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Vertex set: metrics, norms

Similarity measures between u and v are inverses of distance metrics.

Based on paths and random walks

(shortest path).

Katz measure

\(sim(u,v) = \sum_{l=1}^{\infty} \gb^{l}|paths_{u,v}(l)|\). Matrix of scores, \(K = \sum_{i=1}^{\infty} \gb^{i}A^{i}\).

The damping parameter

\(K = (I - \gb A)^{-1} - I\) if the sum converges. Use \(A = U\EW U^{}\): EW decomposition, with \(\ew_{i}\) ordered in descending order. \(\sum_{i=1}^{\infty} \gb^{i}A^{i} = U(\sum_{i=1}^{\infty} \gb^{i}\EW^{i})U^{}\) does not converge \(\forall \gb \geq 0\) and multiplication by \(\infty\) is not well defined. Condition for convergence: \(\gb \ew_{1} < 1\).

But, for \(\gb > 1\) the intuition of weighting longer paths less does not hold.

Variants of Katz

Similarly can use \(e^{-\gb A}\).

Truncated Katz usually used: \(\sum_{l=1}^{k} \gb^{l}A^{l}\): \(O(ln^{2}nz(A))\) op instead of \(O(n^{3})\) inverse finding.

Hitting time

\(-h_{u,v}\); normed by stationary distribution: \(-h_{u,v} \pi_{v}\): to take care of skewing of hitting time due to large \(\pi_{v}\). - Commute time: \(-h_{u,v} - h_{v, u}\); stationary distr normed: \(-h_{u,v}\pi_{v} - h_{v, u}\pi_{u}\).

Rooted PageRank

Random walk can get lost in parts of graph away from u and v; so do random resets and return to u with probability a at each step.

SimRank

A recursive definition: 2 nodes are similar to the extant that they are joined to similar nodes. \(sim(x, x) = 1; sim (x, y) = \frac{\sum_{a \in \nbd(x)}\sum_{b \in \nbd(y)} sim(a, b)}{|\nbd(x)||\nbd(y)|}\).

Common neighbor based

Common neighbors: \(|\nbd(u) \inters \nbd(v)|\): same as taking inner product of rows in adj matrix M corresponding to u and v. (Jaccard): \(\frac{\nbd(u) \inters \nbd(v)}{\nbd(u) \union \nbd(v)}\): pick a feature at random, see probability that it is a feature of both u and v. (Adamic/ Adar): \(\sum_{z \in \nbd(u) \inters \nbd(v)} \frac{1}{\log |\nbd(z)|}\): greater wt to rare features present in both.