5. Newman, Networks 2010 15.2) One of the difficulties with the original small-world model depicted in Fig. 15.3a is that vertices can become disconnected from the rest of the network by the rewiring process. For instance, a single vertex can become disconnected if all of its incident edges around the ring are rewired and it has no shortcuts. a) Show that the probability of this happening to any given vertex is [pe^(−p)]^c. b)
5. Newman, Networks 2010 15.2) One of the difficulties with the original small-world model depicted in Fig. 15.3a is that vertices can become disconnected from the rest of the network by the rewiring process. For instance, a single vertex can become disconnected if all of its incident edges around the ring are rewired and it has no shortcuts. a) Show that the probability of this happening to any given vertex is [pe^(−p)]^c. b)
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5.
Newman, Networks 2010
15.2) One of the difficulties with the original small-world model depicted in Fig. 15.3a
is that vertices can become disconnected from the rest of the network by the rewiring
process. For instance, a single vertex can become disconnected if all of its incident edges
around the ring are rewired and it has no shortcuts.
a) Show that the probability of this happening to any given vertex is [pe^(−p)]^c.
b) Hence, how large must the network be before we expect that one vertex will be
disconnected, if c = 6 and p = 0.01?
Transcribed Image Text:Figure 15.3: Two versions of the small-world model. (a) In the original version of the
small-world model, edges are with independent probability p removed from the circle
and placed between two vertices chosen uniformly at random, creating shortcuts across
the circle as shown. In this example n = 24, c = 6, and p = 0.07, so that 5 out of 72
edges are "rewired" in this fashion. (b) In the second version of the model only the
shortcuts are added and no edges are removed from the circle.
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