Beyond Degree Distribution: local to global structure

Beyond Degree Distribution:
 local to global structure of social contact graphs Stephen Eubank
 Chris Barrett, Jiangzhuo Chen, Maleq Khan, Madhav Marathe, 
 Achla Marathe, Samarth Swarup, Anil Vullikanti NDSSL Tech Report 10-041
 Network Dynamics and Simulation Science Lab Virginia Bioinformatics Institute at Virginia Tech March 30, 2010 2010 International Conference on 
 Social Computing, Behavioral Modeling, and Prediction

Network Dynamics and Simulation Science Laboratory

A multi-method approach to measurement in the social sciences

+ An example application: social network analysis

Network Dynamics and Simulation Science Laboratory

Funded in part by NIGMS Network Dynamics and Simulation Science Laboratory

&

Successful interventions in social systems
 depend on identifying causes •  A ⇒ B ⇒ C C •  A
 B

• 

Causation is too strong; influence

• 

Causation is slippery –  A ⇒ B & A precedes B in time –  A ⇒ B & (not A) ⇒ (not B)

• 

Proximal vs remote cause

• 

Causal feedback loops

• 

Operational definition: 
 cause as control surface

• 

Deductive analysis: consequences of hypothesized causes

• 

A


⇒

C

B
 •  A


⇒

C B

Network Dynamics and Simulation Science Laboratory

Sherlock Holmesʼs Scientific Method
 Doesnʼt Always Work! “When you have eliminated the impossible, 
 whatever remains, however improbable, must be the truth”
 Problems: 1.  “Whatever remains” is rarely unique 2.  Hypotheses are often not obviously impossible: "Is there any other point to which you would wish to draw my attention?”
 "To the curious incident of the dog in the night-time.”
 "The dog did nothing in the night-time.”
 "That was the curious incident," remarked Sherlock Holmes

Network Dynamics and Simulation Science Laboratory

Constructing a synthetic information system:
 choose resolution •  Represent each interactor

Network Dynamics and Simulation Science Laboratory

Constructing a synthetic information system:
 disaggregate observations •  Represent each interactor •  Endow them with attributes

Network Dynamics and Simulation Science Laboratory

Constructing a synthetic information system:
 estimate interactions •  Represent each interactor •  Endow them with attributes •  Model interaction patterns Contact / colocation network

Network Dynamics and Simulation Science Laboratory

Constructing a synthetic information system:
 increase fidelity •  Represent each interactor Contact / colocation network •  Endow them with attributes •  Model interaction patterns •  Model other dynamics Communication network

Network Dynamics and Simulation Science Laboratory

Constructing a synthetic information system:
 across-level agency •  Represent each interactor •  Endow them with attributes •  Model interaction patterns •  Model other dynamics •  Unencapsulated agency

Network Dynamics and Simulation Science Laboratory

Synthetic information systems deduce 
 associations implied by causal hypotheses 1.  Network topology affects outcome:
 e.g. # infected as a function of time


disease

Generates associations between attribute and probability of infection
 Input to contact network model may be remote cause, but network itself is proximal cause

Network Dynamics and Simulation Science Laboratory

Synthetic information systems deduce 
 associations implied by causal hypotheses 2. Other attributes may just be
 “along for the ride”


disease

E.g. Attribute in lower network may be associated with attributes that determine contact network

Network Dynamics and Simulation Science Laboratory

Synthetic information systems deduce 
 associations implied by causal hypotheses 3.  Interactions in one layer may affect network topology in another layer


disease

Associations evolve

information Network Dynamics and Simulation Science Laboratory

Synthetic information systems deduce 
 associations implied by causal hypotheses 1.  Network topology affects outcome:

irreducible complexity 2.  Other attributes may just be
 “along for the ride”:


generating confounders 3.  Interactions in one layer may affect network topology in another layer:

co-evolving networks

Network Dynamics and Simulation Science Laboratory

This approach leads to novel insights The example that follows: •  is not a particular question about interventions in a social system •  Is a new methodology for social network analysis that just scrapes the surface of whatʼs possible

Network Dynamics and Simulation Science Laboratory

Local network structure affects outcomes How do simple interactions at small scales generate 
 complex phenomena at large spatio-temporal scales? (micro/macro, genotype/phenotype, individual/population, …)

Individual transmission

Local network structure affects outcomes How do simple interactions at small scales generate 
 complex phenomena at large spatio-temporal scales? (micro/macro, genotype/phenotype, individual/population, …)

