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Time and Social Networks

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Background:•Most social network research has been static, though there is a growing interest in modeling network dynamics. This is occurring in two related directions:

•Two ways of thinking about time:•Movement of things through a network: Diffusion processes

•Change in the network itself: Structural implications of relation change.

Time and Social Networks

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Time and Social Networks

Historically, time has been incorporated into the network looking at

• changes in the distribution of an item over the population, over time

• the adoption of an innovation• the spread of an idea, etc.)

• different cross-sectional slices of the network

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Network Dynamics

• Limitations• Don’t allow us to explicitly model the changes within the

network, • Explain changes in the distribution of goods as a function of

timing.

• This static bias is built into some views of the network

• What we want is to be able to account for the dynamics of the network in “real time”• to account for changes in relations as a function of changes in

relations occurring around ego.\

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Network Forces

• Reciprocity• If A links to B, B links to A• “…essential for running of complex

societies”

• Dyads:• Reciprocated : A->B, B->A• Unreciprocated: A->B, not B->A• Null: no ties

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Measuring Reciprocity• Assumption: Link strength coded for

“liking” or “friendship”• Reciprocity = difference between sent

and returned liking

• Scale undefined, only in comparison:

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Dyadic Forces

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Transitivity

• Transitive triads:• A->B, B->C, A->C• A->B, not B->C or A->C

• Intransitive triads:• A->B, B->C, not A->C

• Null triads are transitive• No edges

• Complicated to compute on valued graphs

• Count the number of “forbidden triads”

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Enemy of my Friend is an Enemy

B

A

C

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Friend of my Enemy is an Enemy

B

A

C

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Enemy of my Friend is an Enemy

B

A

C

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Enemy of my Enemy is a Friend

B

A

C

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Group Balance

• Combination of friends and enemies:• “A friend of a friend is a friend”• “A friend of an enemy is an enemy”• “An enemy of a friend is an enemy”• “An enemy of an enemy is a friend”

• Assume binary (friend-enemy) or ternary (friend-enemy-neutral) relations

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Closure Force

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Closure Force

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Structural Holes

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Structural Hole Force

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Hierarchy

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Group Balance

• Only allows for 2 opposing groups• Relax rule 4 (“enemy of an enemy”)

• Balanced graph will be partitioned into subsets• Only positive ties within subsets• Only negative ties outside subsets

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Measuring Balance

• Partition into groups• Count departures from balance:

• Positive lines between plus-sets• Negative lines within plus-sets

• Move nodes from one set to another to minimize number of departures

• Classic optimization problem

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Time 1

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Time 5

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Time 10

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Reciprocity

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Transitivity

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40

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20

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01 2 3 4 5 6 7 8 9 10 11 12 13 14

Week

% C

hang

e in

ties

Relational Stability

Relational Stability

• Percentage of ties changing at each step

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 159

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13

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17Extent of Structural Imbalance

Structural Imbalance

Generalized Imbalance

Week

Structural (im)balance

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Result…

• Stable state of small oscillations• Similar to dynamic mechanical systems

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Doreian, Kapuscinski, Krackhardt & Szczypula:A brief history of balance through time.

“…the micro-level processes can be viewed as generating social forces that move the structure toward group balance.”

…negative ties within groups are likely less tolerated than positive ties between groups, as negatives within group may threaten the group in ways that positive ties between groups do not.

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An extension: A balance model of friendship change among adolescents

The basic hypothesis of social balance is that people will make choices that bring the entire group into balance.

But, consider how a given relationship looks from different perspectives: A transition that generates transitivity for one person can generate intransitivity for another. As such, there is no guarantee that friendship change will result in a globally balanced outcome.

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State-Transition diagram of triads

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Testing in simulationWhat are the implications for relationship change if people follow transitivity, from their own point of view?

J. Moody used a simulation, based on data from Add Health, to answer this question.

Rules: For every agent,

Attempt to achieve closureAttempt to achieve transitivity

Resolved through adding and dropping edges

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Strong transitive and intransitive rules:

Results in cliques forming quickly and maintaining over time.

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Moderate values for seeking transitivity and avoiding intransitivity: Results in a fluid network structure.

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Rules for network evolution• Seeking reciprocity (peer approval)

• Seeking transitivity (legitimation)

• Seeking balance (cognitive dissonance)

• Institutional Constraint• Fit to the institutional network

• Institutionalization• Escalation of tie strength - first mover advantage

• Planar optimization

• Feature selection

• Limited number of outgoing ties

• What about seeking power?

• Memory

• Any other forces?

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Friend of my Friend

Becomes my friend (with some probability)

B

A

C

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Balanced Triad

• The most stable configuration of network (Newcomb)

• 3 Simmelian ties (Krackhardt)• Reinforces conformity and similarity• Generates stable social norms of

behavior

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First Phase Transition

• After a small number of links is added, Triadic Closure rule strikes

• After critical mass is reached, transition from linear growth to quadratic

• Critical mass ~= 0.04

• Fully Connected network 40

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Let Conflict Strike

• Conflict inverts an edge from 1 (Friend) to -1 (Enemy)

• Conflict will decay at a linear rate and enemy link will disappear (0) in a short time

• What happens to triads when conflict strikes?

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Friend of my Friend

Can be an enemy???

B

A

C

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...dissonance causes action

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Can Conflict Spread?

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Can Conflict Spread?

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The greater the network density, the greater is the

probability of a catastrophic cascade

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Like the forest fire?

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Phase Transition #2

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alpha = 1.2886

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Degree Distribution

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Some results...

• Mean network density ~= 0.06• Mean density on Russian-language

LiveJournal ~=0.057• Network density is self-regulating;

Rapid increase in density quickly results in a collapse

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Instability - or complexity?

• The model produces viable networks at a set of parameter “sweet spots”

• Collapses otherwise!• e.g. low prior probability of conflict

results in super-critical density of links; when conflict finally strikes, all hell breaks loose!

• Rugged parameterlandscape

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