Diffusion Over Dynamic Networks

1
Diffusion Over Dynamic Networks (plus some social
network intro since Im first) NetSci
Workshop May 16, 2006 James Moody
This work supported by the Network Modeling
Project through the University of Washington NIH
grants DA12831 and HD41877
2
Introduction
We live in a connected world
To speak of social life is to speak of the
association between people their associating in
work and in play, in love and in war, to trade or
to worship, to help or to hinder. It is in the
social relations men establish that their
interests find expression and their desires
become realized. Peter M. Blau Exchange and
Power in Social Life, 1964
3
Introduction
We live in a connected world
"If we ever get to the point of charting a whole
city or a whole nation, we would have a picture
of a vast solar system of intangible structures,
powerfully influencing conduct, as gravitation
does in space. Such an invisible structure
underlies society and has its influence in
determining the conduct of society as a
whole." J.L. Moreno, New York Times, April 13,
1933
These patterns of connection form a social space,
that can be seen in multiple contexts
4
Introduction
Source Linton Freeman See you in the funny
pages Connections, 23, 2000, 32-42.
5
Introduction
High Schools as Networks
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8
Introduction
And yet, standard social science analysis methods
do not take this space into account. For the
last thirty years, empirical social research has
been dominated by the sample survey. But as
usually practiced, , the survey is a
sociological meat grinder, tearing the individual
from his social context and guaranteeing that
nobody in the study interacts with anyone else in
it. Allen Barton, 1968 (Quoted in Freeman
2004) Moreover, the complexity of the relational
world makes it impossible to identify social
connectivity using only our intuitive
understanding. Social Network Analysis (SNA)
provides a set of tools to empirically extend our
theoretical intuition of the patterns that
construct social structure.
9
Introduction
Why do Networks Matter?
Local vision
10
Introduction
Why do Networks Matter?
Local vision
11
Introduction

• Why networks matter
• Intuitive goods travel through contacts
  between actors, which can reflect a power
  distribution or influence attitudes and
  behaviors. Our understanding of social life
  improves if we account for this social space.
• Less intuitive patterns of inter-actor contact
  can have effects on the spread of goods or
  power dynamics that could not be seen focusing
  only on individual behavior.

12
Introduction

• Social network analysis is
• a set of relational methods for systematically
  understanding and identifying connections among
  actors. SNA is
• is motivated by a structural intuition based on
  ties linking social actors
• is grounded in systematic empirical data
• draws heavily on graphic imagery
• relies on the use of mathematical and/or
  computational models. (Freeman, 2004)
• Social Network Analysis embodies a range of
  theories relating types of observable social
  spaces.

13

• Introduction
• Social Network Basics
• Basic data Elements
• Basic data structures
• Network Analysis Buffet
• Networks Diffusion
• Structural constraints on network diffusion
• Reachability
• Distance
• Connectivity
• Closeness centrality
• Temporal Constraints on network diffusion
• Defining dynamic networks
• How order constrains flow
• Reachability variance w. constant structure
• Minimum temporal reachability
• New time-dependent network measures
• Graph-level measures
• Node-level measures

14
Social Network Data Elements

• Social Network data consists of two linked
  classes of data
• Information on the individuals (aka actors,
  nodes, points)
• Network nodes are most often people, but can be
  any other unit capable of being linked to another
  (schools, countries, organizations,
  personalities, etc.)
• The information about nodes is what we usually
  collect in standard social science research
  demographics, attitudes, behaviors, etc.
• Includes the times when the node is active
• b) Information on relations among individuals
  (lines, edges, arcs)
• Records a connection between the nodes in the
  network
• Can be valued, directed (arcs), binary or
  undirected (edges)
• One-mode (direct ties between actors) or two-mode
  (actors share membership in an organization)
• Includes the times when the relation is active

