Dynamics of Collective Decision-Making
Naomi Ehrich Leonard
Mechanical & Aerospace Engineering
Princeton University
[email protected]
www.princeton.edu/~naomi
Alessio Franci, Math Dept., UNAM, Mexico City
Vaibhav Srivastava, MAE Dept., Princeton University
N.E. Leonard – Blochfest– June 30, 2015
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Collective Decision-Making
How to enable a network of distributed agents to decide as a group?
Which alternative is true?
Which action to take?
Which direction to follow?
Has a change been detected?
N.E. Leonard – Blochfest– June 30, 2015
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Role of the Interaction Network Structure in
Collective Decision-Making Dynamics
1.  Speed of convergence
and algebraic connectivity
Jadbabaie, Lin, Morse, 2003
Olfati- Saber and Murray, 2004
Moreau, 2005
2. Accuracy
and effective resistance
Barooah and Hespanha, 2006, 2008
Ghosh, Boyd, Saberi, 2008
Young, Scardovi, Leonard, 2010, 2013
3.  Speed-accuracy tradeoff
and information centrality
Srivastava and Leonard, 2014
4. Optimal selection of leaders
and joint centrality
5.  Controllability
and graph symmetry
Patterson and Bamieh, 2010
Clark and Poovendran, 2011
Fardad, Lin, and Jovanovic, 2011
Fitch and Leonard, 2013
Rahmani, Ji, Mesbahi, and Egerstedt, 2009
Liu, Slotine, and Barabasi, 2011
Olshevsky, 2014
Summers, Cortesi, and Lygeros, 2014
Pasqualetti, Zampieri, and Bullo, 2014
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Animal Groups Rely on Consensus Decision-Making
www.pestipm.org
Nathan Stone
Beneficial to attack?
Which way to go?
Stroeymeyt, Guerrieri, van Zweden, d’Ettorre,
PLoS One, 2010
Couzin, Krause, Franks, Levin, Nature, 2005
Alaska-in-Pictures.com
Good time to migrate?
N.E. Leonard – Blochfest– June 30, 2015
Eikenaar, Klinner, Szostek, Bairlein,
Biol. Lett. 2014
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Nonlinear Decision-Making Dynamics in Schooling Fish
I.D. Couzin et al, Uninformed individuals promote democratic consensus in animal groups, Science, 2011
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Towards a Realization Theory of Collective Decision Making
Animals groups adapt decision making to environment
Case study: House hunting honey bees
Singularities organize group behaviors
Wild About Britain
Singularity theory: Robust bifurcation theory
Proposed model provides realization
Equivalence: connects collective decision making in nature and design
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House Hunting Honey Bees
and the “Waggle Dance”
K. von Frisch, Bees: their vision, chemical senses, and language, 1956.
M. Landauer, Communication among social bees, 1961
T.Seeley and S.Buhrman, Group decision making in swarms of honey bees, Behav Ecol Sociobiol, 1999
K. von Frisch, Nobel Lecture, Dec. 12, 1973
Seeley, Visscher, Passino, Am. Scientist, 2006
Scout communicates: direction, distance, and quality
N.E. Leonard – Blochfest– June 30, 2015
Scott Camazine
vi
of visited site i
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Decision-Making: Deliberations at the Swarm
A vA < vB
B
Scout, commit, recruit
Scout, commit,
recruit, lose interest
Signal decision
when quorum reached
Seeley et al., Am. Scientist, 2006
James Nieh
Scouts apply “stop signal” with head butt to dancers for alternative sites.
Seeley, Visscher, Schlegel, Hogan, Franks, Marshall, Stop signals provide cross
inhibition in collective decision-making by honeybee swarms, Science, 2012.
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Dynamic Model
Seeley et al, Science, 2012.
Decay
Commitment
Recruitment
Stop signal inhibition
B
vA < vB
vA > 1, vB > 1
U
N.E. Leonard – Blochfest– June 30, 2015
A
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Equal Alternatives
B
v=5
v=5
= .2
=5
U
A
D. Pais, P.M. Hogan, T. Schlegel, N.R. Franks, N.E. Leonard, J.A.R. Marshall, A mechanism for value-sensitive
decision-making, PLoS One, 2013.
