'Advanced Network Analysis / Pajek: Large ... - Vladimir Batagelj

V. Batagelj: Analysis and visulization of large networks with Pajek 103
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Complexity of fast sparse matrix multiplication
Let A and B be matrices of networks NA = (I, K, EA, wA) and NB =
(K, J , EB, wB).
Assume that the body of the loops can be computed in the constant time c.
Then we can prove:
If at least one of the sparse networks NA and NB has small maximal degree
on K then also the resulting product network NC is sparse.
And after more detailed complexity analysis:
Let dmin(k) = min(deg A(k), deg B(k)), ∆min = maxk∈K dmin(k),
dmax(k) = max(deg A(k), deg B(k)), K(d) = {k ∈ K : dmax(k) ≥ d},
d ∗ = argmin d(|K(d)| ≤ d) and K ∗ = K(d ∗ ).
If for the sparse networks NA and NB the quantities ∆min and d ∗ are small
then also the resulting product network NC is sparse.
Methodenforum der Fakultät für Sozialwissenschaften, Universität Wien, 21-22. 6. 2007 ▲
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