15
Social Network Data Elements
The unit of interest in a network are the
combined sets of actors and their relations. We
represent actors with points and relations with
lines. Actors are referred to variously
as Nodes, vertices or points Relations are
referred to variously as Edges, Arcs, Lines,
Ties
Example
b
d
a
c
e
16
Social Network Data Elements
In general, a relation can be Binary or
Valued Directed or Undirected
Directed, binary
Undirected, binary
b
d
1
2
1
3
4
a
c
e
Directed, Valued
Undirected, Valued
17
Social Network Data Elements
Social network data are substantively divided by
the number of modes in the data. 1-mode data
represents edges based on direct contact between
actors in the network. All the nodes are of the
same type (people, organization, ideas, etc).
Examples Communication, friendship, giving
orders, sending email. 1-mode data are usually
singly reported (each person reports on their
friends), but you can use multiple-informant
data, which is more common in child development
research (Cairns and Cairns).
18
Social Network Data Elements
Social network data are substantively divided by
the number of modes in the data. 2-mode data
represents nodes from two separate classes, where
all ties are across classes. Examples People
as members of groups People as authors on
papers Words used often by people Events in the
life history of people The two modes of the data
represent a duality you can project the data as
people connected to people through joint
membership in a group, or groups to each other
through common membership There may be multiple
relations of multiple types connecting nodes in
any given substantive setting.
19
Social Network Data Elements
Levels of analysis
Global-Net
Ego-Net
Partial-Network
20
Social Network Data Elements
We can examine networks across multiple levels
1) Ego-network - Have data on a respondent (ego)
and the people they are connected to (alters).
Example 1985 GSS module - May include estimates
of connections among alters
2) Partial network - Ego networks plus some
amount of tracing to reach contacts of contacts
- Something less than full account of
connections among all pairs of actors in the
relevant population - Example CDC Contact
tracing data for STDs
21
Social Network Data Elements
We can examine networks across multiple levels

• 3) Complete or Global data
• - Data on all actors within a particular
  (relevant) boundary
• - Never exactly complete (due to missing data),
  but boundaries are set
• Example Coauthorship data among all writers in
  the social sciences, friendships among all
  students in a classroom
• For the most part, I will be discussing issues
  surrounding global networks.

22
Social Network Data Structures
Visualization
A good network drawing allows viewers to come
away from the image with an almost immediate
intuition about the underlying structure of the
network being displayed. However, because there
are multiple ways to display the same
information, and standards for doing so are few,
the information content of a network display can
be quite variable.
Each of these images represents the exact same
graph information.
23
Social Network Data Structures
Visualization
Network visualization helps build intuition, but
you have to keep the drawing algorithm in mind.
Again, the same graph with two different
techniques
Spring embedder layouts
Tree-Based layouts
(Fair - poor)
(good)
Most effective for very sparse, regular graphs.
Very useful when relations are strongly directed,
such as organization charts.
Most effective with graphs that have a strong
community structure (clustering, etc). Provides
a very clear correspondence between social
distance and plotted distance
Two images of the same network
24
Social Network Data Structures
Visualization
Another example
Spring embedder layouts
Tree-Based layouts
(poor)
(good)
Two layouts of the same network
25
Social Network Data Structures
Visualization
Network visualization helps build intuition, but
you have to keep the drawing algorithm in
mind. Hierarchy Tree models Use optimization
routines to add meaning to the vertical dimension
of the plot. This makes it possible to easily
see who is most central by who is on the top of
the figure. These also include some routine for
minimizing line-crossing. Spring Embedder
layouts Work on an analogy to a physical system
ties connecting a pair have springs that pull
them together. Unconnected nodes have springs
that push them apart. The resulting image
reflects the balance of these two forces. This
usually creates a layout with a close
correspondence between physical closeness and
network distance. In the next slides we give
examples of successful graph layouts
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Social Network Data Structures
Visualization
A spring embedder layout of romantic relations
in a single high school. This image works
because the sparse nature of the graph allows you
to easily trace all of the connections without
any line crossings.
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Social Network Data Structures
Visualization
Using colors to code attributes makes it simpler
to compare attributes and relations. This plot
compares the effectiveness of two different
clustering routines on a school friendship
network. Because the spring-embedder model pulls
communities close, we would expect cohesive
groups to be in the same region of the graph.
This is what we see in the RNM solution at the
bottom.
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Social Network Data Structures
Visualization
29
Social Network Data Structures
30
Social Network Data Structures
31
Social Network Data Structures
Visualization
As networks increase in size, the effectiveness
of a point-and-line display routines diminishes,
because you simply run out of plotting
space. You can still get some insight by using
the overlap that results in from a space-based
layout as information. Here we plot a very large
and dense network (the standard point-and-line
image is in the upper right).
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Social Network Data Structures
Visualization
Adding time to social networks is also
complicated, as you run out of space to put time
in most network figures. One solution is to
animate the network. Here we see streaming
interaction in a classroom, where the teacher
(yellow square) has trouble maintaining
order. The SONIA software program (McFarland and
Bender-deMoll) will produce these figures.
33
Social Network Data Structures
Data Representations
Pictures only take us so far from pictures to
adjacency matrices
Undirected, binary
Directed, binary
34
Social Network Data Structures
Data Representations
From matrices to lists
Arc List
Adjacency List
a b b a b c c b c d c e d c d e e c e d
35
Social Networks Diffusion
Goods flow through networks
36
Social Networks Diffusion

