Conference Agenda

For nonzero vectors x and y, the Salton cosine index is defined as

S(x,y) = <x,y>/(|x|.|y|),

where <x,y> is the inner product and |x| = √<x,x>. It has the following properties: (1) S(x,y) ∈ [-1,1], (2) S(x,y) = S(y,x), (3) S(x,x) = 1, (4) x, y ≥ 0 ⇒ S(x,y) ∈ [0,1], (5) a, b ∈ ℝ ⇒ S(a.x,b.y) = S(x,y).

The Salton index measures similarity. It is usually transformed into a dissimilarity by d(x,y) = 1 - S(x,y) or d(x,y) = arccos(S(x,y))/π. The Salton index is especially useful in analyzing weighted networks because (property 5) it makes comparable nodes of different strengths.

Let W = [w[u,v]] be a matrix representation of a 2-mode network N = ((U, V), L, w); U, V are sets of nodes, L is the set of links, and w : L → ℝ is the weight. We can define a dissimilarity between nodes as D(u,v) = d(w[u,.],w[v,.]); w[u,.] is the matrix row of node u. The idea can not be directly applied to ordinary (1-mode, U = V) networks because in D(u,v) we would compare w[u,u] with w[u,v] and w[v,v] with w[v,u], but we must compare w[u,u] with w[v,v] and w[u,v] with w[v,u]. This is resolved by the corrected Salton index S'(u,v) = (<w[u,.],w[v,.]> + (w[u,u]-w[u,v]).(w[v,v]-w[v,u]))/(|w[u,.]|.|w[v,.]|). The properties 1-5 hold also for S'.

The Salton index can also be generalized to multiway networks, opening a way for new methods for their analysis. The details will be given in the presentation.

Examples from analyses of real-life networks will illustrate the proposed approaches. They are supported by an R package ClusNet available at https://github.com/bavla/Rnet/tree/master/R.