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Predictively Optimal Communities in Dynamic Social Networks
David Darmon
in collaboration with
Erin Uhlfelder, William Rand, and Michelle Girvan31 October 2014
Dynamic Social Networks
▪ The networks are complicated.
Dynamic Social Networks
▪ The networks are complicated.
▪ The individual dynamics are complicated.
Dynamic Social Networks
▪ The networks are complicated.
▪ The individual dynamics are complicated.
▪ The interactions are complicated.
Dynamic Social Networks
Dynamic Social Networks
Dynamic Social Networks
Why Coarsen a Dynamic Network?
▪ Benefits of coarsening:■ Track the activity of collections of
users.
■ Track how collections of users respond to marketing, news events, etc.
■ Track how collections of users interact.
Standard Approachesto Network Coarsening
Coarsen by Structure
▪ Many standard methods coarsen a network based on structural properties of the network.
▪ Ex: Community detection-based methods determine collections of users that are more densely connected inside than outside of a community.
Coarsen by Dynamic Dyadic Relationships
▪ Determine pairwise statistics based on observed dynamics to quantify the interaction between users:
■ Correlation■ Granger causality■ Transfer entropy
▪ Use these statistics to construct a weighted network, and infer communities using theweighted network.
Drawbacks of Standard Approaches
▪ In dynamic social networks, many ‘edges’ are spurious.■ Ex: A single user may follow thousands of
other users, but only interact with a subset of them.
▪ Pairwise dynamical relationships may be very weak.
Coarsen by Predictive Partitioning
▪ Take advantage of both:■ the structure of the network■ the dynamics occurring on the network
to define partitions of the network based on mesoscopic dynamics.
▪ Define these partitions to result in mesoscale view of the network that is predictively optimal.
Sketch ofPredictive Partitioning
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Formulation ofPredictive Partitioning
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Formulation of Predictive Partitioning — Node Dynamics
▪ Consider a network of N nodes.
▪ Let denote the observed activity of node u at time t.
Xu(t)
Xu(t)
Xv(t)
Formulation of Predictive Partitioning — Network Partition
▪ Consider a partition CC of the nodes into G disjoint communities:
C = {C1, C2, . . . , CG}
C1
C2
C3C4
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Formulation of Predictive Partitioning — Community Dynamics
▪ For a community c, define the aggregated activity at time t:
C1C2
C3C4
Ac(t) =X
u2Cc
Xu(t)
= Aggregated activity of nodes in Cc
A2(t)
A4(t)
A1(t)
A3(t)
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Formulation of Predictive Partitioning — Normalization
▪ Let the normalized activity of community c be
A2(t)
A4(t)
C1C2
C3C4
A3(t)
A1(t)
Ac(t) =Ac(t)� sc(t)
�c(t)
=Observed Activity� Seasonal Activity
Seasonal Variability
Formulation of Predictive Partitioning — Predictability
▪ The predictability of the normalized activities are what we wish to optimize by an
appropriate choice of CC.{{Ac(t)}}Cc=1
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A3(t)
A1(t)
Formulation of Predictive Partitioning — Predictability
▪ As a predictability metric, use the redundancy of the normalized activity,
where:
■ is the marginal differential entropy
■ is the differential entropy rate
R⇣Ac
⌘= h
norm
⇣Ac
⌘� h
⇣Ac
⌘
hnorm
⇣Ac
⌘
h⇣Ac
⌘
Formulation of Predictive Partitioning — Predictability
▪ The marginal differential entropy measures how unpredictable the normalized activity is when viewed as Gaussian white noise, ignoring any memory:
▪ The differential entropy rate measures how unpredictable the normalized activity is once we have accounted for any memory in the activity:
h⇣Ac
⌘= h
✓Ac(t)
����Ac(t� 1), Ac(t� 2), . . .
◆
hnorm
⇣Ac
⌘
Formulation of Predictive Partitioning — Predictability
▪ Thus, the redundancy measures how much more predictable the activity is compared to white noise.
R⇣Ac
⌘= h
norm
⇣Ac
⌘� h
⇣Ac
⌘
Formulation of Predictive Partitioning — Predictability
▪ For the partition CC, define the overall redundancy of its induced dynamics by
R(C) =X
c
R⇣Ac
⌘.