• In addition to the dyadic probability that one
  actor passes something to another (pij), two
  factors affect flow through a network
• Topology
• the shape, or form, of the network
• - Example one actor cannot pass information to
  another unless they are either directly or
  indirectly connected
• Time
• - the timing of contact matters
• - Example an actor cannot pass information he
  has not receive yet

37
Social Networks Diffusion
Three features of the networks topology are
known to be important Reachability, Distance
Number of Paths (redundancy)

• Connectivity refers to how actors in one part of
  the network are connected to actors in another
  part of the network.
• Reachability Is it possible for actor i to
  reach actor j? This can only be true if there is
  a chain of contact from one actor to another.
• Distance Given they can be reached, how many
  steps are they from each other?
• How efficiently do ties reach new nodes? (How
  clustered is the network)
• Number of paths How many different paths
  connect each pair?

38
Social Networks Diffusion
Without full network data, you cant distinguish
actors with limited diffusion potential from
those more deeply embedded in a setting.
c
b
a
39
Social Networks Diffusion
Reachability

• Given that ego can reach alter, distance
  determines the likelihood of information passing
  from one end of the chain to another.
• Because flow is rarely certain, the probability
  of transfer decreases over distance.
• However, the probability of transfer increases
  with each alternative path connecting pairs of
  people in the network.

40
Social Networks Diffusion
Reachability
Indirect connections are what make networks
systems. One actor can reach another if there is
a path in the graph connecting them.
a
b
d
a
c
e
f
Paths can be directed, leading to a distinction
between strong and weak components
41
Social Networks Diffusion
Reachability

• Basic elements in connectivity
• A path is a sequence of nodes and edges starting
  with one node and ending with another, tracing
  the indirect connection between the two. On a
  path, you never go backwards or revisit the same
  node twice.
• Example a ? b ? c?d
• A walk is any sequence of nodes and edges, and
  may go backwards. Example a ? b ? c ? b ?c ?d
• A cycle is a path that starts and ends with the
  same node. Example a ? b ? c ? a

42
Social Networks Diffusion
Reachability
Reachability
If you can trace a sequence of relations from one
actor to another, then the two are reachable. If
there is at least one path connecting every pair
of actors in the graph, the graph is connected
and is called a component. Intuitively, a
component is the set of people who are all
connected by a chain of relations.
43
Social Networks Diffusion
Reachability
This example contains many components.
44
Social Networks Diffusion
Reachability
In general, components can be directed or
undirected. For a graph with any directed edges,
there are two types of components Strong
components consist of the set(s) of all nodes
that are mutually reachable Weak components
consist of the set(s) of all nodes where at least
one node can reach the other.
45
Social Networks Diffusion
Distance number of paths
Distance is measured by the (weighted) number of
relations separating a pair
Actor a is 1 step from 4 2 steps from 5
3 steps from 4 4 steps from 3 5 steps from 1
a
46
Social Networks Diffusion
Distance number of paths
Paths are the different routes one can take.
Node-independent paths are particularly important.
There are 2 independent paths connecting a and b.
b
There are many non-independent paths
a
47
Measuring Networks Large-Scale Models
Social Cohesion
White, D. R. and F. Harary. 2001. "The
Cohesiveness of Blocks in Social Networks Node
Connectivity and Conditional Density."
Sociological Methodology 31305-59. Moody,
James and Douglas R. White. 2003. Structural
Cohesion and Embeddedness A hierarchical
Conception of Social Groups American
Sociological Review 68103-127 White, Douglas
R., Jason Owen-Smith, James Moody, Walter W.
Powell (2004) "Networks, Fields, and
Organizations Scale, Topology and Cohesive
Embeddings."  Computational and Mathematical
Organization Theory. 1095-117 Moody, James "The
Structure of a Social Science Collaboration
Network Disciplinary Cohesion from 1963 to 1999"
American Sociological Review. 69213-238

