Journal of Robotics, Networking and Artificial Life, Vol. 3, No. 2 (September 2016), 136-139

Socialized Multi-Agent System Rendezvous via Networks of Networks
Yunzhong Song

College of Electrical Engineering and Automation, Henan Polytechnic University, 2001 Century Avenue, Jiaozuo
454003, P.R.China
Ziyi Fu

College of Electrical Engineering and Automation, Henan Polytechnic University, 2001 Century Avenue, Jiaozuo
454003, P.R.China
Fuzhong Wang

College of Electrical Engineering and Automation, Henan Polytechnic University, 2001 Century Avenue, Jiaozuo
454003, P.R.China
E-mail: songhpu@126.com, fzy@hpu.edu.cn, wangfzh@hpu.edu.cn
www.hpu.edu.cn

Abstract
Networks of networks paradigm was introduced to explore the rendezvous problems of leader-follower multi-agent
system, where the rendezvous actors were categorized into the target object, the leader agents, the informed
follower agents and the isolated follower agents, respectively. Then, three humanized system inspired strategies
were investigated. Results demonstrated that, the democracy strategy could fulfill the rendezvous task in expenses
with a long convergent time and the autarchy democracy could come to the targeted object quickly with the risk offailing in rendezvous, while the mixed strategy cared about both of the convergent speed and the utmost task,
would take autarchy strategy when the connections were available for the follower agents, while the democracy
rules had to be used if the connections were not possible.
Keywords: networks of networks, leader-follower multi-agent network, rendezvous problems, the democracy
strategy, the autarchy strategy, the mixed strategy

1. Introduction
In recent years, the distributed control of multi-agentnetwork has been paid more attentions from different
disciplines. For instance, Ref. 1 analyzed the issue of
how the ratio of leaders to followers affects
connectivity. Ref. 2 discussed the consensus problems
in undirected networks of agents with fixed topology
and switching topology. In Ref. 3, the direct control
laws were used in a heterogeneous network. Ref. 4

provided sufficient conditions for rendezvous in firstorder leader-follower network; Ref. 5 studied the
distributed tracking control of the leader-follower
directed network with noisy measurement. Ref. 6investigated the consensus problems in leader-follower
network with fixed topology and switching topology.
Although the consensus research of multi-agent
networks and leader-follower multi-agent networks
provided valuable results, there still have been someobvious limitations. For instance, the realization
mechanism of the swarm has been considered rarely.

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Yunzhong Song, Ziyi Fu, Fuzhong Wang

But in actual society, the swarm members can interactwith each other in various styles, for example, thefollowers may be kept in the group or be left alone
according to the different utmost objectives; and as thesame way, the leaders may also play with the followers
in the unselfish ways or in the selfish ways, that can beboiled down to the costs and the capabilities of theleaders. Lately, the networks of networks, like Ref.7
which paid much more attention to the coupling among
the networks, was much more powerful in modeling thereal world systems, where interdependence of networks
and the effect of this interdependence on the structuraland functional behavior of the coupled system wereconsidered into detail. So, it is necessary to be clarified
that whether the networks of networks form mechanism
can be verified feasible or not in leader-follower multiagent systems. The already existing results including thedifferent layers built according to the different
relationship among agents, for example, one layer canbe used to express the relationship from the spatial
location of the agents, that is, if they are near each otherenough in spatial physical distance, they can haveconnections with each other; and the another layer canbe assigned to express their social relationships, such asfriendships, and classmates, that is if they are friends orhave been classmates before, they can be categorized
into one layer, and the connection strength can bedetermined with their intimateness. The style we deal
with networks of networks here can enrich the results ofnetworks of networks and it can be referred further. In
this paper, the three strategies namely the democracy
strategy, the autarchy strategy and the mixed strategy
would be used in rendezvous problem of leader-follower
network. Specific to leader-follower multi-agent
network, the democracy strategy means that the motionof agent is affected by the other ones located within thesensing zone of the agent. And the autarchy strategy issuch a strategy, in which the motion of leader is notaffected by any agents, while follower only gets theinfluence of the initial nearest leader within its sensing
zone. The mixed strategy is composed of the autarchy
one and the democracy one. In the mixed strategy, all
the leaders and some followers which are affected byleader initially take the autarchy strategy, while theother followers which are called isolated followers takethe democracy strategy.
Here, we must note that in case of brevity and thelimited space of the paper, only the democracy strategy
is covered.
2. Preliminaries
2

Where the subscript denotes the order number of theagents in the corresponding layer, whereas thesuperscript denotes the order number of the layer thatagents belong to. So, here in (1), N and M denote thebiggest order number of intra-layer and inter-layer. And
the rest can be done in the same manner. In case ofconvenience, the node order number and its
corresponding expression in networks of networks styleis listed in Table 1.
Table1. The node order number and its corresponding
expression in networks of networks style.
V1

