Hierarchical Fuzzy Opinion Networks: Top-Down for Social Organizations and Bottom-Up for Election

1  Abstract — A fuzzy op inion is a Gaussian fuzzy set with th e center representing the opi nion and the st andard deviation representing the uncertainty about the opinion, and a fuzzy opinion network is a conne ction of a number of fuzzy op inions in a structured way. In this paper, w e propose: (a) a top-dow n hierarchical fuzzy opinion network to model how the opinion of a top leader is penetrated into th e members in social orga nizations, and (b) a bo ttom-up f uzzy opinion netw ork to m odel how the opinions of a large number of agents are agglomerated layer- by -layer into a cons ensus or a few opinions in the social processes such a s an election. For the top-down hierarchical fuzzy opinion network, w e prov e that the o pinions of all th e a gents converge to the leader ’ s o pinion, but the uncertainties of the agents in different g roups are g enerally converging to different values. We demon strate that the speed of convergence is grea tly improved b y organizing the agents in a hierarchical structure of small groups. For t he bottom-up hierarchical fuzzy o pinio n network, we si mulate how a wide spectrum of opinions are negotiating and su mmarizing with each other in a layer- by -layer fashion in some typical situations. Index Terms — Opinion dynamics; social hierarchy; fuzzy opinion networks. I. I NTRODUCTIO N Hierarchy i s the most pop ular structure in social organizatio ns such as government, army, company, etc. [1,2,3]. In a com pany, for exam ple, it is typically structured w ith a relatively small top management team, at least one la yer o f middle management, and a large number of lo wer level employees r esponsible for day- to -da y operations [ 4 ]. Why is hierarch y so p ervasive in human societie s acr oss al most all cultures throughout t ime [5 ] , giving the fact that hierarchy i s in dir ect opposition to some of the b est ideas humanity has p roduced such as de mocracy, equality, fairness, and justice [1 ] ? The in terdisciplinar y research on social hierarchy in socio logy [6,7], psy chology [8,9] , management [1 0], ec onomics [ 11 ], and other d isciplines suggest a number of reaso ns. First, hierarch y establis hes social order that is appealing psychologicall y to in dividual s who n eed Li -Xin Wang is w ith the Unive rsity of Chinese Academy of Sciences, Beijing, P.R. Chi na (e-mail: lxw ang@ucas.edu.cn ). safety and stability (peop le come and go, but the system remains). Second, hierarchy provides incentives for individuals in organizations to work hard to obtain higher rank to satisfy material sel f-interest and their need for control that in turn       ( motivating the individuals to work hard for the organizat ion). T hird, hierarchy faci litates coordination and improves e fficiency in comparison to other more egalitaria n structures such as free m arkets ( the rapid            years is an exa mple, which has le d to t he d eglobalization movement in the  laisser- faire capitalis m  economies to reestablish the hierarch y). Fourth, hierarchical differentiation between people fosters more sat isfying working relationships (leaders pro vide the guidance their follo wers need, and followers execute what the leader s want to be realized) . Although hierarch y has bee n s tudied in social sciences for a long time (back to Marx and Engels [ 12 ] in 18 46), the research is largely q ualitative without mathematical modeling. U sually, some concepts or variables are defin ed v erbally, then a theory is developed that describe s t he relationsh ips a mong these variables using natural languages rather tha n mathematical equations [13]. The most mathematically adva nced study related to social hierarchy is perhaps the multidisciplinar y field of opinion d ynamics where the researchers from mathemati cal sociology [ 14 ], economics [ 15 ,16], physics [1 7,18] , social psychology [ 19 ], control [ 20 ], signal p rocessing [2 1], fuzzy systems [ 22 ], etc., join f orces to tack le the problem. A shortcoming of the mainstrea m o pinion dynamics models [ 23 , 24 ] is that the uncertainties of the opinions are not included in the models. H uman opinions ar e inherently u ncertain so that an opinion and its uncer tainty should be co nsidered simultaneously to give the accurate picture of the o pinion. For example, whe n w e are ask ed to review a r esearch paper, w e need to g ive a n o verall r ating for the pap er and, at the sa me time, we must claim our le vel of expertise on the subj ect which is a measure of uncertai