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Cooperative Models of Stochastic
Growth - On a class of reinforced processes with
graph-based interactions

COSTA, MARCELO,ROCHA (2018) Cooperative Models of Stochastic
Growth - On a class of reinforced processes with
graph-based interactions. Doctoral thesis, Durham University.

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Abstract

Consider a sequence of positive integer-valued random vectors denoted by ${\bf x}_n = (x_1(n),\ldots,x_N(n))$ for $n= 0,1,2,\ldots \>$ . Fix ${\bf x}_0$ , and given ${\bf x}_n$ , choose a $\emph{random}$ coordinate $i_{n+1} \in \{1,\ldots,N\}$ . The probability that $\{i_{n+1} = i\}$ for a particular coordinate $i$ is proportional to a non-decreasing function $f_i$ of $\sum_{j =1}^N a_{ij}x_j(n)$ , where $a_{ij} \geq 0$ measures how strongly $j$ cooperates with $i$ . Now, on the event that $\{i_{n+1}=i\}$ , update the sequence in such a way that ${\bf x}_{n+1}={\bf x}_n + {\bf e}_i$ , where ${\bf e}_i$ is the vector whose $i$ -th coordinate is 1 and whose other coordinates are 0. Finally, given $A=(a_{ij})_{i,j=1}^N$ and $f_i, \> i =1,\ldots,N,$ what can one say about $\lim_{n \to \infty} n^{-1} {\bf x}_n$ ?

Item Type:	Thesis (Doctoral)
Award:	Doctor of Philosophy
Keywords:	Stochastic Growth; Urn models; Branching Processes; Stochastic Approximation Algorithm.
Faculty and Department:	Faculty of Science > Mathematical Sciences, Department of
Thesis Date:	2018
Copyright:	Copyright of this thesis is held by the author
Deposited On:	11 May 2018 13:38

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Growth - On a class of reinforced processes with
graph-based interactions

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Cooperative Models of Stochastic Growth - On a class of reinforced processes with graph-based interactions

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Cooperative Models of Stochastic
Growth - On a class of reinforced processes with
graph-based interactions