Q2. What is the definition of a Cayley graph?
The subset S is said to be a generating set for G, and the elements of S are called generators of G, if every element of G can be expressed as a finite product of the powers of the elements in S.
Q3. What is the main purpose of the paper?
In this paper the authors show that Cayley graphs are excellent models for small-world networks, in the sense that with suitable choice of relevant parameters, they can be adapted to possess the distinguishing characteristics of such networks.
Q4. What are the main reasons why the authors believe Cayley graphs are good for small world networks?
The authors believe that their models will have important applications to diverse research fields, including in parallel architectures and communication networks.
Q5. What is the clustering coefficient of node 1?
The set of neighbors of node 1 is S. If s1, s2 ∈ S, then s1 and s2 are adjacent if and only if there is s ∈ S such that s2 = s1s.
Q6. What is the way to get a small world network?
In particular, the method of starting with a regular degree-d network and replacing each of its nodes with the d-node complete graph Kd , which is the only node-symmetric construction proposed in [6], yields only one smallworld network for each starting configuration.
Q7. What are the characteristics of Cayley graphs?
Unlike the probabilistic models of[9], their networks are characterized by closed-form, exact formulas for various properties, which lead to intuitive and fairly precise mechanisms for varying the pertinent parameters.
Q8. What is the clustering coefficient of 1?
For instance, with l = log2 t , or equivalently, t = 2l , the clustering coefficient is:C1 = (t − 1)(t − 2) .(2t − log2 t − 1)(2t − log2 t − 2)Therefore C1 → 14 when t → ∞.
Q9. What is the way to reduce the average internode distance?
which has high clustering but also a large average internode distance, and to randomly rewire the edges so as to reduce the average internode distance.
Q10. What is the meaning of the sentence?
In this case, the authors also say that G is generated by S.The Cayley digraph of a group G and the subset S of G, denoted by Cay(G,S), has vertices that are elements of G and arcs that are ordered pairs (g,gs) for g ∈ G, s ∈ S.
Q11. Why is it important to introduce deterministic models for small-world networks?
One motivation in introducing deterministic models for small-world networks is to facilitate the understanding of their behavior.
Q12. What is the generating set of G?
If S is a generating set of G, then the authors say that Cay(G,S) is the Cayley digraph of G generated by S. If 1 /∈ S (1 is the identity element of G) and S = S−1, then Cay(G,S) is a simple (undirected) graph.
Q13. What is the main difference between the two?
Their models, dubbed “small-world networks” (corresponding to the popular notion of “six degrees of separation”) offer high clustering, like loop networks, yet possess small average internode distances, as in random networks.
Q14. What can be done to make the Cayley graphs more realistic?
By suitably choosing the parameters of the Cayley-graph models, they can be made to mimic many real networks of the types found in social, technological, and biological domains.
Q15. How can the authors obtain different clustering coefficients for 1?
By suitably choosing a, the authors can obtain different clustering coefficients for Γ1, while maintaining a small node degree equal to at + t − l.