Dynamics and control on networks with antagonistic ... · distributed system: ... December 2012...

GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 1/56

Dynamics and control on networks with antagonisticinteractions

Claudio AltafiniSISSA, Trieste http://people.sissa.it/∼altafini

http://people.sissa.it/~altafini

Networks with signed interactions

Dynamics of opinon forming

Balance of large networks

Essentially nonnegative systems

Bipartite consensus

Distributed dynamical sorting

Conclusion


Outline

■ Network with antagonistic relationships: signed graph

■ Dynamics of opinion forming in structurally balanced socialnetworks

■ Structural balance of large-scale social networks

■ Agreement for non-structurally balanced cases: essentiallynonnegative systems

■ Consensus problems on networks with antagonisticinteractions

■ Distributed dynamical sorting





Bipartite consensus


Conclusion



■ an implicit assumption of distributed control:all agents cooperate to achieve a goal

■ cooperation in a model:nonnegative adjacency matrix

■ in many contexts: cooperation and antagonism

BiologicalNetworks

cellsize

Cdh1

Swi5

Clb1,2

Mcm1/SFF

Sic1

Cln1,2

MFB

SBFCln3

+

+

Cdc20

++

+

+

+

++

+

Clb5,6

+

++

_

_

_

_

_

_ _ _

_

_ _

_

++ +

_ _

Social Networks Competing systems





Bipartite consensus


Conclusion



■ positive edge = friendly relationship◆ cooperation◆ alliance◆ trust

■ negative edge = unfriendly relationship◆ competition◆ rivalry◆ mistrust

=⇒ signed adjacency matrix

A = (aij) aij ≶ 0





Bipartite consensus


Conclusion



dynamical problems studied for signed graphs:■ multistationarity: counting the number of equilibrium points

■ periodicity: necessary but not sufficient conditions

■ qualitative stability: asymptotic stabilityfor the "qualitative class" of matrices



● Outline

● Dynamics of opinion forming

● Cooperative systems

● Structural balance

● Monotonicity



Bipartite consensus


Conclusion


Outline









● Outline




● Monotonicity



Bipartite consensus


Conclusion


Opinion forming in signed social networks■ Task:

◆ describe process of opinion forming on a social network◆ information available: neighbors are "friends/enemies"◆ assumption: agents form their opinion based on the

influences of their neighbours



● Outline




● Monotonicity



Bipartite consensus


Conclusion


Opinion forming in signed social networks■ Task:

◆ describe process of opinion forming on a social network◆ information available: neighbors are "friends/enemies"◆ assumption: agents form their opinion based on the

influences of their neighbours

■ Model◆ continuous time dynamical system

x = f(x) x = vector of opinions x ∈ Rn

◆ distributed system: only neighbours can influence theopinion

fi(x) = fi(xj , j ∈ adj(i))

◆ Jacobian terms:

Fij(x) =∂fi(x)

∂xj

= influence of j-th individual on i-th individual



● Outline




● Monotonicity



Bipartite consensus


Conclusion


Opinion forming from social relationships

Assumption: the influence of neighbors reflects their socialrelationship

Assumptions:■ the influence of a friend is positive

∂fi(x)

∂xj> 0 ⇐⇒ aij > 0

■ the influence of an enemy is negative

∂fi(x)

∂xj< 0 ⇐⇒ aij < 0

■ influences do not change sign for different values of x■ negative "self-influence" (forgetting factor)

∂fi(x)

∂xi< 0

=⇒ sgn(Fij) = sgn(aij)



● Outline




● Monotonicity



Bipartite consensus


Conclusion


More specific: distributed influences

Simplifying assumptions for the opinion forming dynamics■ influences are distributed

xi = fi(x) = fi(xi, xj , j ∈ adj(i))



● Outline




● Monotonicity



Bipartite consensus


Conclusion





■ influences are additive

xi = fi(x) =

n∑

j=1

aijψij(xj)− λixi



● Outline




● Monotonicity



Bipartite consensus


Conclusion





■ influences are additive

xi = fi(x) =

n∑

j=1

aijψij(xj)− λixi

■ example of ψij(xj):

ψij(xj) =xj

θj + |xj |

=⇒∂ψj(xj)

∂xj=

θj(θj + |xj |)2

> 0

other possibilities for ψ: odd polynomials, tanh



● Outline




● Monotonicity



Bipartite consensus


Conclusion



■ if aij = {±1}

xi =∑

j∈friends(i)

