Deterministic Multicultural Dynamic Networks: Seeking a Balance between Attractive and Repulsive Forces

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1 Int. J. Communatons, Network and System Senes, 6, 9, ISSN Onlne: ISSN Prnt: Determnst Multultural Dynam Networks: Seekng a Balane between Attratve and Repulsve Fores Krstna B. Hlton, G. S. Ladde Department of Mathemats and Statsts, Unversty of South Florda, Tampa, FL, USA How to te ths paper: Hlton, K.B. and Ladde, G.S. (6) Determnst Multultural Dynam Networks: Seekng a Balane between Attratve and Repulsve Fores. Int. J. Communatons, Network and System Senes, 9, Reeved: Otober, 6 Aepted: Deember, 6 Publshed: Deember 3, 6 Copyrght 6 by authors and Sentf Researh Publshng In. Ths work s lensed under the Creatve Commons Attrbuton Internatonal Lense (CC BY 4.). Open Aess Abstrat An mportant ssue n soety s the attempt to balane ommuntes workng ooperatvely and ohesvely wth one another whle allowng members the ablty to retan ndvdualty and fosterng an envronment of ultural dversty. We seek to study the ohesve propertes of a ulturally dverse dynam soal network. By onsderng a mult-agent dynam network, we seek to model a soal struture and fnd ondtons under whh oheson and oexstene are mantaned. We present a spef llustraton that serves to establsh the framework n whh explt suffent ondtons n terms of system parameters are found for whh the network s ohesve. By utlzng Lyapunov s Seond Method and omparson equatons, we are able to fnd suh ondtons for the gven llustraton. Further, for the llustraton, we deompose the ultural state doman nto nvarant sets and onsder the behavor of members wthn eah set. Moreover, we analyze the relatve ultural affnty between ndvdual members relatve to the enter of the soal network. We also demonstrate how onservatve the estmates are usng Euler type numeral approxmaton shemes based on the gven llustraton. We are then able to onsder how hanges n the varous parameters affet the dynams of the llustrated network. By ganng suh nsght nto the behavor of the llustrated network, we are able to better understand the mpat of both attratve and repulsve nfluenes on the network. Ths leads to establshng a shema for helpng when reatng poles and prates atered to promotng both dversty and oheson wthn a ultural network. Keywords Mult-Agent Network, Cohesveness, Lyapunov Seond Method, Invarant Sets, Soal Network DOI:.436/jns Deember 3, 6

2 K. B. Hlton, G. S. Ladde. Introduton The goal of ths work s to explore the ohesve propertes and behavor of ndvdual members of a dynam soal network. Dynam network systems are often used to model the behavor patterns of anmals, autonomous vehles, the spread of a ontagon, traff flow, and many other types of stuatons. In ths work, we are nterested n modelng the behavor of members wthn a soal network. In partular, we are nterested n the ultural shfts of members wthn ulturally dverse groups. We seek to better understand the nternal and envronmental fators that may foster a sense of ooperaton between members of the network whle allowng ndvdualty and dversty to be mantaned and enhaned. One of the onepts studed usng network dynams s that of onsensus [] [] [3] [4]. In suh models, the ondtons under whh a group olletvely omes to an agreement are studed. Another queston of nterest for suh a network s when the group mght subdvde nto smaller subgroups eah onvergng to a onsensus but never reahng a onsensus as an overall group. Suh dynam network models are useful n many areas beyond just soal networks. For example, work n both bologal networks and ontrol theory onsders suh large-sale dynam models n the ontext of onnetvty, stablty and onvergene [5] [6] [7]. Usng these deas, muh of the work done n these areas look to develop onsensus seekng algorthms and onsder long term o-exstene, oheson, and stablty of the network under onsderaton [8] [9] [] []. Coheson wthn a soal network s a urrent top of great nterest and many authors have done researh wthn ths area [] [3]. The onepts of oheson and ooperaton wthn a group are often mult-faeted, dynam, and omplex but are mportant onepts when tryng to better understand how natons or human groups nterat and funton [4]. As Knoke and Yang note [5], t s soal oheson that enables nformaton to spread and allows a group to at as a unt rather than ndvduals. We often seek to reate stuatons for whh people of dfferent bakgrounds and belefs are able to oexsts and reate a thrvng sense of ommunty. We seek to better understand the group dynams of suh a soety n order to reate poles and prates that enourage a sense of ommunty among ndvduals from a varety of ultural bakgrounds. We use the term multultural soal network to desrbe a soal network n whh the agents have a dverse ultural and/or ethal bakground and are atvely seekng to enhane and to mantan dversty wth harmony and prosperty. In suh a network, the goal of agents s not n approahng a onsensus but rather the ablty to lve and work ooperatvely wth one another for a ommon good and goal. For example, onsder a populaton n an area for whh there exsts a sub-populae of mmgrants. In suh a stuaton, the subgroups or sub-ommuntes of mmgrants desre to be an ntegral part of the ommunty and seek to be respetable produtve members of the ommunty and the soety n general whle retanng ther ultural dversty. In explorng the dynams of a mult-ultural network, we are lookng to better understand the delate balane between a ulturally dverse ohesve soal struture and a soal struture for whh oheson does not exsts. For when oheson s lakng n the soal 583

