Diffusing Information and

Reaching Agreement in Networks

Convergence and Resilience

Shreyas Sundaram Electrical and Computer Engineering

University of Waterloo

Outline

2

Introduction and Network Models

Information Cascades

Linear Iterative Strategies for Distributed Consensus

Resilience to Malicious Nodes

Ongoing Research

Complex Systems and Networks are

Everywhere…

Social Network

Credits: Nature, National Geographic, Sentinel Visualizer

Synchronized Fireflies

Mouse Gene Network

Complex Systems and Networks are

Everywhere…

Sensor Networks for Airplane Health Monitoring

Credits: Sampigethaya et al. (Digital Avionics Conference 2007), Urban Ecoist, Swarmrobots.org

Electrical Power Grid

Robotic Swarms

Information Diffusion

5

Networks arise from interactions between various nodes

Key function: dissemination of information

Social networks: ideas, opinions, knowledge, etc.

Engineered networks: measurements, computations, control signals, etc.

Questions:

How effective is the network at diffusing information quickly, efficiently, reliably, … ?

What effect can a few nodes have on the behavior of the entire network?

Modeling the Network: Topology

6

Network can be modeled as a graph

N nodes {x1, x2, …, xN}

people, sensors, computers, robots, “agents”, …

Edge from xi to xj indicates that xi can influence xj

Modeling the Network: Information

Some (or all) nodes have some personal “information”

Opinion, position, sensor measurement, etc.

This information gets updated over time, based on interactions with other nodes

Model this information as a real number

Denote node xi’s initial information as xi[0]

7

1.2

-0.3

-4.1

2.2

5.0 7.4

20.1

-6.2

8.5

-1.9

5.9

9.8

-6.3

2.7

3.9

-0.5

2.0

Dynamics on Networks

8

Nodes update their values (information) based on the values of their neighbors

This produces dynamics on the network

Exact nature of the dynamics depends on exactly how nodes use their neighbors values

Have to consider both network topology (who talks to whom) and dynamics (what is done with the information) when studying diffusion of information

Diffusion of Information in Networks Studied extensively by various communities

Sociology Epidemiology Physics, Biology and Ecology Economics Communications Computer Science and Engineering …

Many excellent books on this topic Diffusion of Innovations, Rogers, 1962

Dissemination of Information in Communication Networks, Hromkovic et. al., 2005 Communication Complexity, Kushilevitz and Nisan, 1997

Distributed Algorithms, Lynch, 1997

Networks, Crowds, and Markets, Easley and Kleinberg, 2010 …

9

Cascades in Networks

Example: Cascade of a New Idea ([Morris, ’00], [Easley and Kleinberg, ’10])

11

Each node in network can be in state A or state B

e.g., Two competing technologies

Two neighboring nodes get a benefit if they are in the same state, and no benefit if they are in different states

Total benefit to each node is sum of benefits from each neighbor

Rule: Each node periodically looks at states of neighbors

Chooses A if at least a fraction q of its neighbors are A

Chooses B otherwise

q depends on relative benefit from A versus B

Cascades

12

Suppose all nodes start out in state B

Small subset S of nodes change their state to A, and keep it that way

For the specified “threshold” dynamics, under what topological conditions will all nodes eventually adopt A?

x1 x3

x2

x4 x8

x7

x9

x6

x5

Example ([Easley and Kleinberg, ’10])

13

q = ½, S = {x3, x5}

x1 x3

x2

x4 x8

x7

x9

x6

x5

State A State B

Example

14

q = ½, S = {x3, x5}

x1 x3

x2

x4 x8

x7

x9

x6

x5

State A State B

Example

15

q = ½, S = {x3, x5}

x1 x3

x2

x4 x8

x7

x9

x6

x5

State A State B

Example

16

q = ½, S = {x3, x5}

x1 x3

x2

x4 x8

x7

x9

x6

x5

State A State B

Example

17

q = ½, S = {x3, x5}

A spreads throughout network!

x1 x3

x2

x4 x8

x7

x9

x6

x5

State A State B

Example: Different Initial Set

18

q = ½, S = {x1, x3}

x1 x3

x2

x4 x8

x7

x9

x6

x5

State A State B

Example: Different Initial Set

19

q = ½, S = {x1, x3}

x1 x3

x2

x4 x8

x7

x9

x6

x5

State A State B

Example: Different Initial Set

20

q = ½, S = {x1, x3}

Spread of A stops!

