Swarm Collaborative Intelligence: From Networked Control to Trust in MANET

1
Swarm Collaborative Intelligence From Networked
Control to Trust in MANET
John S. Baras Institute for Systems
Research Department of Electrical and Computer
Engineering And Department of Computer Science
University of Maryland, College Park, MD 20742
Workshop on Swarming in Natural and Engineered
Systems Napa Valley, California August 3-4, 2005
2
Thanks to

• Collaborators
• Tao Jiang, George Theodorakopoulos, Xiaobo
  Tan, Wei Xi, Pedram Hovareshti
• Funding sources
• ARL (CTA on CN), ARO, ARO CIP URI (Wireless
  Network Security), ARO MURI (Networked Control
  Systems), DARPA (Dynamic Coalitions)

3
Outline

• Autonomous collaborating vehicles
• A stochastic approach based on MRF
• Analysis
• Convergence study of a simple case
• Convergence speed analysis
• Simulation results
• A hybrid scheme to improve the performance
• Distributed trust in MANET
• Trust (and Mistrust) spreading and dynamics
• Effects of topology on convergence
• Spin glasses and cooperative games
• Collaboration via trust schemes
• Conclusions and future work

4
A Battlefield Scenario

• Mission
• Autonomous, distributed maneuvering of a vehicle
  group to reach and cover a target area
• Constraints
• Desired inter-vehicle distance
• Obstacles avoidance
• Threats (stationary or moving) avoidance
• Requirement
• Using only local or static information

5
Review of Deterministic Gradient-Flow Approach

• Dilemma of the Deterministic gradient-flow
  approach
• Potentials-based approach can accommodate
  multiple objectives and constraints in a
  distributed and computationally effective way
• The system dynamics could be trapped by the local
  minima
• Weighted sum of potential functions
• Target (attraction) potential Jg
• Neighbor (avoidance) potential Jn
• Obstacle potential Jo
• Potential Js due to stationary threats
• Potential Jm due to moving threats

• Gradient flow

6
Being Trapped by Local Minima
Different initial conditions may cause vehicles
to be trapped by local minimum
7
Markov Random Fields

• Markov Random Fields (MRF)
• A collection of random variables XXs, s? S
  with discrete values in phase space ?s
• A neighborhood system on S is a family N Ns,
  s? S, where Ns ? S, and r ? Ns ? s ? Nr
• The marginal probability depends only on
  neighbors phase value
• Gibbs Field (GF)
• A clique is a subset c ? S, such at for all s,r
  ? c, r ? Ns
• The potential energy of a configuration xxs is
  defined as a sum of all clique potentials

8
Gibbs Sampler and Gibbs Distribution

• Gibbs distribution (global description)
• The marginal probability is function of local
  potentials
• Gibbs distribution is function of global
  potentials
• Hammersley-Clifford theorem A MRF on a graph is
  equivalent to a GF
• Gibbs sampler
• Gibbs sampler (MCMC method)defines a Markov chain
  on a Gibbs Field
• The stationary distribution of the MC is the
  Gibbs distribution
• Using simulated annealing algorithm, final
  configuration converges to global minimum with
  probability 1

9
Modeling a Swarm as a GF

• 2D mission space on discrete lattice cells

• Agent s can communicate with neighboring agents
  in Ns which stay within the interaction range Rs
• An agent can go at most Rm within one move, which
  defines the phase space ?s
• Gibbs potential is designed to reflect global
  objective

• Difficulties in applying classical results

• Non-stationary neighborhood system
• Time-varying and state-dependent phase space

10
Gibbs Sampler Based Algorithm

• Algorithm for single vehicle
• Step1. Pick a cooling schedule T(n) and the total
  number N of annealing steps
• Step2. At each annealing step n, conduct a
  location update for the vehicle by performing the
  following
• Determine the set L of candidate locations for
  the next move
• 
  
• 
  
• For each l ? L, evaluate
• Update location by sampling above distribution
• Step 3. Let n n1. If n N, stop otherwise go
  to step 2

11
Convergence Study

• Single vehicle with limited sensing and moving
  range
• Fixed temperature
• Assume accessible area is connected
• Unique stationary distribution
• 
  where
• From any distribution v,
• 
  
  

• Simulated annealing
• Cooling schedule
• Let Qn (PT(n))t
  
• 
  
• 
  where

12
Convergence Rate Study

• Convergence rate of a single vehicle case
• For the single-vehicle case, the convergence rate
  is characterized by
• 
  
  
• where
• 
  

• Using convergence rate bound as a design
  indicator
• Design ?g to maximize the convergence rate
• Potential function
• Empirical distributiondistance

13
Parallel Sampling

• Problems with sequential sampling
• Global indexing is difficult in practice
• Long time for one sweep
• Parallel sampling
• Agents update locations in parallel by sampling
  local characteristics
• Conflicts could be solved by coin-toss.
• Simulation showed the MAS achieve global
  objective with only local strategies.

