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Hierarchy in networksHierarchy in networks
Peter Náther, Mária Markošová, Boris RudolfVyjde : Physica A, dec. 2009
OutlineOutline
Networks in Nature and their properties
Ravasz – Barabási hierarchical network
Vásquez modelHierarchy in the growing scale free
networks with local rules
Properties of natural Properties of natural networksnetworks1. Scale free property2.Small world property3.Hierarchy of nodes
Small world property
4
21
2
i
ii
k
kE
C
Average shortest distance is shortened due to the few shortcuts, while preserving local structure expressed in high clustering coefficient.
Scale free property
1.Network has rich self simillar complex structure.
2.Network has power law degree distribution.
log (k)
log(P(k))=log (N(k)/N)
kN
kNkP
Gamma is tangens of the angle A.
A
N(k) : number of nodes having degree k.
Degree distribution
Structure of the network is a result of its dynamics:
Preferential attachment of new nodes is responsible for the scale free structure of the network.
Can by shown analytically : Barabasi – Albert model
Is some type of dynamics responsible for the hierarchy of nodes in the network?
Sign of hierarchy: power law Sign of hierarchy: power law clustering coefficient distribution clustering coefficient distribution
log (k)
kClog kkC
kC - Average clustering coefficient of nodes with degree k.
RavRaváász - Barabsz - Barabáási hierarchical si hierarchical networknetwork
1. Scale free property coexists with hierarchy of nodes in many real networks (metabolic networks, protein interaction networks, www network, social network).
2. There should be a simple mechanism of creating hierarchical network while maintaining its scale free property.
3. Hierarchy in networks is expressed in the power law scaling of the average clustering coefficient for the nodes with degree k:
4. Hierarchy appears, if certain pattern is added each time unit into the network.
kkC
R-B process of net creation – deterministic version
R-B process of net creation – deterministic version
If the process runs sufficiently long, numerical analysis shows that:
1.Network has scale free character :
2.Network is hierarchical and has power law distribution of clustering coefficients :
3.Average clustering coefficient is constant
161.2, kkP
1, kkC
734.0C
Why ?1
a) Node in the centre of 5 – node modulus has clustering coefficient one and k=4. There are 20 such nodes.
b) Nodes at the centre of 25 node modulus have clustering coefficient 3/19 , k=20, and there are four such nodes.
c) One node at the centre of 125 node modulus has clustering coefficient k=84 and clustering coefficient 3/83.
R – B process, stochastic variant
Several real networks (actor network, semantic word web, www network, internet on the domain level) have hierarchical structure fulfilling approximately the law:
Is this an universal property, or scaling exponent differs from case to case?
Are scaling exponents for the clustering coefficient distribution and the degree distribution in hierarchical scale free network functionally dependant?
1, kkC
Stochastic variant of R-B hierarchical scale free
model.
Pick a p fraction of newly added nodes, and connect each of them independently and preferentially to the nodes of central module .
Preferential attachment means, that the probability of linking new node is proportional to the degree of the central module node .
What shows the numerical analysis of this stochastic model?
Changing p influences both exponents of clustering coefficient and degreedistribution:
decreases with increasing p.
decreases with increasing p.
Network preserves scale free and hierarchical property simultaneously.
, kkC
, kkP
Is an attachment of some regular, or at least of some virtually regular pattern responsible for the hierarchy in growing networks?
VVásquez modelásquez model
Vásques: Hierarchy and scale free property in networks emerges due to the local attachment rules.
Local rules: rules involving node and its nearest neighborhood
Exploring real networks (www, citation network)
Example A : Exploring www 1. One finds new www page using hyperlinks given on already known page. 2. One finds new www page using a search engine.
Example B: Exploring citation network 1. One finds a new paper by following citation list of known paper. 2. One finds a new paper randomly by searching.
Random walking, surfing on graph
Searching the net
Probability that certain node in the network will be visited if we start at randomly chosen node:
ij
outj
ej
outj
jije
ei
J
k
qk
vJq
N
qv ,
1
probability that the surfer decides to follow one link from the node
node out degree
adjacency matrix
Probability of node i being visited from node j by random walk
Probability of node i being visited by random jump from somewhere
inie
ei kq
N
qv
1 Mean field approximation of the
same formula.
