Equivalence of LT Model to Live-edge Model

1. Equivalence of LT Model to Live-edge Model

2. Equivalence between LT and Live Edge Model • Recall LT Model: sum of i/c arc weights • Each node chooses a threshold uniformly at random from • Activates at time iff the net i/c influence weight at the end of step exceeds • Live edge selection process: each node selects a favorite in-neighbor w.p. and no in-neighbor w.p. • Let be the set of seed nodes, which are, by def., active at time

3. Claim: The distributions over (i) sets of nodes active according to the LT model at various time steps and (ii) over sets of nodes reachable at various lengths from the seed nodes in the live edge graph, are identical. • Key proof idea: a node gets active at time iff the net i/c weight from neighbors active at exceeds which is true w.p. = net i/c weight from seeds, since is chosen at random from

4. A node is reachable from a seed via a live edge iff the favorite in-neighbor selects is a seed, which is true w.p. = net i/c weight from seeds. Seeds

5. That was the base case of induction. Now, handle induction in a similar way, except we now consider nodes whose net i/cweight from active in-neighbors at previous time step fell short of What’s the probability that the neighbors of that newly became active push the net inflow of influence over ? Net influence of newly active neighbors Convention: whenever is not an in-neighbor of Max. possible shortfall

6. What’s the probability that is reachable from the seeds in steps, given that it is not reachable in steps? Odds that ’s favorite in-neighbor is reachable from seeds in exactly steps. Convention: whenever is not an in-neighbor of Odds that ’s favorite In-neighbor didn’t come From

7. Nodes reachable in t steps = Nodes activated in t steps.