Finite 33-connected homogeneous graphs

A finite graph \G is said to be {\em $(G,3)$-$($connected$)$ homogeneous} if every isomorphism between any two isomorphic (connected) subgraphs of order at most $3$ extends to an automorphism $g\in G$ of the graph, where $G$ is a group of automorphisms of the graph. In 1985, Cameron and Macpherson determined all finite $(G, 3)$-homogeneous graphs. In this paper, we develop a method for characterising $(G,3)$-connected homogeneous graphs. It is shown that for a finite $(G,3)$-connected homogeneous graph \G=(V, E), either G_v^{\G(v)} is $2$--transitive or G_v^{\G(v)} is of rank $3$ and \G has girth $3$, and that the class of finite $(G,3)$-connected homogeneous graphs is closed under taking normal quotients. This leads us to study graphs where $G$ is quasiprimitive on $V$. We determine the possible quasiprimitive types for $G$ in this case and give new constructions of examples for some possible types

Forwarding and optical indices of 4-regular circulant networks

An all-to-all routing in a graph $G$ is a set of oriented paths of $G$, with exactly one path for each ordered pair of vertices. The load of an edge under an all-to-all routing $R$ is the number of times it is used (in either direction) by paths of $R$, and the maximum load of an edge is denoted by $\pi(G,R)$. The edge-forwarding index $\pi(G)$ is the minimum of $\pi(G,R)$ over all possible all-to-all routings $R$, and the arc-forwarding index $\overrightarrow{\pi}(G)$ is defined similarly by taking direction into consideration, where an arc is an ordered pair of adjacent vertices. Denote by $w(G,R)$ the minimum number of colours required to colour the paths of $R$ such that any two paths having an edge in common receive distinct colours. The optical index $w(G)$ is defined to be the minimum of $w(G,R)$ over all possible $R$, and the directed optical index $\overrightarrow{w}(G)$ is defined similarly by requiring that any two paths having an arc in common receive distinct colours. In this paper we obtain lower and upper bounds on these four invariants for $4$-regular circulant graphs with connection set $\{\pm 1,\pm s\}$, $1<s<n/2$. We give approximation algorithms with performance ratio a small constant for the corresponding forwarding index and routing and wavelength assignment problems for some families of $4$-regular circulant graphs.Comment: 19 pages, no figure in Journal of Discrete Algorithms 201

Bayesian Designs for Early Phase Clinical Trials with Novel Target Agents

My dissertation mainly focus on Bayesian designs for early phase clinical trials with novel target agents. It includes three specific topics: (1) reviewing novel phase I clinical trial designs and comparing their operating characteristics; (2) Proposing a Bayesian optimal phase II clinical trial (BOP2) design with simple and complex endpoints under a unified framework; and (3) extending the BOP2 design to incorporate the durable clinical response as a primary endpoint. A number of novel model-based and model-assisted designs have been proposed to find the maximum tolerated dose (MTD) in phase I clinical trials, but their differences and relative pros and cons are not clear to many practitioners. We review three model-based designs, including the continual reassessment method (CRM), dose escalation with overdose control (EWOC), and Bayesian logistic regression model (BLRM), and three model-assisted designs, including the modified toxicity probability interval (mTPI), Bayesian optimal interval (BOIN), and keyboard designs. We conduct numerical studies to assess their accuracy, safety and reliability, and the practical implications of various empirical rules used in some designs, such as skipping a dose and imposing overdose control. Our results show that the CRM outperforms EWOC and BLRM with higher accuracy of identifying the MTD. For the CRM, skipping a dose is not recommended as it substantially increases the chance of overdosing patients, while providing limited gain for identifying the MTD. EWOC and BLRM appear excessively conservative. They are safe, but have relatively poor accuracy of finding the MTD. The BOIN and keyboard designs have similar operating characteristics, outperforming the mTPI, but the BOIN is more intuitive and transparent. The BOIN yields competitive performance comparable to the CRM, but is simpler to implement and free of the issue of irrational dose assignment caused by model misspecification, thereby providing an attractive approach for designing phase I trials. We propose a flexible Bayesian optimal phase II (BOP2) design that is capable of handling simple (e.g., binary) and complicated (e.g., ordinal, nested and co-primary) endpoints under a unified framework. We use a Dirichlet-multinomial model to accommodate different types of endpoints. At each interim, the go/no-go decision is made by evaluating a set of posterior probabilities of the events of interest, which is optimized to maximize power or minimize the number of patients under the null hypothesis. Unlike most existing Bayesian designs, the BOP2 design explicitly controls the type I error rate, thereby bridging the gap between Bayesian designs and frequentist designs. In addition, the stopping boundary of the BOP2 design can be enumerated prior to the onset of the trial. These features make the BOP2 design accessible to a wide range of users and regulatory agencies, and particularly easy to implement in practice. Simulation studies show that the BOP2 design has favorable operating characteristics with higher power and lower risk of incorrectly terminating the trial than some existing Bayesian phase II designs. The software to implement the BOP2 design is freely available at www.trialdesign.org. Based on the BOP2 design, we propose a BOP2-C design which jointly models the nested efficacy endpoints and the long-term durable clinical response (cure rate) simultaneously. We use a Dirichlet-multinomial model to account for the tumor response CR, PR, SD and PD, and assume Weibull distribution on the time to disease progression in the non-cured patients. At each interim, the go/no-go decision is made based on the posterior estimation of the CR, CR/PR and cure rates, with the optimized design parameters in the posterior probability cutoffs varying with the interim sample size. The BOP2-C design can also explicitly control the type I and type II error rates. Simulation studies show that the BOP2-C design can achieve favorable operating characteristics with high accuracy in identifying the promising treatment, and it is still robust when the true distribution of time to disease progression violates the Weibull assumption

The multiple effects of gradient coupling on network synchronization

Recent studies have shown that synchronizability of complex networks can be significantly improved by asymmetric couplings, and increase of coupling gradient is always in favor of network synchronization. Here we argue and demonstrate that, for typical complex networks, there usually exists an optimal coupling gradient under which the maximum network synchronizability is achieved. After this optimal value, increase of coupling gradient could deteriorate synchronization. We attribute the suppression of network synchronization at large gradient to the phenomenon of network breaking, and find that, in comparing with sparsely connected homogeneous networks, densely connected heterogeneous networks have the superiority of adopting large gradient. The findings are supported by indirect simulations of eigenvalue analysis and direct simulations of coupled nonidentical oscillator networks.Comment: 4 pages, 4 figure