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Can you design a random graph model which can support more than one giant component?   Describe...

Can you design a random graph model which can support more than one giant component?  

Describe what dependence structure between the edges and nodes can bring this about, or

argue why such a graph is impossible. Assess the relevance of plausible existence and

uniqueness of a giant component for a random graph in an applied setting.

Solutions

Expert Solution

Ans : Random graph model :

Standard techniques for analyzing network models usually break down in the presence of clustering. Here we introduce a new analytic tool, the "free-excess degree" distribution, which extends the generating function framework, making it applicable for clustered networks (C>0). The methodology is general and provides a new expression for the threshold point at which the giant component emerges and shows that it scales as (1-C)(-1). In addition, the size of the giant component may be predicted even for more complicated scenarios such as the removal of a fixed fraction of nodes at random.

The field of random graphs is one such area. Recall from college-level math or computer science that an undirected graph is a collection of vertices (also called nodes), with some pairs of vertices connected by edges. (A depiction of an example graph is below.) Originally, those working in graph theory focused on proving many deterministic properties of graphs. For instance, let the degree of a vertex be the number of other vertices it is connected to.

Network theory is a powerful tool for describing and modeling complex systems having applications in widelydiffering areas including epidemiology [16], neuroscience [34], ecology [20] and the Internet [26]. In its beginning, one often compared an empirically given network, whose nodes are the elements of the system and whose edges represent their interactions, with an ensemble having the same number of nodes and edges, the most popular example being the random graphs introduced by Erdos and Renyi [11]. As the field matured, it became clear that the naive model above needed to be refined, due to the observation that real-world networks often differ significantly from the Erdos–Renyi random graphs in having a highly heterogenous non-Poisson degree distribution [5, 15] and in possessing a high level of clustering [33]. Methods for generating random networks with arbitrary degree distributions and for calculating their statistical properties are now well understood.


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