3 K. B. Hlton, G. S. Ladde network, ooperaton may not be as prevalent and we begn to see features suh as segregaton, volene, eonom destablzaton and rme wthn the network. We seek to model suh a stuaton and better understand the soal dynams of a group seekng to fnd suh a balane. In partular, we are lookng to model a dynam soal network for whh there s a balane between onsensus and oheson. We wsh to model a network that s ohesve but for whh there s not a onsensus of ulture, that s to say the network does not develop a sngular ultural dentty. In dong so, we are nterested n better understandng the ohesve propertes of a multultural soal network. We present a prototype of a dynam model for whh we explore the features of suh a network. The presented example s used to exhbt the quanttatve and qualtatve propertes of the network. Further, the tehnques used are omputatonally attratve, easy to verfy and algebraally smple. In addton, the presented results are n terms of network parameters that haraterze the attrbutes of the network. The byprodut of ths provdes tools for plannng and makng poles regardng a dynam network. In Seton, we present an example of suh a network, as well as assumptons and notatons used throughout ths work. In Seton 3, usng Lyapunov s Seond Method and the omparson method [6], we onsder the dynams of ndvdual members wthn the network. In Seton 4, long and short term behavors of group members and nvarant ultural state sets are nvestgated. In Seton 5, we onsder numeral smulatons of the network to better understand the extent that onservatve estmates n Setons 4 and 5 are n a gven example. Fnally, n Seton 5, we onsder parametr varatons wthn the model affetng the dynams of the network. Further, we wll onsder how the model relates to a multultural network.. Problem Formulaton We wsh to model a mult-ultural soal network and therefore desre to apture the behavor of ndvdual agents who are seekng to belong to the group but also wantng to retan ndvdualty and dversty from other agents. In order to do so, we therefore onsder dynam equatons subjeted to both attratve and repulsve fores. In [7], one suh funton onsdered when modelng bologal dynam networks s gven by y g( y) = y a bexp, () n where a, b and are postve onstants and y. The funton g has long-range attraton and short-range repulson. In the followng, we formulate a modfed verson of a network dynam model n whh ndvduals seek to retan a balane between ndvdual member dentty and a group/ommunty membershp. We onsder a network whose dynams are desrbed by norporatng a long-range attraton and short-range repulson smlar to that n Equaton (). I, m =,,, m, Let us onsder a network of m members. For eah member { } 584

4 K. B. Hlton, G. S. Ladde n x s a ultural poston at tme t >. The vetor x t an be representatve of many varous aspets of ulture suh as belefs, behavors, ways of lfe, et. dependng on the network beng onsdered. Further, let us defne a relatve ultural state of th member wth a kth member of the ommunty as xk = x xk, and a enter of ultural state of the network m x = x () k m k = Consder the network whose dynam s gven by m m m x j dx = a xj q x x xj + bsn x x xj exp d t; j= j= j= x. ( t ) = x (3) + The onstant oeffent parameters, a, b, and q represent the weght of the soal moderaton attratveness (q), the repulsve fores (a), the rate of deay of the long range attratveness (), and the long-range attratveness (b) between ndvdual members and soal groups. We say that the network s ohesve f there exst onstants T and M suh that t T t and x ( tt,, x) x M that s to say that the members of the network after some pont n tme reman wthn a ertan dstane of the network enter. Further, we say that the network reahes a onsensus f x tt,, x x as t I, m. In ths ase eah member of the for all network draws loser to eah other and the network enter. Moreover, we defne the term relatve ultural affnty to be x ( t) x( tt,, x ) x ( tt,, x ) between the ultural vetor states of members 3. Charatersts of the Network =, the dstane k k k x and In ths seton, we wsh to explore the dynams of the agents wth the network dynam desrbed by Equaton (3). We wll be onsderng the oheson, qualtatve and quanttatve propertes suh as the overall stablty of the network enter, and varous types of nvarant sets. Whle explorng these deas, we wll also onsder what happens as the sze of the network nreases and what roles the parameters a, b and play wthn the model. Moreover, the presented example s utlzed to exhbt the quanttatve and qualtatve propertes of the network. In order to aomplsh suh a task, we utlze Lyapunov s Seond Method [6]. Ths method s algebraally smple, easy to verfy and omputatonally attratve. Furthermore, the results depend on the system parameters a, b, and q. Let us frst onsder the dynam of the network enter, x (). We note that x =, and k m m dx = a x x q x x x x k= k= ( k) ( k) m ( k ) x. x x k + bsn x x x x exp dt = k = k as defned n Equaton (4) 585