ψij(xj)−∑

j∈advers(i)

ψij(xj)− λixi

=⇒ Fij(x) =∂fi(x)

∂xj=

{

θj(θj+|xj |)2

> 0 if j ∈ friends(i)−θj

(θj+|xj |)2< 0 if j ∈ advers(i)



● Outline




● Monotonicity



Bipartite consensus


Conclusion



■ if aij = {±1}

xi =∑

j∈friends(i)

ψij(xj)−∑

j∈advers(i)

ψij(xj)− λixi

=⇒ Fij(x) =∂fi(x)

∂xj=

{

θj(θj+|xj |)2

> 0 if j ∈ friends(i)−θj

(θj+|xj |)2< 0 if j ∈ advers(i)

■ if all ψij are equal: Persidskii systems

x = Aψ(x)− Λx ψ(x) =

ψ(x1)...

ψ(xn)



● Outline




● Monotonicity



Bipartite consensus


Conclusion


A plethora of behaviors

Converging

0 50 100 150 200 250 300 350 400 450 500−4

−3

−2

−1

0

1

2

3

4

time

opin

ion

"Diverging"

0 50 100 150 200 250 300 350 400 450 500−200

−150

−100

−50

0

50

100

150

200

250

300

time

opin

ion

Periodic

0 50 100 150 200 250 300 350 400 450 500−8

−6

−4

−2

0

2

4

6

time

opin

ion

Simply weird...

0 50 100 150 200 250 300 350 400 450 500−150

−100

−50

0

50

100

time

opin

ion



● Outline




● Monotonicity



Bipartite consensus


Conclusion


A special case: cooperative systems

■ Cooperating agents (all friends): adjacency matrix Anonnegative

■ if x∗ = limt→∞ x(t) then sgn(x∗i ) = sgn(x∗j )

■ the opinion is stronger for more connected people■ even a single xi(0) 6= 0 can steer the whole community■ if xi(0) > 0 and xj(0) < 0 then one of the two will prevail

(the one with higher connectivity)



● Outline




● Monotonicity



Bipartite consensus


Conclusion


Another case: structurally balanced systems

Structurally balanced social networks:■ outcome of the opinion forming process is completely

predictable only from the adjacency matrix A



● Outline




● Monotonicity



Bipartite consensus


Conclusion


The enemy of my enemy...■ in social network theory: certain social relationships

(represented as signed graphs) are "more stressful" thanothers

■ generalization to any graph =⇒structural balance



● Outline




● Monotonicity



Bipartite consensus


Conclusion


Structural balance

Definition A signed graph G(A) = {V , E , A}is said structurally balanced if ∃ partition ofthe nodes V1, V2, V1 ∪ V2 = V, V1 ∩ V2 = 0such that■ aij > 0 ∀ vi, vj ∈ Vq ,■ aij 6 0 ∀ vi ∈ Vq, vj ∈ Vr, q 6= r .It is said structurally unbalanced otherwise.

■ two individuals on the same side of the cut set are "friends"■ two individuals on differend sides of the cut set are

"enemies"

D. Cartwright and F. Harrary, Structural balance: a generalization of Heider’s Theory, Psychological Review, 1956.

D. Easley and J. Kleinberg, Networks, Crowds, and Markets. Reasoning About a Highly Connected World, Cambridge Univ.

Press, 2010



● Outline




● Monotonicity



Bipartite consensus


Conclusion


Examples

Two-partyparlamentary

systems Team sports

Internationalalliances

■ Task: understand why opinon forming is so predictible inthese cases



● Outline




● Monotonicity



Bipartite consensus


Conclusion


Structural balance

Lemma A signed graph G(A) is structurally balanced iff any of thefollowing equivalent conditions holds:1. all cycles of G(A) are positive;2. ∃ a diagonal signature matrix D = diag(±1) such that DAD has all

nonnegative entries;



● Outline




● Monotonicity



Bipartite consensus


Conclusion


Monotone dynamics & structural balance

■ Orthant of Rn determined by the partial order σ = {±1}

Kσ = {x ∈ Rn such that diag(σ)x > 0}

■ partial order generated by σ = {±1}: "6σ"

x1 6σ x2 ⇐⇒ x2 − x1 ∈ Kσ

■ strict partial order (i.e., strict inequality)along all coordinates: "≪σ"

Definition A system x = f(x) is said monotonewith respect to the partial order σ = {±1} if forall initial conditions x1, x2 such that x1 6σ x2 onehas φt(x1) 6σ φt(x2) ∀ t > 0.

