Opinion Consensus of Modified Hegselmann-Krause Models

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1 Opnon Consensus o Moe Hegselmann-Krause Moels Yueheng Yang, Dmos V. Dmarogonas an Xaomng Hu Abstrat We onser the opnon onsensus problem usng a mult-agent settng base on the Hegselmann-Krause (H-K Moel. Frstly, we gve a suent onton on the ntal opnon strbuton so that the system wll onverge to only one luster. Then, moe moels are propose to guarantee onvergene or more general ntal ontons. The overall onnetvty s mantane wth these moels, whle the loss o ertan eges an our. Furthermore, a smooth ontrol protool s prove to avo the ultes that may arse ue to the sontnuous rght-han se n the H-K moel. I. INTRODUCTION The opnon onsensus problem s about opnon ompromse o a ertan event by erent agents. Assume that the opnon s ontnuous, an that all agents have boune onene n the way that they only onser the opnons that are lose to ther own opnon. Agent-base moels o opnon ynams uner these assumptons have been establshe n the begnnng o ths entury by Hegselmann an Krause [] an Wesbuh et al [2]. Both moels lea to lusterng o opnons n a smlar way. In ths paper we wll onser the moel o Hegselmann an Krause (H-K. The prevous stuy about the H-K moel shows that not all ntal postons orresponng to a onnete graph wll lea to onsensus [3], [4], [8]. Ths s beause urng the proess the graph an beome sonnete sne the neghborhoo s base on opnon erenes between pars o agents. The loss o onnetvty an yel several lusters o agent opnons n erent postons. Ths phenomenon s also observe n the ntal paper by Hegselmann an Krause []. However, even or one-mensonal ase, ew theoretal results have been obtane so ar regarng the relatonshp between ths loss o onnetvty an the ntal opnon strbuton. Although the H-K moel an be easly extene to hgher mensonal spaes, ths paper wll manly ous on the one-mensonal ase. On the other han, nstea o mposng onstrants on the ntal strbuton, one an moy the moel to guarantee that onsensus s aheve or any ntal opnons. Ths s relate to the onnetvty mantenane problem n the mult-agent system theory. A way to aheve ths s by usng potental untons. The man ea s that the ore between two agent opnons beomes nntely large when the erene between the opnons beomes bg enough,.e., near the bounary o onene. Ths approah has been use by several researhers n the past ew years, e.g., Yueheng Yang an Xaomng Hu are wth the Optmzaton an Systems Theory vson, Department o Mathemats, Royal nsttute o Tehnology (KTH, Stokholm, Sween. Dmos Dmarogonas s wth automat ontrol lab, Shool o Eletral Engneerng, KTH. Ths work s supporte by SSF, the Swesh Researh Counl an the EU HYCON 2 NoE. [], [2]. Boune ontrollers or onnetvty ontrol are onsere n [0]. The ommon ea n these papers s that no ege s allowe to break urng the proess, thus mposng onstrants n the relatve states o pars o agents that onsttute an ege. However, ths s only a suent onton or onnetvty mantenane beause the loss o some non-rual eges may not nluene the onnetvty. In ths paper we use topologal arguments to guarantee onnetvty nstea o applyng nnte potentals when an ege s boun to break. In partular, nspre by the ea use n [3], we show that ommon neghbors play an mportant role n the problem. I two noes an suh that (, E have some ommon neghbors, then the ege (, an be allowe to break beause they are stll onnete by the ommon neghbors. On the ontrary, they o not have any ommon neghbors, then the ege beomes mportant an shoul not break. Ths approah seems more ntutve rom a soal networks perspetve sne t s more natural to take nto aount the number o ommon neghbors than applyng nnte ores to mantan the graph onnete. The moe moel that we prove n ths paper guarantees opnon onsensus or almost any onnete ntal opnon strbuton, even the rato between the opnon versty an the onene boun s sgnant. Usually one obtans lusterng behavor,.e., sonneteness, o the orgnal H-K moel when ths rato s bg. Ths ssue s overome by usng the moe moel n Seton III. An requrement o our moel s that two or more agents annot have the same ntal opnons. Sne ths s a possble senaro n the real worl, we prove another moel to eal wth ths ase. Furthermore, or the orgnal H-K moel, the rght-han se s not a ontnuous unton o the state x. Ths results to measure zero sets o ntal ontons rom whh the soluton may not be unque. We ntroue n the paper a smooth moaton o the moel n orer to avo ths. The remaner o the paper s summarze as ollows: n Seton II we ormulate the problem uner onseraton. The moe verson o the H-K moel s presente an analyze n Seton III. A smoothe verson o ths s prove n Seton IV. Seton V nlues smulatons that support the erve theoretal results. Fnally, a summary o the results o ths paper as well as possble retons o uture work are nlue n Seton VI. II. MATHEMATICAL PRELIMINARIES A. Bas onepts rom graph theory In ths seton, we revew some onepts rom graph theory that wll be use n ths paper. These entons an

