Hierrhi Assemby of Leder-Asymmetri, Singe-Leder Networks Wseem Abbs nd Mgnus Egerstedt Abstrt The onnetion between eder-symmetry nd ontrobiity in ontroed greement networks provides topoogi, neessry ondition for ontrobiity. In this pper we investigte how to produe hierrhi networks tht, t eh eve in the hierrhy, exhibit the eder-symmetry properties. rph grmmrs re moreover provided for ssembing the eder-symmetri networks of ny size. I. INTRODUCTION In this pper, we investigte how to onstrut network topoogies in hierrhi mnner in suh wy tht they re menbe to extern ontro. In prtiur, the networks wi be omprised of oetion of nodes whose ohesion is ensured through greement-bsed intertion rues. These networks n moreover be ontroed by injeting ontro signs t prtiur input-nodes (so-ed eder-nodes) in the networks. The ontro of suh muti gent systems hs reeived onsiderbe ttention during the st dede nd sever resuts hve been presented regrding the nysis of the underying struture nd hrteristis of these distributed oordintion systems, e.g. [1],[2],[3],[4]. One key question when trying to design ontroers for suh networks is whether or not they re even ontrobe in the first pe. Controbiity issues in these types of networked systems ws first disussed in [5], where onditions for ontrobiity were given in terms of the eigenvetors of the grph Lpin. Lter, more topoogi exportion of the ontrobiity properties in suh eder-foower networks ws given in [6], presenting suffiient ondition for network to be unontrobe in the se of singe eder se using grph symmetries. These onepts were extended in [7] nd through the use of equitbe prtitions. In [8], grph theoreti disussion of the ontrobe subspe of n unontrobe network ws given. Some other resuts reted to ontrobiity of eder-foower systems were presented in [9]. A key onept when studying the ontrobiity of networked systems tht hs emerged is the notion of n extern, equitbe prtition. Suh prtition groups together nodes into es, nd members of the sme e hve been shown to onverge symptotiy to the sme subspe. As suh, neessry ondition for network to be ompetey ontrobe is tht no suh es exist tht shre more thn singe node. We wi refer to suh networks s eder-symmetri nd this pper ddresses the onstrution of eder-symmetri networks through the Emis: wbbs@gteh.edu,mgnus@ee.gteh.edu Shoo of Eetri nd Computer Engineering, eorgi Institute of Tehnoogy, Atnt, A 30332, USA inter-onnetions of mutipe sub-networks tht re themseves eder-symmetri. This proess of inter-onneting networks eds, in turn, to hierrhi struture nd the retionship between the eder-symmetry properties t eh stge of the hierrhy nd the over network is presented. Moreover, these resuts re then used to provide grph grmmr rues for the sef-ssemby of individu nodes into eder-symmetri network of ny size with singe eder. Aso, the mximum distne of ny foower node from the eder in these resuting networks is so given. The reson for defining grph grmmrs for this ssemby tsk is tht they hve been used to mode sef-ssemby proesses invoving rge number of mobie gents in ntur nd diret mnner, s shown in [10],[11]. This pper hs two mjor prts. One is reted to the hierrhi onstrution of eder-symmetri, singe-eder networks nd the other is reted to the grph grmmrs for the sef ssemby of suh networks. Our presenttion strts with system desription in Setion III. In Setion IV, we review some resuts regrding the ontrobiity nd edersymmetry of singe eder networks from grph theoreti point of view. In Setion V, we present resuts reted to the eder-symmetry of interonneted nd hierrhi networks. Setion VI reviews the bsis of grph grmmr onstrutions, nd Setion VII presents the