complex networks - models and applicationskvasnicka/seminar_of_ai/cajagi_maj2013.pdf · complex...

Complex networks introduction and definitionsConnecting nearest neighbour model - CNN

Scale free growing network with hierarchical structureMinimum dominating set problem in scale-free networks

Benchmarking of the MDS problem on big complex SF-NetworksConclusion and discussion

Complex networks - models and applications

Martin CajagiSupervised by: Maria Markosova

KAI

May 6, 2013

Martin Cajagi Complex networks - models and applications




Outline

1 Complex networks introduction and definitions

2 Connecting nearest neighbour model - CNN

3 Scale free growing network with hierarchical structure

4 Minimum dominating set problem in scale-free networks

5 Benchmarking of the MDS problem on big complex SF-Networks

6 Conclusion and discussion





IntroductionTerminology and notationsPropertiesComplex networks properties

Motivation

Complex network is a possible view on relationship betweenentities surrounding us.

For e.g. friends, co-workers, family... can be viewed as ancomplex social network

On the other hand we have networks representing naturalprocesses and structures like functional brain network,metabolic networks, language networks, power-grid...

The interesting part is that most of these networks have somecommon properties.






Simple definition

Complex network is a dynamical graph consisting of :

nodes - this means entities like people, words, cells...

edges - representing the relationship between these entities

processes - changing the structure by adding new nodes andedges or their deletions

The basic difference between standard graph theory and complexnetworks theory is the dynamical property of processes changingthe graph structure.






Used terminology (simplified)

Node degree: Number of edges connected to a node

Neighbourhood (Nb): Set of all vertices connected toanother vertex by an edge

Clustering coefficient: cv = |E(Nb)|(kv

2 )






Clustering coeficient - ilustration

Figure : a) The cv of blue node equals one. b) The cv of blue nodeequals one third. c) The cv of blue node equals zero.






Important analytical properties.

Degree distribution: Degree distribution is a stationaryfunction p(k) representing the probability that a node in agraph has the degree k.

p(k) =

∑v∈V δ(k − kv )

N(1)

Clustering coefficient distribution: Distribution of theaverage clustering coefficient of nodes with the degree k:

c(k) =

∑v∈V δ(k − kv )cv

N(2)






Node degree distribution - ilustration

Figure : On the x-axis is the node degree and on the y-axis is the numberof vertices having degree k. In the left picture we can see the degreedistribution of random networks and on the right picture is the degreedistribution of scale-free networks






Basic statistical properties of complex networks

Complex networks can be:

Scale-free - Means they have scale-free distribution of nodedegrees

Small-world - Means they have small diameter ∼ log(n)where n is the number of nodes in the network

Hierarchical - Means they have scale-free distribution of theaverage clustering coefficient of node having degree k






Loglog plot definition

Figure : loglog plot of scale-free distribution where on the x axis is thevertex degree and on the y-axis the probability of occurrence of thatdegree. The loglog plot can be approximated with a line and α is the tgof the heading angle of that line. The value of α is a typicalcharacteristics of SF-complex networks.






Common properties of real networks I.

Figure :






Common properties of real networks II.

The scaling exponent α is mostly in the interval 2 ≤ α ≤ 3,(Albert,Barabasi, 2002).

These networks usually have small diameter ∼ log(N),(Cohen,Havlin 2003).

They are sparse, number of edges is linear function of the sizeof graph size (Del Genio,Gross,Bassler 2011)

Majority of vertices has k < 2M/N this is a directconsequence of the scale-free distribution as we can see in inprevious illustration.

They have small number of high degree vertices called hubs,this is also direct result of the SF distribution.





IntroductionModel equationsOriginal solutionOur more accurate solution

Introduction

The CNN model has been inspired by social networks(Vasquez).

It is assumed that in such networks two sites with a commonneighbour are connected with greater probability then the tworandomly chosen sites.

This reflects the fact, that in the social network it is moreprobable the two people (nodes) know themselves (areconnected) if they have a common friend (commonneighbour).

An analytical understanding of the CNN model has beenachieved using the notions of potential edges and potentialdegree.






