Kempe-Kleinberg-Tardos Conjecture A simple proof Lecture 2-5 Kempe-Kleinberg-Tardos Conjecture A simple proof Ding-Zhu Du University of Texas at Dallas First, I want to thank you for you presence. ********In this presentation I will try to introduce The social network which is a theoretical structure to study relationships between individuals, groups, organizations, or even entire societies.  It is related to a wide range of disciplines. These disciplines include, but are not limited to information science, biology, economics, geography, communication studies, and so on.. The study of social networks begins with the late eighteenth century, two sociologists (Émile [ei'mi:l] Durkheim and Ferdinand ['fɝdənænd] Fer迪南de Tönnies) foreshadowed the idea of social networks in their theories and research of social groups. Nowadays, we study social networks using network analysis to identify social communities, pick influential person, and design good software.

Outline of KKT Conjecture IM in Threshold Model KKT Conjecture A Simple Proof Applications 1. Brief overview of social networks 2. How to build applications on top of the social network –  Think about a social network being MS Windows, We can build applications on it.

General Threshold Model 1 2 3

LT: Stop! w v Y Inactive Node 0.6 Active Node 0.2 Threshold 0.2 0.3 Active neighbors X 0.1 0.4 U 0.3 0.5 Stop! 0.2 0.5 w v

Inapproximability Theorem Proof Recall

Input size

is monotone submodular. but not submodular.

Outline Threshold Model KKT Conjecture A Simple Proof Applications 1. Brief overview of social networks 2. How to build applications on top of the social network –  Think about a social network being MS Windows, We can build applications on it.

Kempe-Kleinberg-Tardos Conjecture This conjecture is proved by Mossel and Roch in 2007 (STOC’07)

Linear Threshold (LT) Model A node v has random threshold ~ U[0,1] A node v is influenced by each neighbor w according to a weight bw,v such that A node v becomes active when at least (weighted) fraction of its neighbors are active Given a random choice of thresholds, and an initial set of active nodes A0 (with all other nodes inactive), the diffusion process unfolds deterministically in discrete steps: in step t, all nodes that were active in step t-1 remain active, and we activate any node v for which the total weight of its active neighbors is at least Theta(v)

Example Stop! w v Y Inactive Node 0.6 Active Node 0.2 Threshold 0.2 0.3 Active neighbors X 0.1 0.4 U 0.3 0.5 Stop! 0.2 0.5 w v

Outline Threshold Model KKT Conjecture A Simple Proof Applications 1. Brief overview of social networks 2. How to build applications on top of the social network –  Think about a social network being MS Windows, We can build applications on it.

Idea: Piecemeal Growth Seeds can be distributed step by step or altogether, the distribution of final influence set does not change.

Notations ~

Notations ~ ~ ~ ~

Lemma 1 Proof

1st Try ~ ~

1st Try

1st Try Not true!

More Techniques: Antisense Phase and Need-to-Know Representation

Antisense Phase ~ ~

Lemma 2 ~ ~ Proof

2nd Try ~ ~

2nd Try

Lemma 3 Proof

Outline Threshold Model KKT Conjecture A Simple Proof Applications 1. Brief overview of social networks 2. How to build applications on top of the social network –  Think about a social network being MS Windows, We can build applications on it.

Linear Threshold (LT) Model A node v has random threshold ~ U[0,1] A node v is influenced by each neighbor w according to a weight bw,v such that A node v becomes active when at least (weighted) fraction of its neighbors are active Given a random choice of thresholds, and an initial set of active nodes A0 (with all other nodes inactive), the diffusion process unfolds deterministically in discrete steps: in step t, all nodes that were active in step t-1 remain active, and we activate any node v for which the total weight of its active neighbors is at least Theta(v)

Decreasing Cascade Model

Deterministic Model

Independent Cascade

Triggering Model

IC is a special cases of Triggering Model

LT(or MC) is a special cases of Triggering Model

Triggering Model Triggering model is not a general threshold model. When triggering set at every node is fixed, it can be seen as a threshold model. Then # of influenced nodes is a monotone increasing submodular function of seed set. Triggering model is a linear combination of threshold models. Coefficients are probability.

“Only-Listen-Once” Model

References 1 2

1 2 3

1 1 2 3 2 3 1 1 2 3 2 3

1 2 3

Proof of Submodularity

Conjecture

THANK YOU!

Open Problem 1