It is said strongly monotone with respect to thepartial order σ = {±1} if for all initial conditionsx1, x2 such that x1 6σ x2, x1 6= x2 one hasφt(x1) ≪σ φ

t(x2) ∀ t > 0.



● Outline




● Monotonicity



Bipartite consensus


Conclusion



■ monotone system:◆ since order is respected ∀ t > 0, dynamical evolution is

completely predictable◆ outcome is robust to perturbations◆ strong monotonicity = monotonicity + strong connectivity



● Outline




● Monotonicity



Bipartite consensus


Conclusion



■ monotone system:◆ since order is respected ∀ t > 0, dynamical evolution is

completely predictable◆ outcome is robust to perturbations◆ strong monotonicity = monotonicity + strong connectivity

■ graphical characterization◆ Jacobian F (x) = ∂f(x)

∂x

◆ Assumption: Fij(x) sign constant ∀ x◆ Assumption: G(F (x)) is strongly connected ∀ x

Proposition The system x = f(x) is monotone iff any of the followingconditions holds:1. ∃D = diag(±1) such that DF (Dx)D has all nonnegative entries

∀x ∈ Rn;2. all directed cycles of G(F (x)) are positive ∀x ∈ Rn;3. G(F (x)) is structurally balanced ∀x ∈ Rn.

◆ property is independent of the details of the dynamics



● Outline




● Monotonicity



Bipartite consensus


Conclusion


Structural balance for Persidskii systems

■ Persidskii systems x = Aψ(x)− Λx

F (x) = A∂ψ(x)∂x

=⇒ sgn(F (x)) = sgn(A)

=⇒ monotonicity = structural balance of A



● Outline




● Monotonicity



Bipartite consensus


Conclusion




F (x) = A∂ψ(x)∂x



■ dynamics of A and of DAD is the same, up to the opinion signs



● Outline




● Monotonicity



Bipartite consensus


Conclusion




F (x) = A∂ψ(x)∂x



■ dynamics of A and of DAD is the same, up to the opinion signs

■ strength of an opinion: not number of friends but connectivity





Bipartite consensus


Conclusion


Outline











Bipartite consensus


Conclusion


Large-scale signed social networks

■ Question: are real networks structurally balanced?





Bipartite consensus


Conclusion



■ Question: are real networks structurally balanced?■ for social networks: a few large-scale datasets are available

◆ Epinions = trust/distrust network among users of productreview web site Epinions

◆ Slashdot = friend/foes network of the technological newssite Slashdot

◆ WikiElections = election of admin among Wikipedia users

Network nodes edges − edges + edges

Epinions 131513 708507 118619 589888Slashdot 82062 498532 117599 380933

WikiElections 7114 100321 21529 78792





Bipartite consensus


Conclusion



■ Question: are real networks structurally balanced?■ for social networks: a few large-scale datasets are available

◆ Epinions = trust/distrust network among users of productreview web site Epinions

◆ Slashdot = friend/foes network of the technological newssite Slashdot

◆ WikiElections = election of admin among Wikipedia users

Network nodes edges − edges + edges

Epinions 131513 708507 118619 589888Slashdot 82062 498532 117599 380933

WikiElections 7114 100321 21529 78792

■ peculiarity: no "intelligent design" behind, only anaggregation of local choices





Bipartite consensus


Conclusion


Computing structural balance

■ networks are not exactly balance (∃ negative cycles)■ level of structural balance = min. n. of edges whose sign

change renders the network structurally blanced

Example

■ computation is NP-hard■ heuristics:

◆ direct approach: counting cycles −→ unfeasible◆ in statistical physics: computing the ground state of an

Ising spin glass◆ in computer science: MAX-CUT or MAX-XORSAT

problems





Bipartite consensus


Conclusion


Computing the level of structural balance

MAX-XORSAT: maximizing satisfied linear constraints over Z2

Example: n = 5, m = 7

F =

1 1 0 0 0

0 1 1 0 0

0 0 1 1 0

0 0 0 1 1

1 0 0 0 1

1 0 0 1 0

0 1 0 0 1

g =

1

1

0

0

1

1

0

XOR-SAT problem: Fx⊕ g = 0

MAX-XORSAT problem: minx∈Zn2|Fx⊕ g|

xopt =

1

0

0

0

0

=⇒





Bipartite consensus


Conclusion


Evaluate the level of structural balance

■ comparing with null models

Epinions

50000 50500 51000 51500 104000 104500 105000 1055000

5

10

15

20

h(s)

coun

t (%

)