Dynamic process definition

The dynamics of the CNN network is defined by the transitionrates of link states of the node s. Each possible link from s toother nodes in the network can be in three possible states :

- disconnected (d),

- potential edge (p)

- an edge (e). 1

For each node s the transition rates νx→y (s) x , y ∈ {d , p, e} aredefined per link.






Pottential edge definition

A potential edge is defined as follows:- two nodes are connected by a potential edge, if they are notconnected by an edge and they have at least one commonneighbour. For each node s in the growing CNN network a degreeand a potential degree can therefore be defined.

Consider a network with N nodes. Denote the (real) degree andpotential degree of a node s by k(s,N) and k∗(s,N), respectively.






Model equations

∂k(s,N)∂N = νd→e k(s,N) + νp→ek

∗(s,N)− (νe→d + νe→p)k(s,N)

∂k∗(s,N)∂N = νd→p k(s,N) + νe→pk(s,N)− (νp→d + νp→e)k∗(s,N)

k(s,N) = N − k(s,N)− k∗(s,N),






Original Solution

1 All processes leading to edge deletion are neglected:νe→p = νe→d = 0.

2 The transition from a potential edge to an edge has a higherprobability of occurrence then the transition from beingdisconnected to an edge

νp→e =µ1

N, νd→e =

µ0

N2,

where µ1 > 0 and µ0 > 0 are constants .

3 Terms of order 1/N2 are omitted.






Original solution equations

This lead to :

∂k(s,N)∂N = µ0

N + µ1N k∗(s,N)

∂k∗(s,N)∂N = µ0k(s,N)

N − µ1N k∗(s,N)

With solution :

k(s,N) = k0

(Ns

)βk∗(s,N) = k∗0

(Ns

)β,

where : β = µ12

(−1 +

√(1 + 4µ0

µ1

).






More accurate equations

1 Terms of order 1/N2 are not omitted.

We thus start with the system of differential equations:

∂k(s,N)∂N = µ0

N2 (N − k(s,N)− k∗(s,N)) + µ1N k∗(s,N)

∂k∗(s,N)∂N = µ0

N2 k(s,N) (N − k(s,N)− k∗(s,N))− µ1N k∗(s,N).






More accurate solution - fixed point A

k(s,N) = 2(

Ns

)β − 1

k∗(s,N) = 2(µ0β − 1

) (Ns

)β+ 1,

where the exponent β is given by

β = µ12

(−1 +

√1 + 4µ0

µ1

).

This solution provides more information about degree function.






More accurate solution - fixed point B

k(s,N) = µ(

1−(

Ns

)λ3)N + 2

(Ns

)1+λ3 − 1

k∗(s,N) = µµ1

(1 +

(Ns

)λ3)N − 2

(Ns

)1+λ3 − c3v31N1+λ3 + 1

where

v31 = 1 + 1µ1.

The solution B is close to other known models as I will show later.






Region ilustration

0

1

2

3

0 1 2 3 4 5 6 7

µ 0

µ1

A sink, B saddle

A saddle, B sink

Figure : Bifurcation line obtained in the parameter space of the CNNmodel delimiting regions of different stability types of fixed points.Dotted horizontal line indicates an asymptotic behaviour when µ1 →∞.





Introductionmodel Amodel BComments

Base idea

Process for growing network with hierarchy

1 The growth of the network starts from an arbitrary smallgraph.

2 Each time unit one node is injected and labelled by the timeof its addition s.

3 In general, a new node brings m new links. These links areadded to the older nodes by a specific way:

a) The first of the m edges links to the randomly chosen node.b) m − 1 edges link to the neighbours of the selected node.






model A

The model utilizes the knowledge, that the scale free structure is aresult of the preferential linking and the power law degreedistribution of C (k) is a result of the local edge linking. Thedynamics of the absolutely minimal model A is as follows:


2 Each time unit one node is injected and labelled by the timeof its addition s∗.

3 A new node brings m = 2 new links. Then an older edge ischosen randomly and the two links are added to the nodes atboth ends of the randomly selected edge.






Process ilustration

Figure : Node attachment which changes the clustering coefficient of thenode s in the model A.






Analytical results of model A

The analytical solution of the equations derived for model A hasfollowing results :

Clustering distribution :C (k) = 2k

(k+1)2 + 2ln(k)(k+1)2 = 2

k+1 −2

(k+1)2 + 2ln(k)(k+1)2 .