nty abo ut the overall rat ing. I n fact, the uncertainty may be m o re important than the op inion itself in many sit uations, because the uncertainty is more dir ectly related to the p sychological pressure of the agent when the opinion is broa dcasted [2 5]. Hierarchical Fuzzy Opinion Networks: T op-Down for Social Or ganizations and Bottom-Up for Election Li -Xin Wang 2 The fuzzy opinio n networks (FONs) p roposed in [ 22 ] model an op inion by a Gau ssian f uzzy set whose center and stand ard deviation represent the opinion and its uncertainty, respectively, so that the interactio ns between the opinions and their uncertainties ar e s ystematicall y exploited . The goal of this paper is to use the FON fra mework to model social hierarc hy . According to [2 ], social hierarchies can be classi fied into two types: i) f ormal hierarchies that are d elineated by r ule and consensually agreed upon, an d ii) infor mal hierarc hies that are established and subj ectively under stood during t he interactio n among social members. Formal h ierarchies are top -down  a hierarchical structure is designed first an d members at different levels ar e then recruited; informal hierarchies are botto m-up  a hierarchical str ucture emerges after the free interaction a mong the communit y members. We will u se fuzzy op inion net works to m odel both formal and informal hierarchie s . T o model t he formal hierarchy, we first define a basic leader-follo w group of fuzzy agents and stud y its ba sic con vergence pro perties, then we con nect the basic leader-follo wer groups in a hierarchical fashion to ge t the final hierarchical fuzzy opinion network. To model a n in formal hierarchy, we let a large number of fuzzy agents to interact with eac h o ther based on a loca l reference scheme, and we see the initiall y very diver sified opinions are merging gradually in a hierarchical fashio n into a consensus or a number of rep resentative op inions  a proce ss very similar to an election in a de mocratic societ y. This paper is organized as follows. In Sectio n II , the top-do wn hierarch ical fuzzy o pinion network s are construct ed and th eir convergence properties are proved. We also show tha t the speed of convergence to the top lead er  s opinion is greatl y improved b y organizing the follo wers into a hierar chical structure rather than in a flat n onhierarchical fashion. I n Sect ion III, we co nstruct the bottom-up hierarchical fuzzy opinion networks through t he natural process of free in teracting am ong a large number of fuzzy agents based o n t he local reference scheme, and we simulate a number of typical scenario s  consensus reachi ng , polarization, or con verging to multiple ends. Finally, Section IV concludes the paper and the A ppendix contains the proofs of t he theore ms in the paper. II. T OP -D OWN H IERARCHICAL F UZZY O PINION N ETWORK S We start with the definition o f fuzzy opinion network s (FON) and bounded confidence FONs 1 , and then introd uce the basic leader-follower group whic h i s the basic building block of t he top-do wn hierarchica l fuzzy opinion net works . Definition 1: A fuzzy op inion is a Gau ssian fuzzy set  with membership functio n   󰇛  󰇜          where the center    represents the opinion and the sta ndard deviation     characterizes the u ncertainty abo ut the opinion  . A Fuzzy 1 The con cept of F ON was intro duced i n [ 22 ] and the b asic conve rgence properties of bo unded confidence F ONs were studied in [2 6]. Opinion Network (FON) is a connection of a number of Gaussian nodes, where a Gau ssian nod e is a 2-inpu t-1-output fuzzy o pinion   with Gau ssian membership fu nction    󰇛  󰇜            whose center   and standard deviation   are tw o input fuzzy sets to the n ode and the f uzzy set   itself is the output of the node. A Gaussian node is a lso called an agent , a node , or a fuzzy node throughout this paper.  The connection o f the fuzz y nodes can be static o r dynamically c hanging with ti me and the statu s of the nodes. The bounded confidence fuzzy opinion networks , defined below, are FONs with connections that are dynamicall y changing accor ding to the st ates of the nodes  if the fuzz y opinions o f t wo nodes are close enoug h to each o ther, they are connected; otherwise, the y are disco nnected. Definition 2: A boun ded confidence fuzzy opinion network (BCFON) is a dyn amic connection of n fu zzy nodes   󰇛󰇜 (       ) w ith membership functions    󰇛󰇜 󰇛  󰇜       󰇛󰇜      󰇛󰇜 , wh ere the center input   󰇛  󰇜 and the standard deviation input   󰇛    󰇜 to node  at time    (     ) are determined a s