∫∫

∫∫

Slashdots

68000 70000 72000 74000 76000 102000 104000 106000 1080000

5

10

15

20

h(s)

coun

t (%

)

∫ ∫

∫ ∫

WikiElections

14000 14200 14400 14600 14800 20000 20200 20400 20600 20800 210000

10

20

30

40

50

60

70

h(s)

coun

t (%

)

∫∫

∫∫

■ using the Shannon bound of rate-distortion theory(MAX-XORSAT is a lossy source compression channel)

Epinions

0 0.2 0.4 0.6 0.80

0.05

0.1

0.15

0.2Legend

δnullup

δnulllow

δup

δlow

ACHIEVABLE

UNACHIEVABLE

rate

dist

orsi

on

Slashdots

0 0.2 0.4 0.6 0.80

0.05

0.1

0.15

0.2

0.25Legend

δnullup

δnulllow

δup

δlow

ACHIEVABLE

UNACHIEVABLE

rate

dist

orsi

onWikiElections

0 0.2 0.4 0.6 0.80

0.05

0.1

0.15

0.2

0.25Legend

δnullup

δnulllow

δup

δlow

ACHIEVABLE

UNACHIEVABLE

rate

dist

orsi

on





Bipartite consensus


Conclusion


Level of structural balance

Epinions Slashdots WikiElections

■ some nodes have many friends, other many enemies





Bipartite consensus


Conclusion


Apparent disorder and structural balance■ skewed sign distributions implies "apparent disorder": with a

diagonal similarity transformation they disappear

A −→ DAD

■ at the global scale: "order" out of the free local choices





Bipartite consensus


Conclusion


Apparent disorder and structural balance■ skewed sign distributions implies "apparent disorder": with a

diagonal similarity transformation they disappear

A −→ DAD

■ at the global scale: "order" out of the free local choices■ in the imaginary of the national press....





● Outline

Bipartite consensus


Conclusion


Outline











● Outline

Bipartite consensus


Conclusion


Essentially nonnegative matrices

■ main feature of a cooperative system:all agents have the same opinionon a subject =⇒ agreement

■ reason: Perron-Frobenius property

Av = ρ(A)vρ(A) = spectral radius of Av > 0 P.F. eigenvector

■ Question: are there other classes of matrices that have thisproperty?

Definition A matrix A is said eventually nonnegative if it is non-nilpotent and if ∃ ko ∈ N s.t. Ak nonnegative ∀ k > ko

=⇒ ρ(A)I −A is like an M-matrix





● Outline

Bipartite consensus


Conclusion


Essential nonnegativity and agreement

Theorem Given A essentially nonnegative, consider the system

x = −(Λ−A)x, Λ =

λ1. . .

λn

■ λi > ρ(A) ∀ i and λi > ρ(A) for some i =⇒ limt→∞x(t) = 0

■ λi = ρ(A) ∀ i =⇒ limt→∞x(t) = wTx(0)v

■ λi 6 ρ(A) ∀ i and λi < ρ(A) for some i =⇒ limt→∞x(t) = ∞

■ for linear systems: case of critical but not asymptotic stabilityare "rare"

■ divergence for λi 6 ρ(A) may disappear if nonlinear"saturated" functions are used

■ open problem: understand when this happens whilepreserving agreement





● Outline

Bipartite consensus


Conclusion


Example: essentially nonnegative

A =

0 −1 3 −1

1 0 0 0

0 2 0 1

0 0 2 0

s. t. Ak nonnegative for k > 45

■ the linear systemx = ρ(A)I −Ais critically stable

0 2 4 6 8 10−5

−4

−3

−2

−1

0

1

2

t

x

1

x2

x3

x4

0 2 4 6 8 10−6

−4

−2

0

2

4

t

x1

x2

x3

x4

■ the linear systemx = −(Λ−A)xwith λi < ρ(A)is unstable

■ the Persidskii systemx = −Λx+Aψ(x)with λi < ρ(A) appearsto preserve agreement....

