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
Networks with signed interactions
Dynamics of opinon forming
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 2/56
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
Networks with signed interactions
Dynamics of opinon forming
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 3/56
Networks with signed interactions
■ 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
Networks with signed interactions
Dynamics of opinon forming
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 4/56
Networks with signed interactions
■ positive edge = friendly relationship◆ cooperation◆ alliance◆ trust
■ negative edge = unfriendly relationship◆ competition◆ rivalry◆ mistrust
=⇒ signed adjacency matrix
A = (aij) aij ≶ 0
Networks with signed interactions
Dynamics of opinon forming
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 5/56
Networks with signed interactions
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
Networks with signed interactions
Dynamics of opinon forming
● Outline
● Dynamics of opinion forming
● Cooperative systems
● Structural balance
● Monotonicity
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 6/56
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
Networks with signed interactions
Dynamics of opinon forming
● Outline
● Dynamics of opinion forming
● Cooperative systems
● Structural balance
● Monotonicity
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 7/56
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
Networks with signed interactions
Dynamics of opinon forming
● Outline
● Dynamics of opinion forming
● Cooperative systems
● Structural balance
● Monotonicity
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 7/56
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
Networks with signed interactions
Dynamics of opinon forming
● Outline
● Dynamics of opinion forming
● Cooperative systems
● Structural balance
● Monotonicity
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 8/56
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)
Networks with signed interactions
Dynamics of opinon forming
● Outline
● Dynamics of opinion forming
● Cooperative systems
● Structural balance
● Monotonicity
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 9/56
More specific: distributed influences
Simplifying assumptions for the opinion forming dynamics■ influences are distributed
xi = fi(x) = fi(xi, xj , j ∈ adj(i))
Networks with signed interactions
Dynamics of opinon forming
● Outline
● Dynamics of opinion forming
● Cooperative systems
● Structural balance
● Monotonicity
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 9/56
More specific: distributed influences
Simplifying assumptions for the opinion forming dynamics■ influences are distributed
xi = fi(x) = fi(xi, xj , j ∈ adj(i))
■ influences are additive
xi = fi(x) =
n∑
j=1
aijψij(xj)− λixi
Networks with signed interactions
Dynamics of opinon forming
● Outline
● Dynamics of opinion forming
● Cooperative systems
● Structural balance
● Monotonicity
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 9/56
More specific: distributed influences
Simplifying assumptions for the opinion forming dynamics■ influences are distributed
xi = fi(x) = fi(xi, xj , j ∈ adj(i))
■ 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
Networks with signed interactions
Dynamics of opinon forming
● Outline
● Dynamics of opinion forming
● Cooperative systems
● Structural balance
● Monotonicity
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 10/56
More specific: distributed influences
■ 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)
Networks with signed interactions
Dynamics of opinon forming
● Outline
● Dynamics of opinion forming
● Cooperative systems
● Structural balance
● Monotonicity
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 10/56
More specific: distributed influences
■ 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)
Networks with signed interactions
Dynamics of opinon forming
● Outline
● Dynamics of opinion forming
● Cooperative systems
● Structural balance
● Monotonicity
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 11/56
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
Networks with signed interactions
Dynamics of opinon forming
● Outline
● Dynamics of opinion forming
● Cooperative systems
● Structural balance
● Monotonicity
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 12/56
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)
Networks with signed interactions
Dynamics of opinon forming
● Outline
● Dynamics of opinion forming
● Cooperative systems
● Structural balance
● Monotonicity
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 13/56
Another case: structurally balanced systems
Structurally balanced social networks:■ outcome of the opinion forming process is completely
predictable only from the adjacency matrix A
Networks with signed interactions
Dynamics of opinon forming
● Outline
● Dynamics of opinion forming
● Cooperative systems
● Structural balance
● Monotonicity
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 14/56
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
Networks with signed interactions
Dynamics of opinon forming
● Outline
● Dynamics of opinion forming
● Cooperative systems
● Structural balance
● Monotonicity
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 15/56
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
Networks with signed interactions
Dynamics of opinon forming
● Outline
● Dynamics of opinion forming
● Cooperative systems
● Structural balance
● Monotonicity
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 16/56
Examples
Two-partyparlamentary
systems Team sports
Internationalalliances
■ Task: understand why opinon forming is so predictible inthese cases
Networks with signed interactions
Dynamics of opinon forming
● Outline
● Dynamics of opinion forming
● Cooperative systems
● Structural balance
● Monotonicity
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 17/56
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;
Networks with signed interactions
Dynamics of opinon forming
● Outline
● Dynamics of opinion forming
● Cooperative systems
● Structural balance
● Monotonicity
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 18/56
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.