5 K. B. Hlton, G. S. Ladde and x s a statonary enter of the network. We defne the transformaton of the network by z = x x, notng that mz = m x k k = x = z = z z. k k k Therefore, the dynams of the transformed network are gven by dz = d x x = dx m z k = amz qm z z + bsn z zk exp d t; k = ( ) = z t z. (5) (6) Dynam Equaton (6) an be useful n modelng a varety of multultural soal networks. In Equaton (6), the magntude of the repulsve fore s represented by am z and the magntude of the long range attratve fore s desrbed by b z exp z k k. Furthermore, sn z s the sne-ylal nfluene due to the magntude of the devaton of the th agent s ultural state from the enter of the network. Attratve nfluenes an be thought of as attrbutes that brng people to atve membershp wthn the group. Soal aeptane, ganng soal status, eonom opportunty, areer growth, ommon purpose and membershp, personal development, and a sense of mutual respet, trust and understandng are examples of attratve nfluenes wthn a soal ultural network. Repulsve fores are attrbutes that reate some desre for ndvduals to leave or be less nvolved n the group or to preserve some personal dentty from one other wth ther ndvdual magntude of nner repulsve fore. A desre to retan a sense of ndvdualty, eonom or emotonal ost, nterpersonal onflt wthn the group, or dsagreement wth parts of the overall phlosophes of the group are fores that may be onsdered as repulsve fores. In short, eonom, eduatonal, and soal nequaltes oupled wth the rae, gender, ethal and relgous bas are soures of repulsve fores. A balane between the total attraton and repulsve fores attrbutes to a general sense of ndvdual agents mantanng a lve and let lve phlosophy for the greater beneft of the ommunty and the ommon good of soety. In order to better understand the dynams of Equaton (6), we wll use Lyapunov s Seond Method n onjunton wth the omparson method [6]. These methods wll provde a omputatonally attratve means to better understand the movement of members wthn the network. To that end, let us onsder a anddate for energy funton defned by V z = z. (7) Then the dfferental of V along the vetor feld generated by Equaton (7) s gven by 586

6 K. B. Hlton, G. S. Ladde m T 4 T z k dv ( z ) = z dz = am z qm z + bsn z z zk exp dt = LV ( z ) d t, (8) k = where m 4 T z k LV ( z ) = am z qm z + bsn z z zk exp. (9) k = In Subsetons 3. and 3., we wll fnd upper and lower estmates for LV ( z ) respetvely. Usng Lyapunov s Seond Method and the omparson method [6], we wll then use these estmates to onsder the behavor of the agents over tme t. For nstane, we onsder the stablty of the network, and establsh nvarant sets for the network. 3.. Upper Estmate of LV ( z ) In ths subseton, we seek onstrants on a, b,, and q suh that for gven ball, we an establsh an upper estmate of LV ( z ) z outsde of a. We wll then use these assumptons n onjunton wth the Lyapunov method and omparson theorem [6] to establsh the ase for whh r tt u r = u s the maxmal soluton of a omparson dfferental equ- where (,, ), ( t ) aton through ( t, u ). V z r tt,, u, () By onsderng the dervatve of the funton, We note that f r z k r = rexp zk exp has a global maxmum when z = wth a maxmum value of k exp. From Equaton (), Equaton (3), and the fat that sn Equaton (9) redues to: () () (3) z, for I(, m) m 4 z k LV ( z ) am z qm z + b z zk exp k 4 am z qm z + b( m ) z exp 4 4 = am z ( qm ) z z + b( m ) z exp b( m ) exp 4 3 = am z z ( qm ) z z. qm, (4) 587

