Diffusion 1)Structural Bases of Social Network Diffusion 2)Dynamic limitations on diffusion 3)Implications / Applications in the diffusion of Innovations

Two factors that affect network diffusion: Topology - the shape, or form, of the network - simple example: one actor cannot pass information to another unless they are either directly or indirectly connected Time - the timing of contacts matters - simple example: an actor cannot pass information he has not yet received. Diffusion

Connectivity refers to how actors in one part of the network are connected to actors in another part of the network. Reachability: Is it possible for actor i to reach actor j? This can only be true if there is a chain of contact from one actor to another. Distance: Given they can be reached, how many steps are they from each other? Number of paths: How many different paths connect each pair? Diffusion Topology features

Network Toplogy Consider the following (much simplified) scenario: Probability that actor i infects actor j (p ij )is a constant over all relations = 0.6 S & T are connected through the following structure: S T The probability that S infects T through either path would be:

Why Sexual Networks Matter: Now consider the following (similar?) scenario: S T Every actor but one has the exact same number of partners The category-to-category mixing is identical The distance from S to T is the same (7 steps) S and T have not changed their behavior Their partner’s partners have the same behavior But the probability of an infection moving from S to T is: = Different outcomes & different potentials for intervention

Probability of infection over independent paths: The probability that an infectious agent travels from i to j is assumed constant at p ij. The probability that infection passes through multiple links (i to j, and from j to k) is the joint probability of each (link1 and link2 and … link k) = p ij d where d is the path distance. To calculate the probability of infection passing through multiple paths, use the compliment of it not passing through any paths. The probability of not passing through path l is 1-p ij d, and thus the probability of not passing through any path is (1-p ij d ) k, where k is the number of paths Thus, the probability of i infecting j given k independent paths is: Why matter Distance

Probability of infection over non-independent paths: - To get the probability that I infects j given that paths intersect at 4, I calculate Using the independent paths formula.formula

Network Topology: Ego Networks Mixing Matters The most commonly collected network data are ego-centered. While limited in the structural features, these do provide useful information on broad mixing patterns & relationship timing. Consider Laumann & Youm’s (1998) treatment of sexual mixing by race and activity level, using data from the NHSLS, to explain the differences in STD rates by race They find that two factors can largely explain the difference in STD rates: Intraracially, low activity African Americans are much more likely to have sex with high activity African Americans than are whites Interracially, sexual networks tend to be contained within race, slowing spread between races

Network Topology: Ego Networks In addition to general category mixing, ego-network data can provide important information on: Local clustering (if there are relations among ego’s partners. Not usually relevant in heterosexual populations, though very relevant to IDU populations) Number of partners -- by far the simplest network feature, but also very relevant at the high end Relationship timing, duration and overlap By asking about partner’s behavior, you can get some information on the relative risk of each relation. For example, whether a respondents partner has many other partners (though data quality is often at issue).

Network Topology: Ego Networks Clustering matters because it re-links people to each other, lowering the efficiency of the transmission network. Clustering also creates pockets where goods can circulate.

Network Topology: Partial and Complete Networks Once we move beyond the ego-network, we can start to identify how the pattern of connection changes the disease risk for actors. Two features of the network’s shape are known to be important: Connectivity and Centrality. Connectivity refers to how actors in one part of the network are connected to actors in another part of the network. Reachability: Is it possible for actor i to infect actor j? This can only be true if there is an unbroken (and properly time ordered) chain of contact from one actor to another. Given reachability, three other properties are important: Distance Number of paths Distribution of paths through actors (independence of paths)

Reachability example: All romantic contacts reported ongoing in the last 6 months in a moderate sized high school (AddHealth) Male Female (From Bearman, Moody and Stovel, 2004.)

Network Topology: Distance & number of paths Given that ego can reach alter, distance determines the likelihood of an infection passing from one end of the chain to another. Diffusion is never certain, so the probability of transmission decreases over distance. Diffusion increases with each alternative path connecting pairs of people in the network.

Path distance probability Probability of Diffusion by distance and number of paths, assume a constant p ij of paths 5 paths 2 paths 1 path

Path distance probability Probability of Diffusion by distance and number of paths, assume a constant p ij of 0.3

S T S T Return to our first example: 2 paths 4 paths

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 137 people connected through 2 independent paths, core of 30 people connected through 4 independent paths (Node size = log of degree)

Centrality refers to (one dimension of) where an actor resides in a sexual network. Local: compare actors who are at the edge of the network to actors at the center Global: compare networks that are dominated by a few central actors to those with relative involvement equality Network Topology: Centrality and Centralization

Centrality example: Add Health Node size proportional to betweenness centrality Graph is 45% centralized

Centrality example: Colorado Springs Node size proportional to betweenness centrality Graph is 27% centralized

Network Topology: Effect of Structure

Simulated diffusion curves for the observed network. Network Topology: Effect of Structure

The effect of the observed structure can be seen in how diffusion differs from a random network with the same volume Network Topology: Effect of Structure

