DCS/CSCI 2350: Social & Economic Networks Are all the links in a - PDF document

2/26/18 DCS/CSCI 2350: Social & Economic Networks Are all the links in a network the same? What is the effect of different types of links? “The strength of weak ties” edges Reading: Ch 3 of Easley-Kleinberg Mohammad T . Irfan The strength of weak ties u Agenda u Connect local/interpersonal properties to global/ structural properties u Mathematically prove this local to global connection u Show that the “critical” ties are actually weak ties 1

2/26/18 Granovetter’s study (1960s) u Acquaintances, not friends, hold critical information about job opportunities Triadic closure B and C are very likely to become friends (Rapoport, 1953) B C A u Triadic closure increases clustering coeff. (why?) u Reasons why triadic closure happens Opportunity for B and C to meet 1. B and C can trust each other 2. A wants to reduce stress by making B and C friends 3. u Teen suicide <-> low (local) clustering coefficient (Bearman and Moody, 2004) 2

2/26/18 Local bridge u An edge whose endpoints do not have any common friend ó An edge which is not a side of any triangle ó An edge whose deletion causes the distance between its endpoints to be > 2 Q1. Are local bridges important? Why? Q2. Is local bridge a local or a global property? Next: under some condition , every local bridge must be a “weak” tie Background for proving local bridges are weak ties u Tie-strength (simplifying gradation, temporal effect, etc.) u Weak (acquaintance) u Strong (friend) Q. Is tie-strength a local or a global property? u Strong Triadic Closure Property (STCP) Node-level property (Granovetter, 1973) Definition. 3

2/26/18 The strength of weak ties u If a node A satisfies STCP and has at least 2 strong ties, then any local bridge it's involved in must be a weak tie. u Proof. Large-scale social networks (real-world) and the strength of weak ties 4

2/26/18 Weak ties in Facebook (Marlow et al., 2009) Colleagues High school friends Weak ties in Facebook (cont...) 5

2/26/18 Weak ties (passive network) – What’s the use? Twitter (Huberman et al., 2009) 6

2/26/18 Are weak ties really powerful? Gladwell: “weak ties seldom lead to high-risk activism” Counter-argument 7

2/26/18 Counter-argument Counter-argument 8

2/26/18 Other Examples? Community detection in social networks using local bridges Section 3.6 (Advanced) 9

2/26/18 Coauthorship network (Newman- Girvan, 2004) Idea u Delete local bridges one after another u Get connected components u close-knit communities u Divisive graph partitioning (as opposed to agglomerative) 10

2/26/18 But… which local bridge to delete first? Also… what if there’s no local bridge? Need some form of “betweenness” measure for the edges! 11

2/26/18 Solution: Girvan-Newman algorithm (2002) u Calculate the betweenness of each edge u Successively delete the edge(s) with the highest betweenness (and recalculate betweenness) Q. When should we stop? Betweenness of an edge u Every node is sending 1 gallon of water to every other node (total n-1 gallons) u Water will only flow through the shortest paths u Equally distributed among multiple shortest paths u Betweenness of an edge = Quantity of water flowing through it 12

2/26/18 How to compute the betweenness of an edge? 13

2/26/18 Algorithm (A) For each node X do the following: Do BFS starting with X 1. Calculate the # of S.P . from X to every other node 2. Calculate the quantity of water flow through each edge 3. (B) Betweenness of an edge = sum of all water flow through that edge (i.e., sum over all the BFS) A B C D E F G H BFS starting at node A Note: Showing all the edges; it's not BFS tree! A D C B E G F H BFS needs to be done starting at each node (not just node A)! 14

2/26/18 # of S.P . from A to every node A 1 C D B 1 1 This is the # of S.P . Formula: from A to G, not E G 1 2 # of S.P . from A to E = distance from A to G Sum of the # of S.P . from A to each friend of E in the previous level 3 3 F H Calculate water flow on each edge Node A has sent 1 gallon of water to every other node. Water flows only on S.P .– splits evenly on multiple S.P . A 1 C D B 1 1 E G 1 2 2/3 Why? gal. 1/3 gal. 3 H 3 F Bottom-up calculation of water flow 15

2/26/18 Calculate water flow on each edge Node A has sent 1 gallon of water to every other node. Water flows only on S.P .– splits evenly on multiple S.P . A 1 C D B 1 1 E G 1 2 2/3 gal. 1/3 gal. 3 3 F H Calculate water flow on each edge Node A has sent 1 gallon of water to every other node. Water flows only on S.P .– splits evenly on multiple S.P . A 13/6 1 gal. 23/6 gal. gal. 1 C D B 1 1 Why 7/6 gal from D to E? 7/6 5/3 gal. 7/6 Ans. E consumes 1 gallon and gal. gal. passes 2/3 + 2/3 = 4/3 gallons E G 1 2 below it. So, E needs a supply 2/3 of 1 + 4/3 = 7/3 gallons, which 1/3 2/3 gal. is split evenly into two S.P . gal. gal. 1/3 from A to E. gal. 3 H 3 F We are not done yet! For each edge, we need to sum up the water flow from each and every BFS. 16

2/26/18 Karate club (Zachary, 1977) 17