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consider a bipartite user/item graph# taking 2 steps starting at i1 with variable P=1.0 take one step; split P across two users; u1 P=0.5 and u2 P=0.5 take another step; split P further; all of u1's P goes back to i1 whereas u2's P is spread evenly between i1, i2, i3 results in i1 P=0.5+0.16=0.66, i2=0.16 i3=0.16 this dispersion of P calculates the following... starting at i1, talking two random steps what is the probability we end up at itemN? P(i1) = 0.66, P(i2) = 0.16, P(i3) = 0.16 note how these are different depending on the starting point.. starting at i2, talking two random steps what is the probability we end up at itemN? P(i1) = 0.33, P(i2) = 0.33, P(i3) = 0.33 starting at i3, talking two random steps what is the probability we end up at itemN? P(i1) = 0.16, P(i2) = 0.16, P(i3) = 0.66 ## taking 2N steps (large N) if we walk forever we get to a steady state; P(i1) = 0.4, P(i2) = 0.2, P(i3) = 0.4 (and because this is in the limit we get this no matter where we start) ## somewhere in between we saw how 2-steps gives a large difference between items, but doesnt walk the graph very far... and 2N-steps gives no difference between items but provides a better representation of the spread over the graph... as an inbetween we introduce a form of "trace" where we deposit some fraction (BETA) of our "probability" at each item we visit (ie every 2 steps) for BETA=1.0 (ie deposit everything) the trace amounts calculating the 2-step probabilities. ie start with P=1.0; move to user; move to item; deposit 100%; stop cat eg_graph.tsv | ./dispersion.py 1.0 20 {"20": 0.666, "21": 0.166, "22": 0.166} 21 {"20": 0.333, "21": 0.333, "22": 0.333} 22 {"20": 0.166, "21": 0.166, "22": 0.666} for BETA=0.001 (ie deposit a small amount) the trace amounts are approximating the probability of where we would end up after after a walk of 2N steps (for large N) cat eg_graph.tsv | ./dispersion.py 0.0001 20 {"20": 0.25344, "21": 0.12642, "22": 0.25244} 21 {"20": 0.25284, "21": 0.12662, "22": 0.25284} 22 {"20": 0.25244, "21": 0.12642, "22": 0.25344} a more interesting examples lies somewhere in between, say BETA=0.5, where we deposit some probability at "near" items but allow some other fraction to disperse further cat eg_graph.tsv | ./dispersion.py 0.5 20 {"20": 0.57575, "21": 0.18181, "22": 0.24241} 21 {"20": 0.36363, "21": 0.27272, "22": 0.36363} 22 {"20": 0.24241, "21": 0.18181, "22": 0.57575} these traces can then used as features for an item/item comparison matrix where, say, sim(i1,i2) = some f(of their rows) 20 21 22 20 0.666 0.166 0.166 21 0.333 0.333 0.333 22 0.166 0.166 0.666
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