11 Network Level Indicators Bird’s eye view of network Image matrix example of network level Many network level measures Some would argue this is the most appropriate level of analysis

22 Size Number of nodes (people) in the network Matters because as size increases –Density decreases –Clustering increases Reflects network boundary Should always be included as a covariate

33 Density Structural property Given by Should always be included as covariate as well

44 Density & Size Negatively Correlated In STEP study we have data from 24 coalitions at baseline We correlated size and density and discovered a negative association as predicted: R=-0.69

55 Reciprocity (Mutuality, Symmetry) Mutual ties: A  B then B  A Some relations are inherently symmetric or asymmetric –Who did you have lunch with? –Who did you go to for advice? Reciprocity is calculated as the percent of ties that are reciprocated:

66 Triads & Transitivity Holland & Leinhardt introduced the concept of triads and a triad census In a directed graph there are 16 possible triads: –A  B B  C A  C –A  B B  C C  A –…. One can do a triad census of a network calculating the percent of triads of each type in the network

7 MAN (Mutual, Asymmetric, Null) Census D 021U 021C111D111U 030T 030C201120D 120U 120C210300

88 Triads & Transitivity (cont.) Most often concerned with transitivity A transitive triad occurs if: –A  B B  C –Implies –A  C Transitivity implies balance, and balance theory is one of the foundations of many behavioral theories It is believed that people seek balance both toward others and objects (Heider) If a person is imbalanced, this creates cognitive dissonance and people will try to reduce cognitive dissonance (Festinger)

9 Transitive Triad A B C

10 Transitivity The percent of transitive triads provides a measure of cohesion In the STEP study we found an average of 17% of triads were transitive.

11 4 Nodes? One might expect the next level of analysis to increase to 4 nodes, as reciprocity was 2 nodes, and triads 3 nodes, but 4 nodes takes us to groups (this is where cycles come in) And back to the lecture on groups

12 Diameter/Ave. Path Length Diameter: Length of the longest path in the network Ave path length/characteristic path length Average of all the distances between nodes A measure of network size

13 Average and Maximum Change in Cohesion for each Link Removed

14 Cohesion: Measure of how close everyone is, on average, in the network 14

15 Unconnected Nodes Distances are important to calculate in networks What about unconnected nodes Distance equals infinity –Creates intractable math calculations –Substitute some finite number –Defensible on the grounds that if a node is included in a network it is reachable because it is in the same set –Might not be reachable because of measurement error –Might not be reachable because of instrumentation (e.g., 5 closest friends)

16 What to substitute for unconnected nodes? Choices: N-1 –Advantages: is the maximum theoretical distance between nodes in any network N –Advantages: is linearly related to max distance and would be the distance if a node were deleted Max. path length plus 1 –Advantages: is intuitively more meaningful Most Use N-1

17 Clustering Watts re-introduced the clustering coefficient: Average of the individual personal network densities:

18 Personal Network Density PN Density = 1/6 = 16.7%PN Density = 3/6 = 50.0% A z x y z x yB

19 Centralization The degree ties are focused on one or a few people Index ranges from 0 to 1 with 1 being perfectly centralized. Recall: Centralized network are ‘scale free’ networks

20 Examples of Dense Networks (Density=36.4%) Decentralized (9.1%)Centralized (50.9%)

21 Examples of Sparse Networks (Density=18.2%) Decentralized (0.0%)Centralized (87.3%)

22 Centralization Can Be Calculated On All Centrality Measures: Centralization Degree:

23 Centralization (cont.) Similar formulas exist for Centralization Closeness, Betweenness, Integration Can also be calculated by taking the standard deviation of the centrality scores.

24 Core Periphery Structures CP Networks have cores of densely connected people and a Periphery of those loosely connected to the core and to each other Can test whether networks have a C-P structure

25 Core-Periphery Analysis A network with a perfect CP structure will have all core nodes connected and peripheral ones connected only to the core Construct this idealized matrix and correlate the ideal with the empirical. Correlation coefficient is a measure of the CP

26 Children’s Health Insurance of Greater LA (CP=0.29) ▲ Missing ■ Periphery ● Core

27 Network Structure & Behavior Size clearly matters, large networks: –difficult to coordinate & organize –Norms unclear or diffuse –Diffusion takes longer Small networks –Easy to coordinate –Information and behaviors of others are known –Information can travel quickly, but Small networks are not powerful

28 Density We discussed earlier the possible curvilinear relationship Reciprocity: At the individual level, reciprocated relationship should be more likely associated with behavioral transmission: People more likely influenced by reciprocated relationships; On the other hand, advice seeking is asymmetric and one more likely to model those they seek advice from Thus, at individual level, reciprocity affects on behavior depend on relationship and behavior

29 Data from STEP

30 Reciprocity & Transitivity Networks with high levels of reciprocity: –Diffusion within faster; but –Diffusion between groups slower Transitive triads also more likely to: –Increase homogeneity of opinions –Facilitate diffusion within groups, but inhibit diffusion of outside ideas

31 Clustering High rates of clustering are even more indicative of closed subgroups Clustering will inhibit spread between groups but accelerate it within groups Higher clustering will increase the importance of bridges that connect clusters

32 Centralization Centralized networks should/could have fastest diffusion: –Central nodes are key players in the process –Central nodes are gatekeepers –Other properties may interact with centralization

33 Core Periphery Diffusion more likely to occur in the core Take a while for behaviors to filter to the periphery Many innovation may come from the periphery then percolate to the core Core groups can keep infectious diseases endemic to communities – STDs, HIV, etc.

34 2 Mode Data Recall that data on events, organizations, etc. can be used to construct 2 mode networks E.g., in this class students come from different departments Can construct a network based on shared dept. affiliations

35 Transposing a Matrix 35 Event AEvent BEvent C Person 1101 Person 2110 Person 3010 Person N001 Matrix A Person 1Person 2Person 3Person N Event A1100 Event B0110 Event C1001 Matrix A’ (transpose)

36 Excel File IDSPPDASCIPROther

37 Steps Read into UCINET as excel file Input this file Data\affiliations\dept06 Creates 1 mode data person by person And creates 1 mode dept by dept

38 Dept 06 PxP

39 Do They Correlate? Dept affiliations may lead to who knows whom We can correlate the 2 matrices Procedure to do so is know as QAP: Quadratic Assignment Procedure This procedures accounts for the dependencies in the rows and columns QAP Reg. coefficient between knowing and department affiliation is 0.30