrary subsets of vertices. Is there a reduction to matching or flow or some other well understood combinatorial problem? Is there evidence of hardness?
![Page 21: Milena Mihail Georgia Tech. with Stephen Young, Giorgos Amanatidis, Bradley Green](https://reader033.fdocuments.in/reader033/viewer/2022051419/56815a76550346895dc7deaa/html5/thumbnails/21.jpg)
Models with semantics on nodes, and links among nodes with semantic proximity generated by very general probability distributions.
RANDOM DOT PRODUCT GRAPHS KRONECKER GRAPHS
Talk Outline 1. Structural/Syntactic Flexible Models
2. SemanticFlexible Models
Generalizations of Erdos-Gallai / Havel-Hakimi
Generalizations of Erdos-Renyi random graphs
![Page 22: Milena Mihail Georgia Tech. with Stephen Young, Giorgos Amanatidis, Bradley Green](https://reader033.fdocuments.in/reader033/viewer/2022051419/56815a76550346895dc7deaa/html5/thumbnails/22.jpg)
RANDOM DOT PRODUCT GRAPHSKratzl,Mickel,Sheinerman 05Young,Sheinerman 07Young,M 08
![Page 23: Milena Mihail Georgia Tech. with Stephen Young, Giorgos Amanatidis, Bradley Green](https://reader033.fdocuments.in/reader033/viewer/2022051419/56815a76550346895dc7deaa/html5/thumbnails/23.jpg)
SUMMARY OF RESULTS
A semi-closed formula for degree distributionand graphs with a wide variety of densities and degree distributions, including power-laws.
Diameter characterization (determined by Erdos-Renyi for similar average density)
Positive clustering coefficient, depending on the “distance” of the generating distribution from the uniform distribution.
Remark: Power-laws and the small world phenomenon are the hallmark of complex networks.
![Page 24: Milena Mihail Georgia Tech. with Stephen Young, Giorgos Amanatidis, Bradley Green](https://reader033.fdocuments.in/reader033/viewer/2022051419/56815a76550346895dc7deaa/html5/thumbnails/24.jpg)
Theorem [Young, M ’08]
A Semi-closed Formula for Degree Distribution
Theorem ( removing error terms) [Young, M ’08]
![Page 25: Milena Mihail Georgia Tech. with Stephen Young, Giorgos Amanatidis, Bradley Green](https://reader033.fdocuments.in/reader033/viewer/2022051419/56815a76550346895dc7deaa/html5/thumbnails/25.jpg)
Example:
(a wide range of degrees, except for very large degrees)
indicating a power-law with exponent between 2 and 3.
This is in agreement with real data.
![Page 26: Milena Mihail Georgia Tech. with Stephen Young, Giorgos Amanatidis, Bradley Green](https://reader033.fdocuments.in/reader033/viewer/2022051419/56815a76550346895dc7deaa/html5/thumbnails/26.jpg)
Theorem [Young, M ’08]
Diameter Characterization
Re
Remark: If the graph can become disconnected.It is important to obtain characterizations of connectivity as approaches . This would enhance model flexibility
![Page 27: Milena Mihail Georgia Tech. with Stephen Young, Giorgos Amanatidis, Bradley Green](https://reader033.fdocuments.in/reader033/viewer/2022051419/56815a76550346895dc7deaa/html5/thumbnails/27.jpg)
Clustering CharacterizationTheorem [Young, M ’08]
Remarks on the proof
![Page 28: Milena Mihail Georgia Tech. with Stephen Young, Giorgos Amanatidis, Bradley Green](https://reader033.fdocuments.in/reader033/viewer/2022051419/56815a76550346895dc7deaa/html5/thumbnails/28.jpg)
28
Open Problems for Random Dot Product Graphs
Fit real data, and isolate “benchmark” distributions . Characterize connectivity (diameter and conductance) as approaches . Similarity functions beyond inner product (e.g. Kernel functions). Algorithms: navigability, information/virus propagation, etc. Do further properties of X characterize further properties of ?
![Page 29: Milena Mihail Georgia Tech. with Stephen Young, Giorgos Amanatidis, Bradley Green](https://reader033.fdocuments.in/reader033/viewer/2022051419/56815a76550346895dc7deaa/html5/thumbnails/29.jpg)
KRONECKER GRAPHS [Faloutsos, Kleinberg,Leskovec 06]
1
0
1
1
0
0
0
0
1
0
1
1
1
0
1
1
1
0
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
1
1
0
1
1
1
0
1
1
0
0
0
0
1
0
1
1
1
0
1
1
1
0
1
1
0
0
0
0
1
0
1
1
1
0
1
1
1
0
1
1
Another “semantic” “ flexible” model, introducing parametrization.Several properties characterized.
![Page 30: Milena Mihail Georgia Tech. with Stephen Young, Giorgos Amanatidis, Bradley Green](https://reader033.fdocuments.in/reader033/viewer/2022051419/56815a76550346895dc7deaa/html5/thumbnails/30.jpg)
STOCHASTIC KRONECKER GRAPHS
c
a
b
b
ac
aa
ab
ab
bc
ba
bb
bb
cc
ca
cd
cb
bc
ba
bb
bb
aac
aaa
aab
aab
abc
aba
abb
abb
accacd
acb
abc
aba
abb
abb
bac
baa
bab
bab
bbc
bba
bbb
bbb
bcc
bca
bcd
bcb
bbc
bba
bbb
bbb
bac
baa
bab
bab
bbc
bba
bbb
bbb
bcc
bca
bcd
bcb
bbc
bba
bbb
bbb
cac
caa
cab
cab
cbc
cba
cbb
cbb
ccc
cca
ccd
ccb
cbc
cba
cbb
cbb
aca
Several properties characterized (e.g. multinomial degree distributions).Large scale data set have been fit.
[ Faloutsos, Kleinberg, Leskovec 06]
![Page 31: Milena Mihail Georgia Tech. with Stephen Young, Giorgos Amanatidis, Bradley Green](https://reader033.fdocuments.in/reader033/viewer/2022051419/56815a76550346895dc7deaa/html5/thumbnails/31.jpg)
Summary 1. Structural/Syntactic Flexible Models
2. SemanticFlexible Models
Generalizations of Erdos-Gallai / Havel-Hakimi
Generalizations of Erdos-Renyi random graphs
![Page 32: Milena Mihail Georgia Tech. with Stephen Young, Giorgos Amanatidis, Bradley Green](https://reader033.fdocuments.in/reader033/viewer/2022051419/56815a76550346895dc7deaa/html5/thumbnails/32.jpg)
32
Where it all started: Kleinberg’s navigability model
Theorem [Kleinberg]: The only value for which the network is navigableis r =2.
Parametrization is essential in the study of complex networksMoral: ?