7 K. B. Hlton, G. S. Ladde Assumpton H_: Suppose qm >. Let us defne n and let B(, β) = { x : x < β} 3 b( m ) exp β =, qm B(, β ) by B (, β ). For any z B (, β ), I(, m) followng nequalty: (5). Further, let us denote the omplment of the, Equaton (4) yelds the 4 LV z am z z am = 4 V ( z) V ( z). Usng Equaton (6) along wth the omparson theorems [6], we establsh the followng result. Lemma. Let V be the energy funton defned n Equaton (7), z be a soluton of I, m satsfyng the the ntal value problem defned n Equaton (6). For eah dfferental nequalty Equaton (6), t follows that the network s ohesve and where r( t) provded that ( ) ( ) (6) V z t r tt,, u, (7) s the maxmal soluton of the ntal value problem z u ; that s, am du 4u = ud t, r( t ) = u uam z, for t t. am u + uexp am( t t) Proof: Under the assumptons of the lemma and usng the standard argument [6] ombned wth the above dsusson, the proof of the lemma follows from Equaton (6). The ohesveness of the network follows by defnton as the soluton to Equaton (8) s bounded. Remark : We remark that the assumpton H s an alternatve suffent ondton as: From Equaton (4), we have (8) (9) 4 LV ( am + r) z qm z r z + b( m ) z exp ( ) am + r b m = qm z z r z z exp qm r am + r 4 qmv ( z) V ( z), z B (, β ), qm n where B(, β) = { x : x < β}, and () 588

8 K. B. Hlton, G. S. Ladde b m β = exp r, () for any r. 3.. Lower Estmate of LV ( z ) Next, we look to establsh a lower estmate of LV ( z ) suh that LV ( z ) ρ ( t, t, u ), () where ρ ( t) s the mnmal soluton to a omparson equaton through (, ) t u. Imtatng the argument used to arrve at Equaton (4) and notng that, for α >, x < α f and only f α x α I, m, Equaton (9) redues to the nequalty < <, for m 4 LV ( z) am z qm z b z exp j Assumpton H : Let us defne n and B(, β) = { x : x < β} = am z qm z b( m ) z 4 exp b exp 4 = a z qm z + a ( m ) z z. a (3) b exp β =, (4) a, wth ts omplement beng (, ) (, β ), (, ) z B I m, Equaton (3) redues to the followng dfferental nequalty: B β. For 4 LV z a z qm z a = 4 qmv ( z) V ( z). qm Usng Equaton (5) along wth the omparson theorems [6], we establsh the followng result. Lemma. Let V be the energy funton defned n Equaton (7) and z a soluton of I, m satsfyng the the ntal value problem defned n Equaton (6). For eah dfferental nequalty Equaton (5), t follows that ( ) ρ ( ) (5) V z t tt,, u, (6) where ρ ( t) s the mnmal soluton of the ntal value problem a du = 4qmu u d t, u ( t) = u, qm (7) 589

9 K. B. Hlton, G. S. Ladde Provded z t u ; that s, z ( t) au. (8) a qm u + uexp a ( t t) qm Proof: Under the assumptons of the lemma and usng the standard argument [6] ombned wth the above dsusson, the proof of the lemma follows from Equaton (5). Remark : A remark smlar to remark s as follows: From Equaton (3), we have 4 LV ( am r) z qm z r z + b( m ) z exp ( ) am r b m = qm z z + r z z exp qm r am r 4 qmv ( z) V ( z), z B (, β ), qm n where B(, β) = { x : x < β}, where b m β = exp, r (9) (3) For any r >. 4. Long and Short Term Behavor of Members and Invarant Sets After Frst let us note from ρ ( t), the mnmal soluton to the ntal value problem n Equaton (3) n Lemma, we fnd lm ρ t ( t) au = lm t a m u + u exp a t t m a =. qm Smlarly, from the soluton of the omparson dfferental Equaton (7) and Lemma, we note that amu lm r = lm t t am u + u exp a t t am =. Therefore, by Lemmas and, when z B (, β ) B (, β ), t follows that (3) (3) a lm z ( t) am. (33) qm t From Equatons (3), (3) and (33), we onsder one ase and the assoated nvarant sets. Frst, let us onsder the ase for whh β β. That s, let us suppose that 59

10 K. B. Hlton, G. S. Ladde b exp b( m ) exp a qm Let us further suppose that t s the ase that b exp b( m ) exp a a qm qm For β and β, let us defne the followng sets (Fgure ): A= B (, β ) a B = B (, β ) B, qm a C = B, B, qm ( β ) (, β ) (, ) (, ). = D B B am = E B am In the followng, we state and prove a few qualtatve propertes of the soluton proess of the enter of the mult-agent determnst dynam network desrbed by Equaton (3). The followng result exhbts the major nfluene of long range attratve fores (34) (35) (36) Fgure. An example n of the sets defned n Equaton (36). Under the assumptons n Equaton (34), the sets form onentr annul. 59