0 10 20 30 40−6

−4

−2

0

2

4

t0 10 20 30 40

−5

0

5

t





Bipartite consensus

● Laplacian

● Examples

● Bipartite consensus

● Directed graphs

● Stability of L

● Examples

● Nonlinear Laplacians

● Monotonicity


Conclusion


Outline











Bipartite consensus

● Laplacian

● Examples


● Directed graphs

● Stability of L

● Examples


● Monotonicity


Conclusion


Undirected graph case

■ distributed consensus problem on undirected (connected)graph G(A) = {V , E , A}

xi = ui

■ "standard" average consensus solution:

xi =∑

j∈adj(i)

aij(xi − xj)

■ "standard" Laplacian

x = −Lx ℓik =

{

∑

j∈adj(i) aij k = i

−aik k 6= i

■ properties:◆ L has always λ1(L) = 0 as eigenvalue◆ L need not be stable





Bipartite consensus

● Laplacian

● Examples


● Directed graphs

● Stability of L

● Examples


● Monotonicity


Conclusion


Laplacian for signed graphs

■ for a signed graph the following alternative Laplacian can beused

L = (ℓik) s.t. ℓik =

{

∑

j∈adj(i) |aij | k = i

−aik k 6= i.

■ properties:◆ L is always stable: Re[λi(L)] > 0◆ L may or may not have λ1(L) = 0 as eigenvalue





Bipartite consensus

● Laplacian

● Examples


● Directed graphs

● Stability of L

● Examples


● Monotonicity


Conclusion


Laplacian for signed graphs

■ for a signed graph the following alternative Laplacian can beused

L = (ℓik) s.t. ℓik =

{

∑

j∈adj(i) |aij | k = i

−aik k 6= i.

■ properties:◆ L is always stable: Re[λi(L)] > 0◆ L may or may not have λ1(L) = 0 as eigenvalue

■ to show stability: Laplacian potential is still a sum of squares

Φ(x) = xTLx =∑

(vj ,vi)∈E

(

|aij |x2i + |aij |x

2j − 2aijxixj

)

=∑

(vj ,vi)∈E

|aij | (xi − sgn(aij)xj)2

=⇒ 0 6 λ1(L) < λ2(L) 6 . . . 6 λn(L)





Bipartite consensus

● Laplacian

● Examples


● Directed graphs

● Stability of L

● Examples


● Monotonicity


Conclusion


Behavior for different signed graphs

■ to show that L may or may not have λ1(L) = 0 as eigenvalue

Example 1 : structurally balanced

A1 =

0 1 −2

1 0 −4

−2 −4 0

, L1 =

3

5

6

−A1

=⇒sp(L1) = {0, 4.35, 9.65}

Example 2 : not structurally balanced

A2 =

0 1 −2

1 0 4

−2 4 0

, L2 =

3

5

6

−A2

=⇒sp(L2) = {1.2, 2.61, 10.18}





Bipartite consensus

● Laplacian

● Examples


● Directed graphs

● Stability of L

● Examples


● Monotonicity


Conclusion


Behavior for different signed graphs

■ behavior of x = −Lx

Example 1 : structurally balanced

0 5 10 15 20−2

−1

0

1

2

3

4

5

t

x

1

x2

x3

limt→∞

x(t) =

{

+1

−1

Example 2 : not structurally balanced

0 5 10 15 20−4

−2

0

2

4

t

x

1

x2

x3 lim

t→∞x(t) = 0





Bipartite consensus

● Laplacian

● Examples


● Directed graphs

● Stability of L

● Examples


● Monotonicity


Conclusion


Bipartite consensus

Definition The system x = −Lx admits a bipartite consensus solutionif limt→∞ |xi(t)| = α > 0 ∀ i = 1, . . . , n.

■ graphical condition: structural balance

Lemma A connected signed graph G(A) is structurally balanced iff:1. all cycles of G(A) are positive;2. ∃ a diagonal signature matrix D = diag(±1) such that DAD has all

nonnegative entries;3. 0 is an eigenvalue of L.

Example 1 :

DA1D =

1

1

−1

0 1 −2

1 0 −4

−2 −4 0

1

1

−1

=

0 1 2

1 0 4

2 4 0





Bipartite consensus

● Laplacian

● Examples


● Directed graphs

● Stability of L

● Examples


● Monotonicity


Conclusion


Bipartite consensus: theorem

Corollary A connected signed graph G(A) is structurally unbalanced iffany of the following equivalent conditions holds:1. one or more cycles of G(A) are negative;2. ∄ D ∈ D such that DAD has all nonnegative entries;3. λ1(L) > 0 i.e., L > 0.

