Predictively Optimal Communities in
Dynamic Social Networks
David Darmon
in collaboration with
Erin Uhlfelder, William Rand, and Michelle Girvan
31 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 Approaches
to 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 the
weighted 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 of
Predictive Partitioning
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Formulation of
Predictive Partitioning
Formulation of Predictive Partitioning — Node Dynamics
▪ Consider a network of N nodes.
Individual Activity
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▪ Let Xu (t) denote the observed activity of node u at
time t.
0
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Xu (t)
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Individual Activity
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Xv (t)
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Formulation of Predictive Partitioning — Network Partition
▪ Consider a partition C of the nodes into
G disjoint communities:
C = {C1 , C2 , . . . , CG }
C1
C2
C3
C4
Formulation of Predictive Partitioning — Community Dynamics
▪ For a community c, define the aggregated activity at
time t:
X
Ac (t) =
Xu (t)
u2Cc
C2
Community Activity
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Community Activity
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C1
A2 (t)
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A1 (t)
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= Aggregated activity of nodes in Cc
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Time
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C4
Community Activity
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Community Activity
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C3
A4 (t)
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A3 (t)
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Formulation of Predictive Partitioning — Normalization
▪ Let the normalized activity of community c be
Ã3 (t)
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C3
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Ã2 (t)
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C2
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C4
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Normalized Community Activity
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Ã1 (t)
C1
Normalized Community Activity
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sc (t)
Ãc (t) =
c (t)
Observed Activity Seasonal Activity
=
Seasonal Variability
Normalized Community Activity
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Ac (t)
Ã4 (t)
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Formulation of Predictive Partitioning — Predictability
Ã3 (t)
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C3
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Ã2 (t)
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C2
Normalized Community Activity
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C1
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C4
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Normalized Community Activity
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Ã1 (t)
Normalized Community Activity
−3 −2 −1
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▪ The predictability of the normalized activities
{{Ãc (t)}}C
c=1 are what we wish to optimize by an
appropriate choice of C.
Ã4 (t)
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Formulation of Predictive Partitioning — Predictability
▪ As a predictability metric, use the redundancy of the
normalized activity,
⇣
⌘
⇣
R Ãc = hnorm Ãc
⌘
⇣
h̄ Ãc
⌘
where:
⇣
⌘
■ hnorm Ãc is the marginal differential entropy
⇣ ⌘
■ h̄ Ãc is the differential entropy rate
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:
⇣
hnorm Ãc
⌘
▪ The differential entropy rate measures how
unpredictable the normalized activity is once we
have accounted for any memory in the activity:
⇣
⌘
✓
h̄ Ãc = h Ãc (t) Ãc (t
1), Ãc (t
2), . . .
◆
Formulation of Predictive Partitioning — Predictability
▪ Thus, the redundancy measures how much more
predictable the activity is compared to white noise.
⇣
⌘
⇣
R Ãc = hnorm Ãc
⌘
⇣
h̄ Ãc
⌘
Formulation of Predictive Partitioning — Predictability
▪ For the partition C , define the overall redundancy of
its induced dynamics by
R(C) =
X
c
⇣
⌘
R Ãc .
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~c (t + 1)
A
0
2
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−2
~c (t + 1)
A
0
2
−2
−4
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4
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−6
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0
~c (t )
A
2
4
−6
6
0
R(C) > R(C )
−4
−2
0
~c (t )
A
2
4
6
Demonstration with
Synthetic Data
Synthetic Data — Network
User i
▪ Directed Erdös-Rényi network with
N = 160 nodes
p = 0.05 connection probability
User j
1
◆2!1
◆1!4 2
◆1!3
3
◆3!4
4
Synthetic Data — Dynamics
▪ Bass-like Poissonian dynamics:
Xu (t)|X(t
1) ⇠ Poisson(
u (t))
where
X
◆v!u Xv (t
u (t) = r(t) +
1).
v2Pa(u)
▪ i.e. The expected activity of user u at
time t is their background rate r(t)
plus the influence from their inputs.
1
◆2!1
◆1!4 2
◆1!3
3
◆3!4
4
A
r(t) = (sin(!t) + 1)
2
Synthetic Data — Dynamics
▪ Example:
X3 (t)|X(t
1) = X3 (t)|{X1 (t
⇠ Poisson(
1)}
3 (t))
where
3 (t)
= r(t) + ◆1!3 X1 (t
1).
1
◆2!1
◆1!3
◆1!4
3
2
◆3!4
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 .
1
◆2!1
◆1!3
◆1!4
3
2
◆3!4
4
0
Community Activity
50
100
150
200
Community Activity
50
100
150
200
250
250
0
0
Community Activity
50
100
150
200
250
250
50
50
50
0
0
Community Activity
50
100
150
200
Community Activity
50
100
150
200
250
250
100
Time
100
Time
150
100
Time
150
100
Time
150
0
Community Activity
50
100
150
200
0
50
0
0
0
150
vs.
0
50
100
Time
150
0
50
100
Time
150
0
50
100
Time
150
0
50
100
Time
150
0
0
Community Activity
50
100
150
200
Community Activity
50
100
150
200
250
250
Synthetic Data — Experiment
▪ Partition the network into
2, 4, 10, 16, 20, 32, 40
communities.
▪ Use both non-randomized and randomized partitions.
Dynamical Community 1
Synthetic Data — Experiment
▪ Example: 40 communities of size 4.
Non-randomized:
...
Randomized:
...
Dynamical Community 2
Dynamical Community 3
Dynamical Community 4
1.4
Synthetic Data — Results
●
Randomized
Partition
●
●
●
0.8
1.0
●
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●
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0.6
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0.4
Total Redundancy
1.2
Non-randomized
Partition
●
●
●
●●
●●
●
●
● ●
2
4
10
16
20
32
Number of Communities
40
Demonstration with
Data 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 over
7-week period in Summer 2014.
0
Activity Level per Hour
100 200 300 400 500
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50
100
Time (Hours)
150
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
Fast-Greedy
Louvain
Spectral
Total Network
Total Redundancy
0.671
1.050
0.793
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.
Thank you.
ddarmon@math.umd.edu
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.
150
1.0
User i
100
0.8
0.6
50
0.4
0.2
0.0
50
100
User j
150
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Time (Hours)
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3
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per Hour
−2 −1
0
1
2
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50
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Time (Hours)
{Ac (t)
ŝc (t)}Tt=1
150
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100
200
300
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50
100
Time (Hours)
Ac (t) ŝc (t)
ˆc (t)
T
t=1
150
●
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per Hour
−100
0
100
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{Ac (t)}Tt=1
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50
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150
ŝc (t)}Tt=1
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−3
0
Normalized Activity Level
per Hour
−2 −1
0
1
2
0
Activity Level per Hour
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100
Time (Hours)
150
⇢
Ac (t) ŝc (t)
ˆc (t)
T
t=1