Mean number of independent paths

Network Topology: Effect of Structure Clustering Coefficient

Network Topology: Effect of Structure Mean Distance

Network Topology: Effect of Structure

Timing Sexual Networks A focus on contact structure often slights the importance of network dynamics. Time affects networks in two important ways: 1) The structure itself goes through phases that are correlated with disease spread Wasserheit and Aral, “The dynamic topology of Sexually Transmitted Disease Epidemics” The Journal of Infectious Diseases 74:S Rothenberg, et al “Using Social Network and Ethnographic Tools to Evaluate Syphilis Transmission” Sexually Transmitted Diseases 25: ) Relationship timing constrains disease flow a) by spending more or less time “in-host” b) by changing the potential direction of disease flow

Sexual Relations among A syphilis outbreak Jan - June, 1995 Rothenberg et al map the pattern of sexual contact among youth involved in a Syphilis outbreak in Atlanta over a one year period. (Syphilis cases in red) Changes in Network Structure

Sexual Relations among A syphilis outbreak July-Dec, 1995

Sexual Relations among A syphilis outbreak July-Dec, 1995

Data on drug users in Colorado Springs, over 5 years

What impact does this kind of timing have on diffusion? The most dramatic effect occurs with the distinction between concurrent and serial relations. Relations are concurrent whenever an actor has more than one sex partner during the same time interval. Concurrency is dangerous for disease spread because: a) compared to serially monogamous couples, and STDis not trapped inside a single dyad b) the std can travel in two directions - through ego - to either of his/her partners at the same time

Concurrency and Epidemic Size Morris & Kretzschmar (1995) Monogamy Disassortative AssortativeRandom Population size is 2000, simulation ran over 3 ‘years’

Concurrency and disease spread Variable Constant Concurrent K 2 Degree Correlation Bias Coefficient Adjusting for other mixing patterns: Each.1 increase in concurrency results in 45 more positive cases

B C E DF A A hypothetical Sexual Contact Network

B C E DF A The path graph for a hypothetical contact network

Direct Contact Network of 8 people in a ring

Implied Contact Network of 8 people in a ring All relations Concurrent

Implied Contact Network of 8 people in a ring Mixed Concurrent

Implied Contact Network of 8 people in a ring Serial Monogamy (1)

Implied Contact Network of 8 people in a ring Serial Monogamy (2)

Implied Contact Network of 8 people in a ring Serial Monogamy (3)

Timing Sexual Networks Network dynamics can have a significant impact on the level of disease flow and each actor’s risk exposure This work suggests that: a) Disease outbreaks correlate with ‘phase-shifts’ in the connectivity level b) Interventions focused on relationship timing, especially concurrency, could have a significant effect on disease spread c) Measure and models linking network topography to disease flow should account for the timing of romantic relationships

Timing Sexual Networks

Many large networks are characterized by a highly skewed distribution of the number of partners (degree) Large-scale network model implications: Scale-Free Networks Degree or Connectivity?:

Many large networks are characterized by a highly skewed distribution of the number of partners (degree) Large-scale network model implications: Scale-Free Networks Degree or Connectivity?:

Large-scale network model implications: Scale-Free Networks The scale-free model focuses on the distance- reducing capacity of high-degree nodes: Degree or Connectivity?:

Large-scale network model implications: Scale-Free Networks The scale-free model focuses on the distance- reducing capacity of high-degree nodes: Which implies: a thin cohesive blocking structure and a fragile global topography Scale free models work primarily on through distance, as hubs create shortcuts in the graph, not through core-group dynamics. Degree or Connectivity?:

3-Component (n=58) Empirical Evidence Project 90, Sex-only network (n=695) Degree or Connectivity?:

Empirical Evidence:Project 90, Drug sharing network Connected Bicomponents N=616 Diameter = 13 L = 5.28 Transitivity = 16% Reach 3: 128 Largest BC: 247 K > 4: 318 Max k: 12 Degree or Connectivity?:

Empirical Evidence:Project 90, Drug sharing network Multiple 4-components Degree or Connectivity?:

Building on recent work on conditional random graphs*, we examine (analytically) the expected size of the largest component for graphs with a given degree distribution, and simulate networks to measure the size of the largest bicomponent. For these simulations, the degree distribution shifts from having a mode of 1 to a mode of 3. We estimate these values on populations of 10,000 nodes, and draw 100 networks for each degree distribution. * Newman, Strogatz, & Watts 2001; Molloy & Reed 1998 Degree or Connectivity?:

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. Degree or Connectivity?:

Empirical Models for Diffusion Macro-level models Typically model diffusion as a growth rate process over some population. Recent models include more parameters to get better fits: Y is the proportion of adopters, b o a rate parameter for innovation and b 1 a rate parameter for imitation. This is the “Bass Model”, after Bass These models really only work on the rate of change, and assume random mixing.

Empirical Models for Diffusion Were w is a weight matrix for contact between actors. Add peer effects:

Empirical Models for Diffusion