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Transition State Theory for Network Dynamics

Carter T. Butts

Abstract

Many classic questions of structural theory concern discrete changes, such as the formation or dissolution of groups, role turnover, or faction realignment. Here, we consider a basic framework combining prior work on change paths and recent advances in dynamic network modeling with ideas from transition state theory. This framework facilitates both characterizing the process of structural change and, in some cases, predicting it. Notably, this approach allows approximate prediction of network change from cross-sectional models, under limited assumptions regarding the underlying microdynamics. We apply this framework to a simple model of faction realignment in small groups, showing that the process through which realignment occurs can be well-predicted ex ante for a number of different network micro-processes.

Transition State Theory for Network Dynamics

Abstract

Many classic questions of structural theory concern discrete changes, such as the formation or dissolution of groups, role turnover, or faction realignment. Here, we consider a basic framework combining prior work on change paths and recent advances in dynamic network modeling with ideas from transition state theory. This framework facilitates both characterizing the process of structural change and, in some cases, predicting it. Notably, this approach allows approximate prediction of network change from cross-sectional models, under limited assumptions regarding the underlying microdynamics. We apply this framework to a simple model of faction realignment in small groups, showing that the process through which realignment occurs can be well-predicted ex ante for a number of different network micro-processes.
Paper Structure (17 sections, 3 equations, 10 figures)

This paper contains 17 sections, 3 equations, 10 figures.

Table of Contents

  1. A Basic Theory of Network Change
  2. Concepts
  3. Annotating Features of Change Paths
  4. Computation
  5. Application to a Faction Realignment Model
  6. A Simple Model with Competing Solidarities
  7. State Probabilities and the MSP Change Path
  8. The MSPCP Follows the "High Road"
  9. Several Intermediates and Transition States Lie Along the MSPCP
  10. Comparison with Models of Explicit Dynamics
  11. The MSPCP Describes the Most Common Path for All Models
  12. A Single Secondary Change Path Accounts for Residual Behavior
  13. Model Choice Affects Transient Behavior
  14. Discussion
  15. The Steps to Faction Realignment
  16. ...and 2 more sections

Figures (10)

  • Figure 1: Schematic of partner swapping in a heterosexual romantic network (per bearman.et.al:ajs:2004). Alternative pathways require the network to pass through states of differing favorability; those involving highly unfavorable states such as 4-cycles are predicted to be less likely than those involving less unfavorable states (e.g., null graphs).
  • Figure 2: Example of a change path, with annotated intermediate and transition states. The transformation from the source to the target state occurs along the change coordinate; interstitial states of locally maximal probability are known as intermediate states, while local probability minima are transition states. Deep wells between intermediates can act as barriers to progress along the change path.
  • Figure 3: Basis-aligned alliance networks for the faction model. Factions may align along attribute $B^1$ (color) or $B^2$ (shape); affiliation costs frustrate simultaneous alignment along both bases of solidarity.
  • Figure 4: State space of the faction alignment model, projected onto dimensions of edge count and polarization. Darker shaded areas indicate regions of higher state probability. Purple line indicates MSPCP between source and target states.
  • Figure 5: Graph potential (shaded region/right axis) and normalized graph statistics (left axis) for the faction model, along the MSPCP.
  • ...and 5 more figures