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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
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
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
Martin Cajagi Complex networks - models and applications
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
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.
Martin Cajagi Complex networks - models and applications
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
IntroductionTerminology and notationsPropertiesComplex networks 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.
Martin Cajagi Complex networks - models and applications
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
IntroductionTerminology and notationsPropertiesComplex networks properties
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 )
Martin Cajagi Complex networks - models and applications
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
IntroductionTerminology and notationsPropertiesComplex networks properties
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.
Martin Cajagi Complex networks - models and applications
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
IntroductionTerminology and notationsPropertiesComplex networks properties
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)
Martin Cajagi Complex networks - models and applications
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
IntroductionTerminology and notationsPropertiesComplex networks properties
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
Martin Cajagi Complex networks - models and applications
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
IntroductionTerminology and notationsPropertiesComplex networks properties
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
Martin Cajagi Complex networks - models and applications
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
IntroductionTerminology and notationsPropertiesComplex networks properties
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.
Martin Cajagi Complex networks - models and applications
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
IntroductionTerminology and notationsPropertiesComplex networks properties
Common properties of real networks I.
Figure :
Martin Cajagi Complex networks - models and applications
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
IntroductionTerminology and notationsPropertiesComplex networks properties
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.
Martin Cajagi Complex networks - models and applications
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
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.
Martin Cajagi Complex networks - models and applications
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
IntroductionModel equationsOriginal solutionOur more accurate solution
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.
Martin Cajagi Complex networks - models and applications
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
IntroductionModel equationsOriginal solutionOur more accurate solution
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.
Martin Cajagi Complex networks - models and applications
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
IntroductionModel equationsOriginal solutionOur more accurate solution
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),
Martin Cajagi Complex networks - models and applications
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
IntroductionModel equationsOriginal solutionOur more accurate solution
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.
Martin Cajagi Complex networks - models and applications
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
IntroductionModel equationsOriginal solutionOur more accurate solution
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
).
Martin Cajagi Complex networks - models and applications
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
IntroductionModel equationsOriginal solutionOur more accurate solution
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).
Martin Cajagi Complex networks - models and applications
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
IntroductionModel equationsOriginal solutionOur more accurate solution
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.
Martin Cajagi Complex networks - models and applications
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
IntroductionModel equationsOriginal solutionOur more accurate solution
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.
Martin Cajagi Complex networks - models and applications
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
IntroductionModel equationsOriginal solutionOur more accurate solution
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 →∞.
Martin Cajagi Complex networks - models and applications
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
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.
Martin Cajagi Complex networks - models and applications
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
Introductionmodel Amodel BComments
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:
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 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.
Martin Cajagi Complex networks - models and applications
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
Introductionmodel Amodel BComments
Process ilustration
Figure : Node attachment which changes the clustering coefficient of thenode s in the model A.
Martin Cajagi Complex networks - models and applications
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
Introductionmodel Amodel BComments
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.
Martin Cajagi Complex networks - models and applications
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
Introductionmodel Amodel BComments
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.
Martin Cajagi Complex networks - models and applications
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
Introductionmodel Amodel BComments
model B
The model B is more complex then the model A and is based onthese dynamical processes:
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 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.
Martin Cajagi Complex networks - models and applications
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
Introductionmodel Amodel BComments
Changing c(K) - proces ilustration
Figure : Five situations of the node s∗ linking which changes theclustering coefficient of the node s.
Martin Cajagi Complex networks - models and applications
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
Introductionmodel Amodel BComments
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.
Martin Cajagi Complex networks - models and applications
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
Introductionmodel Amodel BComments
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.
Martin Cajagi Complex networks - models and applications
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
Introductionmodel Amodel BComments
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.
Martin Cajagi Complex networks - models and applications
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
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 ?
Martin Cajagi Complex networks - models and applications
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
IntroductionDefinitionHeuristicsMDS search algorithmDataset properties and experiments
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.
Martin Cajagi Complex networks - models and applications
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
IntroductionDefinitionHeuristicsMDS search algorithmDataset properties and experiments
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.
Martin Cajagi Complex networks - models and applications
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
IntroductionDefinitionHeuristicsMDS search algorithmDataset properties and experiments
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)
Martin Cajagi Complex networks - models and applications
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
IntroductionDefinitionHeuristicsMDS search algorithmDataset properties and experiments
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.
Martin Cajagi Complex networks - models and applications
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
IntroductionDefinitionHeuristicsMDS search algorithmDataset properties and experiments
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.
Martin Cajagi Complex networks - models and applications
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
IntroductionDefinitionHeuristicsMDS search algorithmDataset properties and experiments
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.
Martin Cajagi Complex networks - models and applications
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
IntroductionDefinitionHeuristicsMDS search algorithmDataset properties and experiments
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.
Martin Cajagi Complex networks - models and applications
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
IntroductionDefinitionHeuristicsMDS search algorithmDataset properties and experiments
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
Martin Cajagi Complex networks - models and applications
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
IntroductionDefinitionHeuristicsMDS search algorithmDataset properties and experiments
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
Martin Cajagi Complex networks - models and applications
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
IntroductionDefinitionHeuristicsMDS search algorithmDataset properties and experiments
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.
Martin Cajagi Complex networks - models and applications
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
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
Martin Cajagi Complex networks - models and applications
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
IssuesIdea of solutionAlgorithm overview - idea and restrictionsDataset restrictions
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 ?
Martin Cajagi Complex networks - models and applications
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
IssuesIdea of solutionAlgorithm overview - idea and restrictionsDataset restrictions
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.
Martin Cajagi Complex networks - models and applications
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
IssuesIdea of solutionAlgorithm overview - idea and restrictionsDataset restrictions
Graphical ilustration 1
Figure : (a)(b) starting group, (c) leader covers all neighbours,(d)Guarantee is covered,(e) vertices with free edges
Martin Cajagi Complex networks - models and applications
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
IssuesIdea of solutionAlgorithm overview - idea and restrictionsDataset restrictions
Graphical ilustration 2
Figure : (a)groups with free edges, (b) building a tree of groups, (c)finishing remaining edges
Martin Cajagi Complex networks - models and applications
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
IssuesIdea of solutionAlgorithm overview - idea and restrictionsDataset restrictions
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.
Martin Cajagi Complex networks - models and applications
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
IssuesIdea of solutionAlgorithm overview - idea and restrictionsDataset restrictions
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.
Martin Cajagi Complex networks - models and applications
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
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.
Martin Cajagi Complex networks - models and applications
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
SummaryAcknowledgement and Discussion
Thank you for your attention :).
Opening discussion.
Martin Cajagi Complex networks - models and applications