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

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V. Batagelj: Analysis and visulization of large networks with Pajek 103 ✬ ✫ 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 ▲ ❙ ▲ ▲ ● ❙ ▲ ▲ ☛ ✖ ▲ ✩ ✪
V. Batagelj: Analysis and visulization of large networks with Pajek 104 ✬ 2-mode network analysis by conversion to 1-mode network Often we transform a 2-mode network into an ordinary (1-mode) network N1 = (U, E1, w1) or/and N2 = (V, E2, w2), where E1 and w1 are determined by the matrix A (1) = AA T , a (1) uv = � z∈V auz · a T zv. Evidently a (1) uv = a (1) vu . There is an edge {u, v} ∈ E1 in N1 iff N(u) ∩ N(v) �= ∅. Its weight is w1(u, v) = a (1) uv . The network N2 is determined in a similar way by the matrix A (2) = A T A. The networks N1 and N2 are analyzed using standard methods. ✫ Methodenforum der Fakultät für Sozialwissenschaften, Universität Wien, 21-22. 6. 2007 ▲ ❙ ▲ ▲ ● ❙ ▲ ▲ ☛ ✖ ▲ ✩ ✪
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'Advanced Network Analysis / Pajek: Large ... - Vladimir Batagelj

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