14
Stochastic Path Planning Simulation

• Parallel stochastic path exploration based on MRF
  can get around the local minima

• Potential function
• Target (attraction) potential Jg
• Neighbor (avoidance) potential Jn
• Obstacle potential Jo

15
Simulation Gathering

• Potential function
• The first term attracts nodes close to z0
• The second term tends to cluster nodes

16
Simulation Gathering
specified center Z0(25,25)
unspecified center

• 200 nodes on 50 by 50 grid?1 0.05 , ?2 1, ?
  103
• Rm2?2, Rs6?2 T(n)1/(4log(400n))

17
Simulation Line Formation

• Potential function
• ? is scaling factor
• ? is a penalization for node with no neighbor
• mk is the number of neighboring nodes of node k
• ?k,k is the desired angle of the line segment
• dk,k /Rs puts more weight on farther neighbors,
  which encourages the formation of long lines

18
Simulation Line Formation
One line
Three lines

• 200 nodes on 50 by 50 grid
• ?10 , ?5
• Rm2 ?2
• Rs10?2, 6?2, 4?2
• T(n)1/(4log(400n))

Two lines
19
A Hybrid Control Scheme

• Deterministic potential approach
• Pro Save traveling time
• Con May be get trapped by some obstacles
• Stochastic approach based on MRF
• Pro Trouble free. Converge to global minimum
  for sure.
• Con Waste time for path exploration
• Hybrid control scheme combines both advantages
  and may strike the right balance

20
Hybrid Scheme Algorithm

• Step 1. Each vehicle (node) starts with the
  deterministic gradient-flow method and goes to
  Step2
• Step 2. If a vehicle stops moving for d
  consecutive time instants and its location is not
  within the target area, then it switches to the
  simulated annealing method with a predetermined
  cooling schedule
• Step 3. After performing simulated annealing for
  N time instants, the vehicle switches to the
  gradient method and goes to Step 2

21
Impact of Memory

• Hybrid scheme with memory
• Experience can help vehicle to learn the complex
  environments better and thus change its behavior
  to achieve better performance.
• Implementation when a vehicle determines it is
  trapped , it increases the risk level R of that
  spot, and does local sampling as follows

22
Impact of Memory (cont.)

• Hybrid scheme with memory

23
Autonomic Wireless Networks

• Wireless networks, such as mobile ad hoc networks
  (MANET) and sensor networks
• No trusted centralized authority
• Resource (power, bandwidth, computation etc.)
  constraints
• Rapidly and dynamically changing topology and
  connectivity
• Uncertainty incompleteness of trust evidence
  trust values in -1, 1
• Distributed trust computation and locality of
  trust information exchanges
• Unique properties
• Each node is its own authority and it is selfish
• Networking functions (route discovery, packet
  forwarding and etc. ) rely on cooperation between
  nodes
• Cooperation utilizes local information and local
  interactions (between neighbors)

24
Cooperation and Games

• In distributed wireless networks
• Cooperation is restricted to only local
  interactions
• Decision is made by each node individually
• Nodes are self-interested
• Explain and analyze emergent properties
• Game theoretic methods
• Provide a framework for modeling individual
  interactions
• Understand complex global structures and dynamics
  of a system composed of a large number of agents
  with simple local interactions
• Guide for analytical approach
• Examples Ising model, prisoners dilemma
• Goal how to encourage nodes to collaborate in
  games?
• Incentive trust systems to promote cooperation
  and circumvent misbehaving nodes.

