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On the Formation of Community Structures from Homophilic Relationships Katerine Guerrero and Jorge Finke Dpt. Electrical and Computer Engineering Universidad Javeriana Santiago de Cali, Colombia

Thursday, July 5, 12


Networks

2 Thursday, July 5, 12


Networks

Describe a variety of phenomena •

Large amounts of data

Phone call, protein folding, patent citation networks

2 Thursday, July 5, 12


Networks

Describe a variety of phenomena •

Large amounts of data

Phone call, protein folding, patent citation networks

Theoretical foundations for networks •

Capture common patterns

Underlying mechanisms

2 Thursday, July 5, 12


Networks

Describe a variety of phenomena •

Large amounts of data

Phone call, protein folding, patent citation networks

Theoretical foundations for networks •

Capture common patterns

Underlying mechanisms

Validation of mechanisms

2 Thursday, July 5, 12


Networks

Describe a variety of phenomena •

Large amounts of data

Phone call, protein folding, patent citation networks

Theoretical foundations for networks •

Capture common patterns

Underlying mechanisms

Validation of mechanisms

Simple mechanism that drives network structure

2 Thursday, July 5, 12


Related literature Degree distribution M. Newman, S. Strogatz, D. Watts. Random Graphs with Arbitrary Degree Distributions and their Applications, 2001. J. Doyle, J. Carlson. Power Laws, Highly Optimized Tolerance, and Generalized Source Coding, 2000. J. Finke, N. Quijano, K. M. Passino. Emergence of Scale Free Networks from Ideal Free Distributions, 2008.

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Related literature Degree distribution M. Newman, S. Strogatz, D. Watts. Random Graphs with Arbitrary Degree Distributions and their Applications, 2001. J. Doyle, J. Carlson. Power Laws, Highly Optimized Tolerance, and Generalized Source Coding, 2000. J. Finke, N. Quijano, K. M. Passino. Emergence of Scale Free Networks from Ideal Free Distributions, 2008.

Clustering + degree distribution E. Volz. Random Networks with Tunable Degree Distribution and Clustering, 2004. P. Holme, B. Kim. Growing Scale-Free Networks with Tunable Clustering, 2002

3 Thursday, July 5, 12


Related literature Degree distribution M. Newman, S. Strogatz, D. Watts. Random Graphs with Arbitrary Degree Distributions and their Applications, 2001. J. Doyle, J. Carlson. Power Laws, Highly Optimized Tolerance, and Generalized Source Coding, 2000. J. Finke, N. Quijano, K. M. Passino. Emergence of Scale Free Networks from Ideal Free Distributions, 2008.

Clustering + degree distribution E. Volz. Random Networks with Tunable Degree Distribution and Clustering, 2004. P. Holme, B. Kim. Growing Scale-Free Networks with Tunable Clustering, 2002

Community structure J. Leskovec, K Lang, A. Dasgupta, and M. Mahoney. Community Structure in Large Networks: Natural Cluster Sizes and the Absence of Large Well-Defined Clusters, 2009 3 Thursday, July 5, 12


Community structure

4 Thursday, July 5, 12


Community structure

4 Thursday, July 5, 12


Community structure

4 Thursday, July 5, 12


Community structure

•

Sufficient conditions which provides theoretical guarantees

•

Reconstruction algorithm

4 Thursday, July 5, 12


Community structure

Sufficient conditions which provides theoretical guarantees

Reconstruction algorithm

Mechanism: Homophily • •

"Love of the same" Tendencies in relationships of sharing common characteristics (beliefs, values, education, etc.) 4

Thursday, July 5, 12


The model N = {1, . . . , n} G = (N, M ), mij 2 {0, 1} Qi = {j 2 N : mij = 1}, qi = |Qi |

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The model

• • •

Set of agents N = {1, . . . , n}

Directed graph G = (N, M ), mij 2 {0, 1} Constant number of outgoing neighbors

Qi = {j 2 N : mij = 1}, qi = |Qi |

5 Thursday, July 5, 12


The model

• • •

Set of agents N = {1, . . . , n}

Directed graph G = (N, M ), mij 2 {0, 1} Constant number of outgoing neighbors

Qi = {j 2 N : mij = 1}, qi = |Qi |

Two types of agents ti : N ! l, l 2 {0, 1}

Nl = {i 2 N : ti = l}, nl = |Nl |

• •

Same type neighbors Q0i = {j 2 Q : ti = tj }, qi0 = |Q0i | qi0 State of an agent xi = qi

