We study a new method of generating random $d$-regular graphs by repeatedly applying an operation called pegging. The pegging algorithm, which applies the pegging operation in each step, is a method of generating large random regular graphs beginning with small ones. We prove that the limiting joint distribution of the numbers of short cycles in the resulting graph is independent Poisson. We use the coupling method to bound the total variation distance between the joint distribution of short cycle counts and its limit and thereby show that $O(\epsilon^{-1})$ is an upper bound of the $\eps$-mixing time. The coupling involves two different, though quite similar, Markov chains that are not time-homogeneous. We also show that the $\epsilon$-mixing time is not $o(\epsilon^{-1})$. This demonstrates that the upper bound is essentially tight. We study also the connectivity of random $d$-regular graphs generated by the pegging algorithm. We show that these graphs are asymptotically almost surely $d$-connected for any even constant $d\ge 4$. The problem of orientation of random hypergraphs is motivated by the classical load balancing problem. Let $h>w>0$ be two fixed integers. Let $\orH$ be a hypergraph whose hyperedges are uniformly of size $h$. To {\em $w$-orient} a hyperedge, we assign exactly $w$ of its vertices positive signs with respect to this hyperedge, and the rest negative. A $(w,k)$-orientation of $\orH$ consists of a $w$-orientation of all hyperedges of $\orH$, such that each vertex receives at most $k$ positive signs from its incident hyperedges. When $k$ is large enough, we determine the threshold of the existence of a $(w,k)$-orientation of a random hypergraph. The $(w,k)$-orientation of hypergraphs is strongly related to a general version of the off-line load balancing problem. The other topic we discuss is computing the probability of induced subgraphs in a random regular graph. Let $0<s<n$ and $H$ be a graph on $s$ vertices. For any $S\subset [n]$ with $|S|=s$, we compute the probability that the subgraph of $\mathcal{G}_{n,d}$ induced by $S$ is $H$. The result holds for any $d=o(n^{1/3})$ and is further extended to $\mathcal{G}_{n,{\bf d}}$, the probability space of random graphs with given degree sequence $\bf d$. This result provides a basic tool for studying properties, for instance the existence or the counts, of certain types of induced subgraphs.

We study a new method of generating random $d$-regular graphs by repeatedly applying an operation called pegging. The pegging algorithm, which applies the pegging operation in each step, is a method of generating large random regular graphs beginning with small ones. We prove that the limiting joint distribution of the numbers of short cycles in the resulting graph is independent Poisson. We use the coupling method to bound the total variation distance between the joint distribution of short cycle counts and its limit and thereby show that $O(\epsilon^{-1})$ is an upper bound of the $\eps$-mixing time. The coupling involves two different, though quite similar, Markov chains that are not time-homogeneous. We also show that the $\epsilon$-mixing time is not $o(\epsilon^{-1})$. This demonstrates that the upper bound is essentially tight. We study also the connectivity of random $d$-regular graphs generated by the pegging algorithm. We show that these graphs are asymptotically almost surely $d$-connected for any even constant $d\ge 4$. The problem of orientation of random hypergraphs is motivated by the classical load balancing problem. Let $h>w>0$ be two fixed integers. Let $\orH$ be a hypergraph whose hyperedges are uniformly of size $h$. To {\em $w$-orient} a hyperedge, we assign exactly $w$ of its vertices positive signs with respect to this hyperedge, and the rest negative. A $(w,k)$-orientation of $\orH$ consists of a $w$-orientation of all hyperedges of $\orH$, such that each vertex receives at most $k$ positive signs from its incident hyperedges. When $k$ is large enough, we determine the threshold of the existence of a $(w,k)$-orientation of a random hypergraph. The $(w,k)$-orientation of hypergraphs is strongly related to a general version of the off-line load balancing problem. The other topic we discuss is computing the probability of induced subgraphs in a random regular graph. Let $0<s<n$ and $H$ be a graph on $s$ vertices. For any $S\subset [n]$ with $|S|=s$, we compute the probability that the subgraph of $\mathcal{G}_{n,d}$ induced by $S$ is $H$. The result holds for any $d=o(n^{1/3})$ and is further extended to $\mathcal{G}_{n,{\bf d}}$, the probability space of random graphs with given degree sequence $\bf d$. This result provides a basic tool for studying properties, for instance the existence or the counts, of certain types of induced subgraphs.

<p>In this thesis we present a model of the interacting agents on the financial market. The agents are represented by a non-Euclidean random graph, where each agent communicate with another with probability p, and the interaction according to the Ising Model. We investigate properties of the model by direct calculations for small graph sizes, and by perfect simulation for larger graph sizes. We also present a model for asset price variation by using the magnetization of the Ising model.</p>

Exponential Random Graph Models (ERGMs) have been developed for fitting social network data on both static and dynamic levels. However, the lack of large sample asymptotic properties makes it inadequate in assessing the goodness-of-fit of these ERGMs. Simulation-based goodness-of-fit plots were proposed by Hunter et al (2006), comparing the structured statistics of observed network with those of corresponding simulated networks. In this research, we propose an improved approach to assess the goodness of fit of ERGMs. Our method is shown to improve the existing graphical techniques. We also propose a simulation based test statistic with which the model comparison can be easily achieved. / Biostatistics

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In this thesis we present a model of the interacting agents on the financial market. The agents are represented by a non-Euclidean random graph, where each agent communicate with another with probability p, and the interaction according to the Ising Model. We investigate properties of the model by direct calculations for small graph sizes, and by perfect simulation for larger graph sizes. We also present a model for asset price variation by using the magnetization of the Ising model.

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Systemic risk is the risk that an economic shock may result in the breakdown of the fundamental functions of the financial system. It can involve multiple vectors of infection such as chains of losses or consecutive failures of financial institutions that may ultimately cause the failure of the financial system to provide liquidity, stable prices, and to perform economic activities. This thesis develops methods to quantify systemic risk, its effect on the financial system and perhaps more importantly, to determine its cause. In the first chapter, we provide an overview and a literature review of the topics covered in this thesis. First, we present a literature review on network-based models of systemic risk. Finally we end the first chapter with a review on market impact models. In the second chapter, we consider one unregulated financial institution with constant absolute risk aversion investment risk preferences that optimizes its strategies in a multi asset market impact model with temporary and permanent impact. We prove the existence and derive explicitly the optimal trading strategies. Furthermore, we conduct numerical exploration on the sensitivity of the optimal trading curve. This chapter sets the foundation for further research into multi-agent models and systemic risk models with optimal behaviours. In the third chapter, we extend the market impact models to the multi-agent setting. The agents follow a game theoretic strategy that is constrained by the regulations imposed. Furthermore, the agents must liquidate themselves if they become insolvent or unable to meet the regulations imposed on them. This paper provides a bridge between market impact models and network models of systemic risk. In chapter four, we introduce a financial network model that combines the default and liquidity stress mechanisms into a ``double cascade mapping''. Unlike simpler models, this model can quantify how illiquidity or default of one bank influences the overall level of liquidity stress and default in the system. We derive large-network asymptotic cascade mapping formulas that can be used for efficient network computations of the double cascade. Finally we use systemic risk measures to compare the results of including with and without an asset firesale mechanism. / Thesis / Doctor of Philosophy (PhD)