control theory in social networks

Control theory in social networks

By Luke Grymek and Sidun Li 03/06/2013

Introduction

http://www.vitbergllc.com http://www.connect-utb.com

What is a social network?

• Collection of individuals and their interactions in a society, which can be in real life or online

• Examples - Facebook - Citation networks - This class

Facebook

www.connectedaction.net

Example of a Facebook Network Map

Citation networks

http://jasss.soc.surrey.ac.uk/12/4/12.html

EE194 Adv. Control Class

humorgags.blogspot.com

Why study social network?

• $$$ in consumer behavior studies • $$$ in marketing • Help understand/resolve social conflicts

How?

Modeling a social network

• Micro-level: relationships between 2 or a small group of individuals

Modeling a social network

• Meso-level: randomly distributed networks - Exponential random graph model

(Erdős–Rényi : G(n, M), G(n, p) models)

- Scale-free network model

(Barabási–Albert model) http://en.wikipedia.org/wiki/Barab%C3%A1si%E2%80%93Albert_model

http://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93R%C3%A9nyi_model

(p = 0.01)

The network is made of 50 vertices with initial degrees m=1

Modeling a social network

• Macro-level: a combination of meso-level networks

http://en.wikipedia.org/wiki/Social_network

Why controls?

• Too much information = uninformative

Why controls?

• Goal: control “flow of information” in the network, and accurately predict the responses of “inputs” into the system

0

0.2

0.4

0.6

0.8

1

1.2

t=0 t=1 t=2 t=3

x1 x2 x3

Why controls?

• Goal: control “flow of information” in the network, and accurately predict the responses of “inputs” into the system

0 0.5

1 1.5

2 2.5

3 3.5

t=0 t=1 t=2 t=3

x1 x2 x3

Why controls?

• We can also model networks in a structured fashion (nodes and links can have different meanings, but they can have the same topology)

Dispatcher in a network distributing work to the computers below

Professor Khan distributing projects to his students

Why linear dynamics?

• Conclusions drawn from linear dynamics can be extended to nonlinear systems

• If the controllability matrix of the linearized system has full rank at all points, then it is sufficient for most systems to say that the actual nonlinear system is controllable (i.e. small signal model)

http://web.mit.edu

Structural Controllability: Motivation

Structural Controllability: Motivation

Structural Controllability: Motivation

• C = (B, AB, A2B, … ,AN-1B) • If rank(C)=N we have controllability

Ultimate Goal: Minimize (ND ) whose control is sufficient to control the system’s dynamics

Structural Controllability: Motivation

• Minimum ND can be determined by the ‘maximum matching’ of the network

• Structural controllability problem maps into an equivalent geometrical problem on the network

Goal: Minimize (ND ) whose control is sufficient to control the system’s dynamics

Link to System Controllability

• Structural controllable System controllable for almost all parameter values except for a few combinations of values (which have Lebesgue measure zero)

• Strong structural controllable System controllable Goal: Minimize (ND ) whose control is sufficient

to control the system’s dynamics

Structural Controllability

• Undirected network vs. directed network

www.differencebetween.com

0 a12 a13

a21 0 a23

a31 a32 0

0 0 0

a21 0 0

a31 a32 0 a12 = a21 a13 = a31 a23 = a32

Structural Controllability - Example

Structural Controllability Theorem

Matching • Matching: For an undirected graph, a matching M is

an independent edge set without common vertices. – A vertex is matched if it is incident to an edge in the

matching

We gain control over the network if and only if we directly control each unmatched node

• For a directed graph, an edge subset M is a matching if no two edges in M share a common starting vertex or a common ending vertex. – A vertex is matched if it is incident to an edge in the

matching, otherwise it is unmatched

Maximum matching

• Maximum matching: a matching that covers the most amount of vertices in the graph - it’s called perfect if all vertices are matched (i.e. an elementary cycle) - decompose into bipartite graph (a graph whose vertices can be divided into 2 disjoint sets) - solve using Hopcroft-Karp algorithm, which runs in O(V1/2E) time

Minimum Input Theorem

• Theorem: The minimum number of (input) driver nodes (ND) need to fully control a network G(A) is 1 if there is a perfect matching in G(A). Otherwise, it equals the number of unmatched nodes with respective to any maximum matching set. ND = max{N - |M*|, 1}; (N: total # of nodes, M*: set of matched nodes)

• Intuition: Each node must have its own ‘superior’.