25
A Simple Distributed Trust Computation Policy

• Based on simple voting methods
• Voters
• Nodes that qualified as legitimate voters by
  certificates signed by offline servers (have
  trust evidence about node i)
• Assume uniformly distributed in the network
• Policy decision based on threshold
• is the total number of votes node i
  received (signed sum)
• is the decision threshold
• is the number of is neighbors

26
Simple Voting Scheme

• Number of positive votes on node i Vp,i 3
• Number of negative votes on node i Vn,i 1
• Effective votes Vi Vp,i - Vn,i 2
• Given ? 0.3, Vi gt ?Ni 1.8, node i is
  designated trusted

27
Trust Dynamics

• Trust spreading

Initial islands of trusts

• Trust revocation
• Changes in topology, membership, secure paths
• Referees of a node may change, trust evidence for
  a node may change
• Votes timeout or negative votes

28
Trust Graph

• Trust graph GT(VT, ET)
• Induced subgraph of G(V, E) by VT
• VT is the set of nodes which are designated
  trusted by the trust computation algorithm
• ET e e in E and both ends of e are in VT

• Trust metric Psp percentage of trusted pairs
  that are connected by one or more secure paths,
  which are composed of trusted nodes
• NPsecure is the number of trusted pairs that are
  connected by one or more secure paths.
• It is dependent of the cluster size and
  connectivity of GT

29
Random Graph Model

• Erdos and Renyi random graphs (ER model)

• When ? is small
• Most of nodes are considered to be trusted
• Psp is dominated by the edge present probability
  p in ER random graphs
• Zero-one law in random graph theory is present
• Increasing the threshold ? results in
• Reducing the number of trusted nodes
• Increasing critical values
• Smaller Psp

Simulation results of Psp as function of
decision threshold ?
30
Small-world Networks
Psp vs. ? after one iteration
Psp vs. ? in steady states

• Number of trusted paths increases as trust
  spreads with each iteration
• Different curves are with different rewiring
  probability Prw
• Prw 0 represents a regular lattice
• Prw 1 converges to a random graph
• Observe the transition from lattices to random
  graphs
• With a relative small portion of shortcuts,
  small-world networks facilitate the formation of
  secure paths
• The effects of topology are obvious, so any
  distributed trust computation model should take
  into account the topology properties

31
Trust Revocation

• The trust revocation process is initiated
• when topology, membership or secure paths change
• when referees or trust evidence for a node
  changes
• when positive votes are timeout or new negative
  votes are received

• Decision policy of the revocation process
• Revocation on a specific node, say B, usually
  starts from few nodes that have negative
  observations on B
• A node A accepts the revocation on B, if it finds
  that more than a threshold fraction F of its
  neighbors revoke node B
• Question can a revocation be accepted by a large
  fraction of nodes in the network?

32
Phase Transition of Revocation

• Revocation is launched from a randomly chosen
  node in an Erdös-Rényi random graph with average
  degree set as the Y-axis.
• Global cascade area that lie inside of the
  contour represents the percentage of nodes, which
  accept the revocation, is greater than the value
  corresponding to the contour (level surfaces of
  histogram)
• Phase transitions happen suddenly the steep of
  the contours is very sharp, which represents
  phase transitions

33
Previous Work

• Decentralized path-inference protocols
• Combination of trust along and across paths
  (Beth,1994)
• Probability of finding a trust path from source
  to target (Maurer, 1996)
• Local interaction
• EigenTrust (Kamvar, 2003)
• PeerTrust (Xiong, 2004)
• Bayesian methods (Buchegger, 2003)
• Our work is similar with EigenTrust and
  PeerTrust, which provided promising results.
• However, results of EigenTrust and PeerTrust are
  all based on simulations.
• We analyze our local interaction rule using graph
  theory.
• We also provide a theoretical justification for
  network management that facilitates trust
  propagation.

34
Voting Scheme

• Voting rule
• is the trust value of node i
• is the voting value of node j about node i
• Local voting rule
• Function f should satisfy the following
  properties
• The range of f is -1,1.
• Votes from neighbors with higher trust value are
  more credible, so they should carry larger
  weights.
• Policy threshold rule for trustworthiness of the
  target agent
• where is the threshold, which is a
  constant

35
Simple Voting Rule

• We use the weighted average as the voting rule,
  where weights are trust values of voters
• is the degree of node i
• n represents discrete time
• Assume is a constant, i.e. it doesnt
  change with time, which is true when considering
  the steady state
• The voting rule can be written in system
  equationwhere D diagd1 ,d2 ,, dN, T is a
  vector representing trust values of all nodes and
  V is the matrix for votes

36
Convergence of Simple Voting Rule

• Voting without uncertainty
• For each pair (i, j) , if i and j are neighbors,
  then vij 1.
• V A, where A is the adjacency matrix of graph
  G, and D-1A is a stochastic matrix with the
  largest eigenvalue being 1.
• Let be the right eigenvector of D-1A
  corresponding to eigenvalue 1. then
• If , all nodes are
  trusted, and none is trusted otherwise.
  
