Solution to urn models of pairwise interaction with application to social, physical, and biological sciences.

We investigate a family of urn models that correspond to one-dimensional random walks with quadratic transition probabilities that have highly diverse applications. Well-known instances of these two-urn models are the Ehrenfest model of molecular diffusion, the voter model of social influence, and the Moran model of population genetics. We also provide a generating function method for diagonalizing the corresponding transition matrix that is valid if and only if the underlying mean density satisfies a linear differential equation and express the eigenvector components as terms of ordinary hypergeometric functions. The nature of the models lead to a natural extension to interaction between agents in a general network topology. We analyze the dynamics on uncorrelated heterogeneous degree sequence networks and relate the convergence times to the moments of the degree sequences for various pairwise interaction mechanisms.

Analysis of the high-dimensional naming game with committed minorities.

The naming game has become an archetype for linguistic evolution and mathematical social behavioral analysis. In the model presented here, there are N individuals and K words. Our contribution is developing a robust method that handles the case when K=O(N). The initial condition plays a crucial role in the ordering of the system. We find that the system with high Shannon entropy has a higher consensus time and a lower critical fraction of zealots compared to low-entropy states. We also show that the critical number of committed agents decreases with the number of opinions and grows with the community size for each word. These results complement earlier conclusions that diversity of opinion is essential for evolution; without it, the system stagnates in the status quo [S. A. Marvel et al., Phys. Rev. Lett. 109, 118702 (2012)PRLTAO0031-900710.1103/PhysRevLett.109.118702]. In contrast, our results suggest that committed minorities can more easily conquer highly diverse systems, showing them to be inherently unstable.

Solution of the multistate voter model and application to strong neutrals in the naming game.

We consider the voter model with M states initially in the system. Using generating functions, we pose the spectral problem for the Markov transition matrix and solve for all eigenvalues and eigenvectors exactly. With this solution, we can find all future probability probability distributions, the expected time for the system to condense from M states to M-1 states, the moments of consensus time, the expected local times, and the expected number of states over time. Furthermore, when the initial distribution is uniform, such as when M=N, we can find simplified expressions for these quantities. In particular, we show that the mean and variance of consensus time for M=N are 1/N(N-1)(2) and 1/3(π(2)-9)(N-1)(2), respectively. We verify these claims by simulation of the model on complete and Erdos-Rényi graphs and show that the results also hold on these sparse networks.

Solution of the voter model by spectral analysis.

An exact spectral analysis of the Markov propagator for the voter model is presented for the complete graph and extended to the complete bipartite graph and uncorrelated random networks. Using a well-defined Martingale approximation in diffusion-dominated regions of phase space, which is almost everywhere for the voter model, this method is applied to compute analytically several key quantities such as exact expressions for the m time-step propagator of the voter model, all moments of consensus times, and the local times for each macrostate. This spectral method is motivated by a related method for solving the Ehrenfest urn problem and by formulating the voter model on the complete graph as an urn model. Comparisons of the analytical results from the spectral method and numerical results from Monte Carlo simulations are presented to validate the spectral method.

Simultaneous tissue factor expression and phosphatidylserine exposure account for the highly procoagulant pattern of melanoma cell lines.

A correlation between cancer and hypercoagulability has been described for more than a century. Patients with cancer are at increased risk for thrombotic complications, and the clotting initiator protein, tissue factor (TF), is possibly involved in this process. In addition to TF, the presence of negatively charged phospholipids, particularly phosphatidylserine (PS), is necessary to support some of the blood-clotting reactions. There are few reports describing PS exposure by tumor cells. In this study, we characterized the procoagulant properties of the murine B16F10 and the human WM-266-4 melanoma cell lines. Flow cytometry analyses showed constitutive TF expression by both cell lines, in contrast to negative staining observed for the nontumorigenic melanocyte lineage, melan-A. In addition, tumor cells accelerate plasma clotting in a number-dependent manner. For WM-266-4, this ability was partially reversed by an anti-TF antibody but not by aprotinin, a nonspecific serine-protease inhibitor. Furthermore, flow-cytometric analyses showed the presence of PS at the outer leaflet of both cell lines. This phenomenon was determinant for the assembly of the intrinsic tenase (FIXa/FVIIIa) and prothrombinase (FXa/FVa) complexes, resulting in the activation of FX to FXa and prothrombin to thrombin, respectively. As a result, incubation of WM-266-4 with human plasma produces robust thrombin generation. In conclusion, simultaneous TF expression and PS exposure are responsible for the highly procoagulant pattern of the aggressive melanoma cell lines B16F10 and WM-266-4. Therefore, these cell lines might be regarded as useful models for studying the role of blood coagulation proteins in tumor biology.