Netzwerkerin

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Are road maps small-world networks?

Hi,

   I recently read your post in Talk:Small-world network.
   You said that road maps are very possibly NOT small world networks, which I can't really understand. Would you please give me some journal articles or other research which supports your idea? 
   Here is my confusion about this topic. When I calculated the mean shortest path length of an equivalent(here means same numbers of vertices and edges) random graph, I found a problem that when a vertex in a random graph is not connected to vertices in another component, so the sum of shortest path length must exclude this vertex. Here comes the trouble, do I exclude them from denominator? If so, the random graph doesn't have the same number of vertices, which doesn't agree with the definition of such an equivalent random graph, the judgement (actually it resulted a large mean shortest path length which is larger than that of my road map, this supports your idea that road maps are not small-world networks.) would be wrong. If not, i.e. the denominator remains the number of vertices in the graph while their shortest path length(Infinitive) was excluded from the numerator, the judgement comes that this random graph has similar mean shortest path length with the road map and this disapproves your idea.  — Preceding unsigned comment added by 123.163.27.64 (talk) 15:22, 23 January 2014 (UTC) 

Hi there, the problem you address is one of the problems: the corresponding random graph model is not connected in most cases. So, not only do you have single vertices not connected to the rest, you might also get subgraphs that are not connected to the rest. How do you represent your street network? If you make an edge per street and if you do not weight it, it might be that a small street network is more or less "small-wordy" (note that there is no clear definition anyway). But if you would look at the street network of north and south america together, the lenght of any single street between two connections will be much smaller than the maximal distance. In this case, the number of edges between any two cities is linear with their (geodesic) distance. In the small-world model, the average distance is supposed to scale logarithmically with the number of nodes, in the geometric model, it is likely to grow linearly with n. However, all of this is difficult (impossible) to determine for a single network. It is a statement that only makes sense for growing networks or for a set of networks representing the same complex system. Hope this helps. Netzwerkerin (talk) 16:24, 25 January 2014 (UTC)