Local-to-Global Principles for the Hitting Sequence of a Rotor Walk

In rotor walk on a finite directed graph, the exits from each vertex follow a prescribed periodic sequence. Here we consider the case of rotor walk where a particle starts from a designated source vertex and continues until it hits a designated target set, at which point the walk is restarted from the source. We show that the sequence of successively hit targets, which is easily seen to be eventually periodic, is in fact periodic. We show moreover that reversing the periodic patterns of all rotor sequences causes the periodic pattern of the hitting sequence to be reversed as well. The proofs involve a new notion of equivalence of rotor configurations, and an extension of rotor walk incorporating time-reversed particles.Massachusetts Institute of Technology. Undergraduate Research Opportunities ProgramNational Science Foundation (U.S.) (Grant 0644877)National Science Foundation (U.S.) (Grant 1001905)National Science Foundation (U.S.) (Postdoctoral Research Fellowship)National Science Foundation (U.S.). Research Experience for Undergraduates (Program

On the Classification of Universal Rotor-Routers

The combinatorial theory of rotor-routers has connections with problems of statistical mechanics, graph theory, chaos theory, and computer science. A rotor-router network defines a deterministic walk on a digraph G in which a particle walks from a source vertex until it reaches one of several target vertices. Motivated by recent results due to Giacaglia et al., we study rotor-router networks in which all non-target vertices have the same type. A rotor type r is universal if every hitting sequence can be achieved by a homogeneous rotor-router network consisting entirely of rotors of type r. We give a conjecture that completely classifies universal rotor types. Then, this problem is simplified by a theorem we call the Reduction Theorem that allows us to consider only two-state rotors. A rotor-router network called the compressor, because it tends to shorten rotor periods, is introduced along with an associated algorithm that determines the universality of almost all rotors. New rotor classes, including boppy rotors, balanced rotors, and BURD rotors, are defined to study this algorithm rigorously. Using the compressor the universality of new rotor classes is proved, and empirical computer results are presented to support our conclusions. Prior to these results, less than 100 of the roughly 260,000 possible two-state rotor types of length up to 17 were known to be universal, while the compressor algorithm proves the universality of all but 272 of these rotor types

Traversals of Infinite Graphs with Random Local Orientations

We introduce the notion of a "random basic walk" on an infinite graph, give numerous examples, list potential applications, and provide detailed comparisons between the random basic walk and existing generalizations of simple random walks. We define analogues in the setting of random basic walks of the notions of recurrence and transience in the theory of simple random walks, and we study the question of which graphs have a cycling random basic walk and which a transient random basic walk. We prove that cycles of arbitrary length are possible in any regular graph, but that they are unlikely. We give upper bounds on the expected number of vertices a random basic walk will visit on the infinite graphs studied and on their finite analogues of sufficiently large size. We then study random basic walks on complete graphs, and prove that the class of complete graphs has random basic walks asymptotically visit a constant fraction of the nodes. We end with numerous conjectures and problems for future study, as well as ideas for how to approach these problems.Comment: This is my masters thesis from Wesleyan University. Currently my advisor and I are selecting a journal where we will submit a shorter version. We plan to split this work into two papers: one for the case of infinite graphs and one for the finite case (which is not fully treated here

Uniform threshold for fixation of the stochastic sandpile model on the line

We consider the abelian stochastic sandpile model. In this model, a site is deemed unstable when it contains more than one particle. Each unstable site, independently, is toppled at rate $1$, sending two of its particles to neighbouring sites chosen independently. We show that when the initial average density is less than $1/2$, the system locally fixates almost surely. We achieve this bound by analysing the parity of the total number of times each site is visited by a large number of particles under the sandpile dynamics.Comment: 21 pages including bibliography, 19 withou

