School of Information University of Michigan SI 614 Community structure in networks Lecture 17

Outline One mode networks and cohesive subgroups measures of cohesion types of subgroups Affiliation networks team assembly

Why care about group cohesion? opinion formation and uniformity if each node adopts the opinion of the majority of its neighbors, it is possible to have different opinions in different cohesive subgroups

within a cohesive subgroup – greater uniformity

Other reasons to care Discover communities of practice (more on this next time) Measure isolation of groups Threshold processes: I will adopt an innovation if some number of my contacts do I will vote for a measure if a fraction of my contacts do

What properties indicate cohesion? mutuality of ties everybody in the group knows everybody else closeness or reachability of subgroup members individuals are separated by at most n hops frequency of ties among members everybody in the group has links to at least k others in the group relative frequency of ties among subgroup members compared to nonmembers

Cliques Every member of the group has links to every other member Cliques can overlap overlapping cliques of size 3 clique of size 4

Considerations in using cliques as subgroups Not robust one missing link can disqualify a clique Not interesting everybody is connected to everybody else no core-periphery structure no centrality measures apply How cliques overlap can be more interesting than that they exist Pajek remember from class on motifs: construct a network that is a clique of the desired size Nets>Fragment (1 in 2)>Find

a less stingy definition of cohesive subgroups: k cores Each node within a group is connected to k other nodes in the group 3 core 4 core Pajek: Net>Partitions>Core>Input,Output,All Assigns each vertex to the largest k-core it belongs to

subgroups based on reachability and diameter n – cliques maximal distance between any two nodes in subgroup is n 2-cliques theoretical justification information flow through intermediaries

frequency of in group ties Compare # of in-group ties Given number of edges incident on nodes in the group, what is the probability that the observed fraction of them fall within the group? The smaller the probability – the stronger the cohesion within-group ties ties from group to nodes external to the group

considerations with n-cliques problem diameter may be greater than n n-clique may be disconnected (paths go through nodes not in subgroup) 2 – clique diameter = 3 path outside the 2-clique fix n-club: maximal subgraph of diameter 2

cohesion in directed and weighted networks something we’ve already learned how to do: find strongly connected components keep only a subset of ties before finding connected components reciprocal ties edge weight above a threshold

A)all citations between A-list blogs in 2 months preceding the 2004 election B)citations between A-list blogs with at least 5 citations in both directions C)edges further limited to those exceeding 25 combined citations Example: political blogs (Aug 29 th – Nov 15 th, 2004) only 15% of the citations bridge communities

Affiliation networks otherwise known as membership network e.g. board of directors hypernetwork or hypergraph bipartite graphs interlocks

m-slices transform to a one-mode network weights of edges correspond to number of affiliations in common m-slice: maximal subnetwork containing the lines with a multiplicity equal to or greater than m A = slice 1-slice

Pajek: Net>Transform>2- Mode to 1-Mode> Include Loops, Multiple Lines Info>Network>Line Values (to view) Net>Partitions>Valued Core>First threshold and step

Scottish firms interlocking directorates legend: 2-railways 4-electricity 5-domestic products 6-banks 7-insurance companies 8-investment banks

methods used directly on bipartite graphs rare Finding bicliques of users accessing documents An algorithm by Nina Mishra, HP Labs Documents Users

Team Assembly Mechanisms Determine Collaboration Network Structure and Team Performance Roger Guimera, Brian Uzzi, Jarrett SpiroLuıs A. Nunes Amaral Science, 2005 astronomy and astrophysics social psychology economics

Issues in assembling teams Why assemble a team? different ideas different skills different resources What spurs innovation? applying proven innovations from one domain to another Is diversity (working with new people) always good? spurs creativity + fresh thinking but conflict miscommunication lack of sense of security of working with close collaborators

Parameters in team assembly 1. m, # of team members 2. p, probability of selecting individuals who already belong to the network 3. q, propensity of incumbents to select past collaborators Two phases giant component of interconnected collaborators isolated clusters

creation of a new team incumbents (people who have already collaborated with someone) newcomers (people available to participate in new teams) pick incumbent with probability p if incumbent, pick past collaborator with probability q

Time evolution of a collaboration network newcomer-newcomer collaborations newcomer-incumbent collaborations new incumbent-incumbent collaborations repeat collaborations after a time  of inactivity, individuals are removed from the network

BMI data Broadway musical industry 2258 productions from 1877 to 1990 musical shows performed at least once on Broadway team: composers, writers, choreographers, directors, producers but not actors Team size increases from the musical as an art form is still evolving After 1929 team composition stabilizes to include 7 people: choreographer, composer, director, librettist, lyricist, producer

Collaboration networks 4 fields (with the top journals in each field) social psychology (7) economics (9) ecology (10) astronomy (4) impact factor of each journal ratio between citations and recent citable items published A= total cites in 1992 B= 1992 cites to articles published in (this is a subset of A) C= number of articles published in D= B/C = 1992 impact factor

size of teams grows over time

degree distributions data data generated from a model with the same p and q and sequence of team sizes formed

Predictions for the size of the giant component higher p means already published individuals are co- authoring – linking the network together and increasing the giant component S = fraction of network occupied by the giant component

Predictions for the size of the giant component (cont’d) increasing q can slow the growth of the giant component – co-authoring with previous collaborators does not create new edges

network statistics FieldteamsindividualspqfRfR S (size of giant component) BMI social psychology 16,52623, economics14,87023, ecology26,88838, astronomy30,55230, what stands out? what is similar across the networks?

different network topologies economics astronomy ecology

main findings all networks except astronomy close to the “tipping” point where giant component emerges sparse and stringy networks giant component takes up more than 50% of nodes in each network impact factor (how good the journal is where the work was published) p positively correlated going with experienced members is good q negatively correlated new combinations more fruitful S for individual journals positively correlated more isolated clusters in lower-impact journals ecology, economics, social psychology ecology social psychology