Topographic analysis of an empirical human sexual network

1. Topographic analysis of an empirical human sexual network Geoffrey Canright and Kenth Engø-Monsen, Telenor R&I, Norway Valencia Remple, U of British Columbia, Vancouver, Canada

2. Theory meets reality • Here we will combine two things: • The ’topographic’, or ’regions-based’, theoretical approach to analysis of epidemic spreading (ECCS04 and ECCS05) (GSC/KEM) + • A detailed empirical network of human sexual contacts, based on female sex workers (FSW) in Vancouver BC (SNA 2006 + 2007) (VR—’Orchid’)

3. We call the top node for each region the ’Center’ Our topographic picture (briefly) • Eigenvector centrality (EVC)  ‘spreading power’ • High EVC  well connected to well connected nodes • EVC is ‘smooth’  a topographic approach makes sense: for any given graph, we find one or more ‘mountains’ (or ‘regions’), each with its most central node at the top • Region membership is determined by ‘steepest-ascent path’ (on the steepest-ascent graph or SAG) to the Center (top) • Spreading within regions is fairly fast and predictable • Spreading between regions may be neither of these

4. The Vancouver ’Orchid’ dataset • Based on extensive surveys of female sex workers (FSW) and their Clients • Contacts between these, and with partners (and sometimes partners of partners) were recorded • 553 nodes, 1498 links • 2 nodes are HIV positive; other STI’s found in 11 other nodes

5. = male = female = HIV-pos The Orchid graph — regions analysis

6. = male = female The Orchid graph — regions analysis — SAG

7. A purely heterosexual graph is bipartite! • Bipartite graph: two sets of nodes (eg, M and F); all connections are between the two sets (M  F) • We are accustomed to finding only a few regions in the (non-bipartite) graphs we have studied (EX: 10 million nodes, 1 region ...) • In a purely bipartite graph, there are no triangles  the graph is not as ’well connected’ as it could be otherwise • The Orchid graph is ’mostly bipartite’ (only 11/1502 links are homosexual); we conjecture that this is the reason for the many (17) regions that we find • Nevertheless we find the graph to be dominated by 3 large regions (totalling 517/553 nodes)

8. Conjecture: Centers tend to be confined to one gender (M or F) due to bipartite property Here we plot all nodes with at least 20 partners Center = large Here, all Centers are men!

9. Our predictions • When an infection reaches a region, it moves towards the Center (’uphill’), and ’takes off’ when it reaches the Central neighborhood • That is, once the infection reaches the Central neighborhood of a region, the entire region is ’lost’ (ie, rapidly infected) • Movement between regions is heavily dependent on how well connected the regions are • In the Orchid graph, the strongest connections are GreyRedBlue • HIV is found in the Red region (2 hops from Center—bad news), and at the Center of a small region (also 2 hops from the Center of Red region!)   • We expect it to be difficult to protect the Red region; also, the strong connections to the other two are a problem!

10. Spreading simulation start with Red HIV-positive node 233 Total Red Grey Blue fast take off

11. Protecting the Red region is difficult • 237 dominates, but either HIV-pos node infects the Red region fast • Simulations with both infected look like those with just 237 •  if we must prioritize one for protection, it would be 237 • We have immunized the Red Center; no help! • Reason: there remains a very dense Red Central neighborhood •  we find no easy way to protect the Red region • However the graph topology suggests that the Grey region can be protected from infections coming from Red, via protecting the Grey Center (node 117) • We also find that infections from the Grey region are slowed down by immunizing this same node

12. Spreading simulation start with both HIV-positive nodes; immunize Grey Center node No immunization Immunize 117 Grey region takes off later

13. Spreading simulation start node 306 = STI, in Grey region Immunize the Grey Center

14. Conclusions (thus far) • The quasi-bipartite nature of the sexual contact network has made our regions analysis a bit more interesting • However, the main features we found in earlier work are again found here • The role of the region as a unit of analysis is clear • In particular, the whole region is ”lost” once the Central neighborhood is infected • We find it difficult to protect the (big) Red region from the HIV-infected nodes—they are too close to its Center • However, we find that inoculating just one node can significantly hinder GreyRed spreading

15. Future work • Weight the links with realistic infection transmission probabilities per unit time • Since these weights are disease dependent, we will get a distinct adjacency matrix for each disease • The regions analysis is also sensitive to link weights • Thus, using realistic weights will make the regions analysis more realistic, and hence more practical • Using realistic link weights, seek and test promising protection strategies • We have not attempted to do that systematically here, due to 1.b. above • Strategies to be tested need not be limited to those suggested by our analysis, since the simulations are ”agnostic”