2 be otherwse oun n a stanar textbook on graph theory. Conser a set o n noes enote by V {,2,...,n} an a subset E V V. We all G (V,E s a graph wth the set o vertes (or noes V an the set o eges E.In G (V,E, the neghbor set o the vertex s ene by N { V (, E}. ( A graph G (V,E s alle unrete (, E mples (, E. In an unrete graph, there s an ege onnetng two vertes,.e., (, E, then these two vertes, are alle aaent. A graph s alle omplete any two noes are aaent. A path rom a vertex to another vertex s a sequene o stnt vertes startng wth an enng wth, n whh eah vertex s aaent to ts next vertex. Two vertes an are alle onnete there exsts a path rom to. An unrete graph s all a onnete graph any par o vertes s onnete n t. B. Introuton o Hegselmann-Krause moel Conser a system o n autonomous agents labele as,2,...,n, whose opnons are loate n the one-mensonal Eulean spae R. We enote the set o all agents as V {,2,...,n}. For an agent V, the poston o t s opnon s enote by x (t R, whh has the ollowng ynams: ẋ (t u (t, (2 where u (t s onsere as the ontroller o agent. The onsensus problem s to n the ontrollers u (t so that the stak state x(t (x (t x 2 (t x n (t T wll onverge to the subspae generate by the vetor ( T as t. I the ege set E V V s gven, one an then ene a graph G (V, E an generate a bas ontrol protool or the onsensus problem: ẋ (t u (t N (x (t x (t (3 It s a well-known at [7] that the system (3 wll onverge to the equlbrum x (t α,,2,...,n the graph G s onnete, where α n n x (0. Now assume that the graph G (V,E s ene by V {,2,...,n} an E {(, x x } or some > 0. Applyng the same ontrol n (3 wth ths enton o the graph, we obtan the Hegselmann-Krause (H-K moel: ẋ (t (x (t x (t. (4 :x x From now on, we wll all G (V,E the orresponng graph o x(t, where V {,2,...,n} an E {(, x (t x (t }. Sne onnetvty an hange as the system evolves, t annot be guarantee that the system wll onverge to α, where (,,..., T R n an α s ene as above,.e., reah onsensus. On the other han, the graph stays onnete urng the whole proess, then t has been shown that the agents wll nally reah onsensus. Sne the ontrol protool s xe n ths problem, the ntal poston nee etermnes the whole proess. So a natural queston s uner what ntal onton wll the agents reah onsensus?. III. NON-SMOOTH MODEL Conser the stanar Hegselmann-Krause moel: ẋ (t (x (t x (t. :x x Although we have a sontnuous unton on the rghthan se, the almost surely exstene an onvergene o the soluton to ths erental equaton have been prove n [9]. Here the onvergene means the state wll onverge to an equlbrum o the system. However, the equlbrum s not neessarly n the orm α as susse beore. Instea t an orm several lusters [9]. A. Suent Conton or onsensus In ths seton, we gve a suent onton on the ntal states (opnons suh that the system wll onverge to exat one luster. The onept o ommon neghbor wll be use n the theorem. Denton 3.: For an unrete graph G (V,E, the set o ommon neghbors between two noes an s ene as: N {k V (,k E,(,k E} N N. (5 Theorem 3.2: For an ntal onton x(0 R n an the orresponng graph G (V, E, G s onnete an or any par (, E, t hols that n 2 2, then the soluton to (4 wll onverge to α, where α n T x(0. Proo: Beause the ntal graph G s onnete by assumpton, no ege n E s lost urng the proess, the graph G(t wll be always onnete. Then t s well-known that the states wll onverge to the average value o the ntal states. What we nee to show now s that or any par o vertes (,, the stane x (t x (t wll not exee the threshol. Due to the ontnuty o x(t, we onser the stuaton that x (t x (t, an assume that x (t > x (t wthout loss o generalty. Denote N N \ N an N N \ N. t (x (t x (t (ẋ (t ẋ (t (x k (t x (t (x k (t x (t (x k (t x (t (x k (t x (t ( 2(x (t x (t x k (t x (t x k (t x (t ( 2 ( ( 2. (n ( 2 ( 2 (n 2( 2 0.