• The initial
  trust values are very crucial.
• Voting with uncertainty
• vij 1, D-1A is a semi-stochastic matrix.
• We proved ,
  so T?0. Trust cannot be established at
  all!!!

37
Voting with Headers

• We have shown that using the simple voting
  scheme, trust can only be established under
  certain strict conditions
• All votes value are 1 and the initial
  configuration must satisfy
• A single vote with value less than 1 will result
  in failure of trust establishment.
• We introduce the notion of headers
• Headers are pre-trusted agents and only vote for
  nodes that they fully trust.
• If node i is trusted with bi headers, it will
  get bi more votes with value 1. Let B diagb1
  , b2 ,, bN .
• The system equation changes to

38
Convergence of Voting with Headers

• Voting without uncertainty
• V A, and define .
  The system equation changes to
• If there is at least one node i such that bi gt
  0, (DB)-1A goes to 0. Therefore T(n) ? 1
  and all nodes are trusted.
• Voting with uncertainty
• Using the same technique as above, let
  . We are able to find the
  condition such that
• If we let , then all nodes are
  trusted.
• Theorem Given the threshold is ? , the number
  of headers for each node must satisfy
• This theorem proves, as well as provides, a
  network design method to establish a fully
  trusted network by introducing headers

39
Spreading Speed and Topology

• The time for updating rule to reach the steady
  state, i.e., how fast the trust values converge.
• Perron-Frobenius Theorem in algebraic graph
  theory For a stochastic matrix A
• is the largest eigenvalue of A, which is 1
  and is the second largest eigenvalue of
  A.
• The convergence rate of An is of order
• Normalized adjacency matrices are stochastic
  matrices, therefore those with smaller
  converge faster.
• What kind of networks or which network topology
  has smaller second largest eigenvalue

40
Spreading Speed and Topology (cont)

• We consider the small-world model proposed by
  Watts and Strogatz in 1998
• High clustering coefficient and small average
  graphical distance between any pair.
• We use F-model, which is modeled by adding small
  number of new edges into a regular lattice.

• Adding just 1 more edges, spreading finishes in
  10 times less rounds.

41
Ising and Spin Glass Models

• Statistical Physics models for magnetization

• Orientation of each particles spin depends on
  its neighbors
• Ising Model behavior of simple magnets
• Spin Glass Model complex materials
• Math interpretation
• s s1, s2,, sn is a configuration of n
  particle spins, where sj 1 or -1 , spin j
  is up or down
• Hamiltonian, or Energy for configuration s

• Ising Model Jij J for all i, j
• Spin Glass Model Jij depend on i,j and can be
  random processes

42
Ising/SG Models and Games

• Ising and Spin Glass models can be interpreted as
  dynamic (repeated) games each particle selects
  its own spin to maximize its own payoff
• Ising model (Jij J) align its spin with the
  majority of neighbors spin
• High T, conservative agents, not willing to
  change, small payoffs
• Low T, aggressive agents, larger payoffs
• Collection of local decisions reduces the total
  energy of the interacting particles
• Statistical Mechanics primary object of interest
• Recent excitement computation of ground state,
  partition function Z, NP - complete, Replica
  Method
• Application to turbocodes, image restoration,
  neural networks, learning, associative memory,
  SAT, knapsack, SA, number parttioning, graph
  partitioning, CDMA, MIMO,
• Inspires an approach where trust is used as an
  incentive for cooperation
• si represents whether node i cooperates or not
  with neighbors
• Jij can be interpreted as the worth of player j
  to player i
• Cooperate or not based on benefit from
  cooperation and trust values of neighbors