rue sets for sef ssemby of eder-symmetri, singe-eder systems. II. SYSTEM DESCRIPTION In this setion, we show how n we onstrut rge singe-eder network, by onneting together smer singeeder networks in hierrhi wy. The min ide is to grow the network by onneting together individu networks t different stges, where, smer singe eder networks onstitute the first stge of this proess. At the next stge, the eders of these smer subnetworks re onneted together with n extern node ed the super-eder. This supereder serves s the extern input to our system. Throughout this pper, by grph, we men, n undireted grph with no oops nd mutipe edges between the verties. Consider n identi eder-foower networks, i.e. networks where the ontro input is injeted t the eder-node. Eh of these networks (i), where i {1,2,...,n}, hs singe eder nd m foowers. The hierrhi onstrution is now to onnet together the eders of (i) vi nother eder-foower network (), where the eders tke on foower roes, nd new node x s tkes on the eder roe. x s is super eder nd it is so eder node of (). Now, the over network tht is obtined by onneting, (2),..., (n) together vi their eders to n extern
node x s, is so eder-foower network with singe eder x s nd other nodes being foowers. This onstrution is iustrted in Fig. 1 x (i) x s x s x (3) In ight of the seond ft, we wi shift our fous from ontrobiity to the eder-symmetry throughout the reminder of this pper. The reson for this shift is tht the eder-symmetry is purey topoogi ondition, whie ontrobiity is not. It shoud be noted tht it is just neessry ondition for ontrobiity nd no topoogi neessry nd suffiient ondition hs, s of yet, been found for the singe-eder, ontroed greement dynmis. (i) () Fig. 1. (i) is eder-foower network with singe eder x (i) nd m=3 foowers. n = 3 suh networks re onneted together vi their eders through nother eder-foower network () where n extern node x s tkes on eder roe. The resuting inter-onneted network is so eder-foower network with x s s eder nd other nodes being foowers. Wht we woud ike to understnd is how ertin key properties ssoited with the ontrobiity of the individu networks re inherited by the new inter-onneted network. This is the topi of the next setions tht strt with disussion of wht the key topoogi properties re. III. EQUITABLE PARTITIONS AND LEADER-ASYMMETRIC, SINLE LEADER NETWORKS. In this setion we wi review the sope of equitbe prtitions in exmining the ontrobiity of singe eder networks nd stte some resuts from [7],[8] nd [9]. These resuts wi onnet the study of eder-symmetry with tht of ontrobiity (or t est with unontrobiity), whih is onnetion tht wi be pursued throughout the reminder of this pper. Definition 3.1: (Extern Equitbe Prtition): A prtition π of nodes X of grph, with es C 1,C 2,,C r is sid to be n extern equitbe prtition if eh node in C i hs the sme number of neighbors in C j, for i, j {1,2,,r},i j, with r= π, whih denotes the rdinity of prtition. Definition 3.2: (Non-Trivi Extern, Equitbe Prtition): An extern equitbe prtition in whih t est one e hs more thn one nodes is non-trivi extern, equitbe prtition. Definition 3.3: (Leder-Invrint Extern, Equitbe Prtition (LEP)): The LEP is prtition π M = π F πl, where π F ={C M 1,CM 2,,CM s } is the extern equitbe prtition of the foower nodes suh tht the rdinity of π F is minim (i.e. hs the fewest es), nd the eder L beongs to the singeton e C M s+1 ={L} of the prtition π L ={C M s+1 }. Ft 1: Every eder foower network hs unique LEP [9]. Definition 3.4: (Leder-Asymmetri Singe-Leder Network): A eder-foower network is sid to be edersymmetri if its LEP is trivi i.e. every e in its LEP is singeton e. Ft 2: A singe eder