Networks with signed interactions
Dynamics of opinon forming
● Outline
● Dynamics of opinion forming
● Cooperative systems
● Structural balance
● Monotonicity
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 19/56
Monotone dynamics & structural balance
■ monotone system:◆ since order is respected ∀ t > 0, dynamical evolution is
completely predictable◆ outcome is robust to perturbations◆ strong monotonicity = monotonicity + strong connectivity
Networks with signed interactions
Dynamics of opinon forming
● Outline
● Dynamics of opinion forming
● Cooperative systems
● Structural balance
● Monotonicity
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 19/56
Monotone dynamics & structural balance
■ 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
Networks with signed interactions
Dynamics of opinon forming
● Outline
● Dynamics of opinion forming
● Cooperative systems
● Structural balance
● Monotonicity
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 20/56
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
Networks with signed interactions
Dynamics of opinon forming
● Outline
● Dynamics of opinion forming
● Cooperative systems
● Structural balance
● Monotonicity
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 20/56
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
■ dynamics of A and of DAD is the same, up to the opinion signs
Networks with signed interactions
Dynamics of opinon forming
● Outline
● Dynamics of opinion forming
● Cooperative systems
● Structural balance
● Monotonicity
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 20/56
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
■ dynamics of A and of DAD is the same, up to the opinion signs
■ strength of an opinion: not number of friends but connectivity
Networks with signed interactions
Dynamics of opinon forming
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 21/56
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
Networks with signed interactions
Dynamics of opinon forming
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 22/56
Large-scale signed social networks
■ Question: are real networks structurally balanced?
Networks with signed interactions
Dynamics of opinon forming
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 22/56
Large-scale signed social networks
■ 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
Networks with signed interactions
Dynamics of opinon forming
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 22/56
Large-scale signed social networks
■ 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
Networks with signed interactions
Dynamics of opinon forming
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 23/56
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
Networks with signed interactions
Dynamics of opinon forming
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 24/56
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
=⇒
Networks with signed interactions
Dynamics of opinon forming
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 25/56
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
Networks with signed interactions
Dynamics of opinon forming
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 26/56
Level of structural balance
Epinions Slashdots WikiElections
■ some nodes have many friends, other many enemies
Networks with signed interactions
Dynamics of opinon forming
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 27/56
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
Networks with signed interactions
Dynamics of opinon forming
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 27/56
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....
Networks with signed interactions
Dynamics of opinon forming
Balance of large networks
Essentially nonnegative systems
● Outline
Bipartite consensus
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 28/56
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
Networks with signed interactions
Dynamics of opinon forming
Balance of large networks
Essentially nonnegative systems
● Outline
Bipartite consensus
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 29/56
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
Networks with signed interactions
Dynamics of opinon forming
Balance of large networks
Essentially nonnegative systems
● Outline
Bipartite consensus
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 30/56
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
Networks with signed interactions
Dynamics of opinon forming
Balance of large networks
Essentially nonnegative systems
● Outline
Bipartite consensus
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 31/56
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
Networks with signed interactions
Dynamics of opinon forming
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
● Laplacian
● Examples
● Bipartite consensus
● Directed graphs
● Stability of L
● Examples
● Nonlinear Laplacians
● Monotonicity
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 32/56
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
Networks with signed interactions
Dynamics of opinon forming
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
● Laplacian
● Examples
● Bipartite consensus
● Directed graphs
● Stability of L
● Examples
● Nonlinear Laplacians
● Monotonicity
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 33/56
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
Networks with signed interactions
Dynamics of opinon forming
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
● Laplacian
● Examples
● Bipartite consensus
● Directed graphs
● Stability of L
● Examples
● Nonlinear Laplacians
● Monotonicity
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 34/56
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
Networks with signed interactions
Dynamics of opinon forming
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
● Laplacian
● Examples
● Bipartite consensus
● Directed graphs
● Stability of L
● Examples
● Nonlinear Laplacians
● Monotonicity
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 34/56
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)
Networks with signed interactions
Dynamics of opinon forming
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
● Laplacian
● Examples
● Bipartite consensus
● Directed graphs
● Stability of L
● Examples
● Nonlinear Laplacians
● Monotonicity
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 35/56
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}
Networks with signed interactions
Dynamics of opinon forming
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
● Laplacian
● Examples
● Bipartite consensus
● Directed graphs
● Stability of L
● Examples
● Nonlinear Laplacians
● Monotonicity
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 36/56
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
Networks with signed interactions
Dynamics of opinon forming
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
● Laplacian
● Examples
● Bipartite consensus
● Directed graphs
● Stability of L
● Examples
● Nonlinear Laplacians
● Monotonicity
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 37/56
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
Networks with signed interactions
Dynamics of opinon forming
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
● Laplacian
● Examples
● Bipartite consensus
● Directed graphs
● Stability of L
● Examples
● Nonlinear Laplacians
● Monotonicity
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 38/56
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 .