We gain control over the network if and only if we directly control each unmatched node

Matching

• Finding the minimum ND numerically takes O(N1/2L) steps where L denotes number of links (Hopcroft-Karp algorithm)

We gain control over the network if and only if we directly control each unmatched node

Matching Example 1

Directed Path

Max Matching

Matching Example 2

Directed Star

Only 1 link can be part of max matching

Matching Example 3

Only 2 links can be part of max matching

Characterizing and Predicting ND

• ND is determined mainly by the number of incoming and outgoing links each node has and independent of where those links point – Driver nodes tend to avoid hubs

• Denser networks require fewer driver nodes • Larger differences between node degrees

results in more needed drivers • Sparse, heterogeneous networks are the most

difficult to control

ND for different real world networks

Name Nodes Edges ND/N

College Student 32 96 .188

Prison Inmate 67 182 .134

Slashdot 82,169 948,464 .045

WikiVote 7,115 103,689 .666

Epinions 75,888 508,837 .549

Stanford.edu 281,903 2,312,497 .317

Political Blogs 1,224 19,025 .356

Problem solving – System approach

• Model the network using a graph, which by definition is a set of nodes and links

• Apply system control to study the graph: - Given system matrix A and B, is the system controllable? - Study system matrix A to determine the minimum # of non-zero elements in matrix B for controllability

Problem solving – Structural approach

• Model the network • Apply structural control to study the graph :

- Find a maximum matching set and the # of unmatched nodes - Find the driver nodes (the unmatched nodes) - The system is controllable with inputs to the driver nodes

Simulations Goal: identify the minimum # of individuals we need to control to control the whole network

Simulations

- Find a maximum matching set and the # of unmatched nodes - Find the driver nodes (the unmatched nodes) - The system is controllable with inputs to the driver nodes

Simulations

- Find a maximum matching set and the # of unmatched nodes - Find the driver nodes (the unmatched nodes) - The system is controllable with inputs to the driver nodes

Simulations

- Find a maximum matching set and the # of unmatched nodes - Find the driver nodes (the unmatched nodes) - The system is controllable with inputs to the driver nodes

Test in Matlab

• Confirm that the system is controllable A = system topology; B(1,1) = 1;

Simulations

- Find a maximum matching set and the # of unmatched nodes - Find the driver nodes (the unmatched nodes) - The system is controllable with inputs to the driver nodes

Simulations

- Find a maximum matching set and the # of unmatched nodes - Find the driver nodes (the unmatched nodes) - The system is controllable with inputs to the driver nodes

Simulations

- Find a maximum matching set and the # of unmatched nodes - Find the driver nodes (the unmatched nodes) - The system is controllable with inputs to the driver nodes

Simulations

- Find a maximum matching set and the # of unmatched nodes - Find the driver nodes (the unmatched nodes) - The system is controllable with inputs to the driver nodes

Test in Matlab

• Confirm that the system is controllable A = network topology; B(1 , 1) = 1; or B(13 , 1) = 1; or B(1 4, 1) = 1;

Questions?

References

• “Controllability of complex networks”, Liu, Slotine, & Barabási, 2011 Nature

• “Observability of complex networks”, Liu, Slotine, & Barabási, 2013 PNAS

• “Network Medicine: A Network-based Approach to Human Disease”, Barabási, gulbahce, Loscalzo, 2011 Nat Rev Genet.