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Equal Alternatives
Pitchfork singularity at
70% quorum threshold
Pais et al, PLoS One, 2013.
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Sensitivity to Value
Pitchfork bifurcation
Pais et al, PLoS One, 2013.
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Stop Signal as an Adaptive Control Gain
Pais et al, PLoS One, 2013.
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Near-equal Alternatives
Asymmetric model lives in the universal
unfolding of pitchfork singularity
Hysteresis in value difference
Pais et al, PLoS One, 2013.
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Singularities in Bifurcation Theory
Golubitsky and Schaeffer, Springer, 1985
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Pitchfork Singularity and Its Unfolding
y
(y ⇤ ,
⇤
)
Golubitsky and Schaeffer, Springer, 1985
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Honey Bee Model: Organized by Pitchfork
hange coordinates:
= yA
yB ,
= y A + yB
Solutions satisfy:
g( , ) =
⇤
( , , v) =
pv
+ v(1
2 2 v2
⇤
+2
( , , v))
2
v 3 + 4 v 3 + 9v 4 + 2v 2 + 1
2v 2 + v
Recognition of pitchfork singularity:
g=g =g
Unfolding:
= g = 0, g
< 0, g
(
⇤
,
⇤
)=
>0
✓
0,
v2
4v
(v 2
1
3
1)2
◆
G( , , v) = 0
N.E. Leonard – Blochfest– June 30, 2015
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g( , ) ⇠ h(y, )
for all pitchfork singularities
Abstract collective decision-making model organized by pitchfork
()
Motivating work:
()
A. Franci and R. Sepulchre, Realization of nonlinear behaviors from organizing centers, IEEE CDC, 2014
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j
aij
i
Abstract Model
Agent state:
xi 2 R,
i = 1, . . . , N
Slope 1
A. Franci, V. Srivastava, N.E. Leonard, In prep.
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j
aij
i
Abstract Model
Assume: graph is strongly connected and balanced
Linearization at origin for
is consensus dynamic, so singular with 0 eigenvalue
along consensus manifold à bifurcation with center space tangent to consensus manifold
A. Franci, V. Srivastava, N.E. Leonard (2015) A realization theory for bio-inspired collective decision-making,
arXiv:1503.08526 [math.OC]
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Analysis of Abstract Model
Theorem 1 (Uninformed
)
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Proof: Lyapunov-Schmidt Reduction
near
Then
where
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Singularity Analysis and Global Convergence
Can check that equilibria lie on consensus manifold.
Almost global convergence follows since flow induced by system is
strongly monotone and there are no unbounded trajectories.
N.E. Leonard – Blochfest– June 30, 2015
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Analysis of Abstract Model (Informed Dynamics)
Proposition 2 (Informed
):
Assume a complete graph
The Lyapunov-Schmidt reduction around
is
may give the symmetric pitchfork with
has the effect of unfolding the symmetric pitchfork in general
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Abstract Model and Honeybee Decision-Making
30
20
20
10
15
0
u∗2
Sum of opinions
20
25
Sum of opinions
30
10
-10
5
-20
-30
0
1
2
u2
3
4
0
10
0
-10
0
0.2
0.4
0.6
0.8
1
-20
v̄
N.E. Leonard – Blochfest– June 30, 2015
0
1
2
3
4
u2
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Conclusions
Developed mechanistic models for realization of collective decision-making organized by
pitchfork singularity
Rigorously characterized steady-state behavior of these models – will extend to general
graphs and asymmetries in informed individuals
Applicable to investigation of collective decision-making in animal groups:
Captures honeybee nest site selection behavior (of experiments and population models)
Applicable to design of collective decision-making for engineered groups:
Design of dynamics for
and
Exploring effect of graph topology and location of informed agents in network graph
Considering dynamics to realize other singularities
and greater number of alternatives
3 alternatives
Replicator-mutator
dynamics: Hopf
bifurcation
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