48
Measuring Networks Large-Scale Models
Social Cohesion

• Networks are structurally cohesive if they remain
  connected even when nodes are removed. Each of
  these graphs have the exact same density.

0
1
2
3
Node Connectivity
49
Measuring Networks Large-Scale Models
Social Cohesion

• Formal definition of Structural Cohesion
• A groups structural cohesion is equal to the
  minimum number of actors who, if removed from the
  group, would disconnect the group.
• Equivalently (by Mengers Theorem)
• A groups structural cohesion is equal to the
  minimum number of node-independent paths linking
  each pair of actors in the group.

50
Measuring Networks Large-Scale Models
Social Cohesion
Structural cohesion gives rise automatically to a
clear notion of embeddedness, since cohesive
sets nest inside of each other.
2
3
1
9
10
8
4
11
7
5
12
13
6
14
15
17
16
18
19
20
2
22
23
51
Measuring Networks Large-Scale Models
Social Cohesion
Project 90, Sex-only network (n695)
3-Component (n58)
52
Measuring Networks Large-Scale Models
Social Cohesion
Connected Bicomponents
IV Drug Sharing Largest BC 247 k gt 4 318 Max k
12 Structural Cohesion simultaneously gives us a
positional and subgroup analysis.
53
Social Networks Diffusion
Distance number of paths
Probability of transfer
by distance and number of paths, assume a
constant pij of 0.6
54
Social Networks Diffusion
Clustering and diffusion
Arcs 11 Largest component 12, Clustering 0
Arcs 11 Largest component 8, Clustering 0.205
Clustering turns network paths back on already
identified nodes. This has been well known since
at least Rappaport, and is a key feature of the
Biased Network models in sociology.
55
Social Networks Diffusion
Diffusion features on static graphs
56
Social Networks Diffusion
Example on static graphs
57
Social Networks Diffusion
Example on static graphs
Define as a general measure of the diffusion
susceptibility of a graph as the ratio of the
area under the observed curve to the area under
the random curve. As this gets smaller than 1.0,
you get effectively slower median transmission.
58
Social Networks Diffusion
Example on static graphs
59
Social Networks Diffusion
Example on static graphs
60
Social Networks Diffusion
Centrality

• Centrality refers to (one dimension of) location,
  identifying where an actor resides in a network.
  
• For example, we can compare actors at the edge
  of the network to actors at the center.
• In general, this is a way to formalize intuitive
  notions about the distinction between insiders
  and outsiders.

61
Social Networks Diffusion
Centrality

• At the individual level, one dimension of
  position in the network can be captured through
  centrality.
• Conceptually, centrality is fairly straight
  forward we want to identify which nodes are in
  the center of the network. In practice,
  identifying exactly what we mean by center is
  somewhat complicated, but substantively we often
  have reason to believe that people at the center
  are very important.
• Three standard centrality measures capture a wide
  range of importance in a network
• Degree
• Closeness
• Betweenness

62
Social Networks Diffusion
Centrality
A common measure of centrality is closeness
centrality. An actor is considered important if
he/she is relatively close to all other actors.
Closeness is based on the inverse of the distance
of each actor to every other actor in the network.
Closeness Centrality
Normalized Closeness Centrality
63
Social Networks Diffusion
Closeness Centrality in 4 examples
Centrality
C0.0
C1.0
C0.36
C0.28
64
Measuring Networks Flow
Time
Two factors that affect network
flows Topology - the shape, or form, of the
network - simple example one actor cannot pass
information to another unless they are either
directly or indirectly connected Time - the
timing of contacts matters - simple example an
actor cannot pass information he has not yet
received.
65
Measuring Networks Flow
Time
Timing in networks