network exeuting the ontroed greement dynmis is ompetey ontrobe ony if it is eder-symmetri [9]. (see Fig. 2) (2) Fig. 2. hs trivi LEP, so it is eder-symmetri, singe eder network. It is so ompetey ontrobe with s its eder. (2) is not eder-symmetri s it hs non trivi LEP. (3) hs trivi LEP, so it is eder-symmetri but it is not ompetey ontrobe with x (3) s its eder. x (3) (3) IV. HIERARCHICAL LEADER-ASYMMETRIC, SINLE LEADER NETWORKS. Definition 4.1: (Connetion Network nd Interonneted Network) Consider n eder-foower networks, (2),..., (n), eh with singe eder,,...,x (n) respetivey 1, then is network obtined by onneting, (2),..., (n) together vi their eders ony to n extern node ed super eder x s, through onnetion network (), s disussed in the Setion II. The resuting is sid to be n interonneted network, whih is so eder-foower network with singe eder x s nd other nodes being foowers. One question one might sk is whether or not this type of onstrution preserves ertin desirbe properties. In this pper, we wi fous on the issue of eder-symmetry s defined in the Definition 3.4. However, we strt with the question of ontrobiity nd see tht this property is in ft, not preserved when ontrobe networks re interonneted. Lemm 4.1: Let be n interonneted singe-eder network s per Definition 4.1. If the individu networks, (2),..., (n) re ompetey ontrobe with respet to their respetive eders, then ompete ontrobiity of the onnetion network () is neither neessry nor suffiient ondition for the ompete ontrobiity of. Proof: (Counter exmpe)- Let nd (2) be two ompetey ontrobe singe-eder networks with nd s eders respetivey, s shown in the Fig. 3. Interonneted network is obtined by onneting nd (2) through ompetey ontrobe onnetion network (). 1 Note tht we wi, throughout this pper, use x (i) to denote the stte ssoited with node i, but we wi so use it s shorthnd to denote the node itsef, whenever this is er from the ontext.
The resuting interonneted network is not ompetey ontrobe with x s, s its LEP is non-trivi. (see Fig. 3.) (2) x s () x s Fig. 3. Compete ontrobiity of onnetion network () is not suffiient ondition for ontrobiity of interonneted network. Now, et us onnet the eders of sme nd (2) through nother onnetion network () tht is symmetri with respet to x s, nd hene unontrobe. The resuting is ompetey ontrobe with respet to x s even though () is unontrobe. (see Fig. 4.) (2) x s () x s Fig. 4. Compete ontrobiity of onnetion network () is not neessry ondition for ontrobiity of interonneted network. This shows tht the ompete ontrobiity of the individu networks does not ensure the ompete ontrobiity of the interonneted network. Lemm 4.2: Let be pth network with one of the end nodes s eder nd (2) be ny edersymmetri singe eder network with eder. Let be network obtined by onneting the seond end node of with, then is so eder-symmetri, singe eder network with s eder. (2) Fig. 5. Pth network onneted with eder-symmetri, singe eder network gives eder-symmetri. Proof: A pth network with termin node s the ony eder, is ompetey ontrobe nd hene, in π, es ontining the nodes of wi be singetons. Sine is the ony node of (2), onneted with ny node of, so e ontining is so singeton. Sine (2) is itsef eder symmetri nd is in singeton e in π, so other nodes of (2) wi so be in singeton es. Thus, giving trivi π. Hene is eder-symmetri. (see Fig. 5.) Lemm 4.3: Let be n interonneted singe eder network s per Definition 4.1, with subnetworks (i) being eder-symmetri, then, is so eder-symmetri iff es ontining x (i) in re singetons. Proof:( ) If is eder-symmetri, then it hs trivi LEP π by definition with es being singetons. ( ) Let us ssume for the ske of