V2x13x23x33x43V5V6Leaders
V7V8V9Target
V10x14x53x12x32x11x22

Let g={V, E} describe the group topology, thatconsists of a set of vertices, V={1,...,N}, representing
the team members, and a set of edges, E={(i, j)∈V×V｜i∈N} representing the active inter-agent communicationlinks. If g is an undirected graph, there is a line between
node i and j, representing the impact between them ismutual. If g is a directed graph, the line (i, j) is directed,
which consists of starting point i and finishing point j,
representing node i makes an impact on node j.
According to the different functions of the agentsin leader-follower network, the agents can be divided
into two categories, i.e. leaders and followers. The
leaders were capable of knowing the details of theglobal task that is to come to a given target, while thefollowers just need to keep in touch with leaders toreach the target. The agents belong either to the subsetof leaders, Nl, or to the subset of followers, Nf , whereNl∪Nf =N, Nl∩Nf =φ and the number of agents in each
set is given by |Nl|=Nl and |Nf|=Nf respectively. Due tolimited sensing capability of the sensors, each agent has
a limited sensing zone of radius Δ＞0. At any given
time, the set of agents located within the sensing zone ofagent i∈N are referred to as the neighbours of agent i,
Ni={j∈N: |xi－xj|≤Δ}. Each agent has knowledge of therelative coordinates to its neighbouring agents, butcannot detect or communicate with agents outside itssensing zone. Note that both g and Ni are time-varying.
The difference between leaders and followers is thatleaders are aware of the location of the target and cancontrol group but the followers cannot.
We start by introducing a notation for the distance
between two arbitrary agents i and j. Letδ i , j = δ j ,i = xi − x j =k ,lConsider N agents evolving in R . We use single
integrator agents whose motions obey the model:=xij uij , i ∈=N {1,  , N }, j ∈=N {1,  , M }，(1)

Followers
V3V4l ,kkl

(x − x ) (x − x ) ≥ 0kli

jTkli

jSince we are considering a physical system we canassume that xk(t) is a continuous function for any agent

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Socialized Multi-Agent System Rendezvousk∈N. The time derivative δij is not directly defined when

δ ik, ,jl = 0 so here we shall instead consider the time
derivative of (δ i , j )k ,l 2d (d i , j )k ,l2( ==2dd
2 xi − x ji, ji, jk ,ldt

k ,lkl) ( x − x )Tklijeach agent are undirected. The topology of group is thus
described by undirected graph g1={V1, E1}.
The dynamics for an arbitrary follower agent i are givenby(2)To describe the relationship between leaders andtarget, the target attraction function is defined asl
d − xjll,>0 f ( ddj )jlldj
F ( xj , d ) =  0,ld j = 0.
(3)(a)

Topological structure(b)

Position rendezvous

where d is the location of the target and defineld=| xlj − d | representing the agent xlj ’s distance tojl

the target. At any given position x j ≠ d , the direction(l)of F x j , d is towards the target and the magnitude is( )decided by the continuous scalar function f δ j ≥ 0 .
l

In the following analysis, the leader-follower
multi-agent network that consists of four leaders andfive followers is illustrated. The agents are tagged withnumbers, starting at 1 at first, followed with 2, 3 and theother numbers, where agents from 1 to 5 belong tofollower agents, whereas 6 to 9 belong to leader agentsand node 10 is the target. Without loss of generality, we
assume that the target is the origin of coordinates andf(δj)= βδj, β=0.5. The initial configuration of each agent
can be seen in Tab.2.
Table 2. Initial values of the simulation work.

xy1
27.50Followers
234282828-0.2-44532-1617-2Leaders7821.522-10.5

9211Target
1000Fig. 1. Simulation results in the democracy strategy.

−N x + ∑ x∑ (x − x ) =jjxi =−likjjijilkjk ∈N ik ∈N i

j

Where xi denotes the ith agent is in the jth layer, andxkl denotes the kth neighbor of xij , which is in the lthlayer of the networks. We can conclude that the motionsof followers are decided by their neighbors. Considering
the situation of two dimensions, the dynamics can belooked as the resultant vectors of themselves and theirneighbors, where the directions of all the agents are thesame corresponding to the resultant vectors, and thespeeds of the swarm motion members can also bedecided by the norm of the resultant vectors. For anarbitrary leader agent j∈Nl, the dynamics are describedbyx lj =− N lj xlj + ∑ xkm + F ( x lj , d )3. Results and Analysis
In the democracy strategy, the motion of agent isaffected by the other ones located within the sensing
zone of the agent. If agent i has an impact on j, agent i isalso affected by j. In other words, interactions among(4)k∈Nljl(5)mand x j and xk can be defined accordingly. By contrast,
the dynamics for leaders are the same as the followers insome way, except for the impact from the target, which

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Yunzhong Song, Ziyi Fu, Fuzhong Wangis the leaders’ feature. The simulation results in thedemocracy strategy can be seen in Fig.1.In the processof simulation, the distances between the agents from theleader group and the follower group and the target agent
are used to measure the rendezvous time indirectly,
where all the distances are turned into be zero isregarded as the network rendezvous. It is not difficult torecognize that during the democracy strategy, theinformation flow among the different layers is fully
utilized, and the enjoyment of the overall information
via the shared cooperation realized win-win policy of
the agents. That kinds of profit can only be available
depend on the interdependency among the networks ofnetworks. In order to elucidate the democracy in theways of networks of networks further, the paradigm for
democracy of networks of networks is illustrated asFigure 2.

Acknowledgements
This work is partially supported by NSFC Grant(61340041 and 61374079) and the Project-sponsored by
SRF for ROCS, SEM to Yunzhong Song.
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Fig. 2. The paradigm for democracy of networks of networks.

4. Conclusion
In this paper, according to the paradigm of networksof networks we turned the leader-follower agent
systems into four different layers, and each of theselayers can be equivalent to one sub-network, theconnections among them can be viewed as theinterdependency of the networks of networks, and wehave examined and discussed the convergent
characteristics of the democracy strategy. The
simulation of leader-follower multi-agent network
which consisted of four leaders and five followers
demonstrated the potential usage of our suggested
scheme.

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