Applications of random graphs and walks

我的研究是跨学科的。我工作的中心线程集中在组合学和CS理论，算法复杂性和具体应用的问题。概率论在我的大部分工作中扮演着重要的角色，我经常关注随机化，它对问题的影响以及算法的好处。 随机图（网络）和随机游动经常出现在我的工作中。随机网络是普遍存在的，是概率论和组合数学中的经典研究课题。它们的重要性源于作为许多自然现象的模型，从社交网络和社区内的相互作用，通过计算机网络，到种群谱系模型，以及复杂生物系统中蛋白质的相互作用。因此，网络分析在许多领域有着广泛的应用。随机游走同样常见，作为自然发生过程的模型和工具。 我的研究的第二个主题涉及随机设置中的优化问题。虽然一些算法在最坏的情况下可能表现不好，但在典型的设置中可能会做得更好。最近，我们给出了第一个多项式界上的FLIP算法的运行时间寻找一个局部最大值的MAX-CUT图随机边权重。许多基本问题仍然悬而未决：什么是最佳多项式次数？当前边界在1和9.3之间。如果我们不允许一次翻转一个顶点，而是两个呢？只有准多项式边界是已知的。那MAX-K-CUT呢还有许多其他的组合优化问题，其中的复杂性是未知的，我计划研究。 我也研究计算生物学的问题。我们开发了一种基于随机游走的算法，用于从现代单细胞RNA计数中提取有生物学意义的见解。将数据拟合到低维流形上，利用样本的邻近性在样本点上生成图，并分析其结构。许多概率和算法项目出现：设计方法来生成对数据中的各种噪声源更鲁棒的图形，开发新的算法来定位图形中的模式，简化这些，以便分析更大的数据集变得可行等。 另一个主题是强盗问题的变体。代理人每次选择几个行动之一，并获得回报。目标是让代理从观察到的收益中确定最佳行动。这个问题有广泛的文献和应用，但许多自然变化还没有得到很好的理解。假设有几个代理人独立地做出决策，如果他们选择相同的行动，则会受到惩罚。根据代理是否在冲突发生时看到冲突以及是否允许它们进行通信，存在变化。在信息量最少的模型中，即使对于独立同分布，期望的最优后悔也是多项式增长的。支付向量，但最佳指数是未知的。

My research is interdisciplinary. Central threads of my work focus on questions from combinatorics and CS theory, algorithmic complexity and concrete applications. Probability theory plays a role in the vast majority of my work, and I often focus on randomization, its effects on a problem, and benefits in algorithms. Random graphs (networks) and random walks feature often in my work. Random networks are ubiquitous and a classical research topic in probability theory and combinatorics. Their importance stems from being a model of many natural phenomena, from social networks and interaction within communities, through computer networks, to models for genealogy of populations, and interactions of proteins in complex biological systems. Thus network analysis has a multitude of applications in many areas. Random walks are similarly common, as a model for naturally occurring processes and as a tool. A second theme of my research involves problems of optimization in random settings. While some algorithms may behave badly in the worst case, they may do better in typical settings. Recently we gave the first polynomial bound on the runtime of the FLIP algorithm for finding the a local maximum for MAX-CUT on a graph with random edge weights. Numerous fundamental questions remain open: What is the optimal polynomial degree? Current bounds are between 1 and 9.3. What if we allow flipping not one vertex at a time, but two? Only quasi-polynomial bounds are known here. What about MAX-k-CUT? There are numerous other combinatorial optimization problems, where the complexity is unknown, which I plan to study. I also work on problems from computational biology. We developed an algorithm based on random walks for extracting biologically meaningful insights from modern single cell RNA counts. Data is fitted onto a low dimension manifold; Proximity of the samples is used to generate a graph on the sample points, and its structure is analyzed. Many probabilistic and algorithmic projects arise: Devise ways to generate graphs that are more robust to the various sources of noise in the data, develop new algorithms to locate patterns in the graph, simplify these so that analysis of larger data sets becomes feasible etc. These feed back into experimental design. Another topic is variants on the bandit problem. An agent selects at each time one of several actions and receives a payoff. The goal is for the agent to identify from observed payoffs the optimal action. This problem has extensive literature and applications, yet many natural variations are not well understood. Suppose there are several agents who make decisions independently, with a penalty if they select the same action. There are variations depending on whether agents see collisions as they occur, and whether they are allowed to communicate. In the least informative model, the expected optimal regret is known to grow polynomially even for i.i.d. payoff vectors, but the optimal exponent is not known.