5 Thursday, July 5, 12


The model

• • •

Set of agents N = {1, . . . , n}

Directed graph G = (N, M ), mij 2 {0, 1} Constant number of outgoing neighbors

Qi = {j 2 N : mij = 1}, qi = |Qi |

Two types of agents ti : N ! l, l 2 {0, 1}

Nl = {i 2 N : ti = l}, nl = |Nl |

• • •

Same type neighbors Q0i = {j 2 Q : ti = tj }, qi0 = |Q0i | qi0 State of an agent xi = qi State: proportion of neighboring agents of the same type 5

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Example 8 7

1

2

6

5

3 4

: agent of type 0 : agent of type 1

2

0 60 6 61 6 60 m(0) = 6 60 6 60 6 40 0

0 0 0 0 0 1 0 0

0 1 0 0 0 0 0 0

1 0 0 0 1 0 0 0

0 0 0 0 0 0 1 1

0 1 0 1 1 0 1 0

0 0 1 0 0 1 0 1

3

1 07 7 07 7 17 7 07 7 07 7 05 0

1. Identify conditions that lead to patterns in m(t) as t → ∞?

2. What assumptions that enable the formation of communities? 6

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Characterizing communities 1 X xi Index of homophilic relationships hl = nl i2Nl

total segregation 8i xi = 1 hl = 1 7 Thursday, July 5, 12


Characterizing communities 1 X xi Index of homophilic relationships hl = nl i2Nl

total segregation 8i xi = 1 hl = 1

type-neutral 8i xi = 0.5 hl = 0.5 7

Thursday, July 5, 12


Characterizing communities 1 X xi Index of homophilic relationships hl = nl i2Nl

total segregation 8i xi = 1 hl = 1

type-neutral 8i xi = 0.5 hl = 0.5 7

Thursday, July 5, 12

community xi 2 [0, 1] hl > 0.5


Characterizing communities 1 X xi Index of homophilic relationships hl = nl i2Nl

total segregation 8i xi = 1 hl = 1 8 Thursday, July 5, 12


Characterizing communities 1 X xi Index of homophilic relationships hl = nl i2Nl

total segregation 8i xi = 1 hl = 1

N1 homophilic xi 2 [0, 1] h1 = 1 8

Thursday, July 5, 12


Characterizing communities 1 X xi Index of homophilic relationships hl = nl i2Nl

total segregation 8i xi = 1 hl = 1

N1 homophilic xi 2 [0, 1] h1 = 1 8

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N0 heterophilic xi 2 [0, 1] h0 = 0


Assumption 1 (Network)

The network satisfies:

a. G = (N, M) can show total segregation. b. Every i 2 N may connect to both types of agents.

9 Thursday, July 5, 12


Assumption 1 (Network)

The network satisfies:

a. G = (N, M) can show total segregation. b. Every i 2 N may connect to both types of agents. Places bounds on the total number of outgoing neighbors 2  q < min{nl }

9 Thursday, July 5, 12


Assumption 2 (Decision-making)

Establish relationships is stochastic but must satisfy: a. Agent i tends to degrade ties to agents that are unlike itself by disconnecting from an agent j ∈ Qi, ti ≠ tj, and connecting to an agent r ∉ Qi. j i

10 Thursday, July 5, 12


Assumption 2 (Decision-making)

Establish relationships is stochastic but must satisfy: a. Agent i tends to degrade ties to agents that are unlike itself by disconnecting from an agent j ∈ Qi, ti ≠ tj, and connecting to an agent r ∉ Qi. j i

10 Thursday, July 5, 12


Assumption 2 (Decision-making)

Establish relationships is stochastic but must satisfy: a. Agent i tends to degrade ties to agents that are unlike itself by disconnecting from an agent j ∈ Qi, ti ≠ tj, and connecting to an agent r ∉ Qi. j i

j i r

10 Thursday, July 5, 12


Assumption 2 (Decision-making)

Establish relationships is stochastic but must satisfy: a. Agent i tends to degrade ties to agents that are unlike itself by disconnecting from an agent j ∈ Qi, ti ≠ tj, and connecting to an agent r ∉ Qi. j i

j

j i

i r

10 Thursday, July 5, 12

r


Assumption 2 (Decision-making)

Establish relationships is stochastic but must satisfy: a. Agent i tends to degrade ties to agents that are unlike itself by disconnecting from an agent j ∈ Qi, ti ≠ tj, and connecting to an agent r ∉ Qi. j i

j

j i

i r

r

b. Agent i tends to establish ties to agents that are unlike itself by connecting to an agent r ∉ Qi, ti ≠ tr, and disconnecting from an agent j ∈ Qi.