• A focus on contact structure has often slighted
  the importance of network dynamics,though a
  number of recent pieces are addressing this.
• Time affects networks in two important ways
• The structure itself evolves, in ways that will
  affect the topology an thus flow.
• 2) The timing of contact constrains information
  flow

66
Measuring Networks Flow
Time
Drug Relations, Colorado Springs, Year 1
Data on drug users in Colorado Springs, over 5
years
67
Measuring Networks Flow
Time
Drug Relations, Colorado Springs, Year 2 Current
year in red, past relations in gray
68
Measuring Networks Flow
Time
Drug Relations, Colorado Springs, Year 3 Current
year in red, past relations in gray
69
Measuring Networks Flow
Time
Drug Relations, Colorado Springs, Year 4 Current
year in red, past relations in gray
70
Measuring Networks Flow
Time
Drug Relations, Colorado Springs, Year 5 Current
year in red, past relations in gray
71
When is a network?
Source Bender-deMoll McFarland The Art and
Science of Dynamic Network Visualization JoSS
Forthcoming
72
When is a network?

• At the finest levels of aggregation networks
  disappear, but at the higher levels of
  aggregation we mistake momentary events as
  long-lasting structure.
• Is there a principled way to analyze and
  visualize networks where the edges are not
  stable?
• There is unlikely to be a single answer for all
  questions, but the set of types of questions
  might be manageable
• Diffusion and flow (networks as resources or
  constraints for actors)
• The timing of relations affects flow in a way
  that changes many of our standard measures. If
  our interest is in Relational ties as
  channels for transfer or flow of resources (WF
  p.4), then we can use the diffusion process to
  shape our analyses.
• Structural change (networks as dynamic objects of
  study).
• The interest is in mapping changes in the
  topography of the network, to see model how the
  field itself changes over time.
• Ultimately, this has to be linked to questions
  about how network macro-structures emerge as the
  result of actor behavior rules.

73
Network Dynamics Flow
The key element that makes a network a system is
the path its how sets of actors are linked
together indirectly. A walk is a sequence of
nodes and lines, starting and ending with nodes,
in which each node is incident with the lines
following and preceding it in a sequence. A path
is a walk where all of the nodes and lines are
distinct. Paths are the routes through networks
that make diffusion possible. In a dynamic
network, the timing of edges affect whether a
good can flow across a path. A good cannot pass
along a relation that ends prior to the actor
receiving the good goods can only flow forward
in time. A time-ordered path exists between i
and j if a graph-path from i to j can be
identified where the starting time for each edge
step precedes the ending time for the next
edge. The notion of a time-ordered path must
change our understanding of the system structure
of the network. Networks exist both in
relation-space and time-space.
74
Network Dynamics Flow
A time-ordered path exists between i and j if a
graph-path from i to j can be identified where
the starting time for each edge step precedes the
ending time for the next edge. Note that this
allows for non-intuitive non-transitivity.
Consider this simple example Here A can
reach B, B can reach C, and C and reach D. But A
cannot reach D, since any flow from A to C would
have happened after the relation between C and D
ended.
1 - 2
3 - 4
1 - 2
A
B
C
D
75
Network Dynamics Flow
This can also introduce a new dimension for
shortest paths
3 - 4
B
C
5 - 6
1 - 2
A
D
5 - 6
7 - 9
E
The geodesic from A to D is AE, ED and is two
steps long. But the fastest path would be AB,
BC, CD, which while 3 steps long could get there
by day 5 compared to day 7.
76
Network Dynamics Flow
Reachability
Direct Contact Network of 8 people in a ring
77
Network Dynamics Flow
Reachability
Implied Contact Network of 8 people in a ring All
relations Concurrent
78
Network Dynamics Flow
Reachability
2
3
2
1
1
2
2
3
0.57 reachability
Implied Contact Network of 8 people in a
ring Mixed Concurrent
79
Network Dynamics Flow
Reachability
1
8
2
7
3
6
5
4
0.71 reachability
Implied Contact Network of 8 people in a
ring Serial Monogamy (1)
80
Network Dynamics Flow
Reachability
1
8
2
7
3
6
1
4
0.51 reachability
Implied Contact Network of 8 people in a
ring Serial Monogamy (2)
81
Network Dynamics Flow
1
2
2
1
1
2
0.43 reachability
1
2
Which is the minimum possible reachability given
the contact structure.
Minimum Contact Network of 8 people in a
ring Serial Monogamy (3)
82
Identifying the Minimum Path Density of a Graph
A 2-regular graph
line
cycle
83
Identifying the Minimum Path Density of a Graph
A 3-regular spanning tree
l 7g
84
Identifying the Minimum Path Density of a Graph
A 3-regular grid
Each person can reach 4 people indirectly.,
leading again to 7g total arcs per person.
85
Identifying the Minimum Path Density of a Graph
A 3-regular linked clusters
If you count self-loops, one still hits 7l
overall.
86
Reachability as a function of relationship
adjacency
Identified paths
For a regular graph with d(?)T
t1 t2
t1 t3
t2 t3
t1 t2 t3
I think its an open question to define a minimum
reachability graph for non-regular structures.
87
Network Dynamics Flow
In this graph, timing alone can change mean
reachability from 2.0 when all ties are
concurrent to 0.43 a factor of 4.7. In
general, ignoring time order is equivalent to
assuming all relations occur simultaneously
assumes perfect concordance across all relations.
1
2
2
1
1
2
1
2
88
Network Dynamics Flow

