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A common-neighbors-based random graph model for community structure - - PowerPoint PPT Presentation

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A common-neighbors-based random graph model for community structure Emily Fischer Cornell University May 12, 2017 A Exam Emily Fischer Outline 1. Introduction Preferential Attachment (PA) 2. Common Neighbors Model (CN) Degree

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A common-neighbors-based random graph model for community structure

Emily Fischer

Cornell University

May 12, 2017

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A Exam Emily Fischer

Outline

  • 1. Introduction
  • Preferential Attachment (PA)
  • 2. Common Neighbors Model (CN)
  • Degree distribution
  • Community structure
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Preferential Attachment

  • Users prefer to connect to

nodes of high degree

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Preferential Attachment

  • Users prefer to connect to

nodes of high degree

  • Results in heavy-tailed degree

distribution

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Issues with Preferential Attachmment

The LinkedIn graph

  • 1. does NOT have a power law degree distribution
  • 2. has “community structure”
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Log-log plots of degree distribution

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Issues with Preferential Attachmment

The LinkedIn graph

  • 1. does NOT have a power law degree distribution
  • 2. has “community structure”
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What is “community structure”?

  • Strong community

structure

  • More edges within

community than between communities

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What is “community structure"?

  • Preferential attachment
  • One central hub around

high-degree node

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Common Neighbors Model

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Common Neighbors Model

Users prefer to connect to nodes with whom they share many mutual friends

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Common Neighbors Model

Users prefer to connect to nodes with whom they share many mutual friends

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Common Neighbors Model

Sequence of graphs (Gt)t≥0.

  • Given graph Gt with n(t) nodes and m(t) edges
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Common Neighbors Model

Sequence of graphs (Gt)t≥0.

  • Given graph Gt with n(t) nodes and m(t) edges
  • At time t + 1, a new node v arrives with probability α
  • If no new arrival, select v uniformly among existing nodes
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Common Neighbors Model

Sequence of graphs (Gt)t≥0.

  • Given graph Gt with n(t) nodes and m(t) edges
  • At time t + 1, a new node v arrives with probability α
  • If no new arrival, select v uniformly among existing nodes
  • Select receiving node w with probability proportional to number of

common neighbors between v and w

  • Γv(t) is the neighborhood of v at time t
  • Kvw(t) = |Γv(t) ∩ Γw(t)|

P(select w | sender = v) = Kvw(t) + δ

  • u Kvu(t) + δn(t)
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A Exam Emily Fischer

Common Neighbors Model

Sequence of graphs (Gt)t≥0.

  • Given graph Gt with n(t) nodes and m(t) edges
  • At time t + 1, a new node v arrives with probability α
  • If no new arrival, select v uniformly among existing nodes
  • Select receiving node w with probability proportional to number of

common neighbors between v and w

  • Γv(t) is the neighborhood of v at time t
  • Kvw(t) = |Γv(t) ∩ Γw(t)|

P(select w | sender = v) = Kvw(t) + δ

  • u Kvu(t) + δn(t)
  • Form directed edge (v, w).
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Common Neighbors Model

What does Kvw(t) look like? Hard to analyze - feedback

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Common Neighbors Model

What does Kvw(t) look like?

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Common Neighbors Model

What does Kvw(t) look like? Hard to analyze - feedback

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Common Neighbor Process

  • Want to model evolution of Kij(t) on its own.
  • Start at ˜

Kij(0) = 0 for all pairs i, j.

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Common Neighbor Process

  • Want to model evolution of Kij(t) on its own.
  • Start at ˜

Kij(0) = 0 for all pairs i, j.

  • Given ( ˜

Kij(t))i,j≥0, at t + 1,

  • Select i uniformly from existing nodes
  • Choose η = c(n(t))θ nodes, j1, j2, . . . , jη, preferentially with Kijℓ(t) ,

and increase Kijℓ(t + 1) = Kijℓ(t) + 1.

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Common Neighbor Process

  • Want to model evolution of Kij(t) on its own.
  • Start at ˜

Kij(0) = 0 for all pairs i, j.

  • Given ( ˜

Kij(t))i,j≥0, at t + 1,

  • Select i uniformly from existing nodes
  • Choose η = c(n(t))θ nodes, j1, j2, . . . , jη, preferentially with ˜

Kijℓ(t), and increase ˜ Kijℓ(t + 1) = ˜ Kijℓ(t) + 1.

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Common Neighbor Process

Let Ni(t) =

  • j

˜ Kij(t) What is the distribution of Ni(t)?

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Common Neighbor Process

Theorem

Let Ni(t) =

j ˜

Kij(t). Then there exists a random variable Zi such that Ni(t) tθ → Zi in probability, where Zi has characteristic function φZ(z) = exp

  • 1 − α

αθ

αθ

1 t (eitz − 1)dt

  • .
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Common Neighbor Process

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Common Neighbor Process

Result

  • The “total common neighbors” Ni(t) converges when scaled by tθ.

In progress/Future

  • Limiting distribution for ˜

Kij(t).

  • Use these distributions to analyze degree distribution of the graph
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Community Structure

  • How to quantify “strong community structure”
  • Compare community structure of CN and PA.
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Community Structure CN vs. PA

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Modularity

Definition

Given a graph partitioned into c communities, the modularity is Q =

c

  • i=1

(eii − a2

i )

where eii is the fraction of edges with both end vertices in community i, and ai is the fraction of ends of edges with vertices in community i.

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Community Detection

  • Community detection algorithms aim to assign nodes to

communities in a way that is reasonable

  • Some algorithms maximize modularity: Fast-greedy (FG),

Largest-eigenvector (LE)

  • But there are other methods as well: Edge-betweenness (EB),

Walktrap (WC).

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Modularity

Averages of modularity over 100 trials (α = .2, δ = .5) Graph EB FG LE WC CN 500 .450 .472 .423 .401 PA 500 .276 .379 .333 .251 CN 1000 .310 .402 .350 .301 PA 1000 .103 .328 .279 .190 CN 5000 .145 .320 .176 PA 5000 .039 .277 .120

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Conclusion

  • 1. PA mode lacks characteristics of LinkedIn network:
  • Power-law degree distribution
  • Lack of community structure
  • 2. Common Neighbors Model
  • Limiting distribution of Ni(t) in the common neighbors process
  • Better community structure than PA
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Edge Acceptance/Rejection

Node v sends an invitation to a node w.

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Model 1: Edge Acceptance/Rejection

w accepts the invitation with probability pvw(t).

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Edge Acceptance/Rejection

How can acceptance probability achieve goals of (1) non-power law degree distribution and (2) community structure?

  • Rich may choose not to get richer
  • Probability of acceptance based on communities

pvw(t) =

  • p

Cv = Cw q Cv = Cw.

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Edge Acceptance/Rejection

How can acceptance probability achieve goals of (1) non-power law degree distribution and (2) community structure?

  • Rich may choose not to get richer: pvw(t) ↓ 0
  • Probability of acceptance based on communities

pvw(t) =

  • p

Cv = Cw q Cv = Cw.

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Edge Acceptance/Rejection

How can acceptance probability achieve goals of (1) non-power law degree distribution and (2) community structure?

  • Rich may choose not to get richer: pvw(t) ↓ 0
  • Probability of acceptance based on communities:

pvw(t) =

  • p

Cv = Cw q Cv = Cw.

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Edge Acceptance/Rejection

For now, constant acceptance probability pvw(t) = p for all v, w and t ≥ 0.