follo ws: the center input   󰇛  󰇜 is a weighted average of the outp uts   󰇛󰇜 of the n fuzz y nodes at the previous time point  :   󰇛    󰇜     󰇛  󰇜   󰇛  󰇜     󰇛󰇜 with the weights   󰇛  󰇜  󰇱     󰇛  󰇜      󰇛  󰇜      󰇛  󰇜     󰇛󰇜 where   󰇛 󰇜 (       ) is the collection of nodes that are connected to node  at time t , defined as:   󰇛  󰇜  󰇥  󰇝     󰇞  󰇡  󰇛  󰇜    󰇛  󰇜 󰇢    󰇦   󰇛󰇜 where  󰇡  󰇛  󰇜    󰇛  󰇜 󰇢 represents the close ness betwee n fuzzy opinio ns   󰇛  󰇜 and   󰇛  󰇜 ,    󰇟󰇠 are cons tants and    󰇛  󰇜 denotes the number of elements in   󰇛  󰇜 ; and, the standard deviation input   󰇛    󰇜 are determined according to one of the two sche mes: (a) Loca l reference scheme :   󰇛    󰇜   󰈏  󰇛󰇜      󰇛 󰇜       󰇛 󰇜   󰇛󰇜 󰈏   󰇛󰇜 (b) External reference scheme :   󰇛    󰇜       󰇛󰇜    󰇛󰇜    󰇛󰇜 where   󰇛󰇜 denotes the center of fuzzy set   󰇛󰇜 ,   󰇛 󰇜 is an external signal and  is a positive scalin g constant. T he initial fuzzy opinions   󰇛  󰇜 (       ) are Gaussian fuzzy sets 3    󰇛󰇜 󰇛  󰇜             , where the initial opinions      and the initial uncertainties      are given.  It w as pro ved in [ 26] that the opinions   󰇛󰇜 and their uncertainties   󰇛󰇜 of the Gaussian nodes   󰇛󰇜 in the BCFON are evolving according to the follo wing dynamic equations. The Evo lution of BCFON : The fuzzy op inions   󰇛  󰇜 (       ;     ) in the B CFON are Gaussian fuzzy sets:    󰇛  󰇜 󰇛  󰇜        󰇛󰇜      󰇛󰇜    󰇛󰇜 where the opinions   󰇛 󰇜   and their uncertainties   󰇛 󰇜    are evolving according to the following dynamic equations:    󰇛   󰇜     󰇛  󰇜    󰇛  󰇜       󰇛󰇜   󰇛  󰇜     󰇛  󰇜   󰇛  󰇜       󰇛    󰇜 󰇛󰇜 where the weights   󰇛 󰇜  󰇱     󰇛  󰇜      󰇛  󰇜      󰇛  󰇜    󰇛󰇜   󰇛  󰇜         󰇝     󰇞   󰇻    󰇛  󰇜    󰇛  󰇜 󰇻    󰇛  󰇜   󰇛  󰇜            󰇛  󰇜 and the uncertainty inp ut   󰇛    󰇜   󰇻    󰇛󰇜      󰇛 󰇜       󰇛󰇜     󰇛 󰇜 󰇻  󰇛  󰇜 for local referen ce scheme , or   󰇛    󰇜       󰇛󰇜    󰇛 󰇜    󰇛  󰇜 for external reference scheme w ith initial condition    󰇛󰇜    (initial opinion of agent  ) a nd   󰇛󰇜    (uncertainty about the initial op inion), where       ,    ar e constants.  We now d efine the basic lead er-follower group which i s the basic building block of the top-down h ierarchical fu zzy opinion networks of this paper. Definition 3 : A basic leader-follower group (BLFG), illustrated in Fig. 1, consists of a leader node   󰇛 󰇜 with membership functio n    󰇛  󰇜 󰇛  󰇜        󰇛󰇜   󰇛   󰇛󰇜 󰇜  and n follow er nodes   󰇛 󰇜 (        ) with membership functions    󰇛  󰇜 󰇛  󰇜        󰇛󰇜      󰇛󰇜   , where t he leader node passes his opinion   󰇛 󰇜 to each of the n follower nod es and the n follower nod es are connected among the mselves in the bounded co nfidence fashion. Specifically, the leader  s op inion    󰇛󰇜 and its uncertainty   󰇛󰇜 are not influenced b y the n followers, and the opinions   󰇛󰇜 and their uncertainties   󰇛󰇜 of th e n followers are evol ving acco rding to the following dynamic equations:    󰇛    󰇜      󰇛 󰇜    󰇭     󰇛 󰇜    󰇛󰇜    󰇛 󰇜󰇮  󰇛  󰇜   󰇛  󰇜      󰇛󰇜     󰇛󰇜     󰇛 󰇜    󰇛    󰇜 󰇛  󰇜 where   󰇛  󰇜 is given in (10) with       , and the uncertainty input   󰇛    󰇜 is chosen either with the loca l reference scheme ( 11 ), or with the lead er reference scheme :   󰇛    󰇜       󰇛 󰇜    󰇛 󰇜   󰇛  󰇜  We see from (13) that the opinion   of follower i is updated as the average o f the neighbor  s opinion s     󰇛 󰇜    󰇛󰇜 plus the leader  s opinion   󰇛󰇜 . For the uncertainty   of follo wer i , we see from (14) that it is updated as the a verage of the neighbor  s uncertainties     󰇛󰇜     󰇛 󰇜   󰇛󰇜 plus the uncertainty input   󰇛    󰇜 which ta kes eit her the local reference scheme (11) or the lead er reference scheme (15 ). In the loca l reference sche me, agent i vie ws the average o f his neighbor  s opinions as the refe rence, so the closer h is o pinion is Fig. 1:The b asic leader-follower group, where the leader passes his opinion    󰇛󰇜 to each of the n follow ers who are connected among themselves in the bounded confidence f ashion .   