ontrdition tht in π, es ontining x (i) s re singetons but is not eder-symmetri. Then, there must be e C in π ontining more thn one nodes tht either beong to () sme subnetwork (i) or (b) different subnetworks. () is not possibe s eh subnetwork is itsef eder symmetri nd eh x (i) is in singeton e in π. For (b), sine foowers of one subnetwork (i) re not onneted to the eder of nother subnetwork, so nodes in e C ontining foowers of different subnetworks, n never hve sme node to e degree with other es, s required by LEP onstrution. So, (b) is so not possibe. Theorem 4.4: If is n interonneted network s per Definition 4.1 nd, (2),..., (n) re identi, edersymmetri, singe eder networks, then eder-symmetry of the onnetion network () with x s s eder, is suffiient ondition for the interonneted network to be eder-symmetri with the sme eder x s. Proof: Let X be set, ontining the eder nodes of (i) i.e. X = {,,,x (n) }. Aso, et X dir = {x (i) X : x (i) is direty onneted to x s }. Simiry, X not dir = {x (i) X : x (i) is not direty onneted to x s }. Note tht, X = X dir X not dir For proving the bove theorem, we wi prove the foowing ims first. Cim 1: In π, every x d X dir is in singeton e. Proof : Let C s be e in π, ontining the super eder x s. Now ssume for the ske of ontrdition tht there exists x d X dir not ontined in singeton e C d. Then, this C d n ony ontin nother x d X dir s they re the ony nodes in π direty onneted to x s (nd C s ). This requires to be symmetri bout x s. Sine (i) s re identi nd eder-symmetri, so n be symmetri bout x s iff the onnetion network () is symmetri bout x s. But () is eder-symmetri by onstrution nd so, not symmetri bout x s. Thus, is so not symmetri bout x s nd hene, our ssumption is not true, thus, proving the im. Cim 2: In π, every x nd X not dir is so in singeton e. Proof : Let us ssume for the ske of ontrdition tht there exist x nd X not dir tht is not in singeton e C nd in π. Aso, it is direty onneted to some x d X dir. Then there is one of the foowing possibiities tht () C nd so hs some x d X dir. (b) C nd so hs foower node of subnetwork whose eder is x d or () C nd so hs some other x nd X not dir. () is not possibe by im 1. (b) is
not possibe s subnetworks (i) re identi nd so eder-symmetri, so there wi wys be foower node in the subnetwork of eder x d tht n never be ontined in vid e in π. For (), et us ssume tht C nd so ontin x nd X not dir ong with x nd. Sine, x nd is direty onneted to x d tht is in singeton e C d, this requires x nd to be direty onneted to x d so. Now, to mintin the sme node to e degree ondition for vid π, the eder in X not dir tht is direty onneted to x nd wi be ontined in e ong with some other eder tht is direty onneted to x nd. This wi ontinue unti we get e ontining x ndα X not dir nd x ndβ X not dir. Now there must be nother eder in X not dir tht shoud be direty onneted to either one of x ndα or x ndβ but not both, s otherwise the onnetion network () nnot be eder-symmetri. Let tht remining eder be x ndst nd without oss of generity, it is onneted to x ndα. Then for vid π, the e C ndst ontining x ndst must so ontin foower node of subnetwork whose eder is x ndβ nd tht foower node shoud be direty onneted to x ndβ. Sine subnetworks re identi nd eder-symmetri with the the sme number of foower nodes, so foower node in the subnetwork of x ndst wi wys be eft tht n never be ontined in vid e in π. So our ssumption is not vid nd the im is true. From ims 1 nd 2, we get tht in LEP of, eders x (i) of subnetworks (i) wi be ontined in singeton es. Thus, from Lemm 4.3, we get tht is eder-symmetri. x s x (3) Fig. 6. An exmpe iustrting Theorem 4.4. Theorem 4.5: Let, (2),..., (n) be identi nd eder-symmetri singe eder networks with eders,,...,x (n) respetivey. Let be network obtined by interonneting the eders x (i) through network (). If () is eder-symmetri with one of x (i) sy x ( ), then is so eder-symmetri with singe eder x ( ). Proof: Let X = {,,,x (n) }. Aso, ssume tht there exists x ( ) X suh tht () is eder-symmetri with x ( ). Aso x ( ) is eder of subnetwork ( ). Let us do the prtition of X s X = ( ) X dir Xnot dir {x }, where, X dir = {x (i) X : x (i) is direty onneted to x ( ) }. Aso, X not dir = {x (i) X : x (i) is not direty onneted to x ( ) }. Cim: In π, the foower nodes of subnetwork ( ) wi be ontined in singeton es. Proof : Let V be set of foower nodes of ( ) nd V = V d Vnd, where V d V is subset ontining those foowers of ( ) tht re direty onneted to x ( ) nd V nd V is subset ontining the foowers of ( ) not direty onneted to x ( ). Firsty, we show tht, in π, ny v d V d wi be in singeton e. For the ske of ontrdition, et us ssume tht there exists v d1 V d suh tht e C vd1 ontining v d1 is not singeton. Then C vd1 wi so ontin one of the foowing ong with v d1, () some v nd V nd, (b) some other v d V d, () some x nd X not dir, (d) some foower node of subnetwork (i) where (i) ( ). or (e) some x d X dir. Out of these (),() nd (d) re not possibe s none of the nodes in these options is direty onneted to x ( ) whie v d1 C vd1 is direty onneted to x ( ). Now, sine ( ) is eder-symmetri, so it is not symmetri bout x ( ), hene (b) is so not possibe. For (e), ssume tht C vd1 ontins v d1 nd some x d1 X dir, where x d1 is eder of subnetwork (d1). Then v nd1 V nd, where v nd1 is direty onneted to v d1, nd foower node of subnetwork (d1) sy v (d 1) f 1 direty onneted to x d1, must so be ontined in the sme e due to the onstrution rues of π. Simiry, in the next step, v nd2 V nd where v nd2 is direty onneted to v nd1, nd foower node of (d1) direty onneted to v (d 1) f 1, must so be in sme e. This wi ontinue nd sine subnetworks (i) re identi, so we wi wys be eft with foower node in (d1) tht nnot be ontined in vid e in π. So,(e) is so not possibe, nd every v d V d wi be in singeton e in π. Note tht foowers of one subnetwork (i) re not onneted with the eders or foowers of nother subnetwork. Aso, in ( ), every v d V d wi be in singeton e in π. Sine ( ) is so eder-symmetri, so these fts wi direty impy tht every v nd V nd wi so be in singeton e. This proves our im. If we remove the foower nodes of ( ) from, we get, where exty stisfies the onditions in Theorem 4.4 with x ( ) = x s, thus, π is trivi, with nodes being in singeton es. Now dding the foower nodes of ( ) to wi give us. By the bove im, we know tht foower nodes of ( ) n never be use of non trivi π if ( ) is identi to the other subnetworks (i). Combining these fts, we onude tht π is so trivi nd hene, is eder-symmetri with x ( ) s eder. V. RAPH RAMMAR PRELIMINARIES. In this setion, we wi review the bsis of grph grmmrs pproh to mode the tsk of ssembing rge number of sef ontroed prts into presribed formtion. We refer the reders to [10] nd [11] for more detis bout this topi. Definition 5.1: (Rue): A rue is pir of grphs r = (, b ) tht hnges the edge set E( ) of to E( b ) to give b whie keeping the vertex set onstnt, i.e. V( )= V( b ). The size of r is V( )= V( b ).
