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Hierarchy in networks

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Hierarchy in networks

Peter NÃ¡ther, MÃ¡ria MarkoÅ¡ovÃ¡, Boris Rudolf

Vyjde : Physica A, dec. 2009

- Networks in Nature and their properties
- Ravasz â€“ BarabÃ¡si hierarchical network
- VÃ¡squez model
- Hierarchy in the growing scale free networks with local rules

- 1. Scale free property
- Small world property
- Hierarchy of nodes

Small world property

Average shortest distance is shortened due to the few shortcuts, while preserving local structure expressed in high clustering coefficient.

- Scale free property
- Network has rich
- self simillar complex
- structure.
- Network has power law
- degree distribution.

Degree distribution

log(P(k))=log (N(k)/N)

A

Gamma is tangens of the angle A.

log (k)

N(k) : number of nodes having degree k.

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 â€“ Albert model

Is some type of dynamics responsible for the hierarchy of nodes in the network?

log (k)

- Average clustering coefficient of nodes with degree k.

- Scale free property coexists with hierarchy of nodes in many real networks (metabolic networks, protein interaction networks, www network, social network).
- There should be a simple mechanism of creating hierarchical network while maintaining its scale free property.
- Hierarchy in networks is expressed in the power law scaling of the average clustering coefficient for the nodes with degree k:
- Hierarchy appears, if certain pattern is added each time unit into the network.

R-B process of net creation â€“ deterministic version

R-B process of net creation â€“ deterministic version

- If the process runs sufficiently long, numerical analysis shows that:
- Network has scale free character :
- Network is hierarchical and has power law distribution of clustering coefficients :
- Average clustering coefficient is constant

Why

- Node in the centre of 5 â€“ node modulus has clustering coefficient one and k=4. There are 20 such nodes.
- Nodes at the centre of 25 node modulus have clustering coefficient 3/19 , k=20, and there are four such nodes.
- One node at the centre of 125 node modulus has clustering coefficient k=84 and clustering coefficient 3/83.

R â€“ 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?

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 degree

distribution:

decreases with increasing p.

decreases with increasing p.

Network preserves scale free and hierarchical property simultaneously.

Is an attachment of some regular, or at least of some virtually regular pattern responsible for the hierarchy in growing networks?

VÃ¡sques: 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:

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

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.

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).

---------- number of added nodes per time unit,

---------- number of added edges per time unit, where

is the number of surfers walking on net.

number of edges added by one surfer

From equations

we get, that

Number of added nodes per time unit

Number of nodes in the network

Number of surfers

Probability that the visited vertex increases its degree by one

probability that the visited vertex increases its degree by one

number of added nodes per time unit

number of surfers

probability that a surfer follows one of the outgoing links

probability that a visited vertex increases its in degree by one

The structure is expressed in the degree distribution and clustering coefficient distribution.

Degree distribution: How many nodes has degree k?

probability, that node having degree

gains a link.

probability, that node having degree

gains a link.

this is zero for

How the amount of nodes having degree changes with time?

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.

Let , where is a stationary probability of having node with

degree . Incorporating this into previous equation and integrating in we get

Solvable differential equation

Solution of the differential equation

Network is scale free

Clustering coefficient distribution

How changes with time by random walk ?

Probility of folowing one link

Average probability that a vertex pointing to vertex i is visited

Probability, that a visited vertex increases its degree by one

Probability that vertex i is visited

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

Network is hierarchical.

Our model with local rules

NÃ¡ther, MarkoÅ¡ovÃ¡, Rudolf, to appear in Physica A

- Network dynamics:
- We start from the small network, from three totaly interconected nodes.
- Each time unit one node comes from the universe. If it comes at time s,it has a label s. The node brings m>1new edges to the system.
- 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 driven dynamics (CD model)

4

5

3

1

2

Clustering coefficients of all nodes in starting graph is one.

Now clustering coefficients are

Numerical studies of CD model shows, that the network is scale free and hierarchical:

The most simple variant of CD model (SCD â€“ model), in which each new node brings exactly two new edges into the network, is analytically tractable.

- 1. We can show analytically, that the clustering driven node addition is not responsible for the scale free final network structure.

- It is possible to map SCD model to the model of VÃ¡squez and to calculate analytically P(k) and C(k) distributions to show scale free and hierarchical property.

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 .

We should solve an equation

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.

- SCD model can be mapped to the VÃ¡squez model with only one surfer and the probability ( prob. that each visited vertex increases its degree by one). Let us call this V- model.
- Therefore probability, that vertex with degree gains a new link is:

Basic formula

V - model formula

- In our SCD model each time unit two new edges are added. It is therefore comparable to the V â€“ model with .
- We also have clustering driven preference for finding a node by jump and edges are undirected.

Applying to the rate equation

and using

we get

And after the integration

Because as was measured for our network.

And what about hierarchy?

All nodes s with degree k in the SCD model have clustering coefficient given as:

That means, that

CD model, in which more then two nodes are added in one time unit, has been studied numerically.

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.

- 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.
- The fact, that the network is clustering driven is not responsible for the scale free structure, nor for hierarchy of nodes.
- 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.

References

- Ravasz, Barabasi: Hierarchical organization in complex netwoks, cond â€“ mat 026130v2, 2002
- VÃ¡squez: Growing network with local rulesâ€¦Phys. Rev. E 67 (2003) 056104
- NÃ¡ther, MarkoÅ¡ovÃ¡, Rudolf: Hierarchy in the growing scale free network with local rules, Physica A 388 (2009) 5036

Thank you for the attention