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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

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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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0Connectivity, if possible, attained with 2-switches.

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: ?