Publications

Significance: Starting from a proposal of Rubin that every well-defined causal claim specifies a control, this article constructs a formal definition of causality on a Bayesian network. While formal definitions of causality have been proposed by other authors, all of these either decline to require a control and therefore (I suggest) do not have the properties necessary for scientific applications, or are constructed for models that are substantially less general than Bayesian networks and therefore are nonideal for formal philosophical work.  The definition of causality offered in this article overcomes both limitations.

Significance: Much of the social history of ideas can be written as a series of ideological revolutions, with new radical ideas repeatedly overtaking older ones. Moderate positions, by contrast, rarely prevail in this process. Here we investigate whether there might be a dynamical disadvantage to moderate positions. Within a simple model of opinion spreading, we test seven plausible strategies for encouraging a population to embrace a moderate viewpoint and find that only one of these strategies significantly and reliably expands the moderate subpopulation. This helps to identify the nature of the dynamical advantage of committed ideological groups over less extreme positions.

Significance: We propose and analyze a continuous dynamics for Heider's influential theory of structural balance. This dynamics can be written as the simple model, dX/dt = X·X, where X is a matrix of the friendliness or unfriendliness between pairs of nodes in the network and · represents matrix multiplication. We give a closed-form solution to this equation, which we use to predict the Allied and Axis powers of World War II with an accuracy of 15/17, and we prove that initial states of X drawn from a continuous distribution evolve to a maximally balanced state with probability one. To our knowledge, this proof constitutes the first demonstration that a dynamical system of structural balance actually achieves structural balance.

Significance: In sufficiently intense social situations (e.g., a divided company board or a continent embroiled in war), the relationships of the parties involved generally become either friendships or rivalries. Heider's theory of structural balance proposes that some triangles of friendships and rivalries are more stable than others—generalizing the notion that "the enemy of my enemy is my friend." If we assume that relationship statuses change one at a time, we may immediately construct an energy landscape of social balance. The global minima of this landscape are Heider's well-known states of structural balance, but the local minima are much less well understood. Here we characterize these local minima, proving that they have a modular structure that can be used to classify them and deriving bounds on the energies on each group of local minima in this classification.

Significance: Surprisingly, heterogeneous phase oscillators tend to establish collective states of synchrony more reliably than homogeneous (i.e., identical) phase oscillators. This in turn suggests a way in which nature benefits from heterogeneity in populations of otherwise nearly identical individuals – synchrony in groups of Southeast Asian fireflies and clusters of pacemaker cells may in fact be stabilized by the heterogeneity of fireflies and pacemaker cells, respectively. To understand this phenomenon on a deeper level, we characterize of the largest and most commonly studied class of identical phase oscillator systems, showing how it can be succinctly reformulated as Möbius group action. As an application, we use this approach to study the structure of the 3-dimensional submanifolds in the phase space of a series array of Josephson junctions (see below).

Significance: Circuits of Josephson junctions are pervasive in science and engineering, appearing in highly sensitive magnetometers, quantum computing applications, and superconducting switching devices. They are also used for measurement standards; large series arrays of Josephson junctions have been used as the NIST standard for the volt since 1990. The phase space of series arrays of Josephson junctions is naturally foliated into many 3-dimensional submanifolds and one special 2-dimensional submanifold (loosely analogous to how a tree trunk is naturally foliated into many 2-dimensional rings and a central 1-dimensional pith or core). Here we provide an exhaustive characterization of the special 2-dimensional submanifold in this phase space.

Significance: Various forms of calcium and phosphorus can deposit in atherosclerotic lesions along arterial walls to produce large and inflexible mineral deposits within the wall of the artery. This process accelerates stenosis, which eventually causes heart attacks and strokes. The diminished elasticity of the vascular wall may also weaken portions of the artery, leading to aneurysm. We outline a method for identifying the boundaries of the mineralizations in vivo by construction of a three-dimensional feature space from T1W, T2W, and PDW MRI data and analysis of the distribution of data in this feature space. The method would also be also useful as an automated means to identify distinct tissue types and their boundaries within the body.