x ( ) x (3) Fig. 7. An exmpe iustrting Theorem 4.5. Definition 5.2: (Rue Set or rmmr): A rue set (or grmmr) Φ is set of rues tht defines onurrent gorithm for group of individu nodes to foow. Definition 5.3: (System): A system is pir ( 0,Φ) where 0 is n initi grph of the system nd Φ is set of rues ppied on 0. Definition 5.4: (Trjetory): A trjetory of system ( 0,Φ) is (finite or infinite) sequene 0 (r 1,h 1 ) 1 (r 2,h 2 ) 2 (r 3,h 3 ) If the sequene is finite, then there exists termin grph where no rue in Φ is ppibe. We denote trjetory of system by τ nd the set of suh trjetories by T ( 0,Φ). Aso, we use the nottion τ j to denote the j th grph in the trjetory τ j. VI. RAPH RAMMARS FOR PRODUCIN LEADER-ASYMMETRIC SINLE NETWORKS In this setion we wi show how grph grmmrs n be used to produe eder-symmetri, singe eder networks of ny size in deentrized wy. These simpe rues n be used to produe subnetworks of ny size nd then using the previous resuts, we n onstrut bigger networks out of them tht re so eder-symmetri with singe eder. So we n stte our go s, Construt rue set Φ for system ( o,φ) with o s set of isoted nodes, suh tht trjetory of system, τ T ( o,φ), is finite sequene with termin grph s set of eder-symmetri, singe eder networks with p nodes. We the resuting edersymmetri networks with p nodes nd singe eder s ryst. We wi so provide mximum eder to node distne, d in tht ryst, resuting from ( o,φ). We so use the nottion to denote the rdinity of the vertex set of the grph. A. Rues for Crysts of Size p=2 n,n 1 Consider the foowing rue set Φ A { (r0 ) Φ A = 1 (r 1 ) i i i+1 1 i n 1 Cim: Φ A gives eder-symmetri, singe eder rysts of size p=2 n. Proof : Let τ A be trjetory obtined by Φ A nd τ j is the j th grph in this trjetory. Then τ 0 is pth grph with singe node nd singe eder 1. τ 1 is obtined by onneting two τ 0 vi their eders nd mking one of them s new eder 2. By Theorem 4.5, τ 1 is eder-symmetri with eder 2 s τ 0 is eder-symmetri. Aso τ j = 2 τ j 1. This ontinues unti we get termin grph with τ n 1 whih is inft eder-symmetri with singe eder n nd τ n 1 = p=2 n. Here mximum eder to node distne, d = n. B. Rues for Crysts of Size p=k(2) n,k 3,n 0 Consider the foowing rue set Φ B (r 0 ) 1 b 1 (r Φ B = 1 ) b i b i+1 1 i (k 3) (r 2 ) b k 2 (r 3 ) j j j+1 1 j n Cim: Φ B gives eder-symmetri rysts of size p = k(2) n with singe eder. Proof : Proof is exty ike the proof of Φ A with the ony ddition tht in the initi steps, the first three rues r 0,r 1,r 2 re reting pth grph τ k 2 with τ k 2 = k with singe eder 1. Mximum eder to node distne, d = (n+k)-1 in this se. C. Rues for Crysts of Size p=k(2) n + 1,k 3,n 0 Consider the foowing rue set Φ C (r 0 ) ε 1 b 1 (r 1 ) b i b i+1 1 i (k 3) (r Φ C = 2 ) b k 2 (r 3 ) ε 1 1 f in (r 4 ) ε i i i+1 1 i n (r 5 ) ε m ε m ε m+1 1 m (n 1) These rues wi produe the eder-symmetri, singe eder rysts of size p 1=k(2) n, exty the sme wy s in Φ B with n+1 s eder. An extr node, f in is then onneted to n+1 to give ryst of size p=k(2) n. In this se, mximum eder to node distne, d = n+k D. Rues for Crysts of Size p=k(q) n. k,q 3, n 0 Consider the foowing rue set Φ D Φ D = (r 0 ) 1 b 1 (r 1 ) b i b i+1 1 i (k 3) (r 2 ) b k 2 (r 3 ) j j j+1 ε j,1 1 j n (r 4 ) j ε j,m ε j,(m+1) 1 j n, 1 m (q 3) (r 5 ) j ε j,(q 2) 1 j n, Cim: Φ D gives eder-symmetri, singe eder rysts of size p=k(q) n. Proof : Let τ D be trjetory produed by Φ D. Here τ k 2 is pth grph with τ k 2 = k hving singe eder 1 produed by first three rues r 0,r 1 nd r 2. In the next step q of these identi τ k 2 grphs re onneted vi their eders ony, suh tht these 1 s re themseves onneted