C (k) ∝ k−δ

Degree distribution :

k(s, t) = 2(

ts

) 12

P(k) ∝ k−α

where α = 3.0 and δ = 1.






Numerical results of model A

Measured exponent α = 3.0 is the same as is the calculated one.

Figure : Numerically created C (k) distribution for the model A, numberof nodes N = 50000. Scaling exponent δ = 0.98 is in a good accordancewith the analytically calculated δ = 1.0.






model B

The model B is more complex then the model A and is based onthese dynamical processes:


2 Each time unit one node is injected and labelled by the timeof its addition s∗.

3 A new node brings three new links. These links are added tothe older nodes by a specific way:

a) Two of them to the end nodes of a randomly chosen edge.b) The last one to a randomly chosen node s, s < s∗. s is

different from the endpoints of a chosen edge.

The model B is also analytically tractable.






Changing c(K) - proces ilustration

Figure : Five situations of the node s∗ linking which changes theclustering coefficient of the node s.






Analytical results of model B

The analytical solution of the equations derived for model B hasfollowing results :

Clustering distribution :

C (k) ≈ 0.5(k+3)[0.5(k+3)−1]2 −

ln[( k+3

6 )2− 1

3k+3

6

][0.5(k+3)−1]2

C (k) ∝ k−δ

Degree distribution :

k(s, t) = 6(

ts

)β − 3 where β = 13 .

P(k) ∝ k−α

where α = 4.0 and δ = 1.






Numerical results of model B

The exponent δ ∼= 0.86 is a worse approximation of the calculatedscaling exponent δ = 1.0

-10

-5

0

5

10

15

20

1 2 3 4 5 6 7

P(k)

k

Figure : Degree distribution of the numerically simulated model B withN = 800000 nodes. Scaling exponent α = 3.95 matches well thepredicted value α = 4.0.






Comments on models

SF-hierarchical models can be created by simple processes

Comparing model B to model A we can see the increase ofanalytical complexity with adding simple rule

Also we can observe that the threshold for stabilizing thenumerical simulations according to the analytical solutionsgrows rapidly with the number of processes.





IntroductionDefinitionHeuristicsMDS search algorithmDataset properties and experiments

Introduction

The problem of the minimal dominating set (MDS) on varioustypes of graphs arises in many practical areas such ascommunication, civil and electrical engineering, marketing etc.

In general the MDS problem is NP-hard (Raz, R., Safra,1997).

Lot of real networks studied recently have scale-free (SF)structure (Barabasi, Dezso , Ravasz , Oltvai - 2003).

Can appropriate heuristics designed for scale-free networksprovide a fast and accurate solution ?






Motivation example

Advertising problem on social networks:Consider the social network as graph were users are nodes andfriendship between two users is considered as an undirected edge.Let us consider a company who wants to propagate her product.Let as assume that if one user has an banner on his profile all hisfriends will see it. Adding a new banner on a user has a stablecost. The goal is that every users in the network can view theproduct banner and company minimise its expenses.






Minimum dominating set definition

Dominating set (DS): Let be DS be a subset of V such thatfor every v∈V also v∈DS or v is connected by an edge to v∈DS.

Minimum dominating set: Let f(G,DS) be a function thatgenerates all possible dominating sets of a graph G and|f(G,DS)| is the cardinality of such dominating set. Than let γbe the result of f(G,DS) having minimum cardinality.






Greedy by degree

Scale-free networks have several high-degree vertices. In this caseGreedy by k should be efficient. This simple strategy is improvedto greedy by degree (Abhay K. Parekh 1991). This procedureworks in following steps :

1. Create a new graph oriented graph Gt with loops from G.2. Add the node with the actual maximal out-degree into DS.3. Remove all edges pointing to the chosen node and to all itsneighbours4. if there is a node with degree(k) > 0 then go to 2.

In case of greedy by degree heuristics holds that:γ(G)≤ N+1 -

√2M + 1. (Parekh 1991)






Greedy by degree - ilustration

Figure : (a) Is a starting graph where each node is pointing to itsuncovered neighbour and self. (b) Node one was chosen to the DS as thevertex with highest out-degree and lowest ID. Step 3. is executed.






Leaf heuristic

Common property of SF real networks is that they have leafs.