11 K. B. Hlton, G. S. Ladde Theorem. For < <, f for all I$, $ z B (, ) enter x, then Equaton (9) redues to the nequalty Further, f, a neghborhood of the a b LV ( z) 4qmV ( z) V ( z) ( m ). (37) q a q > u, there exsts < for all I(, m), z B (, ). suh that Proof. Let < < and z < for all I. Then, ( ) z t > for t t m 4 z k LV = am z qm z + bsn z z zk + z k exp k m 4 b am z qm z k a b = qm z z m q a b = 4qmV ( z) V ( z) ( m ) q Consderng the non-homogeneous omparson equaton, t follows that where when, (38) a b du = 4qmu u ( m ) d t, u ( t) = v, (39) q u( t), V z (4) u t s the mnmal soluton of Equaton (39) when V ( z ) the soluton of the homogeneous dfferental equaton u. Let û( t ) be a duˆ = 4mquˆ uˆ d t, uˆ ( t ) = u q (4) Then, by usng the method of varaton of parameters, the soluton to the non-homogenous dfferental equaton gven n Equaton (39) s gven by b t u( t) = uˆ ( t) ( m ) ( t, s, uˆ ( s) ) d, s Φ (4) t where uˆ Φ,, =,,. (43) ( tt u) ( tt u) u Usng separaton of varables, the soluton of the homogeneous dfferental equaton s gven by and ua uˆ t =, a q u + uexp am( t t) q (44) 59

12 K. B. Hlton, G. S. Ladde ( ),,. uˆ a exp am t t ( tt u ) = u a 4q u ˆ + u exp am( t t) Therefore, from Equaton (4), u t q ua = a q u + uexp am( t t) q ( ) b t a exp am t t ( m ) t a 4q u ˆ + uexp am( t t) q ua = a q u + uexp am( t t) q b( m ) a α am + u ( exp am( t t ) ) a u( t t) exp am( t t) q a u q ( exp 4am( t t) exp am( t t ) ), am (45) (46) where Let g( t ) be the funton defned as a α = ua + u exp am( t t). q b( m ) a g ( t) = am t t α am ( exp ( ) ) a + u u( t t) exp am( t t) q a u q ( exp 4am( t t) exp am( t t ) ), am (47) (48) We note that g( t ) s ontnuous on [ t ) g( t ),, = and 593

13 K. B. Hlton, G. S. Ladde ( ) b m lm g( t) =. (49) t amu As the lmt as t of g( t ) s fnte, for any gven, suh that ( ) and so g( t ) has an upper bound on on [ t, T ], As g( t ) M to δ > there exsts a T > t b m g( t) < δ, for t > T, (5) amu T,, say g t s ontnuous g t has an upper bound on ths nterval, say =. =, t must be the ase that M > and hene Equaton (4) redues u t M. Further, as M. Let M max { M, M } ua M. (5) a q u + uexp am( t t) q Suppose that t s the ase that a ( q) > u and so the soluton, suh that < a ( q) u t s monotonally nreasng as t. Choosng < < t follows that Equatons (5) has the lower bound z t > for all t t when Thus,, for and < u M, u t > t > t (5) z for all I. Theorem. Let the hypotheses of Lemmas and be satsfed. Then ) the set C D E B, a ( qm) = s ondtonally nvarant relatve to E ; ) the set D s ether self-nvarant or a C D = B qm B am s ondtonally nvarant relatve to D ; (, ) (, ) 3) set C s ether self-nvarant or C D s ondtonally nvarant relatve to C; 4) the set C D s self-nvarant; 5) the set B C D B (, β ) B (, am ) = s ondtonally nvarant relatve to B. Proof. For z E, I(, m) the hypotheses of Lemmas and are satsfed. Thus by the applaton of these Lemmas, we have for t t ( tt ) V( z ( tt z) ) r( tt r) ρ,, ρ,,,,, (53) n >, z B (, am ), where B (, am ) = { x : x > am} and (,, ) ; ρ( tt,, ρ ) r tt r are the mnmal and maxmal solutons of the omparson dfferental equatons gven n Equatons (7) and (8) respetvely. Moreover, for z B (, am ) r = = V z = z and z t = z, the solutons (, ), ρ( tt,, ρ ) are both monotonally dereasng and approahng to a ( qm ) respetvely. Ths yelds wth ρ ( tt ) z( tt z ) r( tt r),, r tt r and am and ρ,, ρ,,,,, (54) 594