3 As we an see, the stane x (t x (t wll not nrease n ths ase, whh proves that the ege (, wll not break 2 n 2 at tme t. Now suppose the rst ege break happens rght ater tme t or the ege (,. Ths means < n 2 2 at tme t. Wth the ntal onton that n 2 2, the number o ommon neghbors o must have erease at some tme beore t. But ths an never happen wthout an ege break, whh ontrats wth the assumpton that (, s the rst ege to break. Thereore, no ege wll break uner the state assumpton. Ths onlues the proo. Remark: The onton n Theorem 3.2 s not a neessary onton or reahng onsensus. Ths an be easly shown by ounter examples. Although t s weaker than the onton o a omplete graph, t s stll strong. Wth the onstrant that eges are ene by stane, most vetors x(0 R n o not satsy ths onton. I we onser nntely many agents unormly strbute on an nterval o length L, then L 2 s requre n Theorem 3.2. B. Weghte Moel As mentone above, Theorem 3.2 hols or a lmte number o ntal ontons. Instea o nng a onton on the ntal states, we moy the moel slghtly to guarantee opnon onsensus or any x(0 wth a orresponng onnete graph. Conser the ollowng moel: ẋ (t N (x (t x (t. (6 Sne a sontnuous weght s ae to the rght-han se o the alreay sontnuous H-K moel (4, t s not obvous that the ollowng statement, whh s vtal or the proo o our man theorem later, s true. Proposton 3.3: Gven an ntal onton x(0 R n satsyng x (0 x (0 or any, we have x (T x (T or any an any T [0,, where x(t s the soluton to the erental equaton (6. Proo: We wll prove the proposton by showng that the stane between any par o noes s larger than the value o some exponental unton whh wll not reah zero n nte tme. Ths s not learly to be true sne the weght ene by the number o ommon neghbors may be erent between erent pars o noes. Wthout loss o generalty, we only nee to nvestgate the stane between two aaent agents an, where s to the rght o. Suppose at tme t 0, the stane between an s x (t x (t δ or some 0 < δ. Dene N {k N x k (t x (t > }, N {k N 0 < x k (t x (t }, N {k N x (t x k (t > } an N {k N 0 < x (t x k (t }. Fg gves a vsualzaton o these entons. It s not har to prove the ollowng ats: ( N N N N { }; ( N N N N {}; ( or any k N, k ; (v or any k N, k ; Fg.. N N N N A graph llustraton o the sets N, N, N, N (v or any k N, k k ; (v or any k N, k k. For example (v s true sne or k N, N k N N k (N N N {} N k ( ( (N N N k (N {} N k (N N \{k} φ N N \{k}, an (v s true sne or k N, N k N (N N k N ( (N ( (N N N { } N k N N k ((N N { } N k N ( (N N { } N k N ( (N N { } N (N k N (N { }\{} N k (N N k { }\{} N k { }\{}. Conser the ervatve o the erene x (t x (t: t (x (t x (t ẋ (t ẋ (t x k (t x (t k x k (t x (t k x k (t x (t k x k (t x (t k (x k (t x (t k x k(t x (t k (x k (t x (t k x k(t x (t k 2 (x (t x (t. (7 For k N, we have k (x k(t x (t k (x k(t x (t k (x k(t x (t k (x k(t x (t k (x k(t x (t k (x k(t x (t k (x (t x (t k k ( k ( k (x k(t x (t

4 k (x (t x (t ( k ( k (x k(t x (t (x (t x (t ( ( δ. The last equalty hols sne k k by (v, an the last nequalty s true beause N N N k {k}, N N k {k} an x k (t x (t δ. I we sum up over k N ( (, we get k (x k(t x (t (x (t x (t k (x k(t x (t ( (x (t x (t ( δ. For k N, we have k (x k(t x (t (x k(t x (t ( δ. ( δ One an get smlar results or k N pluggng all these results to (7, we get t (x (t x (t ẋ (t ẋ (t ( δ ( δ an k N. When ( δ (x (t x (t ( δ (x (t x (t 2 (x (t x (t ( δ ( 2(x (t x (t n(x (t x (t n(x (t x (t. (8 In at, (8 hols or