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Complex networks (both natural and engineered) arise as a result of local interactions between various agents. The efficacy of these networks is often predicated on their ability to diffuse information throughout the network, allowing the agents to reach agreement on an appropriate quantity of interest. In this context, a key metric is the susceptibility (or resiliency) of the network to a few individuals who wish to affect global decisions via their local actions. In this talk, I provide an overview of some recent approaches to analyzing the dynamics of information propagation in networks. I describe how tools from Markov chain theory, linear system theory, and structured system theory can be applied to analyze global behaviour arising from a certain class of linear dynamics, and examine the effect of the network's topology on its resilience to misbehaving agents. I conclude by highlighting some areas for further research.
Citation preview
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!