Network Metrics:
Deep Dive
Network Metrics:
Deep Dive
How to find, create and think about network metrics
Network Metrics
Why?
Measures are tools for thinking.
Network Metrics
How?
Read the literature
Debate the literature
Network Metrics
How?
Network Metrics
Lesson for why to read:
Network Metrics
How?
Network Metrics
How?
Network Metrics
How?
Network Metrics
Where to start?
Other options
Network Metrics
Where to start?
Network Metrics
Where to start?
Network Metrics
Where to start?
Network Metrics
Solution to theoretical problem
Analytically, most of these definitions & operationalizations of cohesion do not distinguish the social fact of cohesion from the psychological or behavior outcomes resulting from cohesion.
Def. 1:
“A collectivity is cohesive to the extent that the social relations of its members hold it together.”
What network pattern embodies all the elements of this intuitive definition?
Example: Cohesion & Clustering
Structural Cohesion
This definition contains 5 essential elements:
Example: Cohesion & Clustering
Structural Cohesion
1) Actors must be connected: a collection of isolates is not cohesive.
Not cohesive
Minimally cohesive:
a single path connects everyone
Example: Cohesion & Clustering
Structural Cohesion
1) Reachability is an essential element of relational cohesion. As more paths re-link actors in the group, the ability to ‘hold together’ increases.
Cohesion increases as # of paths connecting people increases
The important feature is not the density of relations, but the pattern.
Example: Cohesion & Clustering
Structural Cohesion
Consider the minimally cohesive group:
D = . 25
D = . 25
Moving a line keeps density constant, but changes reachability.
Example: Cohesion & Clustering
Structural Cohesion
What if density increases, but through a single person?
D = . 25
D = . 39
Removal of 1 person destroys the group.
Example: Cohesion & Clustering
Structural Cohesion
Cohesion increases as the number of independent paths in the network increases. Ties through a single person are minimally cohesive.
D = . 39
Minimal cohesion
D = . 39
More cohesive
Example: Cohesion & Clustering
Structural Cohesion
Substantive differences between networks connected through a single actor and those connected through many.
Minimally Cohesive Strongly Cohesive
Power is centralized Power is decentralized
Information is concentrated Information is distributed
Expect actor inequality Actor equality
Vulnerable to unilateral action Robust to unilateral action
Segmented structure Even structure
Def 2.
“A group is structurally cohesive to the extent that multiple independent relational paths among all pairs of members hold it together.”
Example: Cohesion & Clustering
Structural Cohesion
Node Connectivity
As size of cut-set
0
1
2
3
Structural Cohesion:
A network’s structural cohesion is equal to the minimum number of actors who, if removed from the network, would disconnect it.
Example: Cohesion & Clustering
Structural Cohesion
0
1
2
3
Node Connectivity
As number of node-independent paths
Structural Cohesion:
A network’s structural cohesion is equal to the minimum number node independent paths connecting each pair.
Example: Cohesion & Clustering
Structural Cohesion
Node connectivity =/= density or edge connectivity!
Example: Cohesion & Clustering
Structural Cohesion
Serendipity
1
2
3
4
Nestedness Structure
Cohesive Blocks
Depth
Sociogram
5
1
2
3
4
5
6
7
8
9
Cohesive Blocking
The arrangement of subsequently more connected sets by branches and depth uniquely characterize the connectivity structure of a network
Example: Cohesion & Clustering
Structural Cohesion
School Attachment
Example: Cohesion & Clustering
Structural Cohesion
Business Political Action
Measuring Networks: Connectivity
Redundancy Global: Structural Cohesion
Hybrid Model of Firm Ownership
Hybrid Model of Firm Ownership
Hybrid Model of Firm Ownership
Hybrid Model of Firm Ownership
Business groups nested within the core
Hybrid Model of Firm Ownership
Core Members are multiply connected & higher in revenue
Hard Problems
Example: Dynamic networks
Example: Dynamic networks
Contact network: Everyone, it is a connected component
Who can “A” reach?
Discussions of network effects on STD spread often speak loosely of “the network.”
There are three relevant networks that are often conflated:
Three relevant networks
Example: Dynamic networks
Exposure network: here, node “A” could reach up to 8 others
Who can “A” reach?
Discussions of network effects on STD spread often speak loosely of “the network.”
There are three relevant networks that are often conflated:
Three relevant networks
Example: Dynamic networks
Transmission network: upper limit is 8 through the exposure links (dark blue). Transmission is path dependent: if no transmission to B, then also none to {K,L,O,J,M}
Who can “A” reach?
Exposable Link (from A’s p.o.v.)
Contact
Discussions of network effects on STD spread often speak loosely of “the network.”
There are three relevant networks that are often conflated:
Three relevant networks
Example: Dynamic networks
Three relevant networks
Discussions of network effects on STD spread often speak loosely of “the network.”
There are three relevant networks that are often conflated:
Example: Dynamic networks
Example: Dynamic networks
A B C D E F
A 0 1 2 2 4 1
B 1 0 1 2 3 2
C 0 1 0 1 2 2
D 0 0 1 0 1 1
E 0 0 0 1 0 2
F 1 0 0 1 0 0
While a is 2 steps from d, and d is 1 step from e, a and e are 4 steps apart.
This is because the shorter path from a to e emerges after the path from d to e ended.