10 Thursday, July 5, 12


Assumption 2 (Decision-making)

Establish relationships is stochastic but must satisfy: a. Agent i tends to degrade ties to agents that are unlike itself by disconnecting from an agent j ∈ Qi, ti ≠ tj, and connecting to an agent r ∉ Qi. j

j

j

i

i

i r

r

b. Agent i tends to establish ties to agents that are unlike itself by connecting to an agent r ∉ Qi, ti ≠ tr, and disconnecting from an agent j ∈ Qi. r i j Thursday, July 5, 12

10


Assumption 2 (Decision-making)

Establish relationships is stochastic but must satisfy: a. Agent i tends to degrade ties to agents that are unlike itself by disconnecting from an agent j ∈ Qi, ti ≠ tj, and connecting to an agent r ∉ Qi. j

j

j

i

i

i r

r

b. Agent i tends to establish ties to agents that are unlike itself by connecting to an agent r ∉ Qi, ti ≠ tr, and disconnecting from an agent j ∈ Qi. r i j Thursday, July 5, 12

10


Assumption 2 (Decision-making)

Establish relationships is stochastic but must satisfy: a. Agent i tends to degrade ties to agents that are unlike itself by disconnecting from an agent j ∈ Qi, ti ≠ tj, and connecting to an agent r ∉ Qi. j

j

j

i

i

i r

r

b. Agent i tends to establish ties to agents that are unlike itself by connecting to an agent r ∉ Qi, ti ≠ tr, and disconnecting from an agent j ∈ Qi. r i j Thursday, July 5, 12

10


Assumption 2 (Decision-making)

Establish relationships is stochastic but must satisfy: a. Agent i tends to degrade ties to agents that are unlike itself by disconnecting from an agent j ∈ Qi, ti ≠ tj, and connecting to an agent r ∉ Qi. j

j

j

i

i

i r

r

b. Agent i tends to establish ties to agents that are unlike itself by connecting to an agent r ∉ Qi, ti ≠ tr, and disconnecting from an agent j ∈ Qi. r i j Thursday, July 5, 12

10


Assumption 2 (Decision-making)

Establish relationships is stochastic but must satisfy: a. Agent i tends to degrade ties to agents that are unlike itself by disconnecting from an agent j ∈ Qi, ti ≠ tj, and connecting to an agent r ∉ Qi. j

j

j

i

i

i r

r

b. Agent i tends to establish ties to agents that are unlike itself by connecting to an agent r ∉ Qi, ti ≠ tr, and disconnecting from an agent j ∈ Qi. r

r

i

i j

Thursday, July 5, 12

10

j


Assumption 3 (Event trajectories)

â&#x20AC;˘

â&#x20AC;˘

Agents are equally likely to establish or degrade ties to agents that are unlike themselves The agents i, j and r are selected randomly from a uniform distribution

11 Thursday, July 5, 12


Theorem Suppose A1-3 hold. Furthermore, suppose the ratio n0/n1 < 1/(q − 1). i. The majority group N1 starting at h1(0) > 0 shows total homophily with probability 1. ii.The minority group N0 starting at h0(0) < 1 − 1/(qn0) shows total heterophily with probability 1. as n0 → ∞.

12 Thursday, July 5, 12


Sketch of the proof Lemma 1 Suppose A1-3 hold and n1 > (qâ&#x2C6;&#x2019;1)n0 + 1. For every agent i â&#x2C6;&#x2C6; N, 0 < xi < 1, disconnecting from agent j to connect to agent r,

Thursday, July 5, 12

P [ti = tj ] < P [ti = tr ],

if ti = 1

P [ti = tj ] > P [ti = tr ],

if ti = 0


Sketch of the proof Lemma 1 Suppose A1-3 hold and n1 > (qâ&#x2C6;&#x2019;1)n0 + 1. For every agent i â&#x2C6;&#x2C6; N, 0 < xi < 1, disconnecting from agent j to connect to agent r, P [ti = tj ] < P [ti = tr ],

if ti = 1

P [ti = tj ] > P [ti = tr ],

if ti = 0

Group size effects agent behavior

Thursday, July 5, 12


Sketch of the proof Lemma 1 Suppose A1-3 hold and n1 > (q−1)n0 + 1. For every agent i ∈ N, 0 < xi < 1, disconnecting from agent j to connect to agent r, P [ti = tj ] < P [ti = tr ],

if ti = 1

P [ti = tj ] > P [ti = tr ],

if ti = 0

Group size effects agent behavior Type l Markov:

