Rupei Xu , Shuzhong Zhang

May 28, 2013

Summer School on Network Science May 20-24, 27-31, 2013

University of South Carolina, Columbia, SC

Agenda

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Introduction: From Graph Theory to Complex Networks

The origin of Graph Theory is commonly attributed to Leonhard Euler who solved the Königsberg bridge problem in 1736.

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Small World Phenomenon

Inspired by the remark of his father that his father was only six handshakes away from the president of the United States, Watts(2003) started the study of "six degrees of separation".

Scale Free Network •

A large number of examples are found to be Scale-free Random Graphs.

Complex Networks

Most social, biological, and technological networks display substantial non-trivial topological features, with patterns of connection between their elements that are neither purely regular nor purely random.

Social Networks

• Nodes (representing individual actors within the network.)

• Ties(which represent relationships between the individuals, such as friendship, knowledge diffusion, crime connections, kinship, organizational position, collaboration relationship, etc.)

Social Networks

Social Networks

• communications, • community, • criminal networks, • diffusion of innovations, • demography, • economic sociology, • health care network, • human ecology,

• language and linguistics,

• literary networks, • organizational studies, • social capital， • virtual social networks

in Facebook, Twitter.

Introduction: Interior Point Method

• In 1984, Karmarkar developed the so-called

interior point method for linear programming. • Nesterov and Nemirovski introduced a special

class of such barriers known as self-concordant barrier functions.

• The class of primal-dual path-following interior point methods is considered the most successful.

Introduction: Sparse Optimization

• In the digital era, machine learning, compressed sensing, data mining, statistics, natural language processing, truss topology design and computational genetics can be readily formulated by optimization problems with tens of thousands or millions of variables.

• The way to find the sparse and other structures of the problem data and solutions has become popular and common in various computational and engineering problems.

• Sparsity not only makes it possible to reconstruct high-dimensional signals and discover its salient information from a small number of measurements, but also makes optimization faster and enables extremely large-scale computation.

Introduction： Compressed Sensing

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Introduction: Bridge Estimator

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Influence Maximization Problem • online social network sites such as Facebook, Myspace and Twitter

have opened a new door for the viral marketing. • people trust the information obtained from their close social circle far

more than the information obtained from general advertisement channels such as TV, newspaper and online advertisement.

• Many people believe that the 'word-of-mouth' marketing is still the most effective marketing strategy.

• Influence Maximization(IM) problem is to determine the initial seed set to make the spread of the influence maximized in the social network.

Influence Maximization Problem •

Influence Maximization Problem •

Influence Maximization Problem • Our influence maximization model is formulated as:

Key Players in Social Networks

• Key Players are those nodes in the network that are considered important with regard to some criteria.

• In this paper, the key players are defined as the seed nodes which can diffuse the influence both efficiently and widely, i.e., they can diffuse the influence to as many nodes as possible in high speed.

Key Players in Social Networks

• If fix the number of the key player only one, then many centrality measures such as highest betweenness centrality, best closeness centrality, highest degree centrality, highest eigenvector centrality etc. can be applied.

• If there are more than one players, simply greedy algorithm according to the centrality measure is not efficient.

• There are also some network entropy methods to solve this problem.

• Borgatti proposed a method to find the key players, defining two problems: the key player problem positive and the key

player problem negative.

Key Players in Social Networks •

Key Players in Social Networks • The objective is to find a way to allocate the budget to the key

players to diffuse the influence to as many nodes as possible in high speed.

An L1 Norm and Lp Norm Relaxation Approaches

• For the Influence Maximization Problem:

(1) (2)

An L1 Norm and Lp Norm Relaxation Approaches

• For the Key Players in Social Networks

(3) (4)

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Simulation Result: Influence Maximization Problem

Random Real Number Initial State and Fixed Threshold

Random Real Number Initial State and Random Threshold

Simulation Result:

Key Players in Social Networks

Conclusion • Formulation: • Not only consider the topology structure of the social network,

but also include the factors that initial state, threshold, negative opinion.

• deterministic Linear threshold model+ nonlinear optimization

• Methodology: • sparse optimization+ interior point method • concave + convex

• Result: • robust and sparse solutions+faster

Discussion

• First order interior point method • Learning effects • Competition • Other approximate algorithm • Diffusion equilibrium