14 K. B. Hlton, G. S. Ladde for t t. From Equaton (54), z E and the defntons of self-nvarant and ondtonally nvarant [8], t follows that statement ) s vald. The proofs of ), 3) and 4) follow by mtatng the argument used n the proof of ). For ρ( tt,, ρ ) s monotonally dereasng and (,, ) a ( qm ) and am as t respetvely. Ths together wth Equaton (54) establshes that (,, ) z D, we note that r tt r s monotonally nreasng to z tt z C D provng statement ). For z C D ρ tt,, ρ s dereasng and the proof of 3) and 4) follows from ) and ). Smlarly, the proof for statement 5) also follows by mtatng the argument used n ). For B, the solutons to the omparson equaton gven by Equaton (7), ( tt,, ) nreasng to ( ), z ρ ρ s monotonally a qm as t. Therefore, by Equaton (54), (,, ) z tt z B C D provng statement 5). Let us expand upon the results of Theorems and. Frst, let us note that these two theorems provde the qualtatve and quanttatve requrements on the ultural state parameters to nsure that the model s ohesve (Theorem ) and smultaneously does not reah a ultural onsensus (Theorem ). We ntrodue the defnton of ultural bound to desrbe the boundary between two ultural sets, dvdng the degree of ndvdual versus ommunty level nteraton domans of the ultural state. Suppose z A. It an be shown that there exsts a neghborhood, B(, ) of the enter suh that for z B(, ), the ndvdual member ultural state s pushed out/repulsed from the ultural state enter x at some tme T dependng on >. Therefore, f the ultural state of the th member x of the network s suh that the relatve ultural affnty between x and the enter, x of the network s suffently lose to zero, then the agent s ultural state s repulsed from the enter. That s to say, the membershp of the soal network wll obtan and then mantan a relatve ultural affnty between members and the enter that s bounded below by a value strtly greater than zero. One the state of the th member z has moved away from the enter, t may be the ase that z remans n A or the ase that the state z moves to the ultural set B, at whh tme the agent s ultural state behavor wll follow that of another ategory of membershp desrbed by the ultural state set B dsussed below. Suppose the ntal value, that s the funton of the magntude of the ultural state, ρ of the omparson equaton s suh that ρ a ( qm). Then the soluton to the lower omparson equaton grows as t grows and approahes asymptotally to the threshold lmt a ( qm ) from below resultng n stronger tes wth the ommunty enter state, x. If the ntal value of the lower omparson equaton s suh that ρ a ( qm), then the soluton deays and asymptotally approahes to the threshold lmt a ( qm ) from above. Therefore, f z s a member of the transformed soal network suh that z B, then by Theorem, over tme, z moves to the ultural bound of the set C. It may also ross the ultural bound or t may be the ase that z approahes asymptotally to the ultural bound of C. Smlarly, f z C, z may stay n C, approahng the ultural bounds of sets B and/or D or t may be the ase that z rosses the ultural bound of D at whh pont the member wll behave 595

15 K. B. Hlton, G. S. Ladde as other members of D. However, f z C, even though t may approah the ultural bound of B, t wll never ross the bound. In terms of a gven soal network, ths mples that members wth a dstnt enough ultural state from the weghted average of ultural states wll retan that dstntveness of ulture. Thus, f the relatve ultural affnty between a member x and the enter of the network s at least a ( qm ) ntally, then the relatve ultural affnty wll always be at least that value. Turnng to the upper omparson equaton, we an onsder the behavor of the transformed network members whose ntal postons are n the sets D and E. Let r be the ntal poston of the soluton r( tt,, r ) to the upper omparson equaton gven n Lemma. If r < ( am), then the soluton r( tt,, r ) grows and approahes asymptotally to the value ( am ) from below. If r > ( am), the soluton deays and approahes asymptotally to the lmt from above. Therefore, f z D, z may approah and ross the ultural boundary of C (but wll reman n C D) or z may approah but not ross the ultural boundary of E. For z E, z may ether ross the ultural boundary of D or the member s ultural state wll approah asymptotally to the ultural boundary of D. Thus, for agents x wthn the network whose ntal relatve ultural affnty wth respet to the enter s suffently large, as t, the relatve ultural affnty wll reman large and the although the agent s attrated bak towards the enter of the network, the relatve affnty s bounded below by a ( qm ). Further, from Lemmas and, f all parameters other than the sze of the network are held onstant, then as the sze of the network nreases, so also the dfferene between the upper and lower bounds on the relatve ultural affnty between agents and the enter of the network nreases. Naturally, nreasng the sze of the network leads to the onept of the rowdng effet. Competton over deology or ultural trats reates a stronger desre for agents to retan more of ther ndvdualty wthn the soety or group. Cultural subgroups that have a hgh degree of separaton n terms of ther relatve ultural affntes are an emergent haraterst of suh large sale multultural networks. In the modelng for members whose ultural state s n (so one aspet of ulture/nterest beng onsdered), we see the network dvdng nto two subgroups wth agents onvergng to states that are symmetr wth respet to the tme axs. One an thnk of stuatons lke a large urban envronment n whh there exst ommuntes eah wth a strong ultural dentty. In suh a ase, agents wthn the ommunty seek to retan ther ultural dversty. Thus, t s expeted that a large relatve ultural affnty between agents of dfferent ommuntes, but a small relatve ultural affnty between agents wthn the same ommunty, s expeted to exst. 5. Numeral Smulaton In ths seton, usng Euler s type numeral to approxmaton sheme appled to Equaton (6), we onsder the numeral smulatons for the network dynams governed by Equaton (6). The goal s to ompare the long term behavor of the smulated soluton wth the theoretal long term behavor gven n Seton 4. We onsder a network on- 596

