Transcript Document

Diffusion Over Dynamic Networks
Stanford University
May 8, 2007
James Moody
Duke University
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.
•These, ultimately, are often features that rest on the diffusion
of some “bit” over the network. We’ll focus today on how
that happens.
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
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 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
*This is a big conditional! – lots of work on how the dyadic transmission rate may differ
across populations.
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
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
Social Networks & Diffusion
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
Social Networks & Diffusion
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
Social Networks & Diffusion
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.
Social Networks & Diffusion
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
Social Networks & Diffusion
Social Cohesion
Project 90, Sex-only network (n=695)
3-Component (n=58)
Social Networks & Diffusion
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
Emergence of multiple connectivity by degree distribution
Emergent Connectivity in low-degree networks
Partner
Distribution
Component
Size/Shape
Social Networks & Diffusion
Emergence of multiple connectivity by degree distribution
Development of STD cores in low-degree networks: rapid transition without stars.
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.
•Centrality affects within-network diffusion likelihood – we’ll not talk
about this much today.
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 2006
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.
Edge timing constraints on diffusion
“Bits” can only flow
forward in time: the finish
time of the next step in a
path must be > the start
time of the last step.
C
A
2-5
8-9
E
B
D
A hypothetical Sexual Contact Network
3-5
F
Edge timing constraints on diffusion
“Bits” can only flow
forward in time: the finish
time of the next step in a
path must be > the start
time of the last step.
A
C
E
D
F
B
The path graph for a hypothetical contact network
Edge timing constraints on diffusion
Edge time structures are characterized by sequence, duration and overlap.
Paths between i and j, have length and duration, but these need not be
symmetric even if the constituent edges are symmetric.
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)
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 concurrency across all relations.
Edge timing constraints on diffusion
4
1
A
B
C
D
E
F
A
0
1
0
0
0
1
B
1
0
1
0
0
0
C
2
1
0
1
0
0
D
2
2
1
0
1
1
2
Path distances need not progress in steps. While (a) is 2
steps from d, and d is 1 step from e, a and e are 4 steps
apart.
This is because a shorter path from a to e emerges after the
path from d to e ended.
E
4
3
2
1
0
0
F
1
2
2
1
2
0
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.
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
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 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.
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.
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.
The Cocktail Party Problem
-
Imagine a typical ‘mixer’ party, where
one of the guests knows a bit of gossip
that everyone would like to know.
-
Assuming that people tell this gossip to
the people they meet at the party:
a) How many people would
eventually hear the gossip?
b) How long would it take to spread
through the group?
The Cocktail Party Problem
-Some specifics to narrow down the problem.
- 30 people invited, party lasts an hour.
-At any given moment in time, you can only carry on
a conversation with 3 other people
-Guests mingle well – they spend a short time
talking to most people, but a long time to a small
number (such as their date).
-Mingling is somewhat space-based – you talk to the
people you bump into, then move on to someone
else after a short time.
-The bit of gossip moves instantaneously across
connected sets (so time-to-diffuse=0).
The Cocktail Party Problem
-Some specifics to narrow down the problem.
A (seemingly) simple network problem: record who
talks to who, and map the network.
Mean distance: 1.99
Diameter: 4 steps
The Cocktail Party Problem
-But such an image conflates many temporally distinct
events. A more accurate image is something like this:
In general, the graphs over which
diffusion happens often:
•
•
•
•
Have timed edges
Nodes enter and leave
Edges can re-occur multiple times
Edges can be concurrent
These features break transmission
paths, generally lowering diffusion
potential – and opening a host of
interesting questions about the
intersection of structure and time in
networks.
The Party Revisited
Question 1: How does the edge timing affect the overall likelihood that
everyone in the party would ultimately hear the gossip?
Simulate a cocktail party, manipulate the “mingling” rate and range and
compare diffusion over both networks.
The Party Revisited
The Party Revisited
Measuring Diffusion Potential with Network Traces:
Cumulative Number of people each node reaches at each step.