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A~c(t+1)
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−6 −4 −2 0 2 4 6
−6
−4
−2
02
46
A~c(t)
A~c(t+1)
R(C) > R(C0)
C C0
Demonstration withSynthetic Data
Synthetic Data — Network
▪ Directed Erdös-Rényi network withN = 160 nodesp = 0.05 connection probability
User j
Use
r i
1
2
3
4
◆2!1
◆1!4◆1!3
◆3!4
1
2
3
4
◆2!1
◆1!4◆1!3
◆3!4
Synthetic Data — Dynamics
▪ Bass-like Poissonian dynamics:
▪ i.e. The expected activity of user u at time t is their background rate plus the influence from their inputs.
Xu(t)|X(t� 1) ⇠ Poisson(�u(t))
where
r(t) =A
2(sin(!t) + 1)
r(t)
�u(t) = r(t) +X
v2Pa(u)
◆v!uXv(t� 1).
1
2
3
4
◆2!1
◆1!4◆1!3
◆3!4
Synthetic Data — Dynamics
▪ Example:
where
X3(t)|X(t� 1) = X3(t)|{X1(t� 1)}⇠ Poisson(�3(t))
�3(t) = r(t) + ◆1!3X1(t� 1).
1
2
3
4
◆2!1
◆1!4◆1!3
◆3!4
Synthetic Data — Dynamical Communities
▪ Assume 4 dynamical communities, each of 40 users.
▪ Set influence inside of a dynamical community to be ten times the influence outside of a dynamical community:
◆inside
= 10⇥ ◆outside
.
0 50 100 150
050
100
150
200
250
Time
Com
mun
ity A
ctiv
ity
0 50 100 150
050
100
150
200
250
Time
Com
mun
ity A
ctiv
ity
0 50 100 150
050
100
150
200
250
Time
Com
mun
ity A
ctiv
ity
0 50 100 150
050
100
150
200
250
Time
Com
mun
ity A
ctiv
ity
0 50 100 150
050
100
150
200
250
Time
Com
mun
ity A
ctiv
ity
0 50 100 150
050
100
150
200
250
Time
Com
mun
ity A
ctiv
ity
0 50 100 150
050
100
150
200
250
Time
Com
mun
ity A
ctiv
ity
0 50 100 150
050
100
150
200
250
Time
Com
mun
ity A
ctiv
ity
vs.
Synthetic Data — Experiment
▪ Partition the network into
2, 4, 10, 16, 20, 32, 40
communities.
▪ Use both non-randomized and randomized partitions.
▪ Example: 40 communities of size 4.
Synthetic Data — Experiment
...
...
Dynamical Community 1
Dynamical Community 2
Dynamical Community 3
Dynamical Community 4
Non-randomized:
Randomized:
Synthetic Data — Results
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Non-randomizedPartition
Randomized Partition
Demonstration withData from Twitter
Twitter Data — The Dataset
▪ Network: 15K network of Twitter users collected via a breadth first search of followers / friends.
▪ Dynamics: Individual statuses collected over7-week period in Summer 2014.
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Twitter Data — Comparison of Community Partitions
▪ Use the partitions inferred by popular community detection algorithms:
■ Fast-Greedy Algorithm for Modularity Maximization■ Louvain Algorithm for Modularity Maximization■ Newman’s Spectral Method
Method Total Redundancy
Fast-Greedy 0.671
Louvain 1.050
Spectral 0.793
Total Network 0.203
Conclusions
Conclusions
▪ The networks are complicated.
▪ The individual dynamics are complicated.
▪ The interactions are complicated.
Conclusions
▪ By taking advantage of both:■ the structure of the network■ the dynamics occurring on the network
we can define partitions of the network based on its mesoscopic dynamics.
▪ By optimizing the predictability of the mesoscopic dynamics, we can detect communities ‘hidden’ from more standard community detection algorithms.
Future Work
▪ Formulate an optimization problem to find predictively optimal communities.
▪ Possible approaches:■ Greedy optimization based on merging of
communities.■ Take advantage of multilevel representations of
structural communities to determine the ‘right’ level from a predictive viewpoint.
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Time (Hours)
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easo
naliz
ed A
ctiv
ity L
evel
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{Ac(t)}Tt=1 {Ac(t)� sc(t)}Tt=1
0 50 100 150
−10
00
100
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300
Time (Hours)
Des
easo
naliz
ed A
ctiv
ity L
evel
per
Hou
r
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{Ac(t)� sc(t)}Tt=1
0 50 100 150
−3
−2
−1
01
23
Time (Hours)
Nor
mal
ized
Act
ivity
Lev
elpe
r H
our
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{Ac(t)}Tt=1 {Ac(t)� sc(t)}Tt=1
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t=1