The Impact of Randomisation in Load Balancing and Random Walks

The real world is full of uncertainties. Classical analyses usually favour deterministic cases, which in practice can be too restricted. Hence it motivates us to add in randomness to make models similar to practical situations. In this thesis, we mainly study two network problems taken from the distributed computing world: iterative load balancing and random walks. An interesting observation is that the problems we study, though not quite related regarding their real world applications, can be linked by the same mathematical toolkit: Markov chain theory. These problems have been heavily studied in the literature. However, their assumptions are mostly \emph{deterministic}, which causes less flexibility and generality to the real world settings. The novelty of this thesis is that we add randomness in these problems in order to observe worst cases vs. average cases (load balancing) and static cases vs. dynamic cases (random walks). For iterative load balancing, the randomness is added on the number of tasks over the entire network. Previous works often assumed worst case initial loads, which may be wasteful sometimes. Hence we relax this condition and assume the loads are drawn from different probability distributions. In particular, we no longer assume the initial loads are chosen by an adversary. Instead, we assume the initial loads on each processor are sampled from independent and identically distributed (i.i.d.) probability distributions. We then study the same problems as in classical settings, i.e., the time needed for the load balancing process to reach a sufficiently small discrepancy. Our main result implies that under such a regime, the time required to balance a network can be much faster. An insightful observation is that the load discrepancy is proportional to the term $t^{-1/4}$ where $t$ is the time used to run the protocol. This implies two main improvements compared with previous works: first, when the initial discrepancy is the same, our regime can reach small discrepancy faster; second, we have established a connection between the time and the discrepancy while previous analyses do not have. For random walks, the randomness is added on the network topologies. This means at each time step (considering discrete times), the underlying network can change randomly. In particular, we want the graph ``evolves'' instead of changing arbitrarily. To model the graph changing process, we adopt a model commonly used in the literature, i.e., the edge-Markovian model. If an edge does not exist between the two nodes, then it will appear in the next step with probability $p$, and if it does then in the next step it will disappear with probability $q$. This model can simulate real world scenarios such as adding friends with each other in social networks or a disruption between two remotely connected computers. Our main contributions regarding random walks include the following results. First, we divided the edge-Markovian graph model into different regimes in a parameterised way. This provides an intuitive path to similar analyses of dynamic graph models. Dynamic models are often hard to analyse in the field because of its complicated nature. We present a possible strategy to reach some feasible solutions by using parameters ($p, q$ above) to control the process. Second, we again analyse the random walk behaviours on such models. We have found that under certain regimes, the random walk still shows similar behaviours especially its mixing nature as in static settings. For the other regimes, we also show either weaker mixing or no mixing results

Studies into the structure and function of various domains of obscurin and titin

Muscles give our bodies the ability to move by stretching and contracting. While contraction is accomplished by the well-known actin-myosin interaction, not much is known about stretch. Two integral muscle proteins involved in stretch are titin and obscurin; both are long rope-like protein molecules that seem to act as molecular springs. Mutations in these two proteins can lead to diseases such as hypertrophic cardiomyopathy and muscular dystrophy, as well as a variety of cancers. In an effort to understand muscle stretch and signaling on a more fundamental level, here we present the high resolution structure of obscurin Ig59, a domain involved in titin/obscurin binding. We also describe how unbound titin moves when stretched. Last, we describe ongoing work in elucidating the high-resolution structures and activation/inhibition mechanisms of obscurin domains Rho-GEF, Rho-GEF-PH, kinase I (KI), and kinase II (KII)

The multi-agent rotor-router on the ring: a deterministic alternative to parallel random walks

International audienceThe rotor-router mechanism was introduced as a deterministic alternative to the random walk in undirected graphs. In this model, an agent is initially placed at one of the nodes of the graph. Each node maintains a cyclic ordering of its outgoing arcs, and during successive visits of the agent, propagates it along arcs chosen according to this ordering in round-robin fashion. The behavior of the rotor-router is fully deterministic but its performance characteristics (cover time, return time) closely resemble the expected values of the corresponding parameters of the random walk. In this work Research partially supported by the ANR Project DISPLEXITY (ANR-11-BS02-014). This study has been carried out in the frame of the Investments for the future Programme IdEx Bordeaux-CP