a1

1

acl −1

2

Thursday, July 5, 12

cl bcl

b2 1 − a1

cl − 1

1 − a2 − b 2

1 − acl −1 − bcl −1

1 − bcl


Sketch of the proof Lemma 1 Suppose A1-3 hold and n1 > (q−1)n0 + 1. For every agent i ∈ N, 0 < xi < 1, disconnecting from agent j to connect to agent r, P [ti = tj ] < P [ti = tr ],

if ti = 1

P [ti = tj ] > P [ti = tr ],

if ti = 0

Group size effects agent behavior Type l Markov: total heterophily hl = 0

a1

1

acl −1

2

Thursday, July 5, 12

cl bcl

b2 1 − a1

cl − 1

1 − a2 − b 2

1 − acl −1 − bcl −1

1 − bcl

total homophily hl = 1


Sketch of the proof Lemma 2 The probability that group Nl starting at 0 < hl(0) < 1 visits hl = 0 before hl = 1 is cX i l 1 Y bj m qm1 = q(m 1)1 where m = a 1+ m i=m j=m j

14 Thursday, July 5, 12


Sketch of the proof Lemma 2 The probability that group Nl starting at 0 < hl(0) < 1 visits hl = 0 before hl = 1 is cX i l 1 Y bj m qm1 = q(m 1)1 where m = a 1+ m i=m j=m j Lemma 3 Suppose G(0) is type-neutral and A1-3 hold. Furthermore, suppose the ratio n0/n1 < 1/(q − 1). Then, i. The majority group N1 shows total homophily before total heterophily with probability 1. ii. The minority group N0 shows total heterophily before total homophily with probability 1. as n0 → ∞ Initial network must have homophily index hl (0) = 0.5 14 Thursday, July 5, 12


Sketch of the proof Lemma 4 All homophily indices 0 < hl < 1 are transient. Lemma 5 The probability that the type l chain starting at 0 < hl (0) < 1 visits hl = 0 is

â&#x2021;˘21 1 +

â&#x2021;˘(cl

if m = cl

m X1

cl 1

1+

i=2 j=2

!

cl 1

i Y

2)1 ,

j

m X1

i Y

i=2 j=2

1.

15 Thursday, July 5, 12

j,

if m 2 {3, . . . , cl

2};


Remarks

8 7

1

2

6

5

3 4

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n0 1 < n1 q 1


Remarks

8 7

1

2

6

n0 1 < n1 q 1 h1 = 1

5

3 4

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Remarks

8 7

1

2

6

5

3 4

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n0 1 < n1 q 1 h1 = 1 h0 = 0


Remarks

8 7

1

n0 1 < n1 q 1

2

6

5

h1 = 1 h0 = 0

3 4

2

0 60 6 60 6 60 m(t) = 6 60 6 60 6 40 0

Thursday, July 5, 12

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

1 1 0 0 0 1 1 1

0 1 0 1 1 0 1 0

0 0 1 0 0 1 0 1

3

1 07 7 17 7 17 7 17 7 07 7 05 0


Remarks

8 7

1

n0 1 < n1 q 1

2

6

5

h1 = 1 h0 = 0

3 4

2

0 60 6 60 6 60 m(t) = 6 60 6 60 6 40 0

Thursday, July 5, 12

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

1 1 0 0 0 1 1 1

0 1 0 1 1 0 1 0

0 0 1 0 0 1 0 1

3

1 07 7 17 7 17 7 17 7 07 7 05 0

Real networks?


Allow hl = 0 and hl = 1 to be transient...

a1 total heterophily hl = 0

1

acl −1

cl − 1

2

bcl

b2 1 − a1

1 − a2 − b 2

1 − acl −1 − bcl −1

17 Thursday, July 5, 12

cl

1 − bcl

total homophily hl = 1


Intermediate stationary states 1

mean{ h0 } mean{ h1 }

0.9

1 0.9

0.9

0.8

0.8

0.8

0.7

0.7

0.7

0.6

0.6

Îľ0

0.6

0.5

0.5 0.5 0.4

0.4

0.4

0.3

0.3 0.3

0.2

0.2

0.2

0.1 0 0.2

0.4

0.6

n 0 /n 1

0.8

Majority group links to minority group member

0.1 0.2

1

0.1 0.4

0.6

0.8

1

0

n0 /n1

Under right incentives the minority group forms a community 18

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Percentage

1


Conclusions

Community structure from distributed decision-making

Group size matters!

Majority group forms a community

Minority group behavior depends on incentives

More than 2 types of agents (analytical results)

Extend for other distributions (power-laws)

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On the formation of community structures