Problem: there is a close-knit cluster, where every node has most of its neighbors inside the cluster

x1 x3

x2

x4 x8

x7

x9

x6

x5

State A State B

Example: Different Initial Set

21

q = ½, S = {x1, x3}

Spread of A stops!

Problem: there is a close-knit cluster, where every node has most of its neighbors inside the cluster

x1 x3

x2

x4 x8

x7

x9

x6

x5

State A State B

Result ([Morris, ’00])

22

Definition: Set of nodes is a cluster of density p if every node in set has at

least a fraction p of its neighbors inside the set

Suppose set S starts with state A, everybody else in state B

Intuition: No node in cluster has enough neighbors outside to allow new information to penetrate

We’ll come back to this concept later

Cascade stops Rest of graph contains a

cluster of density 1-q

Reaching Agreement in

Networks

Reaching Agreement in Networks

24

Previous example studied propagation of a single value

What if there are multiple different values in the network?

e.g., opinions on a topic, sensor measurements, etc.

Objective: get all nodes to reach agreement on some function of these values

Synchronization in biological networks, clocks, multi-agent flocking, distributed optimization, …

Dynamics of Consensus

25

Many mechanisms proposed for emergence of consensus

Here: look at a very simple linear iterative strategy Studied extensively by the control systems community

At each time-step k, each node xi updates its value as a weighted average of its neighbors values

Mathematically:

Special case: wii = wij= 1/(degi +1) (simple averaging)

)(

][][]1[inbrj

jijiiii kxwkxwkx

Example

x1 x3 x2

x1 x3 x2

x1 x3 x2

x1 x3 x2

k = 0

k = 1

k = 2

k

-1 5 8

2 4 6.5

3 4.17 5.25

26

4.14 4.14 4.14

System-Wide Model

27

Local dynamics based on averaging

Need to model system-wide behavior

Evolution of values of all nodes:

Constraints: wij = 0 if node xj is not a neighbor of node xi

W is a stochastic matrix: all elements of W are nonnegative, and each row sums to 1

][

1

1

111

]1[

1

][

][

]1[

]1[

k

NNNN

N

k

N kx

kx

ww

ww

kx

kx

xWx

Previous Example: System-Wide Model

28

Evolution of values: x[k] = Wx[k-1] = W2x[k-2+ = … = Wkx[0]

Thus:

In this example

x1 x3 x2

1 11 12 2

1 1 12 23 3 3

1 13 32 2

[ 1] [ ]

[ 1] 0 [ ]

[ 1] [ ]

[ 1] 0 [ ]

k k

x k x k

x k x k

x k x k

x xW

lim [ ] lim [0]k

k kk

x W x

32 27 7 7

3 32 2 2 27 7 7 7 7 7

32 27 7 7

1

lim 1

1 T

k

k

d

1

W

[0]

lim [ ] [0]

[0]

T

T

kT

k

d x

x d x

d x

Convergence Based on Properties of Markov

Chains

29

Network-wide dynamics: x[k+1] = Wx[k]

Topology of network is captured by sparsity pattern of W

Since W is a stochastic matrix, can view this system as a Markov Chain [deGroot, 1974]

Standard property of Markov Chain: If network is connected and each node has a positive self-weight,

there exists a unique vector dT such that

dT is the left eigenvector of W for eigenvalue 1: dTW = dT

lim

T

k

kT

d

W

d

Proof of Convergence (cont’d)

30

Thus:

Result:

Rate of convergence is geometric, according to second largest eigenvalue of W

Mixing Time of Markov Chain

lim [ ] lim [0] [0]k T

k kk

x W x 1d x

Network connected and all self weights are positive

All nodes reach consensus on dTx[0], where dTW = dT

Time-Varying Networks

31

What happens if the network is changing over time?