4
2
1
Path distances no longer simply add
Example: Dynamic networks
Timing constrains potential diffusion paths in networks, since bits can flow through edges that have ended.
This means that:
Combined, this means that many of our standard path-based network measures will be incorrect on dynamic graphs.
Example: Dynamic networks
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.
Structural measurement implications
Example: Dynamic networks
New versions of classic reachability measures:
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.
Structural measurement implications
Example: Dynamic networks
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. 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
Edge timing constraints on diffusion
When is a network?
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.
Edge timing constraints on diffusion
When is a network?
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. Note min is one of many functions, since the time-to-target speed is really a profile over the duration of t.
Edge timing constraints on diffusion
When is a network?
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.
Edge timing constraints on diffusion
When is a network?
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, not the timing, the most central nodes are nodes 13, 16 and 4.
Since this is a simulation, I permute the observed time-ranges on this graph to test the general relation between the fixed and temporal measures.
Edge timing constraints on diffusion
When is a network?
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.
Edge timing constraints on diffusion
When is a network?
When does timing really matter?
How do structure and dynamics intersect?
Measures
Dependent variable: Reachability in the exposure graph. This is the proportion of pairs in the network that are reachable in time.
Exposure Graph Density
Example: When does concurrency matter?
Measures: Independent variables
Features of the topology. Of key interest is the level of structural cohesion.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
1 | -- | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
2 | 1 | -- | 2 | 2 | 2 | 1 | 1 | 1 | 1 |
3 | 1 | 2 | -- | 2 | 2 | 1 | 1 | 1 | 1 |
4 | 1 | 2 | 2 | -- | 2 | 1 | 1 | 1 | 1 |
5 | 1 | 2 | 2 | 2 | -- | 1 | 1 | 1 | 1 |
6 | 1 | 1 | 1 | 1 | 1 | -- | 2 | 2 | 1 |
7 | 1 | 1 | 1 | 1 | 1 | 2 | -- | 2 | 1 |
8 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | -- | 1 |
9 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -- |
Cell Value = highest k-connected component pair belongs to.
Average = 1.25
Example: When does concurrency matter?
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
1 | -- | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
2 | 1 | -- | 3 | 3 | 3 | 2 | 2 | 2 | 1 |
3 | 1 | 3 | -- | 3 | 3 | 2 | 2 | 2 | 1 |
4 | 1 | 3 | 3 | -- | 3 | 2 | 2 | 2 | 1 |
5 | 1 | 3 | 3 | 3 | -- | 2 | 2 | 2 | 1 |
6 | 1 | 2 | 2 | 2 | 2 | -- | 2 | 2 | 1 |
7 | 1 | 2 | 2 | 2 | 2 | 2 | -- | 2 | 1 |
8 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | -- | 1 |
9 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -- |
Cell Value = highest k-connected component pair belongs to.
Average = 1.6
Measures: Independent variables
Features of the topology. Of key interest is the level of structural cohesion.
Example: When does concurrency matter?
Measures: Independent variables
Features of the topology. Of key interest is the level of structural cohesion.
Average connectivity
Example: When does concurrency matter?
Volume
Distance
Connectivity
Nodes: 148
Mean Deg: 6.16
Density: 0.042
Centralization: 0.187
Nodes: 80
Mean Deg: 5.27
Density: 0.067
Centralization: 0.373
Nodes: 154
Mean Deg: 3.71
Density: 0.025
Centralization: 0.147
Nodes: 128
Mean Deg: 3.39
Density: 0.027
Centralization: 0.205
Mean: 3.59
Diameter: 5
Centralization: 0.312
Mean: 3.02
Diameter: 5
Centralization: 0.413
Mean: 4.99
Diameter: 8
Centralization: 0.259
Mean: 4.55
Diameter: 6
Centralization: 0.301
Largest BC: 0.51
Pairwise K: 1.57
Largest BC: 0.33
Pairwise K: 1.34
Largest BC: 0.08
Pairwise K: 1.07
Largest BC:
Pairwise K: 1.06
Exemplar independent variables
“High Cohesive”
“Low Cohesive”
Example: When does concurrency matter?
Example: When does concurrency matter?
“Low Cohesive”
Example: When does concurrency matter?
Proportion of relations concurrent
Density of the Exposure Network
Example: When does concurrency matter?
Example: When does concurrency matter?
Topology & Time interact: How relational sequencing affects diffusion is conditioned by the structural patterns of relations.
Examples:
- Time limitations mean star nodes can’t interact with everyone at each time tick; effects of high degree are thus limited by schedule/availabiltiy
- If within-cluster ties are also more frequent than between cluster ties, then the effects of communities will be magnified.
-Multiple connectivity should provide routes around breaks built by temporal sequence.
Network Diffusion & Peer Influence
Structural Moderators of Timing Effects
Solution? Turn time into a network!
Time-Space graph representations
“Stack” a dynamic network in time, compiling all “node-time” and “edge-time” events (similar to an event-history compilation of individual level data).
Consider an example:
3. Time Matters
Dynamics of affect dynamics on
So now we:
Solution? Turn time into a network!
3. Time Matters
Dynamics of affect dynamics on
So now we:
Solution? Turn time into a network!
3. Time Matters
Dynamics of affect dynamics on
What’s the point?