2 2 () () e n n x x t t      2 3 2 3 () () e x x t t      2 2 2 2 () () e x x t t      2 1 2 1 () () e x x t t      2 2 () () e a a x x t t    Followers Leader () a xt () a xt () a xt () a xt 4 Fig. 2 : A simulation run of the b asic leader -follower group with local reference scheme, where the top and bottom sub-figures plot the opinions    󰇛 󰇜 and th e uncertainties   󰇛󰇜 of the n =156 followers , respectively; the leader  s opinion    󰇛  󰇜   . to this aver age, the less uncertainty he ha s. In the leader reference sche me, ho wever, agent i views the leader  s o pinion as the reference, so the closer his opinion   󰇛󰇜 is to th e lead er  s opinion   󰇛 󰇜 , the less uncertaint y he has. To get a feel o f t he dynamics of th e opinions and their uncertainties of the agents in the b asic leader-follo wer group, let  s see an exa mple. Example 1 : Consider the basic leader-follower group of Fig. 1 with    followers. With     ,     , the leader  s opinion   󰇛  󰇜   and the initial   (       ) uniformly distributed o ver the interva l [5,25] (       󰇛    󰇜       ) and their uncertainties     for all       , Fig. 2 and Fig. 3 show the si mulation run s of the dynamic model with local reference scheme (11) and lead er reference sche me (1 5), r espectively, where the to p sub-figures of Figs. 2 and 3 plot the opinions   󰇛 󰇜 of the    followers a nd the botto m sub -figures plot the u ncertainties   󰇛󰇜 .  We see from Fi gs. 2 and 3 that for both the lo cal and lead er reference schemes, the opin ions   󰇛󰇜 of all the followers converge to the leader  s op inion    󰇛  󰇜   , but the spee d of convergence i s slo w. I n t he following theor em, we prove that convergence to the leader  s opinion is indeed guaranteed, but the speed of co nvergence i s gr eatly in fluenced b y the n umber of followers in the gro up. Theorem 1: Consid er the basic lead er-follower d ynamics of (13), (14) and (10) with local r eference scheme (1 1) or lead er reference sc heme (15), and suppose t he lead er  s op inion    󰇛  󰇜    is a constant. Starting from arbitrary initial opi nions   󰇛󰇜      and uncertainties   󰇛󰇜       , we have: Fig. 3 : A sim ulation run of the bas ic leader-follower group w ith leader reference scheme, where the top and bottom sub-figures plot the opinions    󰇛 󰇜 and th e uncertainties   󰇛󰇜 of the n =156 followers , respectively; the leader  s opinion    󰇛  󰇜   . (a) th e  followers converge to a consensus in finite time, i.e., there exists   such t hat   󰇛  󰇜    󰇛󰇜 and   󰇛  󰇜   󰇛 󰇜 for all       and all     ; (b) the opinion consensus  󰇛 󰇜 converges to the leader  s opinion   according to   󰇛  󰇜      󰇡     󰇢    󰇛   󰇛     󰇜    󰇜 󰇛  󰇜 where       ; (c) for local reference scheme (11), the uncertaint y consensus  󰇛  󰇜   󰇛    󰇜 (a constant ) for all     ; (d) for lead er reference sche me (15) , the uncertainty consensus  󰇛 󰇜 is changing according to  󰇛  󰇜   󰇛   󰇜     󰇛     󰇜      󰇡     󰇢         󰇛  󰇜 for       , from which we get    󰇛  󰇜   󰇛     󰇜     󰇛     󰇜     󰇛  󰇜 .  The proof of Theorem 1 is g iven in the Appendix. From (16) in Theorem 1 we see that t he opinion co nsensus   󰇛 󰇜 converges to the lead er  s o pinion   with the factor    , i.e., the error  󰇛  󰇜    is r educed by   each time step, so in k tim e steps the erro r   󰇛  󰇜    is reduced b y 󰇡   󰇢    which gives         󰇛  󰇜    󰇛  󰇜 With     (reduce to error to 1%), Fig. 4 plots the k as function of n , from which we see that t he s teps needed to reduce the error increases about li nearly with the number of followers 0 20 40 60 80 100 120 140 160 5 10 15 20 25 O pi ni ons of t h e n = 156 f o l l ow er s w i t h l ocal r ef er ence schem e; t he l eader opi ni on x a= 10 t opi ni ons 0 20 40 60 80 100 120 140 160 0. 9 95 1 1. 0 05 1. 0 1 1. 0 15 1. 0 2 1. 0 25 1. 0 3 1. 0 35 U ncer t a i nt i es of t he n= 156 f o l l ow er s w i t h l oc al r ef e r ence s chem e t uncer t ai nt i es 0 20 40 60 80 100 120 140 160 5 10 15 20 25 O pi ni ons o f t he n = 156 f o l l ow er s w i t h l ea de r r ef e r ence schem e; t he l ea der opi ni on xa= 10 t opi ni ons 0 20 40 60 80 100 120 140 160 1 1. 5 2 2. 5 3 3. 5 4 4. 5 U ncer t a i nt i es of t he n= 156 f o l l ow er s w i t h l eader r ef er ence schem e t uncer t ai nt i es 5 Fig. 4: Plot of (18), the ste ps k nee ded t o re duce the error   󰇛  󰇜     to 1% as function of number of followe rs n in group. in t he group, meaning that larger groups are more difficult to converge to the leader  s opinion than smaller gro ups. The conclusion fro m ( 18) and Fig. 4 is that to speed up the convergence of the follo wers  opinions to the leader  s op inion, reducing the size of the group is crucial. Organizin g t he followers hierarchically in s maller groups, as we will d o next through the top -down hierarc hical fuzzy opinion networks, is an efficient way to speed up the convergence. We now introduce the top-down hierarchical fuzzy opinion networks. Definition 4: A top-down hierarchical fuzzy opinion network ( TD -HFON), illustrat ed in Fig. 5, is constructed from a number of basic leader -follo wer groups of Fig. 1 in a multi-layer stru cture , w here an ag ent    in Level l is a follower to an age nt in Level l +1 and is a leader to som e agents in Level l - 1. I n the notation    , l is the level index (       ), i is the group index (         ), j is the index in t he group (         ), and    is a Gaussian fuzzy set with cen ter     󰇛󰇜 and standard deviation    󰇛 󰇜 .  