Based on “Cascading Behavior in Networks: Algorithmic and Economic Issues” in
Algorithmic Game Theory (Jon Kleinberg, 2007)
and
Ch.16 and 19 of Networks, Crowds, and Markets: Reasoning about a Highly
Connected World (David Easley, Jon Kleinberg, 2010)
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Motivation
Simple Example
Models
Influence Maximization
Similar Work
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What is a network cascade?
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Why do we want to study cascading behavior?
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What are some of the interesting questions to be
raised?
◦ A series of correlated behavior changes
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Social Contexts
Epidemic Disease
Viral Marketing
Covert Organization Exposure
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How can we model a cascade?
What can initiate or terminate a cascade?
What are some properties of cascading behavior?
Can we identify subsets of nodes or edges that have
greater influence in a cascade than others?
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A jar either contains 2 red and 1 blue marble
or 2 blue and 1 red marble
People sequentially come and remove 1
marble and verbally announce which
configuration they believe to be present
(there is an incentive for guessing correctly)
Claim: All guesses beyond the first two are
fixed if they match
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Recall Bayes’ Rule: Pr 𝐴 𝐵 =
Pr A ∗Pr
𝐵𝐴
Pr(𝐵)
Suppose first student draws blue. His guess is blue.
2
(Pr 𝑚𝑎𝑗𝑜𝑟𝑖𝑡𝑦 𝑏𝑙𝑢𝑒 𝑏𝑙𝑢𝑒) = )
3
If the second student draws blue, their choice is trivial. If
it is red, then we have:
1
Pr 𝑚𝑎𝑗𝑜𝑟𝑖𝑡𝑦 𝑏𝑙𝑢𝑒 𝑏𝑙𝑢𝑒, 𝑟𝑒𝑑) = Pr 𝑚𝑎𝑗𝑜𝑟𝑖𝑡𝑦 𝑟𝑒𝑑 𝑏𝑙𝑢𝑒, 𝑟𝑒𝑑) =
2
and they should announce red to break the tie
Third student: suppose first two students drew blue, third
draws red, yet he announces blue
2
◦ Pr 𝑚𝑎𝑗𝑜𝑟𝑖𝑡𝑦 𝑏𝑙𝑢𝑒 𝑏𝑙𝑢𝑒, 𝑏𝑙𝑢𝑒, 𝑟𝑒𝑑) = 3
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All future students have the same information as the third
student and a cascade occurs
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Simple Model
◦ Consider a social network ((𝐺 = 𝑉, 𝐸 such that 𝑉 =
𝑠𝑒𝑡 𝑜𝑓 𝑖𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙𝑠 and 𝑢, 𝑣 ∈ 𝐸 if u and v are
engaged in an activity or friendship
◦ Now consider a situation in which each node has a
choice of behavior – an original behavior, A, or a
new behavior, B with the following incentive given
an edge (v, w) and parameterized by q, 0 ≤ 𝑞 ≤ 1
 If v and w both choose A, they receive a payoff of q
 If v and w both choose B, they receive a payoff of 1-q
 If v and w choose differing behaviors, they receive
nothing
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Let 𝑑𝑣 denote the degree of node v and 𝑑𝑣 𝐴 , 𝑑𝑣 𝐵 denote the
number of neighbors with behavior A and B, respectively
◦ Aside: A node should adopt behavior A if 𝑑𝑣 𝐵 < q𝑑𝑣 and B otherwise
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Each node updates its behavior simultaneously
S - set of nodes adopting behavior B
ℎ𝑞 𝑆 - set of nodes adopting B after 1 update with threshold q
ℎ𝑞𝑘 (𝑆) - set of nodes adopting B after k updates with threshold q
A node w is converted by a set S if for some k, w ∈ ℎ𝑞𝑗 (𝑆) for all j
≥k
A set S is contagious if every node is converted by S
Contagion threshold – the maximum q for which there exists a
finite contagious set (also sometimes called cascade capacity)
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Example
◦ 2-way infinite path
◦ q = ½,
◦ S = {0}
-2
-1
0
1
2
-2
-1
0
1
2
-2
-1
0
1
2
t=0
t=1
t=2
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Example
◦ 2-way infinite path
◦ q = ½,
◦ S = {-1,0,1}
-2
-1
0
1
2
-2
-1
0
1
2
The contagion threshold of this graph is ½:
any set with larger q can never extend!
In fact, we can prove that the maximum
contagion threshold of any graph is ½!
t=0
t=1
Question:
what causes
cascades to
stop?
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Definition: a cluster of density p is a set of
nodes such that each node in the set has at
least a p fraction of its neighbors in the set
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Progressive vs. Non-Progressive
◦ Our prior model was non-progressive – nodes could
change back and forth between states
◦ A progressive model is also interesting – once a
node switches from A to B, it remains B from then
on (consider the behavior of pursuing an advanced
degree)
◦ Intuition: it is easier to find contagious sets with a
progressive model
◦ Actuality: for any graph G, both models have the
same contagion threshold
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Our model thus far is limited
◦ Threshold is uniform for nodes – everyone is just as
predisposed to study algorithms as you are
◦ All neighbors have equal weight – all your facebook
friends are just as important as your immediate
family
◦ Undirected graph – the influence you have on your
boss is the same as he has on you
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We will now introduce several models to
ameliorate these limitations
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Linear threshold model
◦ Goal: allow nodes to weigh influence of their neighbors
differently, and assume that each node’s threshold is
chosen uniformly at random
 Non-negative weights b representing influence
 𝑤∈𝑁(𝑣) 𝑏𝑣𝑤 ≤ 1
 Each node has a threshold θv chosen uniformly at random
from [0,1] indicating the fraction of v’s neighbors that must
adopt the behavior before he does
◦ Definition: a node is activated when it switches from
behavior A to B
 A node becomes active when
𝑎𝑐𝑡𝑖𝑣𝑒 𝑤∈𝑁(𝑣) 𝑏𝑣𝑤
≥ θv
◦ Problem: neighbor influence is strictly additive.