Average probability, that a vertex pointing to vertex i is visited
How to calculate ?
average probability, that vertex having certain out degree and pointing to i node is visited.
average probability to leave this node through one link pointing out of the node i.
outk
v
outk
1
Visiting a new node means sometimes adding a link to it. Therefore, when exploring network by moving in it vertices are visited and links are added in average; being a probability that visited vertex increases its degree by one. (e.g. hyperlink is created to web page in www).
Nv Nvqvvq
Nvqt
Et
N
vs
a
---------- number of added nodes per time
unit,
---------- number of added edges per time unit, where is the number of surfers walking on net.
s
number of edges added by one surfer
a
vsoutin
vs
a
v
Nvqv
N
Ekk
NtvqvE
tvN
outk
vFrom equations
we get, that
a
vsoutin
v
Nvqv
N
Ekk
Nq sv
a
Number of added nodes per time unit
Number of surfersProbability that the
visited vertex increases its degree by one
Number of nodes in the network
Probability of increasing in degree of vertex Probability of increasing in degree of vertex with bywith byoneone
ink
ins
aeev
in
inie
eini
inv
in
kqqqN
kA
kqN
qkv
kvqkA
11
1
probability that a visited vertex increases its in degree by one
probability that a surfer follows one of the outgoing links
number of added nodes per time unit
number of surfers
probability that the visited vertex increases its degree by one
Nq sv
a
How does the network look? What tells us the How does the network look? What tells us the walk about the network structure?walk about the network structure?
The structure is expressed in the degree distribution and clustering coefficient distribution.
Degree distribution: How many nodes has degree k?
0,11 ininininin
in
kakkskksk nAnAt
n
probability, that node having degreegains a link.
1inkprobability, that node having degreegains a link.
ink this is zero for 0inkHow the amount of nodes having degree changes with time?
ink
inkk
sininkkkk
sk
inkkskksk
k
nA
kk
nAnA
t
n
equationthereorganizewe
knAnAt
n
ininininininin
inininin
in
1
0,
11
11
inkk
sk
k
nA
t
n ininin
Thus to get degree distribution, our aim is to solve this differential equation, which gives us a good asymptotic for great networks and long walking times.
inkk
sk
k
nA
t
n ininin
Let , where is a stationary probability of having node with
degree . Incorporating this into previous equation and integrating in we get
inin kkNpn ink
p
ink
in
s
aeevk
kqqqN
A in
11
k
pkCCCp
k
pkqpq
k
pqqp
NpkqqqNkt
Np
in
in
in
in
in
in
in
in
kin
k
kin
s
aek
s
ae
kevska
kins
aeevins
k
321
1
11
Solvable differential equation
e
qk
q
inkinkp ein
11
11
Network is scale free
Solution of the differential equation
Clustering coefficient distribution
ini
outii
ii
ii
kkk
inodeofneighborsamong
edgesofnumberEinodeoftcoefficienclusteringkE
C
,
2
How changes with time by random walk ?iE
ieiniev
i vqkqqt
E
Probability, that a visited vertex increases its degree by one
Probility of folowing one link
Average probability that a vertex pointing to vertex i is visited
Probability that vertex i is visited
ieiniev
i vqkqqt
E
First term denotes the probability, that a
vertex pointing to vertex i is visited, the second one denotes the probability, that a vertex i is visited and the walk follows an out link to its neighbor.
Incorporating and into this equation and using one finds
And by integrating this equation and using the basic formula for the clustering coefficient one finds
iv ink
ini At
k
inike
i Aqt
E
1
1 kC Network is hierarchical.
Our model with local rulesOur model with local rulesNáther, Markošová, Rudolf, to appear in Physica A
Network dynamics:
1.We start from the small network, from three totaly interconected nodes.2.Each time unit one node comes from the universe. If it comes at time s, it has a label s. The node brings m>1 new edges to the system.3.One new edge is linked to an old node by clustering driven preference, that means the linking probability is proportional to the clustering coefficient of the old link. The other m-1 edges is randomly distributed among neighbors of the old node.