• At the graph level, we are interested in two
  properties immediately
• the temporal-implied reachability (perhaps
  relative to minimum)
• b) The asymmetry in reachability. What proportion
  of reachable dyads can mutually reach each other?
• These are directly relevant for overall diffusion
  potential in a network.

1
2
2
1
1
2
1
2
89
Alternative measures
Relative Reach
Conditional Reachability (Harary, 1983)
90
Network Dynamics Flow
The distribution of paths is important for many
of the measures we typically construct on
networks, and these will be change if timing is
taken into consideration Centrality Closeness
centrality Path Centrality Information
Centrality Betweenness centrality Network
Topography Clustering Path Distance Groups
Roles Correspondence between degree-based
position and reach-based position Structural
Cohesion Embeddedness Opportunities for
Time-based block-models (similar reachability
profiles) In general, any measures that take
the systems nature of the graph into account will
differ in a dynamic graph from a static graph.
91
Network Dynamics Flow

• New versions of classic reachability measures
• Temporal reach The ij cell 1 if i can reach j
  through time.
• Temporal geodesic The ij cell equals the number
  of steps in the shortest path linking i to j over
  time.
• Temporal cohesion The ij cell equals the number
  of time-ordered node-independent paths linking i
  to j.
• These will only equal the standard versions when
  all ties are concurrent.
• Duration explicit measures
• 4) Quickest path The ij cell equals the
  shortest time within which i could reach j.
• 5) Earliest path The ij cell equals the
  real-clock time when i could first reach j.
• 6) Latest path The ij cell equals the
  real-clock time when i could last reach j.
• 7) Exposure duration The ij cell equals
  the longest (shortest) interval of time over
  which i could transfer a good to j.
• Each of these also imply different types of
  betweenness roles for nodes or edges, such as a
  limiting time edge, which would be the edge
  whose comparatively short duration places the
  greatest limits on other paths.