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General threshold model
◦ Each node v now has a function gv defined on subsets of
N(v). ∀𝑋 ⊆ 𝑁 𝑣 , 0 ≤ 𝑔𝑣 𝑋 ≤ 1
◦ Furthermore, if 𝑋 ⊆ 𝑌 𝑡ℎ𝑒𝑛 𝑔𝑣 𝑋 ≤ 𝑔𝑣(𝑌)
◦ A node now becomes active when 𝑔𝑣 X ≥ θ𝑣 , 𝑋 =
𝑖 ∈ 𝑁 𝑣 𝑖 𝑖𝑠 𝑎𝑐𝑡𝑖𝑣𝑒}
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Cascade model
◦ Idea of ‘catching’ behavior from your friends
◦ Probabilistic – whenever an edge (u,v) exists such that u
is active and v is not, u is given a chance to activate v
that depends on u, v, and also the set of nodes that have
already tried and failed to activate v.
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Cascade Model, cont.
◦ Replace the g function from the General Threshold
Model with an incremental function that returns the
probability of success of activating a node v given
initiator u and a set of neighbors X that already
attempted and failed
◦ Provably equivalent to general threshold model in
utility
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Independent Cascade Model
◦ Incremental function is independent of X and
depends only on u andv
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Domingos and Richardson – influential work
that posed the question: if we can convince a
subset of individuals to adopt a new product
with the goal of triggering a cascade of future
adoptions, who should we target?
NP-hard, even for many simple special cases
of the models we’ve discussed
Can construct instances of those models for
which approximation within a factor of n is
NP-hard
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Proof of inapproximability relies on ‘knifeedge’ property
Kempe et al. – Submodularity of the influence
function allows approximation within
1
𝑒
1 − − ε (about 63%)
◦ A function is submodular if for all sets 𝑋 ⊆ 𝑌 and all
elements 𝑣 ∉ 𝑌, 𝑓 X ∪ 𝑣 − 𝑓 𝑋 ≥ 𝑓 𝑌 𝑈 𝑣 − 𝑓(𝑌)
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By identifying instances where the influence
function f is submodular and monotone, we
can make use of the following theorem of
Nemhauser, Wolsey, and Fisher:
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Identifying instances in which we have a
submodular influence function
◦ Any instance of the Cascade Model in which the
incremental functions pv exhibit diminishing returns
has a submodular influence function
◦ Any instance of the Independent Cascade Model has
a submodular influence function
◦ Any instance of the General Threshold Model in
which all the threshold functions gv are submodular
has a submodular influence function
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The anchored k-core problem (Bhawalker et al.)
◦ Model – each user has a cost for maintaining
engagement but derives benefits proportional to
the number of engaged neighbors
◦ A k-core is the maximal induced subgraph with
minimum degree at least k
◦ Anchored k-core – the maximal induced subgraph for which
every unanchored vertex has minimum degree at least k
◦ Corresponds with the problem of preventing cascades of
withdrawals
◦ 𝑂(𝑚 + 𝑛log𝑛) exact solution to the 2-core problem
◦ NP-hard to even approximate within a factor of 𝑂(𝑛1−ε )for k ≥
3
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Cascade scheduling (Chierichetti et al.)
◦ Ordering nodes in a cascade to maximize a
particular outcome
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Identifying failure susceptibility (Blume et al.)
◦ Notion of cascading failure
◦ μ-risk – maximum failure probability of any node in
the graph
◦ What about the structure of the underlying graph
causes it to have high μ-risk?
1.
2.
3.
4.
5.
6.
7.
Lawrence Blume, David Easley, Jon Kleinberg, Robert Kleinberg, and Éva Tardos. 2011.
Which Networks are Least Susceptible to Cascading Failures?. In Proceedings of the 2011
IEEE 52nd Annual Symposium on Foundations of Computer Science (FOCS '11). IEEE
Computer Society, Washington, DC, USA, 393-402.
K. Bhawalkar, J. Kleinberg, K. Lewi, T. Roughgarden, and A. Sharma. Preventing Unraveling
in Social Networks: The Anchored k-Core Problem. In ICALP '12.
Flavio Chierichetti, Jon Kleinberg, Alessandro Panconesi. How to Schedule a Cascade in an
Arbitrary Graph. In Proceedings of EC 2012.
Pedro Domingos and Matt Richardson. Mining the network value of customers. In Proc. 7th
ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pages 57–
66, 2001.
D. Easley, J. Kleinberg. Networks, Crowds, and Markets: Reasoning About a Highly
Connected World. Cambridge University Press, 2010.
David Kempe, Jon Kleinberg, and Eva Tardos. Maximizing the spread of influence in a social
network. In Proc. 9th ACM SIGKDD International Conference on Knowledge Discovery and
Data Mining, pages 137–146, 2003.
J. Kleinberg. Cascading Behavior in Networks: Algorithmic and Economic Issues. In
Algorithmic Game Theory (N. Nisan, T. Roughgarden, E. Tardos, V. Vazirani, eds.),
Cambridge University Press, 2007.