16 K. B. Hlton, G. S. Ladde sstng of 5 members wth parameters a =.5, q =.4, b =.4, and =. Further, we note that n ths ase, β =.3, β.5 and a.5 am 5. qm = = (55) In ths example, the ondtons for the nvarant sets gven n Theorems and n Seton (4) are satsfed. Hene, for tme, z.5 Further, for z suh that.3 z suh that.5 z, t s the ase that after some ; that s, the member does not move towards the enter of the network., after some tme,.5 5. Fgure s a plot z of the approxmate solutons for the full membershp of the network. In order to make the dynams of the network learer, Fgure 3 s a plot of the approxmate soluton of Equaton (6) for sx of the members of the network. Next, we onsder the network wth the same ntal values wth the parameters a =.5, b =.4 and q =.4. In ths ase β.6, β.35 and a.35 am qm (56) For z suh that z.35, the member does not move towards the enter of the network and for z suh that z.6, after some tme,.35 z Smlar to above, we have plotted the approxmate soluton for the full network n Fgure 4 and the approxmate soluton for the same sx members as Fgure 3 n Fgure 5. z Fgure. Euler approxmaton of the soluton to the dfferental equaton gven by Equaton (6) wth parameters a =.5, b =.4, and q =.4 yeldng the ultural postons over tme t for the full 5 members of the network. 597

17 K. B. Hlton, G. S. Ladde Fgure 3. Euler approxmaton of the soluton to the dfferental equaton gven by Equaton (6) wth parameters a =.5, b =.4, and q =.4 yeldng the ultural postons over tme t for sx of the network members. Fgure 4. Euler approxmaton of the soluton to the dfferental equaton gven by Equaton (6) wth parameters a =.5, b =.4, and q =.4 yeldng the ultural postons over tme t for the full 5 network members. 598

18 K. B. Hlton, G. S. Ladde Fgure 5. Euler approxmaton of the soluton to Equaton (6) wth parameters a =.5, b =.4, and q =.4 yeldng the ultural postons over tme t for sx of the network members. The last ase we onsdered s the network wth the same ntal postons wth the parameters a =.5, b =.8 and q =.. Thus, wth the gven parameters, β.84, β., and For a. am 5. qm = (57) z suh that z., the member does not move towards the enter of the network and for z suh that z.84, after some tme,. 5. Smlar to above we have plotted the approxmate soluton for the full network n Fgure 6 and the approxmate soluton for the same sx members n Fgure Conlusons We have onsdered requrements on network parameters for long term qualtatve propertes of the network. We develop a model and establsh ondtons on the parameters that ensure a balane between oheson and onsensus. Further, we have onsdered how the ntal ultural state of a network member affets the behavor of that member over tme. The presented ondtons of the system are algebraally smple, easly verfable and omputatonally attratve. The developed results provde a tool for plannng, deson makng, and performane. Furthermore, the presented suffent ondtons are onservatve but robust, verfable, and relable. From the above ondtons, we are able to onsder ertan dynam propertes of the soal networks governed by Equaton (3). z 599

19 K. B. Hlton, G. S. Ladde Fgure 6. Euler approxmaton of the soluton to the dfferental equaton gven by Equaton (6) wth parameters a =.5, b =.8, and q =. yeldng the ultural postons over tme t for full 5 network members. Fgure 7. Euler approxmaton of the soluton to the dfferental equaton gven by Equaton (6) wth parameters a =.5, b =.8, and q =. yeldng the ultural postons over tme t for sx network members. 6

