Abstract
In this paper, we outline a model of graph (or network) dynamics based on two ingredients. The first ingredient is a Markov chain on the space of possible graphs. The second ingredient is a semi-Markov counting process of renewal type. The model consists in subordinating the Markov chain to the semi-Markov counting process. In simple words, this means that the chain transitions occur at random time instants called epochs. The model is quite rich and its possible connections with algebraic geometry are briefly discussed. Moreover, for the sake of simplicity, we focus on the space of undirected graphs with a fixed number of nodes. However, in an example, we present an interbank market model where it is meaningful to use directed graphs or even weighted graphs.
Highlights
The publication of Collective dynamics of ‘small world’ networks by Watts and Strogatz [1] gave origin to a plethora of papers on network structure and dynamics. The history of this scientific fashion is well summarized by Rick Durrett [2]: The theory of random graphs began in the late 1950s in several papers by Erdos and Renyi
In the late twentieth century, the notion of six degrees of separation, meaning that any two people on the planet can be connected by a short chain of people who know each other, inspired Strogatz and Watts [1] to define the small world random graph in which each side is connected to k close neighbors, and has long-range connections
We have discussed a model of graph dynamics based on two ingredients
Summary
The publication of Collective dynamics of ‘small world’ networks by Watts and Strogatz [1] gave origin to a plethora of papers on network structure and dynamics. The history of this scientific fashion is well summarized by Rick Durrett [2]: The theory of random graphs began in the late 1950s in several papers by Erdos and Renyi. At about the same time, it was observed in human social and sexual networks and on the Internet that the number of neighbors of an individual or computer has a power law distribution This inspired Barabasi and Albert [3] to define the preferential attachment model, which has these properties. While this literature is extensive, many of the papers are based on simulations and nonrigorous arguments
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