SETH A. MARVEL

Research Fellow, Michigan Society of Fellows

Department of Mathematics and Center for the Study of Complex Systems

University of Michigan

Department of Mathematics and Center for the Study of Complex Systems

University of Michigan

I am an applied mathematician and general scientist with a specialization in dynamical systems, networks, and algorithms. I use methods from these and other areas of mathematics to study problems in a range of fields, including biology, ecology, sociology, linguistics, engineering, and philosophy. I am especially interested in problems that are both mathematically interesting and of strong fundamental or practical significance in some clear and direct sense. (I find this constraint limiting enough that drawing from a broad range of fields is helpful.) For each problem, I try to arrive at a new solution by building up from fundamental principles.

S. A. Marvel, D. Parise.

*Significance:* Many ranking applications—recommender systems, voting systems, sports rating systems, automated hiring procedures, and election forecasting—must solve problems that can be reduced to a well-known NP-hard problem of computer science called the *minimum feedback arc set* problem. The problem regards a collection of items and a set of pairwise preferences between these items, with the task of ranking the items in a way that minimizes the number of preferences that are inconsistent with the ranking. We have an efficient method for finding near optimal solutions of this problem for large and sparse real-world networks. We do not yet have analytical guarantees of this performance, but in computational tests it substantially outperforms the best available heuristics. The method is naturally extended to the more general problem of linear ordering.

S. A. Marvel, T. Martin, C. R. Doering, D. Lusseau, M. E. J. Newman.

*Significance:* We have discovered that a classic model for the spatial spread of a susceptible-infectious epidemic has its transition between spreading at bounded and unbounded speeds at the same "point"; as the transition from large-world to small-world structure in an influential spatial network model. We believe that this provides a first mathematical foundation for the popular belief that both transitions were simultaneous in history—i.e., that epidemic spreading transitioned from local wave-like propagation to rapid global dispersal just as the network of physical contacts between individuals transitioned from a large to a small world.

S. A. Marvel, H. Hong, A. Papush, S. H. Strogatz.

*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.

S. A. Marvel, J. Kleinberg, R. D. Kleinberg, S. H. Strogatz.

*Significance:* We propose and analyze a 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 for this 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 achieves structural balance.

See also an animation of the process studied in this paper, played out on an artificial social network. Green links indicate friendship, red links indicate rivalry, and gray links indicate neutral relations.

S. A. Marvel, S. H. Strogatz, J. M. Kleinberg.

*Significance:* In sufficiently intense social situations (a divided company board, 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.

S. A. Marvel, R. E. Mirollo, S. H. Strogatz.

*Significance:* Remarkably, phase oscillator dynamics are generally more complicated when the phase oscillators are identical then when they are not. This identifies a way in which nature actually benefits from heterogeneity in populations of otherwise nearly identical individuals – synchrony in groups of Southeast Asian fireflies and clusters of pacemaker cells may actually 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 series arrays of Josephson junctions (see below).

S. A. Marvel, S. H. Strogatz.

*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.

C. L. Higgins, S. A. Marvel, J. D. Morrisett.

*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 to heart attacks and strokes. The diminished elasticity of the vascular wall may also weaken portions of the artery, leading to aneurysm. We describe a method for reliably identifying the boundaries of the mineralizations *in vivo* by combining T1W, T2W, and PDW MRI data and performing clustering analysis (e.g., mean shift) in the feature space of this data. The method is also useful as an automated means to identify distinct tissue types and their boundaries within the body.