92
Network Dynamics Flow
Define time-dependent closeness as the inverse of
the sum of the distances needed for an actor to
reach others in the network.
Actors with high time-dependent closeness
centrality are those that can reach others in few
steps given temporal order. Note this is
directed. Since Dij / Dji (in most cases)
once you take time into account.
If i cannot reach j, I set the distance to n1
93
Network Dynamics Flow
Timing affects the symmetry of a symmetric
contact graph.
8 - 9
C
E
3 - 7
2 - 5
B
A
0 - 1
3 - 5
D
F
Numbers above lines indicate contact periods
94
Network Dynamics Flow
Timing affects the symmetry of a symmetric
contact graph.
E
C
B
A
D
F
95
Network Dynamics Flow
Define fastness centrality as the average of the
clock-time needed for an actor to reach others in
the network
Actors with high fastness centrality are those
that would reach the most people early. These
are likely important for any first mover
problem.
96
Network Dynamics Flow
Define quickness centrality as the average of the
minimum amount of time needed for an actor to
reach others in the network
Where Tjit is the time that j receives the good
sent by i at time t, and Tit is the time that i
sent the good. This then represents the shortest
duration between transmission and receipt between
i and j. Note that this is a time-dependent
feature, depending on when i transmits the good
out into the population. The min is one of many
functions, since the time-to-target speed is
really a profile over the duration of t.
97
Network Dynamics Flow
Define exposure centrality as the average of the
amount of time that actor j is at risk to a good
introduced by actor i.
Where Tijl is the last time that j could receive
the good from i and Tiif is the first time that j
could receive the good from i, so the difference
is the interval in time when i is at risk from j.
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Network Dynamics Flow
How do these centrality scores compare to static
scores? Here I compare the duration-dependent
measures to the standard measures on this example
graph.
Based only on the structure of the ties, not the
timing, the most central nodes are nodes 13, 16
and 4. Since this is a simulation, I simply
randomize the observed time-ranges on this graph
to test the general relation between the fixed
and temporal measures.
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Network Dynamics Flow
How do these centrality scores compare to static
scores? Here I compare the duration-dependent
measures to the standard measures on this example
graph.
Box plots based on 500 permutations of the
observed time durations. This holds constant the
duration distribution and the number of edges
active at any given time.
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Network Dynamics Flow
How do these centrality scores compare? Here I
compare the duration-dependent measures to the
standard measures on this example graph.
Based only on the structure of the ties, this
graph has lots of different centers, depending on
closeness, betweeneess or degree (size). In this
graph, Closeness and Betweenness correlate at
0.64, Closeness and Degree at 0.56, and
Betweeness and degree at 0.71
Node size proportional to degree
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Network Dynamics Flow
How do these centrality scores compare? Here I
compare the duration-dependent measures to the
standard measures on this example graph.
But these edges are timed, since publications
occur at a particular date. Here I treat the
edges as lasting between the first and last
publication date, and animate the resulting
network. Dark blue edges are active, past edges
are ghosted onto the map. Make note of the
fairly high concurrency (some of it necessary due
to two-mode data).
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Network Dynamics Flow
How do these centrality scores compare? At the
individual level, what is the relation between
structural centrality and duration centrality?
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Network Dynamics Flow
How do these centrality scores compare? At the
individual level, what is the relation between
structural centrality and duration centrality?
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Network Dynamics Flow
How do these centrality scores compare? Here I
compare the duration-dependent measures to the
standard measures on this example graph.
Correlation w. Closeness centrality
Box plots based on 500 permutations of the
observed time durations. This holds constant the
duration distribution and the number of edges
active at any given time.
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Network Dynamics Flow
How do these centrality scores compare? What
about at the system level? How do the features
of the temporal ordering affect the overall
asymmetry in reachability and the proportion of
pairs reachable?
Asymmetry
Reachability
Concordance (k3)
Concordance (k3)
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Network Dynamics Flow
How do these centrality scores compare? The
most important actors in the graph depend
crucially on when they are active. The
correlations can range wildly over the exact same
contact structure. Concordance is important, but
not determinant (at least within the range
studied here). We need to extend our intuition
on the global distribution of time in the
graph. The centrality scores described here
are low-hanging fruit simple extensions of
graph-based ideas. But the crucial features
for population interests will be creating
aggregations of these features something like
centralization that captures the regularity,
asymmetry and temporal role-structure of the
network.
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Network Dynamics Flow
How can we visualize such graphs? Animation of
the edges, when the graph is sparse, helps us see
the emergence of the graph, but diffusion paths
are difficult to see Consider an example
Romantic Relations at Jefferson high school
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Network Dynamics Flow
How can we visualize such graphs? Animation of
the edges, even when the graph is sparse, does
not typically help us see the potential flow
space, as its just too hard to follow the
implication paths with our eyes, so it seems
better to plot the implied paths directly.
Consider an example
Plotting the reachability matrix can be
informative if the graph has clear pockets of
reachability
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Network Dynamics Flow
How can we visualize such graphs? Animation of
the edges, even when the graph is sparse, does
not typically help us see the potential flow
space, as its just too hard to follow the
implication paths with our eyes, so it seems
better to plot the implied paths directly.
Consider an example
Plotting the reachability matrix can be
informative if the graph has clear pockets of
reachability
(Good readability example)
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Network Dynamics Flow
How can we visualize such graphs? Animation of
the edges, even when the graph is sparse, does
not typically help us see the potential flow
space, as its just too hard to follow the
implication paths with our eyes, so it seems
better to plot the implied paths directly.
Consider an example
Edges have discrete start and end times, tagged
as days over a 2-year window so first contact
between nodes 10 and 4 was on day 40, last
contact on day 72.
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Network Dynamics Flow
How can we visualize such graphs? Animation of
the edges, even when the graph is sparse, does
not typically help us see the potential flow
space, as its just too hard to follow the
implication paths with our eyes, so it seems
better to plot the implied paths directly.
Consider an example
Here we plot the reachability matrix over the
coordinates for the direct network. . Direct
ties are retained as green lines, if node i can
reach node j, then a directed arrow joins the two
nodes. Here I mark cases where two nodes can
reach each other with red, purely asymmetric with
blue. This is accurate, but hard to read when
reachability paths are long.
(poor readability example)
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Network Dynamics Flow
How can we visualize such graphs? Animation of
the edges, even when the graph is sparse, does
not typically help us see the potential flow
space, as its just too hard to follow the
implication paths with our eyes, so it seems
better to plot the implied paths directly.
Consider an example
Various weightings of the indirect paths also
dont help in an example like this one. Here I
weight the edges of the reachability graph as
1/d, and plot using FR. You get some sense of
nodes who reach many (size is proportional to
out-reach). Here you really miss the asymmetry
in reach (the correlation between number reached
and number reached by is nearly 0).
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Network Dynamics Flow
How can we visualize such graphs? Another tack
is to shift our attention from nodes to edges, by
plotting the line graph (thanks to Scott Feld for
making this suggestion). The idea is to
identify an ordering to the vertical dimension of
the graph to capture the flow through the
network. Consider an example