Dynamic Graph
Static Graph
Nodes that
reach everyone
in 4 steps
Nodes that
never reach
everyone
Node reaches
9 people in 2
steps
Sample “traces” from one run
The Party Revisited
Static Density: 0.21
Static Density: 0.24
Static Density: 0.26
Static Density: 0.28
Static Density: 0.23
Static Density: 0.27
Static Density: 0.31
Static Density: 0.36
Static Density: 0.27
Static Density: 0.32
Static Density: 0.35
Static Density: 0.40
Static Density: 0.30
Static Density: 0.34
Static Density: 0.41
Static Density: 0.42
Average (mean of means) reachability & Distance, all runs
The Party Revisited
Timing always lowers the proportion who could be reached in the network
and lengthens the distances between connected nodes.
This suggests that diffusion over dynamic networks will tend to be slower
than over similar volume static nets.
Note that here we:
a) assumed that diffusion was instant across connected sets
b) assumed complete cliques among conversation groups
c) everyone started at the same time
d) a small group (30 nodes).
If the group is larger, the proportional effects are more dramatic.
If diffusion takes time, edges expire before traversed.
Question 2: Since old paths can’t be joined when actors make
new contacts, will the “small world” rewiring effect work?
Small World Mechanisms on Dynamic Graphs
Small World Mechanisms on Dynamic Graphs
Simulation setup:
1. Generate a 200 node ring lattice, where every node has 6 ties.
2. Assign starting times to edges as a random draw from a uniform
distribution. Mean concurrency levels are set by compressing or
stretching the starting-time distribution.
3. Each edge is given a duration drawn from a skewed distribution.
4. Once edge-times are set, randomly rewire the graph by reassigning
one end of the edge to a node chosen at random.
5. Calculate the reachability and mean distance scores for each rewiring.
6. Repeat 4-5 many times, increasing the number of edges rewired.
Simulation varies the proportion of edges rewired and the level of graph
concurrency in the network.
Small World Mechanisms on
Dynamic Graphs
Small World Mechanisms on Dynamic Graphs
•The rapid shortening of distance we typically see in small-world
simulations does not occur in dynamic networks.
•The initial distances are much higher, since many nodes are not
reachable.
•The rapid decreasing marginal returns to rewiring are much slower
•When concurrency is relatively low, the effects of rewiring are
nearly linear
•When concurrency is relatively high, the characteristic curve
starts to emerge, but is much less steep.
•Note all of these concurrency levels are non-trivial. Even
when only 4% of two-paths in the graph are concurrent, nearly
50% of nodes have at least 1 concurrent edge.
•Why?
Small World Mechanisms on Dynamic Graphs
Why do we see this pattern?
•Long distant out-reach is rare:
•Consider a set of typical reach-paths in a dynamic network with timedisjoint edges:
Time 
e1
e1
p12 = p(e2 > e1)
e2
e3
e4
e2
p23 = p(e3 > e2)
e3
p34 = p(e4 > e3)
P34 < p23 < p12 < 1.0
Small World Mechanisms on Dynamic Graphs
Why do we see this pattern?
•Long distant out-reach is rare:
•If we allow concurrency & lengthen the duration of edges
(proportionate to the observation window):
Time 
e1
e1
p12 = p(e2 > e1)
e2
e3
e4
e2
p23 = p(e3 > e2)
e3
p34 = p(e4 > e3)
Pij is still decreasing, but not as rapidly.
Small World Mechanisms on Dynamic Graphs
Why do we see this pattern?
•Long distant out-reach is rare:
•If we allow concurrency & lengthen the duration of edges
(proportionate to the observation window):
Time 
e1
e1
p12 = p(e2 > e1)
e2
e3
e2
p23 = p(e3 > e2)
e4
e3
p34 = p(e4 > e3)
Pij is constant
Small World Mechanisms on Dynamic Graphs
Why do we see this pattern?
•Overlapping paths does not imply joint reach
Two starting nodes
Medium-concurrency graph
Small World Mechanisms on Dynamic Graphs
Two starting nodes
Low-concurrency graph
Temporal structure and reachability
Structure and Variability
Examples thus far lack meaningful network structure.