all t suh that 0 < x (t x (t. We know that the soluton to the erental equaton: ẏ(t ny(t s y(t y(0e nt. I we enote (t x (t x (t, then (t n (t. In other wors, (t (0e nt > 0, sne (0 > 0. Thus, x (T x (T or all T [0,. We wll also use the ollowng onept rom graph theory: Denton 3.4: For an unrete graph G (V, E, an ege (, E s alle rual N φ,.e. there oes not exst k V suh that (,k E an (k, E smultaneously. Theorem 3.5: (Man Theorem For any ntal onton x(0 R n suh that: the orresponng graph s onnete; x (0 x (0 or any, the system (6 wll onverge to the equlbrum α, where α n T x(0,.e., onsensus s reahe. Proo: I no rual ege breaks urng the proess, the graph wll keep onnete. Thereore, we ust nee to hek when a rual ege s gong to break,.e., x (t x (t, an N φ. Assumng that x (t > x (t wthout loss o generalty, we get t (x (t x (t ẋ (t ẋ (t x k (t x (t k x k (t x (t k \{} \{ } \{} k (x k(t x (t k (x k(t x (t 2(x (t x (t k k 2. \{ } Now we only nee to show that \{} k \{ } 2, (9 k n orer to prove t (x (t x (t 0. We wll now show that or all k N \{}, we have that k 2, 2. Due to the assumpton we have mae, there s no par o agents wth the same ntal opnon. Aorng to Proposton 3.3, there wll not be two agents reahng the same state n nte tme. So we have x (t x (t or any an t <. I (, s a rual ege wth x (t x (t, then agent has only one neghbor to ts let whh s agent. Then all the other neghbors o must be loate to ts rght. I 2, every s rght neghbor s a ommon neghbor o an another rght neghbor. Ths s beause x (t x (t an x (t x (t max{x (t x (t,x (t x (t}. We have / N k aorng to the enton o a rual ege. Thereore, we have N k N \{,k}, whh mples k 2 or k N \{}. Equvalently one an get k 2 or k N \{ } 2. By pluggng these results nto the let-han se o the

5 nequalty (9, we get \{} \{} k \{ } \{ } 2. k Note that, whh means N {} an N \{} φ, then (9 s also true sne the rst term on the let-han se s equal to 0. However, n Theorem 3.5, t s requre that no two agents have the exat same ntal opnon. Although ths s a set o measure zero n the state spae, t s a ommon senaro n realty. To aommoate ths senaro an at the same tme avo some numeral ultes enountere when the agents are very lose to eah other, one an treat all the agents wth the same state as one new agent. Ths s equvalent to sayng that we stll treat agents wth the same state separately but wth a weght ve by the number o agents at that poston. I M s ene as the number o agents whh have the same poston as agent, then (6 an be rewrtten as: ẋ (t N ( N M (x (t x (t, (0 where N s the number o ommon opnon lusters between an. Here two agents belong to the same luster an only they have the same opnon. We an show that N {, } /M k 2. It s not har to prove the ollowng orollary: Corollary 3.6: For any ntal onton x(0 R n whose orresponng graph s onnete, the ontrol protool (0 wll guarantee onsensus. Nevertheless, on the rght-han se o (0 the weght s not symmetr. So n Corollary 3.6 the equlbrum s atually not the ntal average. Ths s n some sense a shortomng sne we gnore the weght o those agents who have the same opnon. IV. SMOOTHED MODEL Another ssue or the orgnal H-K moel (4 s the sontnuous rght-han se. In the theory o erental equatons, the onton o Lpshtz ontnuty s essental or the exstene an unqueness o the soluton. As state n [9], the onvergene o the soluton to (4 s guarantee or almost all ntal ontons, whh mples there an exst a set (wth measure zero o sngular