20 K. B. Hlton, G. S. Ladde In ths paper, we explored the features of a mult-ultural network wth dynams desrbed by a spef dfferental equaton and the long term stablty and behavors of ndvdual members wthn suh a network. We are nterested n further explorng soal networks n the ontext of better understandng the relatve ultural affnty between agents x j and not just the ultural affnty between an agent and the enter of the network. Our hopes are to better understand what fators may lead to preservng a lower bound on the relatve ultural affnty x j that s strtly greater than zero as t. In modelng suh a network, we are lookng to better understand how dversty between all members may be mantaned over the long term wthn a ulturally dverse network. Further, we are explorng the effet of nose on the network by onsderng smlar dynams and stohast dfferental equatons. The goal for both suh questons s to better understand the mpat of perturbatons/mpulses, both nternal and external, on the behavor and dversty of mult-ultural networks. Aknowledgements The authors would lke to aknowledge the researh support by the Mathematal Senes Dvson, US Army Researh Offe, Grants No. W9NF---9 and W9NF Referenes [] Degroot, M.H. (974) Reahng a Consensus. Journal of the Ameran Statstal Assoaton, 69, [] Fredkn, N.E. (4) Complex Objets n the Polytopes of the Lnear State-Spae. arxv: [3] Aemoglu, D., Como, G., Fagnan, F. and Ozdaglar, A. (3) Opnon Flutuatons and Dsagreement n Soal Networks. Mathemats of Operatons Researh, 38, [4] Ma, H. (3) Lterature Survey of Stablty of Dynamal Mult-Agent Systems wth Applatons n Rural-Urban Mgraton. Ameran Journal of Engneerng and Tehnology Researh, 3, 3-4. [5] Ladde, Gangaram S. and Sljak, Dragoslav D. (975) Connetve Stablty of Large-sale Stohast Systems. Internatonal Journal of Systems Sene, 6, [6] Ladde, G.S. and Lawrene, B.A. (995) Stablty and Convergene of Large-Sale Stohast Approxmaton Proedures. Internatonal Journal of Systems Sene, 6, [7] Anabtaw, M., Sathananthan, S. and Ladde, G.S. () Convergene and Stablty Analyss of Large-Sale Parabol Systems under Markovan Strutural Perturbatons-I. Internatonal Journal of Appled Mathemats,, [8] Cao, Y., Yu, W., Ren, W. and Chen, G. (3) An Overvew of Reent Progress n the Study of Dstrbuted Mult-Agent Coordnaton. IEEE Transatons on Industral Informats, 9, [9] Zhu, Y.-K., Guan, X.-P. and Luo, X.-Y. (3) Fnte-Tme Consensus for Mult-Agent Systems va Nonlnear Control Protools. Internatonal Journal of Automaton and Computng,,

21 K. B. Hlton, G. S. Ladde [] Hu, H.-X., Yu, L., Zhang, W.-A. and Song, H. (3) Group Consensus n Mult-Agent Systems wth Hybrd Protool. Journal of the Frankln Insttute, 35, [] Huang, M. and Manton, J.H. (9) Coordnaton and Consensus of Networked Agents wth Nosy Measurements: Stohast Algorthms and Asymptot Behavor. SIAM Journal on Control and Optmzaton, 48, [] Fredkn, N.E. (4) Soal Coheson. Annual Revew of Soology, 3, [3] Bruhn, J. (9) The Group Effet, Soal Coheson and Health Outomes. Sprnger, New York. [4] Axelrod, R.M. (997) The Complexty of Cooperaton: Agent-Based Models of Competton and Collaboraton. Prneton Unversty Press, Prneton. [5] Knoke, D. and Yang, S. (8) Soal Network Analyss. Sage, Thousand Oaks. [6] Lakshmkantham, V. and Leela, S. (969) Dfferental and Integral Inequaltes: Ordnary Dfferental Equatons. Aadem Press, Pttsburgh. [7] Gaz, V. and Passno, K.M. (3) Stablty Analyss of Swarms. IEEE Transatons on Automat Control, 48, [8] Ladde, G.S. and Leela, S. (97) Analyss of Invarant Sets. Annal d Matemata Pura ed Applata, 94, Submt or reommend next manusrpt to SCIRP and we wll provde best serve for you: Aeptng pre-submsson nqures through Emal, Faebook, LnkedIn, Twtter, et. A wde seleton of journals (nlusve of 9 subjets, more than journals) Provdng 4-hour hgh-qualty serve User-frendly onlne submsson system Far and swft peer-revew system Effent typesettng and proofreadng proedure Dsplay of the result of downloads and vsts, as well as the number of ted artles Maxmum dssemnaton of your researh work Submt your manusrpt at: Or ontat jns@srp.org 6