Theorem Consider a connected signed graph G(A). The systemx = −Lx admits a bipartite consensus solution iff G(A) is structurallybalanced. In this case

limt→∞

x(t) =1

n

(

1TDx(0)

)

D1

where D ∈ D renders DAD nonnegative.If instead G(A) is structurally unbalanced then limt→∞ x(t) = 0 .





Bipartite consensus

● Laplacian

● Examples


● Directed graphs

● Stability of L

● Examples


● Monotonicity


Conclusion


Extension to directed graphs

■ structural balance for digraphs: look at directed cycles!

Lemma A strongly connected, signed digraph G(A) is structurallybalanced iff any of the following equivalent conditions holds:1. all directed cycles of G(A) are positive;2. ∃D ∈ D such that DAD has all nonnegative entries;3. 0 is an eigenvalue of L.

Theorem Consider a strongly connected, signed digraph G(A). The sys-tem x = −Lx admits a bipartite consensus solution iff G(A) is struc-turally balanced. In this case

limt→∞

x(t) = νTDx(0)D1

where D ∈ D is such that DAD nonnegative. When G(A) is weightbalanced limt→∞ x(t) = 1

n

(

1TDx(0)

)

D1. If instead G(A) is structurallyunbalanced then limt→∞ x(t) = 0





Bipartite consensus

● Laplacian

● Examples


● Directed graphs

● Stability of L

● Examples


● Monotonicity


Conclusion


Examples

Example : structural balanced digraph

0 200 400 600 800 1000−5

0

5

t

limt→∞

x(t) =

{

+1

−1

Example : structural unbalanced digraph

0 200 400 600 800 1000−5

0

5

t

limt→∞

x(t) = 0





Bipartite consensus

● Laplacian

● Examples


● Directed graphs

● Stability of L

● Examples


● Monotonicity


Conclusion


Stability of L and linear algebra

■ conditions on the eigenvalues:1. Geršgorin theorem: eigenvalues are in the union of the

disks{

z ∈ C s.t. |z − ℓii| 6∑

j 6=i

|aij | = ℓii}

.

2. L is diagonally dominant (by rows) i.e.,

|ℓii| >∑

j 6=i

|ℓij |, i = 1, . . . , n

Consequence: z = 0 cannot be in the interior of theGeršgorin disks

=⇒z = 0 always on the boundary of the Geršgorin disks,regardless of structural balance

=⇒nonsingularity and asymptotic stability of L cannot bedetected by standard linear algebra tools! (Geršgorin thm,diagonal dominance, Cassini ovals, Levy-Desplanques thm,etc.)





Bipartite consensus

● Laplacian

● Examples


● Directed graphs

● Stability of L

● Examples


● Monotonicity


Conclusion


Examples

Example 1 : −L1 is critically stable

L1 =

3 −1 2

−1 5 4

2 4 6

Example 2 : −L2 is asymptotically stable

L2 =

3 −1 2

−1 5 −4

2 −4 6

■ matrices have the same Geršgorin disks and the samecomparison matrix





Bipartite consensus

● Laplacian

● Examples


● Directed graphs

● Stability of L

● Examples


● Monotonicity


Conclusion


Stability of diagonally equipotent matrices

■ L is said diagonally equipotent (by rows) i.e.,

|ℓii| =∑

j 6=i

|ℓij |, i = 1, . . . , n

■ diagonally equipotent matrices can be

◆ singular and critically stable⇔ all cycles are positive

◆ nonsingular and asymptotically stable⇔ at least one cycle is negative

■ classification is complete for diagonally equipotent matrices

■ purely graphical condition, like in qualitative stability (butrequires diagonal equipotence)





Bipartite consensus

● Laplacian

● Examples


● Directed graphs

● Stability of L

● Examples


● Monotonicity


Conclusion


Nonlinear Laplacian feedback schemes

■ ∃ several possible nonlinear generalizations of the Laplacianfeedback scheme◆ absolute Laplacian flow

xi = −∑

j∈adj(i)

|aij | (hij(xi)− sgn(aij)hij(xj))

◆ relative Laplacian flow

xi = −∑

j∈adj(i)

|aij |hij(xi − sgn(aij)xj)

where hij is in the class of infinite sector nonlinearities

S ={

h : R → R, (h(ξ)− h(ξ∗)) (ξ − ξ∗) > 0 if ξ 6= ξ∗, h(0) = 0,

and∫ ξ

ξ∗(h(τ)− h(ξ∗))dτ → ∞ as |ξ − ξ∗| → ∞

}





Bipartite consensus

● Laplacian

● Examples


● Directed graphs

● Stability of L

● Examples


● Monotonicity


Conclusion



■ Consider a monotone dynamics

x = f(x)

■ Jacobian F (x) = ∂f(x)∂x

■ Assumption: Fij(x) sign constant ∀ x■ Assumption: G(F (x)) is strongly connected ∀ x

Proposition The system x = f(x) is monotone iff any of the followingconditions holds:1. ∃D ∈ D such that DF (Dx)D has all nonnegative entries ∀x ∈ Rn;2. all directed cycles of G(F (x)) are positive ∀x ∈ Rn;3. G(F (x)) is structurally balanced ∀x ∈ Rn.