To see ho w fast the hierar chical str ucture can speed up the convergence to t he leader  s op inion, we reorganize t he n =156 followers in Example 1 into a 3-level and a 4-level TD-HFONs in the following e xample. Example 2: Consider the 3-level and 4 -level T D-HFONs in Fig. 6. In t he 3 -level T D-HFON (left i n Fig. 6), L evel-1 consists of 12 groups w i th 1 2 agents in each group, L evel -2 consists o f a single gro up of 12 agents with ea ch age nt b eing t he leader of one the 12 g roups in L evel-1, and Level-3 is the top leader w ho is the leader of the 1 2-agent g roup in Le vel-2. W ith      ,     , the to p lead er  s opinion   󰇛  󰇜   and the initial opinions of the 12 agents in a group     󰇛󰇜 (                      ) uniformly distributed over the interval [5,25] (    󰇛󰇜     󰇛     󰇜       ) and all their uncertainties    󰇛󰇜   , Fig s . 7 and 8 show the simulation runs o f t he dynamic m odel with local ref erence scheme (11) and lead er reference scheme (15), r espectively , where the top sub-figures of Figs. 7 and 8 plot the op inions     󰇛󰇜 of the    agents i n Levels 1 and 2 and the b ottom sub-figures plot the cor responding uncertai nti es    󰇛 󰇜 . Similarly, in the 4 -level T D-HFON (rig ht in Fig. 6), Level- 1 consists o f 25 groups with 5 agents in ea ch group, Level - 2 consists of 5 groups with 5 agents in each group and these 2 5 agents are the lead ers o f the 25 groups in Level -1, Level- 3 consists of a single group of 5 agents wh o are the leaders of the 5 groups in Level -2, and Level-4 is the top leader who is the leader of the 5-agent group in Level-3. With     ,     , the to p lead er  s opinion   󰇛  󰇜   and the initial opinions o f the 5 age nts in a group    󰇛󰇜 (                              ) uniformly distributed over t he interval [5,25 ] (    󰇛󰇜     󰇛    󰇜      ) and all their uncertainties    󰇛󰇜   , Fig s . 9 and 10 sho w the simul ation runs of the d ynamic model with local reference sc heme ( 11) and lead er reference scheme (15), respectively, where the top sub-figures o f Figs. 9 and 10 plot the opinio ns    󰇛󰇜 of the    agents in Levels 1, 2 and 3 and the bottom sub -figures plot the cor responding uncertainties    󰇛󰇜 .  Comparing Fig. 2 and 3 with Figs. 7-10, w e have the following observations: (a) The op inions    󰇛 󰇜 of all the agents conver ge to the leader  s o pinion no matter the agents are organized hierarchically in small groups or in one large group. (b) The speed of convergen ce to the leader  s opinion is greatly improved when the agents are organized hierarchicall y in small groups; the more the levels or the smaller the group s, the faster t he convergence will b e (comparing the top sub-figures of Figs. 2, 7 and 9 for the lo cal reference scheme, and the top sub -figures of F igs. 3, 8 and 10 for the lead er reference scheme). (c) A lthough t he opinions    󰇛 󰇜 of all the agents converge to the leader  s opinion, their uncertaintie s    󰇛󰇜 in general converge to d ifferent values f or agents i n different g roups, reflecting the different p rocesses that the agents in differ ent groups were experie ncing durin g the convergence to the leader  s opinion. Indeed, we w ill prove in the following theorem that the observations above are true i n general. 0 20 40 60 80 100 120 140 160 180 200 0 100 200 300 400 500 600 700 800 900 1000 S t eps ne ede d t o r educe er r or t o 1% as f unct i on of f o l l ow er num ber num ber of f ol l ow er s n i n a gr o up st eps n eede d t o r edu ce t h e e r r or t o 1% 6 Fig. 5:The top-down hierarchical fuzzy opinio n networks. Fig. 6: Reorganizing the age nts in Example 1 into a 3-level TD-HFON w ith 12 followers in each group (lef t) and a 4-level TD-HFON with 5 followers in each group (right). 11 X l Level 1 Level L (Top Leader) Level 3 Level 2 3 X l l n 2 X l l n 1 X l l n 22 X l 21 X l 13 X l 12 X l Level l Level l +1 Level L-1   2 2 () () e a a x x t t    () a xt () a xt () a xt () a xt 11 () l xt 3 () l l n xt 2 () l l n xt 1 () l l n xt 22 () l xt 21 () l xt 13 () l xt 12 () l xt 1 1 X L  1 1 X L L n   1 3 X L  1 2 X L  1 1 () L xt  1 1 () L L n xt   1 2 () L xt  1 3 () L xt  1 11 X l  1 12 X l  1 11 () l xt  1 12 () l xt  1 11 () l xt  1 12 () l xt  Level 1 Level 3 Level 2 12 agents 12 agents 12 agents 12 agents 5 agents 5 agents 5 agents 5 agents 5 agents 5 agents 25 agents in 5 groups 5 agents Level 4 156 agents in a 3-level TD-HFON with 12 followers each group 155 agents in a 4-level TD-HFON with 5 