CSCI 3210: Computational Game Theory Cascading Behavior in - - PDF document
11/17/19 CSCI 3210: Computational Game Theory Cascading Behavior in Networks Ref: [AGT] Ch 24 Mohammad T . Irfan Email: mirfan@bowdoin.edu Web: www.bowdoin.edu/~mirfan Course Website: www.bowdoin.edu/~mirfan/CSCI-3210.html Cascading
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Mohammad T . Irfan Email: mirfan@bowdoin.edu Web: www.bowdoin.edu/~mirfan Course Website: www.bowdoin.edu/~mirfan/CSCI-3210.html
u Analogy: epidemic u Starts with a few sick
people
u Propagates
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Alice
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u Studied in sociology since 1940s u Examples
u Indirect/informational effects
u Adoption of new medical or agricultural innovations u Photos/video going viral u Sudden success of new products u Rise of celebrities
u Direct-benefit effects
u Technology adoption– Xbox/PS4, phone, fax, email,
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u What is the process by which these happen?
u Models u Depart from one-shot game setting
u Who are the most influential entities?
u Algorithms
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u Stephen Morris (2000)
u Initial adopters: A set of nodes adopting B at
the start
u Time steps 1, 2, 3, ... u At each time step
u Every node simultaneously updates their behavior
in response to the current behaviors of neighbors u 2 versions
u Nonpregressive: Nodes may switch between A & B u Progressive: Once switch to B, stay at B
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u Complete cascade: everyone gets
"converted"
u Initial adopters are called contagious set
u Depends on
u Initial adopters u Network structure u Threshold q
u Contagious threshold: Max q s.t. there is a finite
contagious set
u Infinite line graph with q = 1/2 u … – A – A – A – B – A – A – A – …
Vs
u … – A – A – B – B – B – A – A – …
Oscillations Complete cascade with contagious threshold = 1/2
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Complete contagion with q = 1/2
u Setting: graph has countably infinite nodes,
with finite maximum degree
equivalent w.r.t. a finite contagious set
any (infinite) graph
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u Granovetter (1978) u Kempe, Kleinberg, Tardos (2003)
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u Actions of node i, xi is 0 for old
behavior A and 1 for new behavior B
u Threshold of each node i, bi
u Assume: bi is chosen randomly
between 0 and 1 u Influence level from j to i, wji
u Assume: wji >= 0 and
u Decision rule
u Adopt B iff
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wji
wjix j ≥ bi
j∈N (i)
j bi i
wji ≤1
j∈N (i)
u The influence on a node is any general
function of its neighbors
u Assume: this function is (weakly) monotonous
u Influence can't decrease as more people adopt B
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j
gi(x N (i)) ≥ bi ?
bi i
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u Starts with the initial adopters of B u Progressive: no switching back to A u Time steps: 1, 2, 3, ...
(same as before)
u Probabilistic model u Dynamics
u For any edge u à v s.t. xu = 1, xv = 0
u u is given one chance to convert v u u's success probability is a function of u, v, and Fv
u Fv is set of neighbors of v that have already tried and
failed u Equivalent to the general threshold model!
(Kempe et al., 2005)
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u u's success probability is a function of
u Independent of other nodes
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u Posed by Domingos and Richardson [2001] u Kleinberg et al’s formulation [2003]: u Input
u Graph instance and cascade model u Integer (budget) k >= 1 u Function f(S), giving the expected number of
nodes converted by initial adopters S. u We want to
u Find a set S of k nodes such that f(S) is maximized
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u NP-hard to find the optimal set S u NP-hard to approximate
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u The function f is submodular
u Shows the property of
diminishing marginal return u Greedy hill-climbing search gives
0.63 approximation (Nemhauser+, 1978)
u Greedily selecting k nodes will lead to
0.63 times the maximum spread u Greedy algorithm:
u For iteration 1 to k:
u Pick the node that increases f the most
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|S| f(S)
Larger set Smalle r set
u Non-probabilistic model, instantiated using data u Something is more general
u Allows switching back and forth u Allows negative influences
(Why is negative influence troublesome in LTM?)
u Threshold values not required to be in [0, 1]
u A model focused on outcome (LTM focuses on process) u Most influential nodes problem should be w.r.t. a
desirable outcome
u Must ensure stable outcomes (LTM allows unstable
initial adopters)
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