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Milena MihailGeorgia Tech.
with
Stephen Young, Giorgos Amanatidis, Bradley Green
Flexible Models for Complex Networks
The Internet is constantly growing and evolving giving rise to new models and algorithmic questions.2
degree
4 102 100
freq
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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.
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 …
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“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.
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
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
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.
connected, mincost, random
The (well studied) Degree Sequence Realization Problem is:
Definitions
The Joint-Degree Matrix Realization Problem is:connected, mincost, random
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
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Reduction toperfect matching:
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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.
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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:
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[ Havel-Hakimi ] Construction Algorithm: Greedy: any unsatisfied vertex is connected withthe vertices of highest remaining degree requirements.
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add
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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 .
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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 .
Theorem, Joint Degree Matrix Realization [Amanatidis, Green, M ‘08]:
Proof [sketch]:
Balanced Degree Invariant:
Example Case Maintaining Balanced Degree Invariant:
deletedelete
add
add add
Note: This may NOT be asimple “augmenting” path.
Theorem, Joint Degree Matrix Connected Realization [Amanatidis, Green, M ‘08]:
Proof [sketch]:
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 .
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.)
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9 available edges & 11 vertices.
There are not enough edges to connect all the vertices!
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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?
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
RANDOM DOT PRODUCT GRAPHSKratzl,Mickel,Sheinerman 05Young,Sheinerman 07Young,M 08
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.
Theorem [Young, M ’08]
A Semi-closed Formula for Degree Distribution
Theorem ( removing error terms) [Young, M ’08]
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.
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
Clustering CharacterizationTheorem [Young, M ’08]
Remarks on the proof
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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 ?
KRONECKER GRAPHS [Faloutsos, Kleinberg,Leskovec 06]
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Another “semantic” “ flexible” model, introducing parametrization.Several properties characterized.
STOCHASTIC KRONECKER GRAPHS
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Several properties characterized (e.g. multinomial degree distributions).Large scale data set have been fit.
[ Faloutsos, Kleinberg, Leskovec 06]
Summary 1. Structural/Syntactic Flexible Models
2. SemanticFlexible Models
Generalizations of Erdos-Gallai / Havel-Hakimi
Generalizations of Erdos-Renyi random graphs
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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: ?