# A Beckman-Quarles type theorem for finite desarguesian by Benz W.

By Benz W.

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An undirected geometric random graph with N nodes is denoted as Gp(rij ) (N ), where p(rij ) is the probability of having a link between two nodes i and j (or j and i) at distance rij from each other. In this graph the expected number of edges or links between nodes is by deﬁnition: N N L= p (rij ) . i=1 j=i+1 Let us assume that N nodes are uniformly distributed over a 2-dimensional area with size Ω. To derive the average number of links over all possible conﬁgurations, E[L], we have used a dissection technique and assumed that area Ω is covered with m > N small squares (or placeholders) of size ∆Ω.

6, bottom part). n−1 ] = n−1 k Pr [h = k] = k=0 k=0 n−1 2k(n − k) = . n(n + 1) 3 In the 2-dimensional lattice of size n × m, that has n nodes in horizontal direction and m nodes in vertical direction, we have: hn×m = hhorizontal + hvertical E [hn×m ] = E [hhorizontal ] + E [hvertical ] For each occurrence of hn×m , either hhorizontal or hvertical can be 0 but not both simultaneously. 6). 2) we note that in lattice graphs the hopcount growth is polynomial with respect to increasing network size N , while in random graphs the expected hopcount is only logarithmic in N .

4. 5 5 Fig. 4. Growth of the giant component size as function of the mean nodal degree in a random graph. Because clustering coeﬃcient is the percentage of neighbors of a node that are connected to each other, and in a random graph links between nodes are established independently with probability p, we may expect the clustering coeﬃcient in a random graph to be: CG = p. This result has been proved in both [50] and [44]. 2 Regular lattice graph model A regular lattice graph is constructed with nodes (vertices) placed on a regular grid structure.