Networks with signed interactions
Dynamics of opinon forming
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
● Laplacian
● Examples
● Bipartite consensus
● Directed graphs
● Stability of L
● Examples
● Nonlinear Laplacians
● Monotonicity
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 39/56
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
Networks with signed interactions
Dynamics of opinon forming
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
● Laplacian
● Examples
● Bipartite consensus
● Directed graphs
● Stability of L
● Examples
● Nonlinear Laplacians
● Monotonicity
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 40/56
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
Networks with signed interactions
Dynamics of opinon forming
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
● Laplacian
● Examples
● Bipartite consensus
● Directed graphs
● Stability of L
● Examples
● Nonlinear Laplacians
● Monotonicity
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 41/56
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.)
Networks with signed interactions
Dynamics of opinon forming
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
● Laplacian
● Examples
● Bipartite consensus
● Directed graphs
● Stability of L
● Examples
● Nonlinear Laplacians
● Monotonicity
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 42/56
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
Networks with signed interactions
Dynamics of opinon forming
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
● Laplacian
● Examples
● Bipartite consensus
● Directed graphs
● Stability of L
● Examples
● Nonlinear Laplacians
● Monotonicity
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 43/56
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)
Networks with signed interactions
Dynamics of opinon forming
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
● Laplacian
● Examples
● Bipartite consensus
● Directed graphs
● Stability of L
● Examples
● Nonlinear Laplacians
● Monotonicity
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 44/56
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∗| → ∞
}
Networks with signed interactions
Dynamics of opinon forming
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
● Laplacian
● Examples
● Bipartite consensus
● Directed graphs
● Stability of L
● Examples
● Nonlinear Laplacians
● Monotonicity
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 45/56
Monotone dynamics & structural balance
■ 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.
Networks with signed interactions
Dynamics of opinon forming
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
● Laplacian
● Examples
● Bipartite consensus
● Directed graphs
● Stability of L
● Examples
● Nonlinear Laplacians
● Monotonicity
Distributed dynamical sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 46/56
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)
Networks with signed interactions
Dynamics of opinon forming
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
● Sorting in Rn
● Sorting in finite time
● Sorting in Rn
● Linear algebra behind sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 47/56
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
Networks with signed interactions
Dynamics of opinon forming
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
● Sorting in Rn
● Sorting in finite time
● Sorting in Rn
● Linear algebra behind sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 48/56
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)
Networks with signed interactions
Dynamics of opinon forming
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
● Sorting in Rn
● Sorting in finite time
● Sorting in Rn
● Linear algebra behind sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 48/56
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
Networks with signed interactions
Dynamics of opinon forming
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
● Sorting in Rn
● Sorting in finite time
● Sorting in Rn
● Linear algebra behind sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 49/56
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}
Networks with signed interactions
Dynamics of opinon forming
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
● Sorting in Rn
● Sorting in finite time
● Sorting in Rn
● Linear algebra behind sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 49/56
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}
■ consider p ∈ Rn+ and write P = diag(p)
L1 = 0 ⇐⇒ LP−1P1 = LP−1p = 0
Networks with signed interactions
Dynamics of opinon forming
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
● Sorting in Rn
● Sorting in finite time
● Sorting in Rn
● Linear algebra behind sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 49/56
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}
■ 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}
Networks with signed interactions
Dynamics of opinon forming
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
● Sorting in Rn
● Sorting in finite time
● Sorting in Rn
● Linear algebra behind sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 50/56
Distributed sorting in Rn+
■ 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)
Networks with signed interactions
Dynamics of opinon forming
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
● Sorting in Rn
● Sorting in finite time
● Sorting in Rn
● Linear algebra behind sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 51/56
Distributed sorting in Rn+
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
Networks with signed interactions
Dynamics of opinon forming
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
● Sorting in Rn
● Sorting in finite time
● Sorting in Rn
● Linear algebra behind sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 52/56
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"...)
Networks with signed interactions
Dynamics of opinon forming
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
● Sorting in Rn
● Sorting in finite time
● Sorting in Rn
● Linear algebra behind sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 53/56
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
Networks with signed interactions
Dynamics of opinon forming
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
● Sorting in Rn
● Sorting in finite time
● Sorting in Rn
● Linear algebra behind sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 54/56
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
Networks with signed interactions
Dynamics of opinon forming
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
● Sorting in Rn
● Sorting in finite time
● Sorting in Rn
● Linear algebra behind sorting
Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 55/56
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)
Networks with signed interactions
Dynamics of opinon forming
Balance of large networks
Essentially nonnegative systems
Bipartite consensus
Distributed dynamical sorting
Conclusion
● Conclusion
GIPSA, Grenoble, December 2012 Dynamics and control on networks with antagonistic interactions – p. 56/56
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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