followers each group 7 Fig. 7: The opinions (top) and their uncertainties (bottom) of the n=156 agents in the 3-level TD-HFON of Fig. 6 with lo cal reference scheme. Fig. 8: The opinions (t op) and their uncertainties (bottom) of the n=156 agents in the 3-level TD-HFON of Fig. 6 with leader reference scheme. Theorem 2 : Consider the gener al T D-HFON in Fig. 5 with dynamics o f all the gro ups fo llowing ( 13), (14) and (10) with local reference sche me (11) or leader reference scheme (15), and suppo se the top leader  s opinion   󰇛  󰇜    is a co nstant. Starting fro m arbitrar y ini tial opinions    󰇛󰇜   and uncertainties    󰇛󰇜    , we have: (a) th e o pinions    󰇛 󰇜 o f all th e agents (                          ) converge to the leader  s opinion   ; (b ) the uncertainties    󰇛󰇜 of the follo wers in the same leader-follower gro up conver ge to a consta nt, but different groups in general conver ge to different values.  The proof of T heorem 2 is given in the Appendix. We now move to t he next se ction to study the b ottom -up hierarchical fuzz y opinion networks. Fig. 9: The opinions (t op) and their uncertainties ( b ottom) of the n=155 agents in the 4-level TD-HFON o f Fig. 6 with local reference scheme. Fig. 10 : The opinions (top) and their uncertainties (bottom) of th e n=155 agents in the 4-level TD-HFON of Fig. 6 with leader reference scheme. III. B OTTOM -U P H IERARCHICAL F UZZY O PINI ON N ETWORKS As we discussed in the Intro duction that althoug h so cial hierarchy is prevalent throug hout culture and time, hierarchy is against some of the best val ues of humanity  hierarchy is undemocratic, u nequal, unfair, and unjust. So , if we have to choose hierarchy to govern a lar ge population such as a na tion, we should have some cou nter measures to preve nt t hose in the higher levels to abuse their po wer. Election b y the g eneral public is the way of choice o f most co untries in the world to select their top leaders. In the election scenario, the opinions of the large populat ion are i nitially very di versified and many small leader s are emerging to represent different interest groups, then these small leaders have to compromise with each other to select the middle-level lead ers, this p rocess continues level- by - level in a bottom-up fashion until so me co nsensuses are reached. We now propose the b ottom-up hierarchical fuzzy opinion networks to model such p rocesses. 0 10 20 30 40 50 60 70 80 90 100 5 10 15 20 25 O pi ni ons of t he n =15 6 a gen t s i n 3- l evel T D - H FO N w i t h l ocal r ef er ence schem e; t h e l ead er opi ni on xa=1 0 t opi ni ons 0 10 20 30 40 50 60 70 80 90 100 1 1. 0 05 1. 0 1 1. 0 15 1. 0 2 1. 0 25 U ncer t ai nt i es of t he n= 156 ag ent s i n 3- l ev el T D - H F O N w i t h l ocal r ef er ence schem e t uncer t ai nt i es 0 10 20 30 40 50 60 70 80 90 100 5 10 15 20 25 O pi ni ons of t he n =15 6 a gen t s i n 3- l evel T D - H FO N w i t h l eade r r ef er ence schem e; t he l ead er opi ni on x a=1 0 t opi ni ons 0 10 20 30 40 50 60 70 80 90 100 1 1. 1 1. 2 1. 3 1. 4 1. 5 1. 6 1. 7 1. 8 U ncer t ai nt i es of t he n= 156 ag ent s i n 3- l ev el T D - H F O N w i t h l ead er r ef er ence schem e t uncer t ai nt i es 0 10 20 30 40 50 60 70 80 90 100 5 10 15 20 25 O pi ni ons of t he n =15 5 a gen t s i n 4- l evel T D - H FO N w i t h l ocal r ef er ence schem e; t h e l ead er opi ni on xa=1 0 t opi ni ons 0 10 20 30 40 50 60 70 80 90 100 1 1. 0 5 1. 1 1. 1 5 U ncer t ai nt i es of t he n= 155 ag ent s i n 4- l ev el T D - H F O N w i t h l ocal r ef er ence schem e t uncer t ai nt i es 0 10 20 30 40 50 60 70 80 90 100 5 10 15 20 25 O pi ni ons of t he n =15 5 a gen t s i n 4- l evel T D - H FO N w i t h l eade r r e f er ence s chem e; t he l ead er opi ni on x a= 10 t opi ni ons 0 10 20 30 40 50 60 70 80 90 100 1 1. 1 1. 2 1. 3 1. 4 1. 5 U ncer t ai nt i es of t he n= 155 ag ent s i n 4- l ev el T D - H F O N w i t h l ead er r ef er ence schem e t uncer t ai nt i es 8 Definition 5: A bottom- up hierarchical fuzzy op inion network (BU-HFON), illustrated in Fig. 1 1, is the lay ered connection o f a number of bo unded confide nce fuzzy o pinion networks (BCFON) with local reference sche me (Definition 2), where the converged opinions o f a lo wer level B CFON a re passed to the upper level BCFON as the i nitial opinions .  