- The party simulation is a (space-constrained) random network
- Lattices make all nodes structurally equivalent in the contact pattern
Question 3: How does time shape diffusion potential in realistic graphs?
a) How much does the contact structure matter?
- Minimum possible time-risk
- Variance in the variability of time-risk
- Individual position vs. network totals
b) How much of the diffusion potential can be explained with local rules?
Temporal structure and reachability
Structure and Variability
Time ordering for the minimum path-density, 2-regular graph.
t2
t
t2
2
t1
t1
t2
t1
t1
t2
Minimize by weaving early – late – early in paths.
Temporal structure and reachability
Structure and Variability
Simulate time structure on a small sample of real graphs.
- These graphs are small walks (~100 nodes) from the soc coauthor network.
- Construct times and durations just as in the SW study
- Record the overall reachability and correlation between node-level centrality
- Examine the reachability pattern relative to minimum possible
- See if we can use some systematic features of the resulting time order to
predict reachability
Temporal structure and reachability
Structure and Variability
5 example coauthor graphs. (Some of you are in this figure).
Temporal structure and reachability
Structure and Variability
Min
reachability
Proportion of pairs reachable through time
Temporal structure and reachability
Structure and Variability
Relative Reach – Reachability over minimum possible
Temporal structure and reachability
Structure and Variability
Relative Reach – Reachability over minimum possible
Temporal structure and reachability
Structure and Variability
Volume
Distance
Connectivity
Nodes: 83
Mean Deg: 3.04
Density: 0.037
Centralization: 0.237
Mean: 0.398
Diameter: 6
Centralization: 0.321
Largest BC:0.16
Pairwise K: 1.07
Nodes: 148
Mean Deg: 6.16
Density: 0.042
Centralization: 0.187
Mean: 3.59
Diameter: 5
Centralization: 0.312
Largest BC: 0.51
Pairwise K: 1.57
Nodes: 80
Mean Deg: 5.27
Density: 0.067
Centralization: 0.373
Mean: 3.02
Diameter: 5
Centralization: 0.413
Largest BC: 0.33
Pairwise K: 1.34
Nodes: 154
Mean Deg: 3.71
Density: 0.025
Centralization: 0.147
Mean: 4.99
Diameter: 8
Centralization: 0.259
Largest BC: 0.08
Pairwise K: 1.07
Nodes: 128
Mean Deg: 3.39
Density: 0.027
Centralization: 0.205
Mean: 4.55
Diameter: 6
Centralization: 0.301
Largest BC:
Pairwise K: 1.06
Temporal structure and reachability
Structure and Variability
K=1
N=10
K=2
N=9
K=3
K=4
K=3
N=4
K=2
K=4
N=5
1
2
3
4
5
6
7
8
9
10
1
.
3
3
3
2
2
2
2
2
1
2
3
.
3
3
2
2
2
2
2
1
3
3
3
.
3
2
2
2
2
2
1
4
3
3
3
.
2
2
2
2
2
1
5
2
2
2
2
.
4
4
4
4
1
6
2
2
2
2
4
.
4
4
4
1
7
2
2
2
2
4
4
.
4
4
1
8
2
2
2
2
4
4
4
.
4
1
9
2
2
2
2
4
4
4
4
.
1
K=1
Average K = 2.38
0
1
1
1
1
1
1
1
1
1
.
Temporal structure and reachability
Structure and Variability
Kcon: 2.95
Net1
Kcon: 1.55
Net3
Kcon: 2.43
Net2
Kcon: 1.36
Net4
4 clustered networks w. different global connectivity
Temporal structure and reachability
Structure and Variability
Relative (to min) Reachability
Temporal structure and reachability
Structure and Variability
Interaction of Structure and Time
7
Mean Relative Reach
6
5
4
3
2
1
1
1.5
2
Pairwise k Connectivity
2.5
3
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.
The Mingle Mixing Problem Space