Download - 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
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 core4 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 probabilitythat 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
1 23
4 567
8
910
111213
1415
16
1718
19
20
21
22 2324
2526
27
2829 30
3132
3334 35 36
37 38 39
40
1 DigbysBlog2 JamesWalcott3 Pandagon4 blog.johnkerry.com5 OliverWillis6 AmericaBlog7 CrookedTimber8 DailyKos9 AmericanProspect10Eschaton11Wonkette12TalkLeft13PoliticalWire14TalkingPointsMemo15Matthew Yglesias16WashingtonMonthly17MyDD18JuanCole19Left Coaster20BradfordDeLong
21 JawaReport22VokaPundit23Roger LSimon24TimBlair25Andrew Sullivan26 Instapundit27BlogsforBush28 LittleGreenFootballs29BelmontClub30Captain’sQuarters31Powerline32 HughHewitt33 INDCJournal34RealClearPolitics35Winds ofChange36Allahpundit37MichelleMalkin38WizBang39Dean’sWorld40Volokh(C)
(B)
(A) 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 29th – Nov 15th, 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 =
1 1 1 1 0
1 1 1 1 0
1 1 2 2 0
1 1 2 4 1
0 0 0 1 1
1 1
1 2
1
2 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 AmaralScience, 2005
astronomy andastrophysics
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 1877-1929 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 1990-91 (this is a subset of A) C= number of articles published in 1990-91 D= B/C = 1992 impact factor
degree distributionsdata
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
Field teams individuals p q fR S (size of giant component)
BMI 2258 4113 0.52 0.77 0.16 0.70
social psychology
16,526 23,029 0.56 0.78 0.22 0.67
economics 14,870 23,236 0.57 0.73 0.22 0.54
ecology 26,888 38,609 0.59 0.76 0.23 0.75
astronomy 30,552 30,192 0.76 0.82 0.39 0.98
what stands out?
what is similar across the networks?
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