Transcript Document

Diffusion Over Dynamic Networks
(plus some social network intro since I’m 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
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
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:
Introduction
Source: Linton Freeman “See you in the funny pages” Connections, 23, 2000, 32-42.
Introduction
High Schools as Networks
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.
Introduction
Why do Networks Matter?
Local vision
Introduction
Why do Networks Matter?
Local vision
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.
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.
1.
2.
3.
Introduction
Social Network Basics
a. Basic data Elements
b. Basic data structures
c. Network Analysis Buffet
Networks & Diffusion
a. Structural constraints on network diffusion
a. Reachability
b. Distance
c. Connectivity
d. Closeness centrality
b. Temporal Constraints on network diffusion
a. Defining dynamic networks
b. How order constrains flow
c. Reachability variance w. constant structure
d. Minimum temporal reachability
c. New time-dependent network measures
a. Graph-level measures
b. Node-level measures
d. Visualizing Diffusion potential in time-dependent
Graphs
Social Network Data Elements
Social Network data consists of two linked classes of data:
a) 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
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
a
d
c
e
Social Network Data Elements
In general, a relation can be:
Binary or Valued
Directed or Undirected
b
b
d
a
c
a
e
c
1
a
b
d
1
3
c
Undirected, Valued
e
Directed, binary
Undirected, binary
b
d
d
2
4
e
a
c
Directed, Valued
e
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).
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.
Social Network Data Elements
Levels of analysis
Global-Net
Ego-Net
Partial-Network
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
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.
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.
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
Social Network Data Structures
Visualization
Another example:
Spring embedder layouts
Tree-Based layouts
(poor)
(good)
Two layouts of the same network
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 linecrossing.
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
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.
2
12
9
63
Male
Female
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.
Social Network Data Structures Visualization
Social Network Data Structures
Social Network Data Structures
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).
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.
Social Network Data Structures
Data Representations
Pictures only take us so far:
from pictures to adjacency matrices
b
b
d
a
c
e
Undirected, binary
a
b
1
a
b 1
c
1
d
e
c
d
1
1
c
e
a
1
a
b 1
c
1
d
e
1
1
a
e
Directed, binary
1
1
d
b
1
c
1
d
e
1
1
1
Social Network Data Structures
Data Representations
From matrices to lists
a
a
b 1
c
d
e
b
1
c
d
e
1
1
1
1
1
1
1
1
Adjacency List
ab
bac
cbde
dce
ecd
Arc List
ab
ba
bc
cb
cd
ce
dc
de
ec
ed
Social Networks & Diffusion
“Goods” flow through networks:
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
Social Networks & Diffusion
Three features of the network’s 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?
Social Networks & Diffusion
Without full network data, you can’t distinguish actors with limited diffusion
potential from those more deeply embedded in a setting.
c
b
a
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.
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.
b
a
a
d
c
b
e
f
c
f
d
e
Paths can be directed, leading to a distinction between “strong” and “weak”
components
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
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.
Social Networks & Diffusion
Reachability
This example
contains many
components.
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.
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
Social Networks & Diffusion
Distance & number of paths
Paths are the different routes one can take. Node-independent paths are
particularly important.
b
There are 2 independent
paths connecting a and
b.
There are many nonindependent paths
a
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 31:305-59.
Moody, James and Douglas R. White. 2003. “Structural
Cohesion and Embeddedness: A hierarchical Conception of
Social Groups” American Sociological Review 68:103-127
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. 10:95-117
Moody, James "The Structure of a Social Science
Collaboration Network: Disciplinary Cohesion from
1963 to 1999" American Sociological Review. 69:213238
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
2
1
Node Connectivity
3
Measuring Networks: Large-Scale Models
Social Cohesion
Formal definition of Structural Cohesion:
(a) A group’s structural cohesion is equal to the minimum number of actors who,
if removed from the group, would disconnect the group.
Equivalently (by Menger’s Theorem):
(b) A group’s structural cohesion is equal to the minimum number of nodeindependent paths linking each pair of actors in the group.
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
5
11
7
12
13
6
14
15
17
16
18
19
20
2
22
23
Measuring Networks: Large-Scale Models
Social Cohesion
Project 90, Sex-only network (n=695)
3-Component (n=58)
Measuring Networks: Large-Scale Models
Social Cohesion
IV Drug Sharing
Largest BC: 247
k > 4: 318
Max k: 12
Structural Cohesion
simultaneously gives
us a positional and
subgroup analysis.
Connected
Bicomponents
Social Networks & Diffusion
Distance & number of paths
Probability of transfer
by distance and number of paths, assume a constant pij of 0.6
1.2
1
probability
10 paths
0.8
5 paths
0.6
2 paths
0.4
1 path
0.2
0
2
3
4
Path distance
5
6
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.
Social Networks & Diffusion
Diffusion features on static graphs
Social Networks & Diffusion
Example on static graphs
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.
Social Networks & Diffusion
Example on static graphs
Table 2. OLS Regression of Relative Diffusion Ratio on Network Structure
Variable
Model 1
Model 2
Model 3 Model 4
***
***
Intercept
1.62
1.90
1.02***
1.81***
Connectivity
Distance
-0.207***
-0.179***
Independent Paths
-0.077***
-0.056***
0.023***
0.015***
Distance x Paths
Clustering
Clustering Coefficient
-0.692***
-0.653***
Grade Homophily
-0.026**
-0.007
***
Peer Group Strength
-0.868
-0.141
Degree Distribution
Degree Skew
-0.023
-0.007
*
Assortative Mixing
-0.189
-0.059
Control Variables
Network Size/100
0.005***
-0.005***
-.005***
0.004*
Proportion Isolated
-0.007
-1.106***
-.984*** -0.300*
Non-Complete
-0.006
-0.052*
-.078**
-0.006
2
Adj- R
0.85
0.76
0.60
0.90
N
124
124
124
124
Model 5
1.71***
-0.171***
-0.052***
0.016***
-0.454***
-0.009*
-0.146
-0.002
-0.071
0.002**
0.058
0.018
0.93
121
Social Networks & Diffusion
Example on static graphs
Figure 4. Relative Diffusion Ratio
By Distance and Number of Independent Paths
1.2
Observed / Random
1
k=8
0.8
k=6
k=4
0.6
k=2
0.4
2.3
2.8
3.3
3.8
4.3
4.8
Average Path Length
5.3
5.8
6.3
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.
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
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:


Cc (ni )   d (ni , n j )
 j 1

g
1
Normalized Closeness Centrality
CC' (ni )  (CC (ni ))(g 1)
Social Networks & Diffusion
Centrality
Closeness Centrality in 4 examples
C=0.0
C=1.0
C=0.36
C=0.28
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.
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:
1) The structure itself evolves, in ways that will affect the topology an
thus flow.
2) The timing of contact constrains information flow
Measuring Networks: Flow
Time
Drug Relations, Colorado Springs, Year 1
Data on drug users in
Colorado Springs, over
5 years
Measuring Networks: Flow
Time
Drug Relations, Colorado Springs, Year 2
Current year in red, past relations in gray
Measuring Networks: Flow
Time
Drug Relations, Colorado Springs, Year 3
Current year in red, past relations in gray
Measuring Networks: Flow
Time
Drug Relations, Colorado Springs, Year 4
Current year in red, past relations in gray
Measuring Networks: Flow
Time
Drug Relations, Colorado Springs, Year 5
Current year in red, past relations in gray
When is a network?
Source: Bender-deMoll & McFarland “The Art and Science of Dynamic Network Visualization” JoSS Forthcoming
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” (W&F 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 macrostructures emerge as the result of actor behavior rules.
Network Dynamics & Flow
The key element that makes a network a system is the path: it’s 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.
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:
A
1-2
B
3-4
C
1-2
D
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.
Network Dynamics & Flow
This can also introduce a new dimension for “shortest” paths:
B
3-4
C
D
A
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.
Network Dynamics & Flow
Reachability
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Direct Contact Network of 8 people in a ring
1
1
Network Dynamics & Flow
Reachability
1
1
2
2
2
2
2
1
1
2
2
2
2
2
2
1
1
2
2
2
2
2
2
1
1
2
2
2
2
2
2
1
1
2
2
2
2
2
2
1
1
2
Implied Contact Network of 8 people in a ring
All relations Concurrent
2
2
2
2
2
1
1
1
2
2
2
2
2
1
Network Dynamics & Flow
Reachability
3
2
1
2
1
1
1
1
1
2
2
3
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
= 0.57 reachability
Implied Contact Network of 8 people in a ring
Mixed Concurrent
Network Dynamics & Flow
Reachability
8
1
1
2
7
3
6
5
4
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
= 0.71 reachability
Implied Contact Network of 8 people in a ring
Serial Monogamy (1)
1
1
1
1
1
1
1
Network Dynamics & Flow
Reachability
8
1
1
2
7
1
1
1
1
1
1
1
1
3
6
1
4
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
= 0.51 reachability
Implied Contact Network of 8 people in a ring
Serial Monogamy (2)
1
1
1
Network Dynamics & Flow
1
2
1
1
1
1
2
1
1
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
= 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)
Identifying the Minimum Path Density of a Graph
A 2-regular graph
t2
t2
t2
t1
t1
t2
t1
t1
t2
l  3g  4
line
l  3g
cycle
Identifying the Minimum Path Density of a Graph
A 3-regular spanning tree
t1
t1
t3
13
t2
11
16
t1
t1
t2
4
2
t3
t3
t2
t1
t2
t1
t2
t2
t3
3
t1
t2
t1
22
t3
t3
t2
19
t1
20
t1
t1
18
t3
21
t2
t3
t3
8
9
17
7
t3
1
t1
t2
t3
6
5
10
t1
15
t3
t1
t2
t2
14
t2
12
t3
t2
t3
t2
t3
t2
l = 7g
Identifying the Minimum Path Density of a Graph
A 3-regular grid
t2
t1
t3
t2
t1
t2
t1
t2
t3
t2
t1
t3
t2
t1
t2
t1
t1
t3
t2
t1
t3
t3
t2
t3
t3
t2
t2
t1
t3
t3
t3
t1
t3
t2
t1
t1
t1
t2
t1
t3
t2
t3
t3
t3
t1
t2
t1
t3
t3
t2
Each person can reach 4
people indirectly., leading
again to 7g total arcs per
person.
t3
t3
t3
t2
t1
t3
t3
t2
t1
t3
Identifying the Minimum Path Density of a Graph
A 3-regular linked clusters
t1
2
t3
t2
t3
1
t2
3
t1
4
t1
t3
6
t2
t3
5
t2
7
t1
8
t1
t3
10
t2
t3
9
t2
If you count self-loops, one still hits 7l overall.
11
12
t1
t3
Reachability as a function of relationship adjacency
Identified paths:
t1
For a regular graph with d()=T
t2
T (T  1)(T  2)...(T  l  1)
Pi  
l!
l 2
T
t1
t3
t2
t3
t1
t2
t3
I think it’s an open question to define a minimum reachability graph for non-regular structures.
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.
2
1
1
2
2
1
1
2
In general, ignoring time order is equivalent to
assuming all relations occur simultaneously –
assumes perfect concordance across all relations.
Network Dynamics & Flow
1
2
At the graph level, we are interested in two
properties immediately:
2
1
a)
1
2
1
1
1
1
1
1
1
1
1
1
1
1
the temporal-implied reachability (perhaps
relative to minimum)
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
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.
Alternative measures:
Relative Reach
P 
R
min( P  )
Conditional Reachability (Harary, 1983)
P   min( P  )