We now simulate the BU -HFON to see ho w the initial fuzzy opinions ar e a gglomerated layer- by -layer in so me t ypical situations. Example 3: Consider a 5-level BU -HFON of Fi g. 11 ( L =5) with    agents in Level 1 whose initial op inions    󰇛󰇜 and initial uncertainties    󰇛󰇜 ( i =1,2,  , n ) are randomly distributed over the interval s [5,25] and ( 0,1), respectively. The five BCFONs in the fi ve leve ls are evolving accordin g to the dynamic equatio ns (7)-(11 ), where      for Level 1 BCFON,      for Level 2 BCFON,      for Level 3 BCFON,      for Level 4 BCFON and      for Level 5 B CFON. The meaning of these    s are explai ned as follows. For the Level 1 agents (the general public), we choose a large   (=0.95) because the g eneral public has no obligation to reach some consensuses so that they ca n show little sign of compromise (a lar ge   means talking only to those whose opinions are very close to each other). For the Level 2 agents (the loca l r epresentatives of the general public), they have to show some sign of co mpromise i n order for th e process to continue, so we choo se a littl e smaller   (=0.7) to m odel the situation. Then, the Level 3 agents must be even more compromising in order to reach so me rough co nsensuses, so we choose a even smaller   (=0.45) for the se middle level a gents. This pr ocess continues with s maller and s maller    s for the upper level a gents (     for Level 4 an d      for Level 5) because the higher the le vel they are in, the more pressure the y have to rea ch the final consen sus (this is wh y many elected agents fall to realize their election promises when they ar e in the o ffice, bec ause they have to consider m an y different concerns when t hey are in the higher lev els). With b =0.5 for all the B CFONs and each BCFON evolving 40 time steps, i.e., th e Level 1 BCFON is operating from t =0 to t =40, then followed b y the Level 2 BCFON which is operating from t =41 to t =8 0 with the converged fuzz y opinio ns of the Level 1 BCFON as the initial values, this process continues with t he Le vel 3 BCFON op erating from t=81 to t=12 0, the Level 4 B CFON operating fro m t=121 to t=160 and th e Lev el 5 BCFON operating fro m t=160, Fig. 12 sho ws a s imulation run in a typical situation, where the to p and bottom sub -figures in Fig. 11 :The bottom-up hierarchical f uzzy opinion networks. Level 1 Level 2 1 1 X0 （ ） 1 X0 n （ ） 1 -1 X0 n （ ） 1 -2 X0 n （ ） 1 5 X0 （ ） 1 4 X0 （ ） 1 3 X0 （ ） 1 2 X0 （ ） BCFON 1 2 1 X0 （ ） 2 4 X0 （ ） 2 3 X0 （ ） 2 2 X0 （ ） 2 2 X0 n （ ） 2 2 -1 X0 n （ ） 3 1 X0 （ ） 3 3 X0 （ ） 3 2 X0 （ ） 3 3 X0 n （ ） 3 3 -1 X0 n （ ） Initial fuzzy opinions Converged fuzzy opinions from Level 1 Converged fuzzy opinions from Level 2 BCFON 2 Level L L 1 X0 （ ） 2 X0 L （ ） X0 L L n （ ） 1 X m X Converged fuzzy opinions from Level L-1 Final fuzzy opinions BCFON L 9 Fig. 12: The o pinions (top) of the agents in different levels and their uncertainties (bottom). Fig. 13: The opinions (top) of the agents in different levels and their uncertainties (bottom). Fig. 12 show the op inions (   󰇛󰇜 of (7)) of the agents and their uncertainties (   󰇛 󰇜 of (8)), respectively. We see from to p sub-figure of Fig. 12 that the Level 1 ge neral public (    ) converge to a large number o f op inions due to the lar ge   (=0.95), then with a s maller   (=0.7) the Level 2 agents converge to about 17 o pinions , which are further co mbined by the Level 3 agents ( with      ) into 11 opinions, and continuing with     the Level 4 agents reac h 5 opinions, finally, t he top level agents have to adopt a very small      to reach a single consens us. The bottom s ub- figure of Fig. 12 shows that the uncertainties are gett ing larger and larger for the higher level ag ents, reflect ing the fact that t he higher level agents m ust demonstrate m ore compromises which result in more uncertainties ab out their decisio ns. Fig. 14: The opinions (top) of the agents in different levels and their uncertainties (bottom). Fig. 15: The opinions (top) of the agents in different levels and their unc ertainties (bottom). Figs. 1 3-15 show t he si mulation runs in other typical situations, where a consensus is reached in Fig. 13, but in the situations of Figs. 14 and 15, a consensus cannot be rea ched after five rounds of negotiatio ns. Comparing the bo ttom sub-figures o f Figs. 12 -15 we see that the uncertainties of t he Level 5 agents are high if the y converge to a single co nsensus (Figs. 12 and 13) , but if they converge to t wo consensuse s (Fig. 14), their uncertaintie s are much lo wer, and if the y are allowed to keep three dif ferent opinions ( Fig. 15), their uncertainties are even lower. This d emonstrates that the uncertainty   󰇛 󰇜 in our HFON model pr ovides a good measure for the p sychological pressures of the agents in di fferent levels.  