Some Comments on the Some Comments on the Foundations of Network - PowerPoint PPT Presentation

Some Comments on the Some Comments on the Foundations of Network Analysis Foundations of Network Analysis Carter T. Butts Carter T. Butts Department of Sociology and Department of Sociology and Institute for Mathematical Behavioral Sciences Institute for Mathematical Behavioral Sciences University of California, Irvine University of California, Irvine Prepared for the August 25, 2009 UCI MURI AHM. This work was Prepared for the August 25, 2009 UCI MURI AHM. This work was supported by DOD ONR award N00014-8-1-1015. supported by DOD ONR award N00014-8-1-1015.

Background Background  Many foundational Many foundational issues in network issues in network analysis analysis Vertex, edge set definition, Vertex, edge set definition,  time scales, etc. time scales, etc.  Often taken for granted Often taken for granted Things "everyone knows" Things "everyone knows"  - but impact is not well- - but impact is not well- understood! understood!  Today: some comments Today: some comments from a recent review, and from a recent review, and thoughts on how this thoughts on how this affects our work affects our work

Choosing the Vertex Set Choosing the Vertex Set  Most basic issue - whence Most basic issue - whence the vertex set? the vertex set?  Not always obvious Not always obvious Selection/boundary issues Selection/boundary issues  Choice of scale in multiscale Choice of scale in multiscale  systems systems Subordination among Subordination among  organizations organizations Containment/ recombination Containment/ recombination  in households in households  Different choices can Different choices can greatly affect network greatly affect network measures measures

Effect of Vertex Aggregation Effect of Vertex Aggregation Katrina EMON Data (Butts et al., 2009)

Valued Edges and Thresholding Valued Edges and Thresholding  Well-recognized (but not well- Well-recognized (but not well- understood) issue: dealing understood) issue: dealing with valued edges with valued edges Most concepts, models Most concepts, models  dichotomous, but not all relations dichotomous, but not all relations are are  Usual approach is Usual approach is thresholding, but this has non- Phenomenal Impact thresholding, but this has non- obvious consequences.... obvious consequences.... Can be reasonable if edge Can be reasonable if edge  behavior sigmoidal and threshold behavior sigmoidal and threshold is well-chosen is well-chosen Otherwise, same data can lead to Otherwise, same data can lead to  completely different results completely different results Edge Value

Effect of Threshold Selection Effect of Threshold Selection ( C. Elegans Data (Watts, Strogatz, 1998)

Edge Timing and Network Edge Timing and Network Processes Processes Often, networks assumed as Often, networks assumed as  substrate for a social process substrate for a social process Process Time Scale May need to consider network May need to consider network Random Mixing  dynamics... dynamics... Edge, vertex turnover Edge, vertex turnover  Episodic Interaction ...but nature of dynamics ...but nature of dynamics (Hazard Based)  depends on relative relative time time depends on scales of network, process scales of network, process Process/Network Coevolution evolution evolution Not whether network "is" static Not whether network "is" static  Evolution on Fixed Graph or dynamic in isolation or dynamic in isolation Right model can vary from fixed Right model can vary from fixed  Edge Time Scale network to random mixing network to random mixing Different models for different Different models for different  purposes purposes

Illustration: Diffusion on an Illustration: Diffusion on an Evolving Network Evolving Network  Common process of  How does edge timing Common process of How does edge timing interest: diffusion affect diffusion? interest: diffusion affect diffusion? Simple example: Illustrative simulation: Simple example: Illustrative simulation:   Once "infected," vertices Two sample networks Once "infected," vertices Two sample networks   "infect" neighbors w/iid "infect" neighbors w/iid Mean duration, std dev of Mean duration, std dev of  exponential waiting times exponential waiting times onset time varied onset time varied Process continues until all Process continues until all  Poisson diffusion on Poisson diffusion on  no available hosts left no available hosts left dynamic network (starting dynamic network (starting Adding edge dynamics at time 0) w/unit infection at time 0) w/unit infection Adding edge dynamics  rate rate Each relationship begins Each relationship begins  Basic outcome: expected Basic outcome: expected after iid exp waiting time, after iid exp waiting time,  fraction of population fraction of population has iid exp duration has iid exp duration infected by a single, random infected by a single, random Intuition: edge dynamics Intuition: edge dynamics  "seed" "seed" affect permeability of affect permeability of network to diffusion network to diffusion How powerful are timing How powerful are timing  effects? effects?

WTC Radio Data (Butts et al., 2007) Sexual Contact Data (Potterat et al., 2002)

● Time scales determine diffusion behavior ● Three basic regimes ● Near-complete diffusion ● Incomplete WTC Radio Data diffusion (Butts et al., 2007) ● Minimal diffusion ● Behavior similar across networks ● Differences in degree distribution, clustering, cohesion matter less Sexual Contact Data than timing! (Potterat et al., 2002)

Some Conclusions and Project- Some Conclusions and Project- Related Comments Related Comments  General conclusions:  Project-specific General conclusions: Project-specific recommendations: recommendations: Need to be attentive to the Need to be attentive to the  basics basics Simple models for valued Simple models for valued  data? data? "Any old network" may be "Any old network" may be  OK for algorithm testing, OK for algorithm testing, "Threshold regression" "Threshold regression"  but not for serious analysis but not for serious analysis ERGMs? ERGMs? Need to learn more about Need to learn more about  Models for vertices Models for vertices  robustness of methods to robustness of methods to w/containment or w/containment or "network specification "network specification hierarchical structure hierarchical structure error" error" Not sure that block- Not sure that block-  May need models for May need models for hierarchical ERGMs hierarchical ERGMs  alternate data enough, but a start enough, but a start alternate data representations representations Keep pushing on Keep pushing on  dynamics! dynamics!