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Evolutionary Models and Dynamical Properties of Complex Networks

Name: Jianguo Liu

University of Shanghai for Science and Technology

2010-3-24

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Outline

1. Complex networks analysis by Citespace

2. Network evolution models3. Dynamical properties on scale-

free networks4. Personalized recommendation

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1999 年 -2010 年发表的以“ complex networks” 为主题词的 SCI 论文数

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1999 2001 2003 2005 2007 2009

publ i shed SCIpapers

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Citespace 软件介绍 CiteSpace ：由美国德雷赛尔大学信息科学与技术学院的陈超美开发。该程序可以登录到 cluster.cis.drexel.edu/~cchen/citespace后免费使用。

利用 Citespace 寻找某一 学科领域的研究进展和当 前的研究前沿，及其对应 的基础知识。

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复杂网络论文作者合作网 (1999-2010)

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复杂网络研究小组状况 (1999-2010)

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复杂网络各个国家研究状况 (1999-2010)

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利用引文分析观察当前的研究热点 (1999-2010)

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Top cited authors(1999-2010)

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各研究领域之间的关系 (1999-2010)

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个性化推荐的知识图谱

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Top cited authors

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目前的研究热点

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Outline

1. Background introduction2.2. Network evolution modelsNetwork evolution models3. Dynamical properties on

scale-free networks4. Personalized

recommendation

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2.Scale-free Network Evolution Models

1. Multistage random growing small-world networks with power-law degree distribution

2. Growing scale-free network model with tunable assortative coefficient

3. Self-learning mutual selection model for weighted networks

4. Random evolving networks under the diameter and dverage connectivity constraint

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2.1.Multistage random growing small-World networks with power-law degree distribution

Liu Jian-Guo, Dang Yan-Zhong and Wang Zhong-Tuo, Chinese Physics Letters 23(3) 746-749 (2006)

•One node is added in each time step ；•Select the node u according to the preferential mechanism ；•Select a neighbor node of node u ；

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1. One node is added in each time step ；2. Select the node u according to the preferential mechanism ；3. Select a neighbor node of node u according to ps ；

2.2. Growing scale-free network model with tunable assortative coefficient

Qiang Guo, Tao Zhou, Jian-Guo Liu et al., Physica A 371 814-8

22 (2006)

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Two parameters: attractive factor p, the number of candidates m

2.3 Self-learning mutual selection model for weighted networks

Jian-Guo Liu et al., DCDIS B Supplement, Complex Networks, 14 (S7)

33-36, (2007).

1

1

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s ap

s a

s s p

1

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1

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5m=2

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2.4 Random Evolving Networks Under the Diameter and Average Connectivity Constraint

The growth of random networks under the constraint that the diameter, defined as the average shortest path length between all nodes, and the average connectivity remains approximately constant is studied. We showed that, if the network maintains the form of its degree distribution and the maximal degree is a N-dependent cutoff function, then the degree distribution would be approximately power-law with an exponent between 2 and 3.

Jian-Guo Liu et al., Journal of System Science and System Engi

neering 16(1) 107-112 (2007).

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Motivation

In the biological networks, the constant diameter may be related to important properties of these biological networks, such as the spread and speed of responses to perturbations. In the Internet backbone network, the average distance is one of the most important factors to measure the efficiency of communication network, and it plays a significant role in measuring the transmission delay. These constraints can be thought of as the environmental pressures, which would select highly efficient structure to convey the packets in it.

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Motivation

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Construction of the model

The expression for the diameter d of a random network with arbitrary degree distribution was developed

Where is the average degree,

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In order to seek a degree distribution that maintains its distribution and has an approximately constant diameter independent of N. The parameter N can be accomplished by imposing a N-dependent cutoff function

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The distribution p(k) can be determined by writing this equation for and Algebraic manipulation yields the relation

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Using an integral approximation , a more explicit formulation can be written as following.

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When the numerically calculated

degree distributions for various values of

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Discussion of part two

We have presented a reason for the existence of power-law degree distribution under the diameter constraint observed in the Internet backbone network where there are evolutionary pressures to maintain its diameter. Our analysis shows that, if the maximal degree is a N-dependent cutoff function, the form of a robust network degree distribution should be power law to maintain its diameter, while the average connectivity per node affect the distribution exponent slightly.

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Outline

1. Background introduction2. Network evolution models3.3. Dynamical properties on Dynamical properties on

complex networkscomplex networks4. Personalized

recommendation

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3.1 Structural effects on synchronizability of scale-free networks

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3.1 How to measure the synchronizability

Where Q is the ratio of the eigenvalues. The synchronizability would be increased as Q decreases, vice verse.

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The edge exchange method is introduced to adjust the network structure, and the tabu search algorithm is used to minimize the eigenvalue ratio Q

min

Qiang Guo, Liu Jian-Guo, et al, Chinese Physics Letters 24 (8) (200

7) 2437-2440.

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In summary, using the tabu optimal algorithm, we have optimized network synchronizability by changing the connection pattern between different pairs of nodes while keeping the degree distribution. Starting from scale-free networks, we have studied the dependence between the structural characteristics and synchronizability. The numerical results suggest that a scale-free network with shorter path length, lower degree of clustering, and disassortive pattern can be easily synchronized.

