Emergence of Scaling and Assortative Mixing by Altruism

Li Ping

The Hong Kong PolyU

2010.07.29

Outline

Motivation and background Network model Statistical properties Summary

Motivations

Networks are evolving in time Some topological features arising:

Power – law degree distributionsAssortativity mixing patterns……

What’s the origin of these properties?

Barabasi-Albert Model

Rules:Growth &Preferential attachment

Results:Power-law degree distribution!

Beyond Scaling-Assortativity

Assortativity a negative or positive correlation in

adjacent nodes with respect to a certain property

Measurement

€

r ∝ kik j − ki k jor

€

knn (k) = k 'P(k ' | k)k '∑

Assortative (disassortative) Mixing in Real Networks

Internet

AS1998 AS2003 AS2008 -0.198 -0.14 -0.13

Erdos collaboration networks

Erdos971 Erdos981 Erdos991

0.2245 0.2251 0.2243

AssortCoeff

AssortCoeff

Existing Model for Social Networks

BA growth mechanismAdding new nodes with probability

MixingAdding new edges between node 1

and 2 with probability 1-

M. Catanzaro et al. Physica A 338(2004) 119-124

€

pkik j

j

∑

€

(1− p)k1k j

j

∑P(k2 | k1)

€

p

€

p

Our Model

Growth rule: Each time, a new node is added in the existing network

Altruistic attachmentPreferentially choose a node to be an

agent according to its degreeRandomly choose a node to be the

target from the neighborhood of the agent

agent

New node

N ＝ 150 ， m ＝ 3

AA or BA？

Statistical Properties-Degree Distribution

N ＝ 2000 ， m ＝ 3 ， with assortativitycoefficient ＝ 0.081

Power Law Exponent for Different Network Size

Assortativity

BA AA

Fragility and Robustness

Randomly remove nodes

Targeted remove nodes

AA

BA

Clustering Coefficients

Conclusion

Assortativity and scale － free properties can be reproduced by introducing altruistic attachment mechanism

AA network model shows some differences with respect to various statistical properties by numerical simulations

More things that AA tells us…

Thank you for your attention!