Neighbors of each node change over time

Model for linear strategy:

Weight matrix is now time-varying, but still stochastic at each time-step

View this as a non-homogeneous Markov chain

1 11 1 1

1

[ 1] [ ] [ ]

[ 1] [ ] [ ] [ ]

[ 1] [ ] [ ] [ ]

N

N N NN N

k k k

x k w k w k x k

x k w k w k x k

x W x

Result

32

As long as the network is connected over time, and there is a lower bound on the weights,

dT is some vector, no longer left eigenvector of some matrix

Depends on how the network changes over time

Still an open question to characterize final consensus value in time-varying networks (except for some special cases)

References: Tsitsiklis 1984, Jadbabaie et al., 2003, Ren et al., 2005, Moreau, 2005, Xiao et al., 2004, Olfati-Saber et al., 2007, …

lim [ ] [0]T

kk

x 1d x

Potential for Incorrect Behavior

33

So far: assumed that all nodes in the network behave as expected

“What happens if some nodes don’t follow the specified strategy?”

Simplest case: suppose some node keeps its value constant

Result [Tsitsiklis ‘84, Jadbabaie et al. ‘03, Gupta et al ‘05, …+

As long as the network is connected over time, if some node keeps its value constant, all nodes will converge to the value of that node

Linear strategy is easily influenced by stubborn/malicious agents (?)

Robustness of Linear Iterative

Strategies to Malicious Nodes

Exchanging Information in Arbitrary

Networks: Building Intuition

Node x1 wants to obtain x3’s value

Node x2 is malicious and pretends x3[0] = 9

Node x4 behaves correctly and uses x3[0] = 5

Node x1 doesn’t know who to believe i.e., is node x3’s value equal to 9 or 5?

Node x1 needs another node to act as tie-breaker

35

5]0[3 x

x1

x2 x3

x4

Graph Connectivity

The connectivity of a graph is the maximum number of node-disjoint paths between any two nodes

Menger’s Theorem: If a graph has connectivity k, there is a set of k nodes that disconnects the graph This set of nodes is called a vertex cut

Connectivity: 1 Connectivity: 2 Connectivity: 3

36

x1

x2

x3

x4

x5 x1

x2

x4

x3

x1

x2

x3

x4

Main Results

37

In fixed networks with up to f malicious nodes, we show:

Node xi has 2f or fewer node-disjoint paths from

some node xj

f malicious nodes can update their values in such a way that xi cannot calculate any function of

xj’s initial value

Node xi has 2f+1 or more node-disjoint paths from

every other node

xi can obtain all initial values after running linear strategy for

at most N time-steps with almost any weights

Main Results

38

In fixed networks with up to f malicious nodes, we show:

Node xi has 2f or fewer node-disjoint paths from

some node xj

f malicious nodes can update their values in such a way that xi cannot calculate any function of

xj’s initial value

Node xi has 2f+1 or more node-disjoint paths from

every other node

“Easy”

“Tricky”

xi can obtain all initial values after running linear strategy for

at most N time-steps with almost any weights

Modeling Faulty/Malicious Behavior in

Linear Iterative Strategies

39

Linear iterative strategy for information dissemination:

Correct update equation for node xi:

Faulty or malicious update by node xi:

ui[k] is an additive error at time-step k

Note: this model allows node xi to update its value in a completely arbitrary manner

)(

][][]1[inbrj

jijiiii kxwkxwkx

( )

[ 1] [ ] [ ] [ ]i ii i ij j i

j nbr i

x k w x k w x k u k

Modeling the Values Seen by Each Node

40

Each node obtains neighbors’ values at each time-step

Let yi[k] = Cix[k] denote values seen by xi at time-step k

Rows of Ci index portions of x[k] available to xi

][

1000

0100

0001

][

][

][

][

3

4

3

1

3 k

kx

kx

kx

k xy

C

For node x3:

x4

x3

x2

x1

Linear Iteration with Faulty/Malicious

Nodes

41

Let S = {xi1, xi2

, …, xif} be set of faulty/malicious nodes

Unknown a priori, but bounded by f

Update equation for entire system:

1

2

1 2

1 11 1 1

1

[ 1] [ ]

[ ]

[ ][ 1] [ ]

[ ]

[ 1] [ ][ ]