Individual transmission

Simple group transmission

Local network structure affects outcomes How do simple interactions at small scales generate 
 complex phenomena at large spatio-temporal scales? (micro/macro, genotype/phenotype, individual/population, …)

Individual transmission Courtesy David Nadeau, SDSC

Complex population transmission

Simple group transmission

Local network structure affects outcomes How do simple interactions at small scales generate 
 complex phenomena at large spatio-temporal scales? (micro/macro, genotype/phenotype, individual/population, …)

The network is as important a part of the dynamics as transmission parameters are. Individual transmission Courtesy David Nadeau, SDSC

Complex population transmission

Simple group transmission

We know social networks exhibit structure …

Comparing observed features of a sociogram with the predictions of a random graph is as old as sociometry itself. Moreno's “sociodynamic effect'' (Moreno, 1934) states that the observed distribution of choices received is more variable than that predicted by chance – there are more “stars" and “isolates" than expected in a random graph.



– P. Holland and S. Leinhardt (eds.), Perspectives on social network research. New York: Academic Press. 1979.

Network Dynamics and Simulation Science Laboratory

… but we donʼt know what it is

As for the taxonomy of large sociograms, this apparently involves problems of great complexity. It would seem offhand that a taxonomy of “nets” … would arise naturally from the consideration of the statistical parameters, e.g. as a continuum of nets in the parameter space. But the statistical parameters themselves are singled out on the basis of taxonomic considerations, which have yet to be clarified.

– Anatol Rapoport and William Horvath, 


Behav Sci. 1961, 6, 279–291

Network Dynamics and Simulation Science Laboratory

Consider the space of all graphs

2

There are 2N graphs on N vertices. When the vertices are distinct, many of these graphs are topologically distinct.

Network Dynamics and Simulation Science Laboratory

Some of these graphs have a 
 particular degree distribution

Scale-free

Proofs about properties of scale-free graphs (e.g. “no epidemic transition in a scale-free graph”) describe mean outcomes across the hexagonal region, sometimes variances.

Network Dynamics and Simulation Science Laboratory

Some of these could represent social networks

Scale-free

Social nets The set of social networks may overlap the set of scale free graphs, but may lie in an out-of-the-way corner. Most of the weight in an average is not from realistic social networks.

Network Dynamics and Simulation Science Laboratory

Degree, Clustering, Assortativity, etc.
 are measures of local network structure # vertex

24

edge

26

Deg Deg

#

Degree

#

Clustering

#

1

7

0

21

1

2

2

2

8

2/3

2

1

3

4

3

8

1

1

1

5

1

4

0

2

2

3

5

1

2

3

6

2

5

2

3

3

6

3

5

2

Color

Color

#

Red

Red

1

Red

Yellow

4

Red

Green

7

Red

Blue

3

Yellow Yellow

2

Yellow Green

2

Yellow Blue

2

Green Green

0

Green Blue

3

Blue

2

Blue

Create “more random” graphs 
 by swapping edge endpoints

Network Dynamics and Simulation Science Laboratory

Constrained edge swaps maintain 
 local structural invariants

Network Dynamics and Simulation Science Laboratory

Different swapping rules maintain 
 different structural constraints

This choice maintains degree distribution invariant

Network Dynamics and Simulation Science Laboratory

Different swapping rules maintain 
 different structural constraints

This choice maintains assortativity by color invariant

Network Dynamics and Simulation Science Laboratory

Different swapping rules maintain 
 different structural constraints

?

No choice maintains assortativity by degree invariant

Network Dynamics and Simulation Science Laboratory

Different swapping rules maintain 
 different structural constraints

Any choice maintains clustering distribution invariant

Network Dynamics and Simulation Science Laboratory

Edge swapping is one step in a Markov chain

Markov chain

Our Markov chain mixes quickly, sampling from all graphs with the constrained property. It simulates the entire distribution of outcomes across the space – not just mean and variance – and demonstrates where the starting point lies in that distribution. Network Dynamics and Simulation Science Laboratory

The chainʼs starting point is important

Similar studies have gone the other direction, introducing structure into random graphs. There is no guarantee the result is relevant to social networks.

Network Dynamics and Simulation Science Laboratory

age assortative

infections

degree preserving

days cumulative infections

days

Conclusions •  Local structure of interaction networks – beyond degree distribution – has important effects on dynamics in social systems •  High resolution representations of social systems suggest innovative techniques, e.g. an edge-swapping Markov Chain for network analysis •  Synthetic information systems provide a new methodology to
 deduce associations implied by multi-theory, multi-perspective, 
 causal hypotheses

Network Dynamics and Simulation Science Laboratory