in pth grph now with 2 s their eder to give τ k+q 3. Now by the diret ppition of Theorem 4.5, τ k+q 3 is so eder-symmetri with 2 s eder. Aso τ k+q 3 = kq. In the next step q identi τ k+q 3 re onneted vi their eders 2 tht re onneted in pth grph, thus giving us τ (k 2)+2(q 1) with 3 s eder nd τ (k 2)+2(q 1) = q(kq)=k(q) 2. Agin τ (k 2)+2(q 1) is eder-symmetri by the diret ppition of Theorem 4.5. This ontinues n times unti we get termin grph τ (k 2)+n(q 1) whih is inft eder-symmetri with n+1 s eder nd τ (k 2)+n(q 1) = k(q) n. Here, mximum eder to node distne, d = n(q-1)+k-1. VII. EXAMPLE AND ENERAL ALORITHM An exmpe showing the onstrution of rysts of size p=8, using the rue sets of Setion VI-A re shown in the Fig. 8. A rues in the rue sets in Setion VI re binry 2. An gorithm behind the grph grmmrs of bove ses is presented beow. Agorithm I Require: be grph, suh tht () is eder-symmetri with singe eder (b) = p 1 : Ftorize p s p=k(q) n 2 : Mke Pth rphs 1 initiy with singe eder 1 nd 1 = k 3 : for i=1 to i=n 4 : i+1 = Connet q no. of i s together vi their eders i, s.t. these i s re onneted in pth grph with the end node s new eder i+1 of i+1. 5 : i=i+1 6 : end 7 : n+1 is required with singe eder n+1 Here the ftoriztion step of p = k(q) n is importnt s it is not unique. The mximum eder to node distne d depends on the speifi hoie of k,q nd n for the sme p. It turns out tht for sme p=k(q) n, ftoriztion with rger vue of n produes ryst of size p with smer d, where d is the mximum eder to node distne, if we use the bove sheme. Aso, for sme p, if two ftoriztions hve sme n, then the one with rger q produes ryst with smer d. VIII. CONCLUSIONS In this pper, we disussed the onstrution of hierrhi eder-foower networks through the interonnetion of mutipe subnetworks tht re themseves eder-foower. 2 Rues, whose vertex sets hve two verties re binry. Fig. 8. p=8 3 1 3 r 0 r 2 1 1 1 2 1 r 1 2 1 2 2 An exmpe ssemby sequenes for produing rysts of size We investigted the eder-symmetry property of suh interonneted networks whih is neessry ondition for their ontrobiity, in terms of their subnetworks. We so gve suffiient ondition for the interonneted hierrhi networks to be eder-symmetri. Moreover, these resuts re used to design the rues for sef-ssemby of isoted nodes into eder-symmetri, singe eder networks of ny size in deentrized mnner. REFERENCES [1] A. Jdbbie, J. Lin nd A.S. Morse, Coordintion of groups of mobie utonomous gents using nerest neighbor rues, IEEE Trns. Automt. Contr.,, vo. 48, no.6, pp. 988-1001, 2003. [2] M. Mesbhi, On stte-dependent dynmi grphs nd their ontrobiity properties, IEEE Trns. Automt. Contr.,, vo. 50, no.3, pp. 387-392, 2005. [3] M. Ji, A. Muhmmd nd M. Egerstedt, Leder-bsed muti-gent oordintion: Controbiity nd optim ontro, Amerin Contro Conferene,, Minnepois, pp. 1358-1363, June 2006. [4] R. Ofti-Sber, J.A. Fx nd R.M. Murry, Conensus nd oopertion in networked-mutigent systems, IEEE Proeedings, vo. 95, no.1, pp. 215-233, 2007. [5] H.. Tnner, On the ontrobiity of nerest neighbor interonnetions, IEEE Conferene on Deision nd Contro, pp.2467-2472, De. 2004. [6] A. Rhmni nd M. Mesbhi, On the ontroed greement probem, Amerin Contro Conferene, Minnesot, pp. 1376-1381, June 2006. [7] M. Ji nd M. Egerstedt, A grph-theoreti hrteriztion of ontrobiity for muti-gent systems, Amerin Contro Conferene, New York, pp. 4588-4593, Juy 2007. [8] S. Mrtini, M. Egerstedt nd A. Bihi, Controbiity deompositions of networked systems through quotient grphs, IEEE Conferene on Deision nd Contro, Cnun, pp. 5244-5249, De. 2008. [9] A. Rhmni, M. Ji nd M. Egerstedt, Controbiity of muti-gent systems from grph theoreti perspetive, SIAM Journ on Contro nd Optimiztion, vo. 48, no. 1, pp. 162-186, 2009. [10] E. Kvins, R. hrist nd D. Lipsky, rph grmmrs for sef ssembing roboti systems, IEEE Interntion Conferene on Robotis nd Automtion, New Orens, pp. 5293-5300, Apri 2004. [11] E. Kvins, R. hrist nd D. Lipsky, A rmmti Approh to Sef-Orgnizing Roboti Systems, IEEE Trns. Automt. Contr., vo. 51, no. 6, pp. 949-962, June 2006. 1 1