DCS/CSCI 2350: Social & Economic Networks What can we say about - - PDF document

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DCS/CSCI 2350: Social & Economic Networks

What can we say about a network having friends as well as enemies? “Structural Balance” Reading: Ch 5 EK

Mohammad T . Irfan

Agenda

u Merge: different types of edges + different

types of nodes

u Network of friends and enemies

u Balanced/stable configurations and their

characterization

u Local (a few nodes) à global (whole network)

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Structural balance

u First model (will be relaxed later)

u Complete graph (edge between each pair of nodes) u Each edge: either + or – u Example: sports team, countries, etc.

u Founders

u Fritz Heider (1940s) u Cartwright and Harary (1950s)

Exercise

u Graph with just 2 nodes

– +

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• Def. Balanced triangle

u Graph with just 3 nodes

– + – – + + – + + + – –

US UK SK US NK SK

? ?

Balanced Unbalanced

• Def. Balanced complete graph

A complete graph is balanced if and only if every triangle in it is balanced

u Q1. Can you draw a balanced and an

unbalanced complete graph?

u Q2. Is this definition a little extreme?

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Characterization of balanced network

u Balance Theorem (Harary, 1953)

A labeled complete graph is balanced if and only if either (i) all edges are + or (ii) graph consists of a bipolar faction

u Local (triangle) à Global (either everyone gets

along or there are two battling factions)

u Proof. (on board)

Extensions

• 1. Relax strict definition of balanced network

u

Weakly balanced network

• 2. General network (non-complete)

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Ext.# 1: Weakly balanced network

– + – – + + – + + + – –

US UK SK US NK SK

?

Balanced Unbalanced

US ISIS Syria

Characterization of weak balance

u Complete network is weakly balanced

if and only if either (i) all + or (ii) multiple battling factions

u Proof. Very similar to the previous

characterization (with strict def. of balance)

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Ext.# 2: General network

u Signed graph u Def. of balance

u A general signed graph is balanced if it consists of

a bipolar faction

u Definition is not based on triangles

u Characterization

u Balanced if and only if there exists no cycle with

• dd number of –ve edges

u Q. What goes wrong otherwise?

Real-world examples

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Liberation of Bangladesh in 1971 (Michael Moore, Journal of Social Psychology, 1978)

USA Pakistan India Bangladesh China

+ + – – – ? + – –

USSR

Structural balance & World War I

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Structural balance & World War I Social network among characters Graham Sack, 2013

AI question: generate narrative from the evolution of stability