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

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

Exchange and Power in Social Life, 1964 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 There’s a long history of intuition about the claim that ‘networks matter’ – from the micro perspective of association with others (Blau) to a macro perception that the entire world is ‘connected’ in some sense. While Moreno was speaking pure wish, we now have the methods -- and oftentime the data – to identify the sturucture of this ‘vast solar system’ which underlies social space.

What are Social Networks? Source: Linton Freeman “See you in the funny pages” Connections, 23, 2000, 32-42.

What are Social Networks?

What are Social Networks? Information exchange network: Email exchanges within the Reagan white house, early 1980s NSOLN: Oliver North NSJMP: John Poindexter NSRBM: ROD B. MCDANIEL NSAGK: Alton G. Keel NSPBT: PAUL THOMPSON NSWRP: BOB PEARSON NSCPC: Craig P. Coy NSRLE: Robert L. Earl NSVMC: Vincent Cannistraro NSJRS: JAMES R. STARK NSHRT: Howard Teicher NSRKS: Ron Sable NSRCM: ROBERT C. MCFARLAN NSDGM: David Major NSDBR: Dennis Ross NSPWR:Peter Rodman Source: Author’s construction from Blanton, 1995

What are Social Networks?

What are Social Networks? Overlapping Boards of Directors Largest US Manufacturing firms, 1980. 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 Consider the following example. Here we have sampled respondents (red dots) reporting on their interaction with romantic partners. A classic local network module would ask about their characteristics and behaviors, then attempt to relate those characteristics to ego’s behavior. All of these sampled nodes have the exact same number of partners.

Why do Networks Matter? Global vision But these nodes are situated in dramatically different parts of the real underlying global network. Here some of them (lower left) are truly local isolates, but most are embedded in a larger network structure. These are real data from Add Health, on romantic involvement.

Why do Networks Matter? The spread of any epidemic depends on the number of secondary cases per infected case, known as the reproductive rate (R0). R0 depends on the probability that a contact will be infected over the duration of contact (b), the likelihood of contact (c), and the duration of infectiousness (D). Given what we know of b 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)

Three answers based on network structure 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. If we focus on the broad features of global networks, there are 3 broad models. The first two are really re-discoveries of old ideas, which have made a large splash in the popular science and physics world. While many of these treatments are over-stated, the general point is good. The first of these models is the “small world” model, built on Milgram’s early work (and that of Kochen, de sola pool, etc.), showing that people in the united states were, on average, separated by about 6 acquaintances.

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

Three answers based on network structure Small World Networks 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. One way (and there are likely others) to explain this fact is that a small number of people in the population form bridges who have ties outside of the local structure. Much of the recent work on small world graphs have assumed this structure (namely the bottom figure), focusing on the implications for diffusion (will go in fits and starts) and searchability (impossible for a pure SW, but possible when mixed w. stars). Note the substantive difference between the graph as theorized and the graph as studied – the studied lattice has a strong node-connectivity, built around the ring. This affects diffusion, as we will see.

Three answers based on network structure 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. The second model focuses not on the average path distance, but on local network involvement.

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) It turns out that many observed networks have a characteristic involvement (degree) distribution, where a very small number of people have many ties, and most people have very few. So most of us have 1 or 2 lifetime sex partners (far left of the graph), while a few NBA stars have many more (the far right).

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) This distribution is usually so skewed, that it makes sense to plot the histogram in log-log form, where the characteristic distribution then becomes clear. A power-law distribution often emerges, which has this functional form.

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 l = -1.3 But connectivity does not depend on the hubs. An empirical example of interest to population types: The sexual contact network among high-risk actors in colorado springs. Most (mainly johns) have only one partner, but a few high-activity pros have many.

Three answers based on network structure Structural Cohesion White, D. R. and F. Harary. 2001. "The Cohesiveness of Blocks in Social Networks: Node Connectivity and Conditional Density." Sociological Methodology 31:305-59. James Moody and Douglas R. White. “Structural Cohesion and Embeddedness: A hierarchical Conception of Social Groups” American Sociological Review 68:103-127                                                         The third model has it’s sources directly in the social sciences and is not a restatement of long-known ideas (though it does build on a rich literature about social cohesion). Since Doug and I don’t have a flashy book jacket, you’ll have to make do with our mug shots. For expanded detail, see the ASR article. Instead of focusing on the local involvement of actors (that their friends are friends with each other, or that there are a few with many ties), we ask what relational configuration is required for a cohesive network? This is, in many ways, theoretically opposite of Barabasi – we focus directly on the minimum requirements for holding a network together.

Three answers based on network structure Structural Cohesion Formal definition of Structural Cohesion: 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): A group’s structural cohesion is equal to the minimum number of independent paths linking each pair of actors in the group. Working through the mathematical requirements for the intuitive notion of ‘widely distributed’ ties, we come up with the following two equivalent definitions of structural cohesion (read).

Three answers based on network structure Structural Cohesion Networks are structurally cohesive if they remain connected even when nodes are removed To illustrate, consider these four networks, each with an identical volume of social ties. The graphs become more difficult to separate, and the number of independent paths increase. This is illustrated in the far left, where we can always trace at least 3 completely independent paths between every pair of people in the net. 1 2 3 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. 2 1 3 9 4 8 10 11 5 7 12 13 6 14 15 (hide? Check for time). One of the most important aspects of our conception of structural cohesion is that the conception results in a scale-able measure of embeddedness, since cohesive sets are hierarchically nested. Here, for example, we see that the whole network is connected in a component (through node 7), but that within each BC that 7 links, further embedded sets emerge. This means that our conception: a) subsumes the substantive importance of connecting individuals highlighted by Barabasi (I.e. nodes like 7) b) allows us to differentiate classic ‘primary’ groups from the overall structure of the network. 17 16 18 19 20 2 22 23

Three answers based on network structure Structural Cohesion Epidemic Gonorrhea Structure Contrast this with a separate study of gonorrhea in the same area. Here they find that the largest connected component comprises 71% of the total sampled. Note the large cycle evident in this image. 578 individuals identified, 410 (71%) in largest component (A) and 107 (18.5%) form a biconnected core. G=410 Source: Potterat, Muth, Rothenberg, et. al. 2002. Sex. Trans. Infect 78:152-158

Three answers based on network structure Structural Cohesion Epidemic Gonorrhea Structure Here we see what they call the ‘core’ of the network, which includes the largest biconnected component of the network (plus that little blip on the left of the figure). Source: Potterat, Muth, Rothenberg, et. al. 2002. Sex. Trans. Infect 78:152-158

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

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 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.

Development of STD Cores in Low-degree networks?

Development of STD Cores in Low-degree networks?

Development of STD Cores in Low-degree networks?

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