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

10/28/19 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 1 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) 2 1

10/28/19 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) 3 Def. Balanced triangle u Graph with just 3 nodes US + + + + ? UK SK + – US + – ? – – NK SK – – Balanced Unbalanced 5 2

10/28/19 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? 6 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) 7 3

10/28/19 Extensions 1. Relax strict definition of balanced network Weakly balanced network u 2. General network (non-complete) 8 Ext.# 1: Weakly balanced network US + + US UK SK + + + ? – – US – – + – Syria ISIS NK SK – Unbalanced Balanced 9 4

10/28/19 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) 10 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 negative edges u Q. What goes wrong otherwise? 11 5

10/28/19 Real-world examples 12 Liberation of Bangladesh in 1971 (Michael Moore, Journal of Social Psychology, 1978) USA – USSR ? – + + China Bangladesh + – Pakistan – – India 13 6

10/28/19 Structural balance & World War I 14 Structural balance & World War I 15 7

10/28/19 Social network among characters Graham Sack, 2013 AI question: generate narrative from the evolution of stability 16 8