max( P  )  min( P  )
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.
Network Dynamics & Flow
New versions of classic reachability measures:
1) Temporal reach: The ij cell = 1 if i can reach j through time.
2) Temporal geodesic: The ij cell equals the number of steps in the shortest path
linking i to j over time.
3) Temporal cohesion: The ij cell equals the number of time-ordered nodeindependent 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.
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.*
CTDCloseness
1

T
( Dij )
j
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 n+1
Network Dynamics & Flow
Timing affects the symmetry of a symmetric contact graph.
C
A
2-5
8-9
E
B
D
Numbers above lines indicate contact periods
3-5
F
Network Dynamics & Flow
Timing affects the symmetry of a symmetric contact graph.
A
C
E
D
F
B
Network Dynamics & Flow
Define fastness centrality as the average of the clock-time needed
for an actor to reach others in the network:
C fast 
1
N 1
 max( time)  time
ij
j
Actors with high fastness centrality are those that would
reach the most people early. These are likely important for
any “first mover” problem.
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:
Cquick 
1
N 1
 min( T
jit
 Tit )
j
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.
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.
Cexposure 
1
N 1
 (T
ijl
 Tijf )
j
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.
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
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).
Network Dynamics & Flow
How do these centrality scores compare?
At the individual level, what is the relation between structural centrality and duration
centrality?
Network Dynamics & Flow
How do these centrality scores compare?
At the individual level, what is the relation between structural centrality and duration
centrality?
Network Dynamics & Flow
How do these centrality scores compare?
Correlation w. Closeness centrality
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.
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)
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.
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
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 it’s 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:
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 it’s 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)
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 it’s 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.
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 it’s 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)
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 it’s 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 don’t 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 outreach).
Here you really miss the
asymmetry in reach (the
correlation between number
reached and number reached by
is nearly 0).
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:
1) Convert every edge to a node
2) Draw a directed arc between
edges that (a) share a node and
(b) precede each other in time.
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:
1) Convert every edge to a node
2) Draw a directed arc between edges
that (a) share a node and (b) precede
each other in time.
3) 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.
Network Dynamics & Flow
Further Complications, that ultimately link us back to the question of
“When is a network”
1) 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.
2) 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.
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.
4) 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.