It is clear either the leaf or its neighbour (parent) has to be inγG . If not then the leaf could not be covered.

Parent could have more leaves so the first heuristic rule is toput all parents into the DS.

~~

~~ ~Blue nodes are leafs.






Flower heuristic

Flower is a node with a having specific neighbourhood Subset Swhere :

∀u ∈ S |u ∈ Nb(v) ∧ (∀w ∈ Nb(u)|(w ∈ Nb(v)) ∨ (w = v))

JJJJJ~~~ ~

Red node = flower, Blue nodes ∈ Sflower

Flower is a superset of Parent.






Sweeping heuristic

The idea is to use a quasi-leaf. I define it as a node withstarting out-degree > 2, that has actual out-degree equal to 1and it is not a neighbour of node chosen to DS.

This heuristic is the first tie-breaker in case having twovertices having the same maximal out-degree.

~ ~~~ ~

~~ ~~

yRed nodes are in DS, blue nodes are neighbours of DS, greennodes are quasi-leafs, red-blue node was chosen by this heuristic.






Proposed MDS search algortihm

1 calculate the flower-set in G.

2 ∀v ∈ flower-set add v to DS.

3 ∀v∈Sflowers remove all edges pointing to (v ∨ u |u∈ Nb(v))

4 while(∃ v| deg(v)>0) greedy by degreesweeping






Dataset properties

network nodes edges c leafs

Amazon 334,863 925,872 0.4297 71742EmailEnron 36,692 367,662 0.4970 11211ca-AstroPh 18,772 396,160 0.6306 1282as-skitter 1696415 11095298 0.2963 293468

TexasRoad 1,379,917 3,843,320 0.0470 251082






Experimental results

network gbds parents flowers gbdt parentt flowert

Amazon 147338 142613 460 424EmailEnron 3096 3809 3082 1 1 1ca-AstroPh 2474 2559 2428 1 1 1as-skitter 293505 279007 2478 2526

TexasRoad 444225 5035

Legend : gbd means greedy by degree, parent means gdbimproved by leaf heuristic, flower means gbd improved by flowerheuristic. The subscripted s means solution and t means time.





IssuesIdea of solutionAlgorithm overview - idea and restrictionsDataset restrictions

Practical problems on benchmarking complex networks

shortage of datasets of the same class

shortage of implemented algorithms MDS problem

complexity problem with larger graphs

unknown size of optimal solution for input datasets






Expectation

If there is a luck of datasets in a class then generate new datasetsof the same class.Expected properties of the output:

The same degree distribution

Logarithmic diameter

Prescribed minimal dominating set

- The method as general as possible and should run in O(N2)maximalHow to do this ?






Idea of solution

1 Choose a graph from desired class for the input.

2 Run a fast Algorithm for the MDS problem on that graph.This creates un upper bound on the prescribed MDS.

3 Depending on the error estimation of previous algorithm setthe cardinality of the prescribed MDS.

4 Copy the degree distribution of the chosen graph as an inputparameter.

5 Run an algorithm with input(γ, Degree distribution) andoutput having these parameters.






Graphical ilustration 1

Figure : (a)(b) starting group, (c) leader covers all neighbours,(d)Guarantee is covered,(e) vertices with free edges






Graphical ilustration 2

Figure : (a)groups with free edges, (b) building a tree of groups, (c)finishing remaining edges






Comments to the process

All selects are chosen uniformly randomly from an appropriateset.

There is simple set of check after each step, that has besatisfied. If some check fails, then the set for taking nodes hasto adjusted.

The conditions ensure that if the input is correct the methodwill end.






Input restrictions informall

We expect that the input was taken from an existing graph but itis not a necessity.

The sum of degrees of highest vertices is high enough

The sum of degrees of lowest vertices is low enough

The sum of degrees of all vertices is high enough and thedegrees has appropriate distribution to ensure connectivity ofthe final graph.





SummaryAcknowledgement and Discussion

Summary

I showed improved analytical solution for an existing model(CNN)

I showed two new models of growing SF-hierarchical networks(models A,B)

I proposed new method for the MDS problem on SF-network.

I proposed an algorithm for creating benchmarking datasets.





SummaryAcknowledgement and Discussion

Thank you for your attention :).

Opening discussion.


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