43
Spin Glass Cooperative Game

• Spin Glass model as a cooperative game (spin
  glass game)
• In
  , the weights wij frustrate the system
• Both positive and negative local feedback (e.g.
  wij?-1, 1)
• Interaction topology (i.e. the matrix J Jij )
  moderates effects pos. and neg. fback
• S ? N 1, 2, , N is a coalition, in which
  all nodes cooperate
• v(S) value of characteristic function of the
  game , v 2N?R maximum payoff S can get
  without cooperation from other nodes N /S.
• Model can be used to find what form or policy
  for Jij can induce all (or most) nodes to
  cooperate maximize the coalition

G (N, v)
6
2
J21
J12
3
Subset S1,2,3,4 v(S)J12J21J14J41J43J34
-J36 -J15
1
5
J34
J14
J41
4
J43
44
Cooperative Games and Dynamic Coalitions

• Have a number of players, some can be coalitions
  themselves
• How do they negotiate an acceptable DC security
  policies set?
• What are the properties of the final result the
  negotiated policy set?
• Is there an efficient scheme that gets us there?

• Cooperative games allow us to set up different
  types of games between the players, examine
  different concepts of solutions and values
• Can prove mathematically properties of the
  solution and value e.g. minimizes maximum
  dissatisfaction, is anonymous, is stable
• Can get iterative methods to get to solution
  (negotiation schema), can use all kinds of
  constraints, invariance to aV b scaling
  (preferences)
• Working on extensions to partial information,
  learning, robustness to uncertainties

45
Spin Glass Cooperative Game Properties

• Spin Glass game is a convex and superadditive
  game iff (net pos. effects)
• Shapley value
  in the core
• Not well understood in the regime of both
  negative and positive net effects
• Effects of interaction matrix structure
  (sparsity, neighborhood structure, range of
  interactions, strength of interactions) not well
  understood Topology effects in network analog
• Oriented Spin Glass Game G(N,v) where v now
  depends on both an interaction matrix J and a
  preference vector L a pair of char. fcns
• Replica method can be used to analyze various
  problems under various models and constraints on
  J and L

46
Cooperative Games with Negotiation

• Consider G (N, v), N as before but with
• G (N, v) convex, superadditive, if
• Theorem G (N, v) has a nonempty core. The
  payoff allocation to node i ,
  
  is in the core. Compute
  as follows
• This payoff allocation indicates a way to
  encourage cooperation
• Players with positive gain can negotiate with
  their neighbors by sacrificing certain gain
  (offering their partial gain ?ijxji )

47
Trust as Mechanism to Induce Collaboration

• Trust is an incentive for collaboration
• Nodes who refrain from cooperation get lower
  trust values
• They will be eventually penalized because other
  nodes tend to only cooperate with highly trusted
  ones.
• Assume, for node i, that the loss for not
  cooperating with node j is a nondecreasing
  function of xji as f (xji), and the new
  characteristic function is
• Theorem if ,
  the core is nonempty and
  
• is a feasible payoff
  allocation in the core.
• By introducing a trust mechanism, all nodes are
  induced to collaborate without any negotiation

48
Dynamics of Cooperation

• System model

• Two linked dynamics
• Trust propagation
• Game evolution

• The network is modeled as a discrete-time system

j all neighbors of i vij trust value node i
votes for node j
49
Game Evolution

• Strategy of node i
• ?ij 1 ( 0) represents that i cooperates (does
  not cooperate) with
• its neighbor j
• Payoff for node i when interacting with j
  xij Jij ?ij ?ji
• xij gt 0 (lt 0) positive link (negative link)
• Node selfishness ? cooperate with neighbors on
  positive links
• Strategy updates node i chooses ?ij 1 only if
  all of the following are satisfied
• Neighbor j has not been revoked
• Neighbor j is cooperative
• xij gt 0, or the cumulative payoff of i is less
  than the case when it unconditionally conducts
  ?ij 1.
• Trust propagation
• The threshold is chosen to ensure global
  revocation propagation
• Reestablishing period t once a node is revoked,
  in order to reestablish trust the revocation has
  to be nullified for t consecutive time steps

50
Results of Game Evolution

• Theorem
  , there exists t0, such that for a reestablishing
  period t gt t0
• The iterated game converges to Nash equilibrium
• In the Nash equilibrium, all nodes cooperate with
  all their neighbors.
• Comparison of games with (without) trust
  mechanism, strategy update

Percentage of cooperating pairs vs negative links
Average payoffs vs negative links
51
Conclusions and Future Research