ponts. Moreover, some numeral problems may arse rom ths sontnuty when one wants to mplement the moel. For example, the relatve error wll be unboune the stane s aroun the threshol n the H-K moel. A ommon remey or these problems s to approxmate the orgnal unton by a ontnuous (even erentable n some ases unton. The approxmaton has the same value as the orgnal unton exept aroun the ponts where the sontnuty ours. Aroun those ponts, a smoothng unton s use to replae the orgnal unton, e.g., [5]. We rewrte the orgnal moel as: where ẋ (t ρ (x (t x (t, ( {, x (t x ρ (t, 0, x (t x (t >. We an moy ρ (x n the ollowng way: we enote by β x x 2 the stane between agent an agent an ntroue a potental unton between an as: r(β β, 0 β 2, ϕ(β, 2 < β ( ε 2,, ( ε 2 < β <. (2 where s a postve onstant an ϕ s a hosen monotonally nreasng unton on the nterval [( 2, ε 2 ] to make r(β erentable or any β (0, (e.g., hgh orer polynomals. Then let ρ r(β β A. Convergene, 0 β 2, ϕ (β, 2 < β ( ε 2, 0, ( ε 2 < β <. (3 I we onser the moel ( wth the hoe o ρ n (3, then the rght-han se s a Lpshtz ontnuous unton, whh wll ensure the exstene an the unqueness o the soluton to the erental equaton. But onvergene may not be guarantee n ths ase. Thereore, the ollowng theorem s requre. Theorem 4.: For any ntal onton x(0 R n, there exsts a vetor x R n suh that the soluton x(t to the erental equaton ( wth the hoe o ρ n (3 wll onverge to x as t. Remark: The man ea o the proo s to use LaSalle s nvarane prnple. We skp the proo here uo to the spae lmtaton. However, one an easly gure out that or the equlbrum x o the system, ether x x or x x ε hols. Both stuaton may our n the largest nvarant subset when one apples LaSalle s nvarane prnple. It s the reason why more than one lusters may our or some ntal states. B. Suent Conton or onsensus Smlarly to seton 3., there s a suent onton or the ntal states to guarantee onsensus by usng the ontrol protool (. When the smoothe moel ( s use, the weght between a par o noes (, ontnuously ereases to zero when the stane exees. Although one onsers (, to be an ege when there s a postve weght on t n general, we wll stk on the prevous enton o the ege set, whh s E {(, x x }. Then we wll get the smoothe verson o Theorem 3.2.

6 (a (b (a (b Fg. 2. Tme evoluton o 5 agent opnons aorng to (ase (a moel (4 an (ase (b moel (6. Intal opnons are unormly spae on an nterval o length 5. The nteraton raus s hosen to be Theorem 4.2: For an ntal onton x(0 R n, the orresponng graph G (V,E s onnete an or any par (, E, 2 n 2, then the soluton to ( wll onverge to α, where α n T x(0. Proo: The proo s smlar to that o Theorem 3.2 an s thus omtte. V. SIMULATIONS We wll present some smulaton results o the weghte non-smooth moels n ths seton. In the rst example, 5 agent opnons are ntally unormly spae on an nterval o length 5. The nteraton raus s hosen to be 0.98 to avo some sngularty rom the sontnuous rght-han se n the non-smooth moels. We use both the orgnal H-K moel (4 an the weghte moel (6. Fg. 2 shows the smulaton result. The orgnal H- K moel verges to three lusters (n (a an the moe moel (6 reahes onsensus (n (b. In the seon example, we want to show how the onene o ntal opnons aets the smulaton result by usng the two moe moels: (6 an (0. 