Affine Equivalence and Saddle Connection Graphs of Half-Translation Surfaces

To every half-translation surface, we associate a saddle connection graph, which is a subgraph of the arc graph. We prove that every isomorphism between two saddle connection graphs is induced by an affine homeomorphism between the underlying half-translation surfaces. We also investigate the automorphism group of the saddle connection graph and the corresponding quotient graph.

Kazhdan-Lusztig Polynomials of Matroids Under Deletion

We present a formula which relates the Kazhdan–Lusztig polynomial of a matroid $M$, as defined by Elias, Proudfoot and Wakefield, to the Kazhdan–Lusztig polynomials of the matroid obtained by deleting an element, and various contractions and localizations of $M$. We give a number of applications of our formula to Kazhdan–Lusztig polynomials of graphic matroids, including a simple formula for the Kazhdan–Lusztig polynomial of a parallel connection graph.

Network of interacting neurons with random synaptic weights

Since the pioneering works of Lapicque [17] and of Hodgkin and Huxley [16], several types of models have been addressed to describe the evolution in time of the potential of the membrane of a neuron. In this note, we investigate a connected version of N neurons obeying the leaky integrate and fire model, previously introduced in [1–3,6,7,15,18,19,22]. As a main feature, neurons interact with one another in a mean field instantaneous way. Due to the instantaneity of the interactions, singularities may emerge in a finite time. For instance, the solution of the corresponding Fokker-Planck equation describing the collective behavior of the potentials of the neurons in the limit N ⟶ ∞ may degenerate and cease to exist in any standard sense after a finite time. Here we focus out on a variant of this model when the interactions between the neurons are also subjected to random synaptic weights. As a typical instance, we address the case when the connection graph is the realization of an Erdös-Renyi graph. After a brief introduction of the model, we collect several theoretical results on the behavior of the solution. In a last step, we provide an algorithm for simulating a network of this type with a possibly large value of N.

Exploiting Graph Regularized Multi-dimensional Hawkes Processes for Modeling Events with Spatio-temporal Characteristics

Multi-dimensional Hawkes processes (MHP) has been widely used for modeling temporal events. However, when MHP was used for modeling events with spatio-temporal characteristics, the spatial information was often ignored despite its importance. In this paper, we introduce a framework to exploit MHP for modeling spatio-temporal events by considering both temporal and spatial information. Specifically, we design a graph regularization method to effectively integrate the prior spatial structure into MHP for learning influence matrix between different locations. Indeed, the prior spatial structure can be first represented as a connection graph. Then, a multi-view method is utilized for the alignment of the prior connection graph and influence matrix while preserving the sparsity and low-rank properties of the kernel matrix. Moreover, we develop an optimization scheme using an alternating direction method of multipliers to solve the resulting optimization problem. Finally, the experimental results show that we are able to learn the interaction patterns between different geographical areas more effectively with prior connection graph introduced for regularization.