Semi-Autonomous NetworksNetwork Resilience and Adaptive Trees

Airlie Chapman and Mehran Mesbahi

Distributed Space Systems Lab (DSSL)

University of Washington

Airlie Chapman and Mehran Mesbahi (Distributed Space Systems Lab (DSSL))Semi-Autonomous Networks University of Washington 1 / 16

Motivation

Network structure links to system properties withinnetworked multi-agent system

Particularly strong in consensus-type coordinationalgorithms

Semi-autonomous systems are formed fromconsensus-type systems by introducingleaders/in�uencing agents

�How does network structure e�ect theperformance of semi-autonomous systems?�

�Can you dynamically rewire the network structureto improve or degrade the performance ofleaders/in�uencing agents?�

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Airlie Chapman and Mehran Mesbahi (Distributed Space Systems Lab (DSSL))Semi-Autonomous Networks University of Washington 2 / 16

Outline

Form the problem as a input-output dynamic system

Introduce measures to quantify the system's performance to in�uencingagents

Relate these measures to the network structure via an equivalent electricalnetwork

Form local edge swap protocols to alter these measures

Use game theoretic techniques to quantify the protocol's success

Airlie Chapman and Mehran Mesbahi (Distributed Space Systems Lab (DSSL))Semi-Autonomous Networks University of Washington 3 / 16

Graphs and Consensus Model

Graph structure: G = (V ,E ), where V ∈ Rn and E ∈ ReMatrix representation L(G) ∈ Rn×n where

[L(G)]ij =

di i = j

−1 {i , j} ∈ E0 otherwise

and di is the degree of node vi .

Consensus Model

ẋi (t) = ∑{i ,j}∈E

(xj(t)−xi (t)) ⇐⇒ ẋ(t) =−L(G)x(t)

Airlie Chapman and Mehran Mesbahi (Distributed Space Systems Lab (DSSL))Semi-Autonomous Networks University of Washington 4 / 16

Semi-Autonomous Model

In�uencing node setR= (R,ER), |ER |= rEach agent in R is attachedto exactly one agent in V

Common mean, gaussian input

Dirichlet Matrix

A(G,R) =−(L(G) +B(R)B(R)T

)

In�uence Model

ẋ(t) = A(G,R)x(t) +B(R)u(t)

Example:

A(G,R) =

−3 1 1 01 −2 1 01 1 −3 10 0 1 −2

,

B(R) = 1 00 00 00 1

Airlie Chapman and Mehran Mesbahi (Distributed Space Systems Lab (DSSL))Semi-Autonomous Networks University of Washington 5 / 16

Nomenclature

Attachment points: π (ER) is theagents that the in�uence agentsdirectly attach.

Main Path agents: M is the set ofagents that lie on any of the shortestpaths between agents in R.Main neighbors: Γ(vi ) is the closestagents to vi that is inM.

E�ective resistance: Ee� (vi ) is thepotential drop between vi and R,when a 1 Amp current source isconnected across them.

π(ER) = {v1,v3},M= {v1,v2,v3}Γ(v4) = Γ(v5) = Γ(v6) = v2

1r

2r

1v2v

3v

4v

1 Amp

1r

2r

1v2v

3v

4v

Airlie Chapman and Mehran Mesbahi (Distributed Space Systems Lab (DSSL))Semi-Autonomous Networks University of Washington 6 / 16

In�uence Measures

Mean tracking measure is the average mean ofthe error to steer the entire network to the originover an in�nite time horizon:

Jµ (G,R) = 2E‖z(0)‖=1∫ ∞0

z(t)T z(t)dt.

Variance damping measure is the averagevariance of the error due to external agentsinjecting white noise as t→ ∞:

Jσ (G,R) =2

nlimt→∞

Ez(t)T z(t).

Error signal z(t) = x(t)−uc1, where uc ∈R is thecommon mean of u(t).

Airlie Chapman and Mehran Mesbahi (Distributed Space Systems Lab (DSSL))Semi-Autonomous Networks University of Washington 7 / 16

Mean Tracking Measure

Jµ (G,R) := 2E‖z(0)‖=1∫ ∞0

z(t)T z(t)dt.

Equivalent expression

Jµ (G,R) =1

ntr(−A(G,R)−1

),

and another related to the graph structure through the e�ective resistance as

E�ective Resistance Representation

Jµ (G,R) =1

n

n

∑i=1

Ee� (vi ) .

Airlie Chapman and Mehran Mesbahi (Distributed Space Systems Lab (DSSL))Semi-Autonomous Networks University of Washington 8 / 16

Tree networks

A distance interpretation

Jµ (T ,R) =1

n( ∑vi∈M

Ee� (vi )

+ ∑vi /∈M

[Ee� (Γ(vi )) +d (vi ,Γ(vi ))

]).

A single in�uence agent attached to vi

Jµ (T ,Ri ) =1

n

(n

∑j=1

d (vi ,vj) +n

)

=n−1n

c (vi ,T ) +1.

where c (vi ,G) is the closeness centrality of vi , i.e., the average distance betweenvi and all other nodes in the graph.