26 agent opnons are unormly sprea on the nterval o length 5, whle the other 20 opnons are all loate at poston ntally. s hosen to be 0.98 agan. The ntal opnon average α s approxmately.85. Although the ntal strbuton oes not ulll the onton n Theorem 3.5, the system oes onverge to the ntal opnon average by usng the ontrol protool (6 (n Fg. 3(a. I only the opnon luster s onsere, these 20 agent opnons s n some sense gnore sne there s also one agent opnon postone at among the rst 26 agent opnons. So usng moel (0, we get a symmetr result n Fg. 3(b, an the ompromse opnon s 2 n the en. VI. CONCLUSIONS AND FUTURE WORK In ths paper, we rst gave a suent onton or opnons onsensus or the orgnal Hegselmann-Krause moel, as well as or a moel wth a ontnuous rght-han se. Furthermore, we prove two moe versons o the Hegselmann-Krause moel suh that onsensus s guarantee or any ntal onguraton orresponng to a onnete graph. Future work wll examne the ase o hgher mensonal spaes, an n partular the two mensonal spae. It s not Fg. 3. Tme evoluton o 46 agent opnons aorng to (ase (a moel (6 an (ase (b moel (0. Intal opnons o 26 agents are unormly spae on an nterval o length 5, whle the rest 20 agents are all ntally postone at. The nteraton raus s hosen to be har to see that Theorem 3.2, 4., an 4.2 an be extene to any nte mensonal spaes R m the absolute value s moe to the Eulean norm, an wth an approprate use o the Kroneker prout. However, t shoul be note that Theorem 3.5 s not extenable to the hgher mensonal ase n a straghtorwar manner sne the lne struture s use n the proo. So the extenson to hgher mensons may requre more eort n ths ase. REFERENCES [] R. Hegselmann an U. Krause, Opnon ynams an boune onene moels, analyss, an smulaton, Journal o Artal Soetes an Soal Smulaton, vol. 5, no. 3, [2] G. Wesbuh an G. Deuant an F. Amblar an J. Naal, Meet, suss, an segregate!, Complexty, vol. 7, no. 3, 2002, pp [3] J. Lorenz, Consensus strkes bak n the Hegselmann-Krause moel o ontnuous opnon ynams uner boune onene, Journal o Artal Soetes an Soal Smulaton, vol. 9, no., pp. 8. [4] J. Lorenz, A stablzaton theorem or ynams o ontnuous opnons, Physa A: Statstal Mehans an ts Applatons, vol. 355, ssue, 2005, pp [5] R. Olat-Saber an R.M. Murray, Flokng wth obstale avoane: ooperaton wth lmte ommunaton n moble networks, Proeengs o the 42n IEEE Conerene on Deson an Control, vol. 2, 2003, pp [6] R. Olat-Saber an R.M. Murray, Consensus problems n networks o agents wth swthng topology an tme-elays, IEEE Transatons on Automat Control, vol. 49, ssue 9, 2004, pp [7] R. Olat-Saber an R.M. Murray an J.A. Fax, Consensus an Cooperaton n Networke Mult-Agent Systems, Proeengs o the IEEE, vol. 95, no., 2007, pp [8] V.D. Blonel an J.M. Henrkx an J.N. Tstskls, On Krause s Mult-Agent Consensus Moel Wth State-Depenent Connetvty, IEEE Transatons on Automat Control, vol.54, no., 2009, pp [9] V.D. Blonel an J.M. Henrkx an J.N. Tstskls, Contnuoustme average-preservng opnon ynams wth opnon-epenent ommunatons, SIAM J. Control an Optmzaton, 200, pp [0] D.V. Dmarogonas an K.H. Johansson, Deentralze onnetvty mantenane n moble networks wth boune nputs,, 2008 IEEE Internatonal Conerene on Robots an Automaton, pp [] M.C. DeGennaro an A. Jababae, Deentralze ontrol o onnetvty or mult-agent systems, 45th IEEE Con. Deson an Control, 2006, pp [2] M. J an M. Egerstet, Conneteness preservng stbute oornaton ontrol over ynam graphs, 2005 Ameran Control Conerene, pp [3] T. Gustav an D.V. Dmarogonas an M. Egerstet an X. Hu, Suent ontons or onnetvty mantenane an renezvous n leaer-ollower networks, Automata, vol. 46, ssue, 200, pp