Microscopic Evolution of Social Network

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• Microscopic Evolution of Social Network

Zheng Jiangchuan
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Outline
Motivation
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Introduction
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Method Overview
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Evaluation
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Motivation

• Conventional social network study
• Primarily focus on the static structure of social
  network
• Reveal statistical network properties observed in
  real-world data power-law degree distribution,
  small world property, community.

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Motivation

• What is missing or rarely studied
• How does the social network we have observed come
  about
• What force drives the social network to exhibit
  the noted static macroscopic structural
  properties?
• How would the social network evolve in the
  future?

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Introduction

• Basic Intuition
• The answer lies in the laws governing the
  temporal evolution of social network
• Since the social network is a self-organized
  network, such laws are primarily hidden in the
  temporal behaviors of individual nodes

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Introduction

• Temporal Behaviors of Individual Node

At what rate does a new node arrive?
Node Arrival Process
How long will a new node stay active during its
life time?
When a node creates a new edge, which target will
this node most likely connect to?
Edge Initiation and Selection Process
How long will a node sleep before creating a new
edge?
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Introduction

• Individual node behavior overview
• Node arrives at some rate
• The newly arrived node decides its active
  lifetime
• The node initiates its first edge to a specific
  target node
• The node goes to sleep for some time
• The node wakes up, if its life time has not
  expired, then it selects a target node to connect
  to.
• Each node carries out the above process
  simultaneously, collectively leading to the
  macroscopic evolution of social network

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Introduction

• Basic Task
• Develop a generative model that is capable of
  describing the evolution of a social network
• This model is specified by the process in the
  previous slide at an intuitive level
• Need to quantify every step in this process
  mathematically from empirical observations

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Introduction

• What can this generative model be used for
• Provide mathematical insight into the question of
  how the social network with the static properties
  we have observed come about
• Predict the future evolution of the current
  social network
• Explain the temporal behaviors of humans in
  social domain

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Method Overview

• How to quantify each step in the generative
  process mathematically?
• Figure out the mathematical expression for each
  step by estimating from empirical temporal social
  network data based on Maximum Likelihood
  Estimation principle.
• For each step, select a model with certain
  parameters that maximize the likelihood of the
  data we have observed

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Data Set

• Four data sets with temporal information
• Flickr (03/2003-09/2005)
• Delicious (05/2006-02/2007)
• Answers (03/2007-06/2007)
• LinkedIn (05/2003-10/2006)

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Edge Attachment Process

• Basic method
• Evolve the network edge by edge, and for every
  edge arriving into the network, measure the
  likelihood that the particular edge endpoints
  would be chosen under some given model
• Pick the model and associated parameters that
  maximize the sample likelihood

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Edge Attachment Process

• Candidate models
• Before proceeding to the MLE experiments, need to
  propose some candidate models
• Edge attachment by degree
• Edge attachment by age of the node
• Carry some simple experiments to justify the
  effectiveness of the proposed models
  qualitatively

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Edge Attachment Process

• Edge attachment by degree
• The probability that a new edge connects to a
  specific node is proportional to the degree of
  that node at the moment
• This is intuitively consistent with common sense
  as people are more likely to know those
  influential individuals

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Edge Attachment Process

• Edge attachment by degree
• Simple experiments for justification, not MLE.
• Plots the probability that a new edge connects to
  a node with a certain degree

The experiments match well with the degree
preferential attachment model
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Edge Attachment Process

• Edge attachment by age of the node
• The probability that a new edge connects to a
  specific node is proportional to age of that node
  at the moment
• The intuition is older, more experienced users
  of a social network are also more engaged and
  thus absorb more edges

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Edge Attachment Process

• Edge attachment by age of the node
• Depicts how many number of edges are absorbed by
  nodes of specific age normalized by the number of
  nodes that have achieved that age

The experiments do not match well with the
proposed model, but anyway it is a possible choice
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Edge Attachment Process

• Maximum likelihood estimation
• Four models
• D degree preferential attachment
• DR combination of degree preferential attachment
  and uniformly at random attachment
• A age preferential attachment
• DA combination of degree preferential attachment
  and age preferential attachment
• For each model and for each data set, plot the
  sample likelihood w.r.t model parameters

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Edge Attachment Process

• Maximum likelihood estimation

Conclude that model D performs reasonably well
compared to more sophisticated variants based on
degree and age
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Locality of edge attachment

• Basic Intuition
• While the degree preferential attachment model
  appears to be a reasonable model, it fails to
  take into account the locality of edge attachment
• Intuitively, people are more likely to connect to
  people with common friends, that is, a new edge
  tends to span a small number of hops

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Locality of edge attachment

• Experiments that empirically justify this
  intuition
• Plots the probability that a newly created edge
  spans a certain number of hops

The experiments on real data do not match well
with PA model in terms of decreasing rate
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Locality of edge attachment

• Insight from the experiments
• The double exponential decrease of
  suggest that newly created edges are very likely
  to span only a small number of hops, forming
  triangles
• So the degree preferential attachment model
  should be replaced by triangle-closing models,
  i.e., each new edge connects to a node two hops
  away

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Locality of edge attachment

• Mathematical model of triangle-closing
• The edge creating process can be decomposed into
  two steps select the neighbor by some random
  rule, then select the neighbors neighbor by
  possibly another rule
• There are many possible triangle-closing models,
  depending on how to select neighbor at each step

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Locality of edge attachment

• Select best triangle-closing model using MLE

On average, the random-random triangle-closing
model performs relatively well, and will be used
to describe the evolution.
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Node life time

• Selected Model
• By performing similar maximum likelihood
  estimation experiments, found that node lifetimes
  are best modeled by an exponential distribution

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Time gap between edges

• Selected Model
• Intuitively, individuals with more friends are
  likely to make new friends in a shorter time,
  meaning the gap distribution for nodes with
  different degrees should be different.
• More precisely, the gap distribution should be
  conditional on the degree of the node

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Time gap between edges

• Selected Model
• For a specific data set, by estimating and
  for each using maximum likelihood estimation,
  we are able to find the possible function of
  and with , respectively.

While a is a constant, independent of d, k is a
linear function of d, although the linear
coefficient b varies among different data sets
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Complete Model
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Evaluation

• Very novel and rigorous evaluation method
• Basic ideas For a specific data set, if the
  evolution model is correct, then the static
  properties of the final network that are computed
  mathematically by such an evolution model should
  be close to what we have observed in the data set
  

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Evaluation

• Very novel and rigorous evaluation method
• 1. Based on the proposed evolution model,
  analytically derive a mathematical expression for
  the degree distribution of the final network as a
  function of the parameters in the evolution model
• 2. For a specific temporal social network data
  set, estimate the parameters needed by the
  evolution model.
• 3.Substitute the estimated parameters into the
  mathematical expression for the degree
  distribution of the final network
• 4. Compare the result with the true degree
  distribution observed in the final snapshot of
  this social network data set.

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Evaluation

• Analytic derivation

Estimate and from the temporal social
network dataset, respectively
Compare this with the parameter of the degree
distribution directly estimated from the real
data set
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Evaluation

• Results

Surprisingly Similar!
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Thank you!
Q A