Clustering coefficients of all nodes in starting graph is one.
1 2
3
4
Now clustering coefficients are 1,3
2,3
2,1 4321 CCCC
5
Clustering driven dynamics (CD model)
Numerical studies of CD model shows, that the network is scale free and hierarchical:
The most simple variant of CD model (SCD – model), in which each new node brings exactly two new edges into the network, is analytically tractable.
1
3
kkC
kkP
1. We can show analytically, that the clustering driven node addition is not responsible for the scale free final network structure.
2. It is possible to map SCD model to the model of Vásquez and to calculate analytically P(k) and C(k) distributions to show scale free and hierarchical property.
Clustering driven node addition is Clustering driven node addition is not responsible for the scale free not responsible for the scale free net structure.net structure.
In the SCD model each step creates a triangle of connected nodes Therefore for each node s the number of edges among its neighbors is k(s)-1, and its clustering coefficient is.
Let us therefore solve the model, where each time unit a node comes and links to the older node with probability proportional to .
sksksk
sksksk
sC2
1
12
2
1
1k
We should solve an equation
t
dstsk
tsk
t
tsk
1
1
1
,
,,
tgsfbtsk ,Let us look for the solution in a form:
Giving this into main integro - differential equation we have:
And from the initial conditions k(t,t)=1, we get f(s)=1/2-g(s). Seeking g(t) in a form we finally have:
which does not lead to the power law degree distributionm, and clustering driven network is not scale free.
2
1
2
1
2,
tgsftsk
ttg alog
2
1
2
1
log2
12,
s
ttsk a
Remapping of the SCD model to the Remapping of the SCD model to the VVásquez modelásquez model
1. SCD model can be mapped to the Vásquez model with only one surfer and the probability ( prob. that each visited vertex increases its degree by one). Let us call this V- model.
2. Therefore probability, that vertex with degree gains a new link is:
1vq
ink
in
s
aeevk
kqqqN
A in
11 Basic formula
inaeekkvqq
NA in 1
1 V - model formula
1s
3. In our SCD model each time unit two new edges are added. It is therefore comparable to the V – model with .
4. We also have clustering driven preference for finding a node by jump and edges are undirected.
2
1eq
kvq
CkqN
NA aeek
21
1
Applying to the rate equation
and using
we get
And after the integration
kA 0,11
knAnAt
nkkkk
k
kk Npn
kqCk
q
Ckq
q
kpt
kp
aee
eea
21
121 2
eq
kkp1
1, Because as was measured for our network.
3,2
1 eq
sksksk
sksksk
sC2
1
12
2
1
And what about hierarchy?
All nodes s with degree k in the SCD model have clustering coefficient given as:
That means, that
1, kkC
CD model, in which more then two nodes are added in one time unit, has been studied numerically.
CD modelCD model
SCD model is a simplified variant of CD model, in which one node and more then two edges are added in one time unit. One edge is linked by clustering driven preference to the certain old node s, other edges are distributed randomly among neighbors of the node s. Numerical studies of such model shows that the network has scale free property and hierarchy:
m is a number of edges added in each time unit.
mkkC
kkP
,
3,
SummarySummary
1. Local rules are responsible for the hierarchy in the networks. They lead to the fact, that in average certain pattern is added each time unit.
2. The fact, that the network is clustering driven is not responsible for the scale free structure, nor for hierarchy of nodes.
3. Virtual preferential attachment is responsible for the scale free property. If we link a node with clustering driven preference (or even with random preference ) the addition of another edges to the neighbor nodes ids in fact preferential. The higher degree has the neighbor, the higher probability it has to catch a new link.
ReferencesReferences1. Ravasz, Barabasi: Hierarchical organization in complex netwoks,
cond – mat 026130v2, 20022. Vásquez: Growing network with local rules…Phys. Rev. E 67 (2003)
0561043. Náther, Markošová, Rudolf: Hierarchy in the growing scale free
network with local rules, Physica A 388 (2009) 5036
Thank you for the attention