"Social Networks, Cohesion and Epidemic Potential" James Moody Department of Sociology Department of Mathematics Undergraduate Recognition Ceremony May 5, 2004

1)What are Social Networks Examples of networks all around us 2)Why do networks matter? Conduits for diffusion 3)Structure and Diffusion: 3 network features to explain STD prevalence Small changes make big differences 4)Future directions for bright young mathematicians Modeling network dynamics "Social Networks, Cohesion and Epidemic Potential"

What are Social Networks? “To speak of social life is to speak of the association between people – their associating in work and in play, in love and in war, to trade or to worship, to help or to hinder. It is in the social relations men establish that their interests find expression and their desires become realized.” Peter M. Blau Exchange and Power in Social Life, 1964

What are Social Networks? Source: Linton Freeman “See you in the funny pages” Connections, 23, 2000,

What are Social Networks?

exchanges within the Reagan white house, early 1980s Information exchange network: What are Social Networks? Source: Author’s construction from Blanton, 1995

What are Social Networks?

Overlapping Boards of Directors Largest US Manufacturing firms, Source: Author’s construction from Mizruchi, 1992

What are Social Networks? Paul Erdös collaboration graph Erdös had 507 direct collaborators (Erdös # of 1), many of whom have other collaborators (Erdös #2). (My Erdös # is 3: Erdös  Frank Harary  Douglas R. White  James Moody) Source: Valdis Krebs

Why do Networks Matter? “Goods” flow through networks:

Why do Networks Matter? Local vision

Why do Networks Matter? Global vision

Why do Networks Matter? The spread of any epidemic depends on the number of secondary cases per infected case, known as the reproductive rate (R 0 ). R 0 depends on the probability that a contact will be infected over the duration of contact (  ), the likelihood of contact (c), and the duration of infectiousness (D). Given what we know of  and D, a “homogenous mixing” assumption for c would predict that most STDs should never spread. The key lies in specifying c, which depends on the network topography.

Structure and Diffusion: What aspects matter? Reachability in Colorado Springs (Sexual contact only) High-risk actors over 4 years 695 people represented Longest path is 17 steps Average distance is about 5 steps Average person is within 3 steps of 75 other people (Node size = log of degree)

Small World Networks Based on Milgram’s (1967) famous work, the substantive point is that networks are structured such that even when most of our connections are local, any pair of people can be connected by a fairly small number of relational steps. Three answers based on network structure

High probability that a node’s contacts are connected to each other. Small average distance between nodes C=Large, L is Small = SW Graphs Small World Networks Three answers based on network structure

In a highly clustered, ordered network, a single random connection will create a shortcut that lowers L dramatically Watts demonstrates that small world properties can occur in graphs with a surprisingly small number of shortcuts Disease implications are unclear, but seem similar to a random graph where local clusters are reduced to a single point. Three answers based on network structure Small World Networks

Scale-Free Networks Across a large number of substantive settings, Barabási points out that the distribution of network involvement (degree) is highly and characteristically skewed. Three answers based on network structure

Many large networks are characterized by a highly skewed distribution of the number of partners (degree) Three answers based on network structure Scale-Free Networks

Many large networks are characterized by a highly skewed distribution of the number of partners (degree) Three answers based on network structure Scale-Free Networks

The scale-free model focuses on the distance- reducing capacity of high-degree nodes: Three answers based on network structure Scale-Free Networks

The scale-free model focuses on the distance-reducing capacity of high-degree nodes, as ‘hubs’ create shortcuts that carry the disease. Three answers based on network structure Scale-Free Networks

Colorado Springs High-Risk (Sexual contact only) Network is power-law distributed, with = -1.3 But connectivity does not depend on the hubs. Three answers based on network structure Scale-Free Networks

White, D. R. and F. Harary "The Cohesiveness of Blocks in Social Networks: Node Connectivity and Conditional Density." Sociological Methodology 31: James Moody and Douglas R. White. “Structural Cohesion and Embeddedness: A hierarchical Conception of Social Groups” American Sociological Review 68: Structural Cohesion Three answers based on network structure

Formal definition of Structural Cohesion: (a)A group’s structural cohesion is equal to the minimum number of actors who, if removed from the group, would disconnect the group. Equivalently (by Menger’s Theorem): (b)A group’s structural cohesion is equal to the minimum number of independent paths linking each pair of actors in the group. Three answers based on network structure Structural Cohesion

Networks are structurally cohesive if they remain connected even when nodes are removed Node Connectivity Three answers based on network structure Structural Cohesion

Structural cohesion gives rise automatically to a clear notion of embeddedness, since cohesive sets nest inside of each other Three answers based on network structure Structural Cohesion

Connectivity Pairwise Connectivity profile Three answers based on network structure Structural Cohesion

Path distance probability Probability of infection by distance and number of paths, assume a constant p ij of paths 5 paths 2 paths 1 path Three answers based on network structure Structural Cohesion

Epidemic Gonorrhea Structure Source: Potterat, Muth, Rothenberg, et. al Sex. Trans. Infect 78: G=410 Three answers based on network structure Structural Cohesion

Source: Potterat, Muth, Rothenberg, et. al Sex. Trans. Infect 78: Epidemic Gonorrhea Structure Three answers based on network structure Structural Cohesion

3-Component (n=58) Project 90, Sex-only network (n=695) Three answers based on network structure Structural Cohesion

Connected Bicomponents IV Drug Sharing Largest BC: 247 k > 4: 318 Max k: 12 Three answers based on network structure Structural Cohesion

Development of STD Cores in Low-degree networks? While much attention has been given to the epidemiological risk of networks with long-tailed degree distributions, how likely are we to see the development of potential STD cores, when everyone in the network has low degree? Low degree networks are particularly important when we consider the short-duration networks needed for diseases with short infectious windows. Three answers based on network structure

Development of STD Cores in Low-degree networks?

Very small changes in degree generate a quick cascade to large connected components. While not quite as rapid, STD cores follow a similar pattern, emerging rapidly and rising steadily with small changes in the degree distribution. This suggests that, even in the very short run (days or weeks, in some populations) large connected cores can emerge covering the majority of the interacting population, which can sustain disease.

Future Directions: Network Dynamics