Bipartite consensus

● Laplacian

● Examples


● Directed graphs

● Stability of L

● Examples


● Monotonicity


Conclusion


Monotonicity-based bipartite consensus

■ any monotone system can be used to obtain a feedback lawguaranteeing bipartite consensus

Theorem Given a strongly monotone x = f(x), the corresponding Lapla-cian flow

xi = −∑

j∈adj(i)

(|Fij(x)|xi − Fij(x)xj)

= −∑

j∈adj(i)

|Fij(x)| (xi − sign(Fij(x))xj) ,

is globally converging to a bipartite consensus.

■ Example: Persidskii system

xi = −∑

j∈adj(i)

|aij∂ψ(xj)

∂xj| (xi − sign(aij)xj)





Bipartite consensus


● Sorting in Rn

● Sorting in finite time

● Sorting in Rn

● Linear algebra behind sorting

Conclusion


Outline











Bipartite consensus


● Sorting in Rn


● Sorting in Rn


Conclusion


Sorting numbers with a dynamical system

■ analog computation: continuos-time dynamical systems thatexecute an algorithm

■ benchmark task: sorting numbers

Problem Given the order vector p ∈ Rn, construct a dynamicalsystem

x = f(x, p)

s. t. ∀x(0) x∗ = limt→∞ x(t) is "aligned" with the order p

■ most famous example: Brockett double bracket

H = [H, [H, N ]]

where N = N(p)





Bipartite consensus


● Sorting in Rn


● Sorting in Rn


Conclusion


Sorting numbers with a dynamical system

■ analog computation: continuos-time dynamical systems thatexecute an algorithm

■ benchmark task: sorting numbers

Problem Given the order vector p ∈ Rn, construct a dynamicalsystem

x = f(x, p)

s. t. ∀x(0) x∗ = limt→∞ x(t) is "aligned" with the order p

■ most famous example: Brockett double bracket

H = [H, [H, N ]]

where N = N(p)

■ Task: distributed sorting◆ order vector p is not shared by the agents◆ local information: i-th agent knows only pi





Bipartite consensus


● Sorting in Rn


● Sorting in Rn


Conclusion


Distributed sorting in Rn+

■ "standard" consensus problem

x = −Lx ℓik =

{

∑

j 6=i aij k = i

−aik k 6= i

P.F. theorem: L1 = 0 =⇒ limt→∞ x(t) = x∗ ∈ span{1}





Bipartite consensus


● Sorting in Rn


● Sorting in Rn


Conclusion




x = −Lx ℓik =

{

∑

j 6=i aij k = i

−aik k 6= i


■ consider p ∈ Rn+ and write P = diag(p)

L1 = 0 ⇐⇒ LP−1P1 = LP−1p = 0





Bipartite consensus


● Sorting in Rn


● Sorting in Rn


Conclusion




x = −Lx ℓik =

{

∑

j 6=i aij k = i

−aik k 6= i


■ consider p ∈ Rn+ and write P = diag(p)

L1 = 0 ⇐⇒ LP−1P1 = LP−1p = 0

■ calling H = LP−1 =⇒ sorting problem

x = −Hx hik =

{

∑

j 6=i aij/pi k = i

−aik/pk k 6= i

P.F. theorem: Hp = 0 =⇒ limt→∞ x(t) = x∗ ∈ span{p}





Bipartite consensus


● Sorting in Rn


● Sorting in Rn


Conclusion



■ passing from L to H = LP−1:the " agrement subspace" is titled

−10 0 10

−10

−5

0

5

10

x1

x 2

← span(p)

← span(1)

■ practical meaning: right weights on A

xi = −∑

j 6=i

aij

(

xipi

−xjpj

)

■ state transmitted: xj

pj=⇒ scheme is distributed

(weight is known only to the agent that transmits it)





Bipartite consensus


● Sorting in Rn


● Sorting in Rn


Conclusion



Theorem Consider a strongly connected graph G(A) and an order vectorp ∈ Rn+. The system

x = −Hx, H = LP−1

with P = diag(p), is such that

x∗ = limt→∞

x(t) = γp

where γ = ξTx(0), with ξ a left eigenvector of H relative to the 0 eigen-value. For all x(0) ∈ Rn, if γ 6= 0 then x∗ is a p-sorted vector such thatx∗i > 0, i = 1, . . . , n, if γ > 0, and x∗i < 0, i = 1, . . . , n, if γ < 0.