General Bounds for n-tree are 2− 1n≤ Jµ (T ,Ri )≤ 12 (n+1) .

Airlie Chapman and Mehran Mesbahi (Distributed Space Systems Lab (DSSL))Semi-Autonomous Networks University of Washington 9 / 16

Variance Damping Measure

Jσ (G,R) :=2

nlimt→∞

Ez(t)T z(t).

Equivalent expression

Jσ (G,R) =2

ntr(P(G,R)),

where P (G,R) is the controllability gramian

P (G,R) :=∫ ∞0

eA(G,R)τB (G,R)B (G,R)T eA(G,R)T τdt,

and another related to the graph structure through the e�ective resistance as

E�ective Resistance Representation

Jσ (G,R) =1

n ∑vi∈π(ER )

Ee� (vi ) .

A single external agent is attach to vi for any G then, Jσ(G,Ri

)= 1

n.

Airlie Chapman and Mehran Mesbahi (Distributed Space Systems Lab (DSSL))Semi-Autonomous Networks University of Washington 10 / 16

Network Design Problem

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Maintain connectedgraph

Decentralized, parallel,asynchronous

Local agent informationof the graph structure

I (vi ) is the neighborsof vi closer to somein�uence agent than vi .

Airlie Chapman and Mehran Mesbahi (Distributed Space Systems Lab (DSSL))Semi-Autonomous Networks University of Washington 11 / 16

Balancing Measures

Relationship between measures

Jµ (G,R) = Jσ (G,R) +1

n ∑vi /∈π(ER )

Ee� (vi ) .

For security, large Jµ (G,R) (resistance to external control) and smallJσ (G,R) (external noise damping) is favorable.Approach: For trees - increase ∑vi /∈π(ER )Ee� (vi ) while keeping Jσ (G,R)small.

Compacts the main path (Protocol 3)Elongates the remaining graph (Protocol 1)

Airlie Chapman and Mehran Mesbahi (Distributed Space Systems Lab (DSSL))Semi-Autonomous Networks University of Washington 12 / 16

Distributed Protocol for Trees

If ∃vj ,vk ∈N (vi ) , vj 6= vk ,Protocol 1: {(vj ,vk /∈ I (vi )} - Increases Jµ (T ,R)Protocol 3: {vj ,vk ∈ I (vi ) and vi /∈ π (ER)} - Decreases Jσ (T ,R)Protocol 5: either condition of Protocol 1 and 3 - Guarantees?

Then remove edge eij and add edge ejk .

Airlie Chapman and Mehran Mesbahi (Distributed Space Systems Lab (DSSL))Semi-Autonomous Networks University of Washington 13 / 16

Simulation of Protocol 5

0 20 40 602

4

6

8

10

12

14

Number of Edge Swaps

Jµ(T

,R)

0 20 40 600.085

0.09

0.095

0.1

0.105

0.11

0.115

Number of Edge Swaps

Jσ(T

,R)

Protocol 1Protocol 3Protocol 5Optimal Graph

0 20 40 602

4

6

8

10

12

14

Number of Edge Swaps

Jµ(T

,R)

0 20 40 600.085

0.09

0.095

0.1

0.105

0.11

0.115

Number of Edge Swaps

Jσ(T

,R)

Protocol 1Protocol 3Protocol 5Optimal Graph

Airlie Chapman and Mehran Mesbahi (Distributed Space Systems Lab (DSSL))Semi-Autonomous Networks University of Washington 14 / 16

hybrid.mp4Media File (video/mp4)

Price of Stability (PoS) and Anarchy (PoA)

Protocol 5 is a �nite signed potential game =⇒ at least one Nash equilibrium.

De�nition

With the objective to minimize some function value then

PoS =Best Equilibrium Value

Optimal Valueand PoA =

Worst Equilibrium Value

Optimal Value.

For protocol 5:

With cost 1/Jµ (T ,R) the PoS= 1 and PoA≤ r .With cost Jσ (T ,R) the PoS= 1 and PoA< 11

√5

20 ≈ 1.23.(Consequence: For r = 1, protocol 5 will always reach the optimal value for1/Jµ (T ,R).)

Airlie Chapman and Mehran Mesbahi (Distributed Space Systems Lab (DSSL))Semi-Autonomous Networks University of Washington 15 / 16

Conclusion

Provided links between the e�ciency of semi-autonomous systems and theunderlying network structure via measures Jµ (G,R) and Jσ (G,R).Proposed local protocols involving adjacent edge swaps that predictably alterthese measures.

Pending Questions: How about...

simultaneous friendly and malicious in�uence agents?

generalized graphs?

intelligent in�uencing agents?

Airlie Chapman and Mehran Mesbahi (Distributed Space Systems Lab (DSSL))Semi-Autonomous Networks University of Washington 16 / 16