• A stochastic potential based approach guarantees
  global objective can be achieved by simple local
  strategies
• The parallel sampling algorithm saves running
  time compared with the sequential sampling
  algorithm
• Emergent behaviors of self-organized swarms are
  observed in simulations
• A hybrid scheme is proposed to achieve better
  performance by combining deterministic
  gradient-flow approach and stochastic potential
  based approach
• Convergence study of the distributed parallel
  algorithm
• Tighter convergence rate bound estimation and
  parameters estimation of the hybrid scheme
• Cooperative learning to further improve the
  performance of the hybrid scheme
• Convergence analysis when only partially observed
  potential functions available due to imperfect
  sensors
• Schedule of measurements due to sensor power
  constraints

52
Conclusions and Future Research

• Analyzed and evaluated fundamental methods to
  induce collaboration in wireless networks with
  mobile nodes
• Focused on distributed schemes using only local
  interactions
• Developed and analyzed a cooperative game
  framework and showed that negotiation between
  agents can induce collaboration
• We developed a distributed trust establishment,
  propagation and maintenance scheme for such
  networks and showed that it can also induce
  collaboration
• Showed that trust propagation displays phase
  transitions
• Investigated the linked dynamics of trust
  propagation and game evolution and showed the
  benefits in inducing collaboration
• Methods inspired from statistical physics of spin
  glasses
• Future directions include analysis of networks
  with dynamic topologies, robustness of these
  collaboration inducing mechanisms, identification
  of parameters (including topology types) that
  influence the dynamics and qualities of
  collaborative behavior

53
Publications

• Tao Jiang and John S. Baras, Ant-based Adaptive
  Trust Evidence Distribution in MANET,
  Proceedings of 2nd International Workshop on
  Mobile Distributed Computing, in conjunction with
  the Intern. Conference on Distributed Computing
  Systems, Tokyo, Japan, March 2004.
• John S. Baras and Tao Jiang, Dynamic and
  Distributed Trust for Mobile Ad-Hoc Networks,
  Proceedings of 24th Army Science Conference,
  Orlando, Florida, December 2004.
• John S. Baras and Tao Jiang, Cooperative Games,
  Phase Transitions on Graphs and Distributed Trust
  In MANET, invited paper, Proceedings 2004 IEEE
  Conference on Decision and Control, December
  2004, Bahamas.
• John S. Baras and Tao Jiang, Managing Trust in
  Self-organized Mobile Adhoc Networks, invited
  paper, Wireless and Mobile Security Workshop,
  Network and Distributed Systems Security
  Symposium, February 2005, San Diego, USA.
• Tao Jiang and John S. Baras, Autonomous Trust
  Establishment, 2nd International Network
  Optimization Conference (INOC), February 2005,
  Lisbon, Portugal.
• John S. Baras and Tao Jiang, Cooperation, Trust
  and Games in Wireless Networks, invited paper,
  in Proceedings of Symposium on Systems, Control
  and Networks, honoring Professor P. Varaiya,
  Birkhauser, June 2005.
• Tao Jiang and John S. Baras, Graph Algebraic
  Interpretation of Trust Establishment in
  Autonomic Networks, submitted to Wiley Journal
  of Networks (special issue)

54
Publications

• J.S. Baras, X. Tan and P. Hovareshti,
  Decentralized Control of Autonomous Vehicles, in
  Proc. of 42nd IEEE Conference on Decision and
  Control, Hawai, Dec 2003.
• W. Xi, X. Tan, and J. S. Baras, A stochastic
  algorithm for self-organization of autonomous
  swarms, to appear in Proc. 44th IEEE Conference
  on Decision and Control.
• J. S. Baras and X. Tan, Control of autonomous
  swarms using Gibbs sampling, in Proceedings of
  the 43rd IEEE Conference on Decision and Control,
  Atlantis, Paradise Island, Bahamas, 2004, pp.
  47524757.
• W. Xi, X. Tan, and J. S. Baras, Gibbs
  sampler-based path planning for autonomous
  vehicles Convergence analysis, in Proceedings
  of the 16th IFAC World Congress, Prague, Czech
  Republic, 2005.
• 4W.Xi, X. Tan, and J.S. Baras, A hybrid scheme
  for distributed control of autonomous swarms,
  2005, in Proc. of 24th American Control
  Conference.