By A Grothendieck
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Additional info for A general theory of fibre spaces with structure sheaf (National Science Foundation research project on geometry of function space : report)
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 .
The best way to determine the most probable value range for ξ is through extensive measurements. To our knowledge reliable and extensive measurements of this type for typical wireless ad-hoc network environments are not available yet. 5. 5 5 Fig. 10. Link probability in lognormal geometric random graph model for diﬀerent ξ values. In the case ξ = 0 the lognormal model reduces to the pathloss model with circular coverage per node. 11) with a simple step function as link probability: lim p(rij ) = ξ→0 1 if rij < 1 .
The service area is much larger than coverage area of a single node, and 2. the node density is low. A relatively large service area is equivalent to a low link density. 4)). Considering this, we can say that the border eﬀect is negligible and the degree distribution is binomial when the mean node degree is low. In the remainder of this section we justify this statement and try to quantify conditions for its validity through simulations. 4 shows the degree distribution found through simulations for ξ = 3 and diﬀerent number of nodes uniformly distributed over an area of 20 × 20.