Example: natural orderpi = i, i = 1, . . . , 10

0 10 20 30 40−0.4

−0.2

0

0.2

0.4

t0 10 20 30 40

−0.4

−0.2

0

0.2

0.4

0.6

t





Bipartite consensus


● Sorting in Rn


● Sorting in Rn


Conclusion


Sorting in finite time

■ for continuous-time consensus problem: ∃ nonlinear schemeable to show convergence in finite time

■ adapting these schemes to the sorting problem:

Theorem Consider a d-symmetrizable irreducible A and an order vectorp ∈ Rn+. The system

x = −∑

j 6=i

aijsgn

(

xipi

−xjpj

)∣

∣

∣

∣

xipi

−xjpj

∣

∣

∣

∣

α

, 0 < α < 1

is such that x(t) converges in finite time to γp for γ = ξTx(0), where ξis a left eigenvector of L relative to the 0 eigenvalue such that ξT p = 1.

■ d-symmetrizable A: diag(ω)A = ATdiag(ω) for some ω > 0

■ =⇒ finite-time analog sorter (a "premiere"...)





Bipartite consensus


● Sorting in Rn


● Sorting in Rn


Conclusion


Distributed sorting in Rn

■ when instead p ∈ Rn =⇒ negative weights in H = LP−1

■ must use sign similarities as before

If d = sgn(p), D = diag(d) =⇒ A→ Ad = DAD

Ld s. .t. ℓd,ik =

{

∑

j 6=i |ad,ij | k = i

−ad,ik k 6= i

Theorem Consider a strongly connected graph G(A) and an order vectorp ∈ Rn, pi 6= 0, i = 1, . . . , n. The system

x = −Hdx, Hd = LdP−1

with P = diag(|p|), Ld = DLD is such that

x∗ = limt→∞

x(t) = γp,

where γ = ξTx(0), with ξ is a left eigenvector of Hd relative to the 0eigenvalue. For all x(0) ∈ Rn, if γ 6= 0, x∗ is a p-sorted vector such thatsgn(x∗) = sgn(p) if γ > 0 and sgn(x∗) = −sgn(p) if γ < 0





Bipartite consensus


● Sorting in Rn


● Sorting in Rn


Conclusion


Distributed sorting in Rn

■ if pi < 0 =⇒ put xi on the other side of a diagonalsimilarity partition

xi = −∑

j 6=i

aij

(

xi|pi|

− didjxj|pj |

)

■ corresponding state is "negated", as asked in the sortingproblem

0 10 20 30 40−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

t0 10 20 30 40

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

t





Bipartite consensus


● Sorting in Rn


● Sorting in Rn


Conclusion


Linear algebra behind sorting

■ Ld is diagonally equipotent (by rows)

|ℓd,ii| =∑

j 6=i

|ℓd,ij |, i = 1, . . . , n

■ Q = diag(1/|p|) =⇒Hd = LdQ is generalized diagonally equipotent (by rows)

|hd,ii|qii =∑

j 6=i

|hd,ij |qii, i = 1, . . . , n

■ =⇒ −Hd singular

■ =⇒ −Hd diagonally semistable (but not diagonally stable)

■ =⇒ ker(Hd) = span(p)





Bipartite consensus


Conclusion

● Conclusion


Conclusion

■ when antagonism is present in a distributed system, thepossible dynamical behaviors become usually unpredictable,except for special cases where Perron-Frobenius-likearguments are applicable:

1. structurally balanced graphs2. essentially nonnegative matrices (plus diagonal

similarities)3. inverse positive matrices (plus diagonal similarities)

■ bipartite agreement/consensus stays toagreement/consensus just like a monotone system stays to acooperative system

■ structural balance property is purely graphical−→ "qualitative stability" -like condition

■ even when structural balance is not exact, there are chancesto get (bipartite) agreement and perhaps also (bipartite)consensus

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