[ ] [ ]

f

S

f

S

i

Ni

i i i

N N NN N

ik k

k

i i

u kx k w w x k

u k

x k w w x ku k

k k

B

x W x

u

e e e

y C x

Constraint: weight wij = 0 if node xj is not a neighbor of node xi

Recovering the Initial State

42

System model for linear iteration with malicious nodes

This is a linear dynamical system

Use tools from linear system theory to analyze behavior

[ 1] [ ] [ ]

[ ] [ ]

S S

i i

k k k

k k

x Wx B u

y C x

Using Structured System Theory to Prove

Resilience of Linear Iterations

43

To prove resilience, we use the following approach:

Linear strategy is resilient to f malicious nodes in a given network

Network has connectivity 2f+1

A linear system has some property “P”

(Structured System Theory)

Background on

Linear and Structured

System Theory

Properties of Linear Systems

45

Controllability: can the state be driven to any desired value using some sequence of inputs?

Observability: does the output trajectory uniquely specify the state of the system, when the inputs are known (or zero)?

Strong Observability: does the output trajectory uniquely specify the state of the system, when the inputs are unknown?

Invertibility: does the output trajectory uniquely specify the input of the system, when the state is known?

][][

][][]1[

kCxky

kBukAxkx

State: x 2 n,

Output: y 2 p,

Input: u 2 m

Properties of Linear Systems

46

Controllability: can the state be driven to any desired value using some sequence of inputs?

Observability: does the output trajectory uniquely specify the state of the system, when the inputs are known (or zero)?

Strong Observability: does the output trajectory uniquely specify the state of the system, when the inputs are unknown?

Invertibility: does the output trajectory uniquely specify the input of the system, when the state is known?

][][

][][]1[

kCxky

kBukAxkx

State: x 2 n,

Output: y 2 p,

Input: u 2 m

Standard Approach: Use algebraic tests to determine if properties hold

Linear Structured Systems

47

System is structured if every entry of the matrices (A,B,C) is either zero, or an independent free parameter

Used to represent and analyze dynamical systems with unknown/uncertain parameters *Lin ‘74, Dion et al., ‘03+

Structured system theory: determines properties of systems based on the zero/nonzero structure of matrices

][][

][][]1[

kCxky

kBukAxkx

mpn uyx ,,

Structural Properties

48

“Structured system has property P”: Property P holds for at least one choice of free parameters in the matrices (A, B, C)

Structural properties are generic!

Use graph based techniques to determine if structural properties hold

Structured system has property P

Structured system will have property P for almost any choice

of free parameters

Example of Structured System and

Associated Graph

49

Structured system can be represented as a graph

Structured system:

Associated graph H:

1 2 7

3 4 5

8

6

0 0 0

0 0 0[ 1] [ ] [ ],

0 0 0 0 0

0 0 0 0 0

x k x k u k

][

000

000

000

][

11

10

9

kxky

1

2

3 4 Output

vertices: Y

State vertices: X

Input

vertices: U

Example: Test for Structural Invertibility

50

Theorem [van der Woude, ’91]:

e.g.,

System is structurally invertible

Graph H has m node-disjoint paths from inputs to outputs

J. W. van der Woude, Mathematics of Control Systems and Signals, 1991

1

2

3 4 Output

vertices: Y

State vertices: X

Input

vertices: U

Example: Test for Structural Invertibility

51

Theorem [van der Woude, ’91]:

e.g.,

System is structurally invertible

Graph H has m node-disjoint paths from inputs to outputs

J. W. van der Woude, Mathematics of Control Systems and Signals, 1991

1

2

3 4 Output

vertices: Y

State vertices: X

Input

vertices: U

Structurally Invertible

References on Structured Systems

52

C. T. Lin, “Structural Controllability”, IEEE TAC, 1974

K. J. Reinschke, Multivariable Control: A Graph-Theoretic Approach, 1988

J-M. Dion, C. Commault and J. van der Woude, “Generic Properties and Control of Linear Structured Systems: A Survey”, Automatica, 2003

D. D. Siljak, Decentralized Control of Complex Systems, 1991

Sundaram & Hadjicostis, CDC 2009, ACC 2010 (Structural properties over finite fields, upper bound on generic controllability/observability indices)