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3.1 Structural effects on synchronizability

min

max

Combining the tabu search (TS) algorithm and the edge exchange method, we enhance and weaken the synchronizability of scale-free networks with degree sequence fixed to find the structural effects of the scale-free network on synchronizability

Liu Jian-Guo, et al, International Journal of Modern Physics C 18(7)

1087-1094 (2008).

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The numerical results indicate that D, C, r and Bm influence synchronizability simultaneously. Especially, the synchronizability is most sensitive to Bm.

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Effect of the loop structure on synchronizability

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Outline

1. Background introduction2. Network evolution models3. Dynamical properties on

complex networks4.4. Personalized recommendationPersonalized recommendation

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Personalized recommendation

Improved collaborative filtering algorithm based on information transaction.

Ultra accuracy recommendation algorithm by considering the high-order user similarities

Effect of user tastes on personalized recommendation

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Why recommend

We face too much data and sources to be able to find out those most relevant for us. Indeed, we have to make choices from thousands of movies, millions of books, billions of web pages, and so on. Evaluating all these alternatives by ourselves is not feasible at all.

As a consequence, an urgent problem is how to automatically find out the relevant objects for us.

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Collaborative filtering algorithm

Herlocker et al., ACM Trans. Inf. Syst. 22: 5-53 (2004)

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Content-based algorithm

The user will be recommended items similar to the ones this user preferred in the past

Pazzani & Billsus, LNCS 4321: 325-341 (2007)

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Improved collaborative filtering algorithm based on information transaction

In traditional CF algorithm, Firstly, the user similarity is computed based on the Pearson coefficient

)}(),(min{1

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Then give the predicted score to the uncollected objects based on the user similarities

m

lli

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By using the diffusion process to compute the user similarity to improve CF algorithm

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Iliustration of the diffusion-based user similarity

The target user is activated and be assigned a unit recommendation power, then the mass is diffused from the target user to the objects he has collected, then the it’s diffused back from the objects to the users.

Jian-Guo Liu et al, International Journal of Modern Physics C 20 285 (2009) .

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Two improved algorithms

We argue that the potential role of the object degrees should be taken into account to regulate the user similarity

Only the top-N most similar users’ opinion are taken into account to save the memory and increase the computation speed.

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Numerical results of the average ranking score

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Diversity

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Average ranking score of the top-N algorithm

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Conclusion and discussions Using the diffusion process to compute the user similarity

could improve CF algorithmic accuracy The computational complexity of the presented algorithm is

much less than that of the standard CF. Both the two modified algorithm can further enhance the

accuracy. With properly choice of the parameter N, top-N algorithm

can simultaneously reduces the computational complexity and improves the algorithmic accuracy.

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2,Ultra accuracy recommendation algorithm by considering the high-order user similarities

The main idea

Figure 1 shows an illustration, where 1 denotes the mainstream preference shared by all A, B and C, and 2 is the specific preference of A and C. Both 1 and 2 contribute to the correlation between A and C. Since 1 is the mainstream preference, it also contributes to the correlations between A and B, as well as B and C. Tracking the path A to B to C, the preference 1 also contributes to the second-order correlation between A and C. Statistically speaking, two users sharing many mainstream preferences should have high second-order correlation, therefore we can depress the influence of mainstream preferences by taking into account the second-order correlation.

Jian-Guo Liu et al, Physica A 389 881 (2010) .

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The presented algorithm

Denote the user similarity matrix is S

where H is the newly defined correlation matrix, S={ } is the first-order correlation defined as Eq.(2), and is a tunable parameter. As discussed before, we expect the algorithmic accuracy can be improved at some negative .

ijs

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Numerical results

r

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Effect of user tastes on personalized recommendation

We study the effects of user tastes on the mass-diffusion-based personalized recommendation algorithm, where a user's tastes or interests are defined by the average degree of the objects he has collected. We argue that the initial recommendation power located on the objects should be determined by both of their degree and the users' tastes. By introducing a tunable parameter, the user taste effects on the configuration of initial recommendation power distribution are investigated.

Jian-Guo Liu et al, International Journal of Modern Physics C 20 285 (2009) .

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Numerical results

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Achievements Selected Publications All publications has been cited over 118. H-index is 7

Phys. Rev. E 80, 017101(2010); 74, 056109 (2006). Physica A 389, 881 (2010); 371, 861 (2007); 366, 578 (2006); 371, 814 (2006); 377, 302 (2

007). European Physics Journal B 66, 557 (2008); Int. J. Mod. Phys. C 18 1087 (2008); 20, 285(2009); 20, 1925 (2009); 18, 1087 (2007); 18,

294 (2007). Chin. Phys. Lett. 25, 773 (2008); 25, 2718 (2008); 23, 746 (2006); 377, 302 (2007); 23, 232

7 (2006); 24, 2437 (2007); Mod. Phys. Lett. B 19, 785 (2005); 20 815 (2006); Int. J. Syst. Sci. and Syst. Engi. 16, 107 (2007); J. Comp. & Appl. Math. 181, 252 (2005) PNAS ， 107 ， 4511 (2010)

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Future works

实证数据分析 大规模数据分析，例如在线社区，手机通讯等。细致研究数据的统计特性，例如胡海波发现虚拟社会网络的度度相关性是负值，明显不同于真实的社会网络。张国清和周涛则发现 AS 系统的增长也满足 Moore 定律，互联网内核比较稳定。

网络通缉特性与个性化推荐效果之间的关联关系研究 度量二部分网络的统计指标，找到这些指标，尤其是推荐多样性之间的关系。

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Many thanks