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

2/26/18 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) 1

2/26/18 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 – + 2

2/26/18 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? 3

2/26/18 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 Weakly balanced network u 2. General network (non-complete) 4

2/26/18 Ext.# 1: Weakly balanced network US + + US UK SK + + + ? – – US – – + – Syria ISIS NK SK – Balanced Unbalanced 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) 5

2/26/18 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 odd number of –ve edges u Q. What goes wrong otherwise? Real-world examples 6

2/26/18 Liberation of Bangladesh in 1971 (Michael Moore, Journal of Social Psychology, 1978) USA – USSR ? – + + China Bangladesh + – Pakistan – – India Structural balance & World War I 7

2/26/18 Structural balance & World War I Social network among characters Graham Sack, 2013 AI question: generate narrative from the evolution of stability 8