Application to Resilient

Information Dissemination

Recovering the Initial State

54

System model for linear iteration with malicious nodes

Objective: Recover initial state x[0] from outputs of the system, without knowing uS[k]

Almost equivalent to strong observability of system

The set S is also unknown here

Only know it has at most f elements

[ 1] [ ] [ ]

[ ] [ ]

S S

i i

k k k

k k

x Wx B u

y C x

Recovering the Initial State

55

Want to ensure that the output trajectory uniquely specifies the initial state

Same output trajectory must not be generated by two different initial states and two (possibly) different sets of f malicious nodes

By linearity, can show:

[ 1] [ ] [ ]

[ ] [ ]

S S

i i

k k k

k k

x Wx B u

y C x

Linear system

is strongly observable for

any known set Q of 2f nodes

Can recover initial state in system

for any unknown set S of f nodes

[ 1] [ ] [ ]

[ ] [ ]

Q Q

i i

k k k

k k

x Wx B u

y C x

Structural Strong Observability

56

For any set Q of 2f nodes, strong observability of

is a structural property

Graph of system is given by graph of original network, with additional inputs and outputs

Using tests for structural strong observability, we show:

[ 1] [ ] [ ]

[ ] [ ]

Q Q

i i

k k k

k k

x Wx B u

y C x

xi has 2f+1 node-disjoint paths from every other

node

Above system will be strongly observable for any set Q of 2f

nodes

By generic nature of structural properties:

How long will each node need to wait before the values that it receives uniquely specifies the initial state?

If a linear system is strongly observable, outputs of system over N time-steps are sufficient to determine initial state

Any node can obtain x[0] after at most N time-steps

Robustness of the Linear Iterative Scheme

57

Network is 2f+1 connected

For almost any W, any node can recover all initial values despite actions

of f malicious nodes

Summary

58

We know linear iterative strategies are quite powerful if:

Network is fixed

All nodes know the entire network

All nodes know the linear combinations used by other normal nodes

Each node can store a lot of data and do extensive computations

Would like to relax these conditions

Study more “natural” mechanisms to produce robustness

Nodes should require no knowledge of network

Ongoing Research

59

Tradeoff between knowledge of network and ability to overcome malicious behavior

New (relaxed) objective:

“All normal nodes should reach consensus on some value that is between the smallest and largest initial values of

the normal nodes”

Malicious nodes should not be able to bias the consensus value excessively

f-Local Model

60

For large networks, also want to allow the possibility of a large number of malicious nodes

f-local malicious model: allow up to f malicious nodes in every neighborhood

Natural strategy: have each normal node be “suspicious” of extreme values in its neighborhood

Remove the f highest and lowest values in its neighborhood, and take weighted average of remaining values

( )

[ 1] [ ] [ ] [ ] [ ]i ii i ij j

j nbr i

x k w k x k w k x k

Neighbors after removing

extreme values

Convergence

61

Under what conditions will this strategy work?

Fact: 2f+1 connectivity is no longer sufficient

Connectivity of graph is n/2, but no node ever uses a value

from opposite set

Fully-connected graph with n/2 nodes

Initial value 0

Fully-connected graph with n/2 nodes

Initial value 1

One-to-one edges between sets

Similarities with Cascade Problem

62

Similar phenomenon to “clusters” seen earlier in the cascade problem

Graph contains sets where no node in the set has enough neighbors outside

Key differences:

Every neighbor of every node might have a different value in our setting: “Filtering” rule versus “Threshold” rule

No malicious nodes in the cascade problem

Robust Graphs

63

We introduce the following definitions

A set S is r-reachable if it has a node that has at least r neighbors outside the set

A graph S is r-robust if for any two disjoint subsets, at least one of the sets is r-reachable

Preliminary results:

Graph is (3f+1)-robust Normal nodes will reach

consensus despite actions of any f-local set of malicious nodes

Graph is less than (f+1)-robust

Normal nodes will not reach consensus for some initial values

Ongoing Research

64

Try to narrow the gap between sufficient and necessary robustness conditions

Characterize robustness of typical complex networks

Erdos-Renyi networks

Scale-free networks

Use “r-robust” networks to characterize behavior of other information diffusion dynamics

Thanks!