• So now we
• Convert every edge to a node
• Draw a directed arc between edges that (a) share
  a node and (b) precede each other in time.

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Network Dynamics Flow
How can we visualize such graphs? Another tack
is to shift our attention from nodes to edges, by
plotting the line graph (thanks to Scott Feld for
making this suggestion). The idea is to
identify an ordering to the vertical dimension of
the graph to capture the flow through the
network. Consider an example

• So now we
• Convert every edge to a node
• Draw a directed arc between edges that (a) share
  a node and (b) precede each other in time.
• Concurrent edges (such as 13-8 and 13-5 or
  1-16,2-16 will be connected with a bi-directed
  edge (they will form completely connected
  cliques) while the remainder of the graph will be
  asymmetric ordered in time.

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Network Dynamics Flow

• Further Complications, that ultimately link us
  back to the question of
• When is a network
• Range of temporal activity
• When the graph is globally sparse (like the
  example above), the path-structure will also be
  sparse. Increasing density will lead to lots of
  repeated interactions, and thus reachability
  cycles.
• Consider email exchange networks or classroom
  communication networks vs. sexual networks. In
  sexual or romantic networks, returning to a
  partner once the relation has ended is rare, in
  communication networks it is common.
• Observed vs. Real
• - We will often have discrete observations
  of real-time processes. How do we account for
  between-wave temporal ordering? What are the
  limits of observed measures to such inter-wave
  activity?
• - The Snijders et. al. Siena modeling approach
  is an obvious first step here.

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Network Dynamics Flow

• Further Complications, that ultimately link us
  back to the question of
• When is a network
• 3) Temporal reachability as higher-order model
  feature
• As the capacity of ERGM models continue to
  expand, we can start to build temporal sequence
  rules in to the local models (such as
  communication triplets, or avoidance of past
  relations once ended), which then makes it
  sensible to ask whether the models fit the
  time-structure of the data.
• Optimal observation windows
• Either for data collection or visualization, we
  often have to decide on a time-range for our
  analyses. What should that range be?
• 5) Relational temporal asymmetry. For many
  types of relations, it is difficult to decide
  when relations end. This taps a distinction
  between activated and potential relations.

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