0 20 40 60 80 100 120 140 160 180 200 5 10 15 20 25 O pi ni ons of t he ag ent s i n di f f e r ent l evel s of t he B U - H F O N t opi ni on s 0 20 40 60 80 100 120 140 160 180 200 0 1 2 3 4 5 6 U ncer t ai nt i es of t he age nt s i n di f f er ent l evel s of t h e B U - H FO N t uncer t ai nt i es Level 1 Level 2 Level 3 Level 4 Level 5 d= 0. 9 5 d= 0. 7 d= 0. 4 5 d= 0. 2 d= 0. 0 5 0 20 40 60 80 100 120 140 160 180 200 5 10 15 20 25 O pi ni ons of t he ag ent s i n di f f e r ent l evel s of t he B U - H F O N t opi ni on s 0 20 40 60 80 100 120 140 160 180 200 0 1 2 3 4 5 6 U ncer t ai nt i es of t he age nt s i n di f f er ent l evel s of t h e B U - H FO N t uncer t ai nt i es Level 1 Level 2 Level 3 Level 4 Level 5 d= 0. 9 5 d=0 . 7 d= 0. 4 5 d=0 . 2 d= 0. 0 5 0 20 40 60 80 100 120 140 160 180 200 5 10 15 20 25 O pi ni ons of t he ag ent s i n di f f e r ent l evel s of t he B U - H F O N t opi ni on s 0 20 40 60 80 100 120 140 160 180 200 0 1 2 3 4 5 6 U ncer t ai nt i es of t he age nt s i n di f f er ent l evel s of t h e B U - H FO N t uncer t ai nt i es Level 1 Level 2 Level 3 Level 4 Level 5 d= 0. 9 5 d= 0. 7 d= 0. 45 d= 0. 2 d= 0. 05 0 20 40 60 80 100 120 140 160 180 200 5 10 15 20 25 O pi ni ons of t he ag ent s i n di f f e r ent l evel s of t he B U - H F O N t opi ni on s 0 20 40 60 80 100 120 140 160 180 200 0 1 2 3 4 5 6 U ncer t ai nt i es of t he age nt s i n di f f er ent l evel s of t h e B U - H FO N t uncer t ai nt i es Level 1 Level 2 Level 3 Level 4 Le vel 5 d= 0. 9 5 d=0 . 7 d= 0. 45 d=0 . 2 d= 0. 05 10 IV. C ONCL UDING R EMARKS The top-do wn a nd bottom-up hierarchical fuzzy opinion networks (HFON) developed in this paper provide a mathematical framework to model the dynamical propagation and for mation of opinions and their uncertainties through t he hierarchical structures. For the to p-down HFON, we prove that the opinions of all followers throughout the hierarchy converge to the top leader  s o pinion, but the uncertainties of the follo wers in different groups are different, which means that alt hough all the follo wers have to f ollow the top leader  s opinion, their psychological acceptance (the uncertainty) for the top leader  s opinion is differen t. W e s how that the iterations needed to reduce the tracking erro r between t he followers and lead er  s opinions b y a certain percenta ge is pr op ortional to the numb er of followers in the group, this means that organizing the followers hierarchicall y can greatl y improve the efficiency; for example, i f we orga nize 155 followers in a 4 -level top-down HFON with five follo wers in each leader -follo wer group, then the speed o f convergence to the top leader  s opinion is approximately     ti mes fa ster than o rganizing t he 155 followers in a single flat group . For the bottom-up HFON, we show that the ps ychological p ressure (the u ncertainty) of the agents in the higher le vels is greater than those in t he lower levels because t he higher le vel agents h ave to m ake mor e compromises (tougher decisions), also we sho w that the uncertainties are lo wer if the higher level age nts are allo wed to keep different opi nions. In the fut ure resear ch, we will apply t he HFO N models to some real organizations a nd real election scenario s. A PPENDIX Proof of Theore m 1: (a) Let  󰇛  󰇜  󰇛    󰇛  󰇜      󰇛  󰇜    󰇛󰇜 󰇜  and  󰇛  󰇜     󰇛󰇜  󰇛  󰇜 󰇛󰇜 with   󰇛󰇜  󰇱     󰇛  󰇜        󰇛  󰇜             󰇛  󰇜   󰇛󰇜 for       ,         ,  󰇛  󰇜  󰇛  󰇜    󰇛󰇜 for         and  󰇛  󰇜 󰇛 󰇜 󰇛  󰇜    󰇛󰇜 Then, with the leader  s opinio n   󰇛  󰇜    being a constant, the opinion dynamic equation (13) can be r ewritten in the matrix form:  󰇛    󰇜   󰇛󰇜  󰇛  󰇜  󰇛󰇜 We need the follow Lemma from [ 27 ] to continue our pro of. Lemma : If the row-stochastic matrix  󰇛 󰇜 in (A4 ) satisfies the following three conditio ns: i) the diagonal of  󰇛  󰇜 is p ositive, i.e.,   󰇛  󰇜   for         , ii) there is    such that the lowest p ositive entr y of  󰇛  󰇜 is greater than  , and iii) any t wo nonempty saturated sets for  󰇛 󰇜 have a nonempty intersection , where   󰇝     󰇞 is saturated for  󰇛 󰇜 if   󰇛  󰇜   and    i mplies    , then a consensus is r eached for   󰇛  󰇜      󰇛  󰇜 in finite time. We now show that the  󰇛  󰇜 of ( A1)-(A3) satisfies the three conditions i n the Lemma. Since     󰇛󰇜 acco rding to the definition of   󰇛  󰇜 in (10) , we have   󰇛  󰇜      󰇛  󰇜     for       ; with  󰇛   󰇜 󰇛󰇜 󰇛  󰇜   , condition i) in th e Lemma is satisfied. Since   󰇛  󰇜    , it follows that a ny positive   󰇛  󰇜      󰇛  󰇜         , hence condition ii) of the Le mma is sati sfied. To ch eck condition iii), notice from (A1) that   󰇛 󰇜 󰇛  󰇜      󰇛  󰇜     for       , which implie s that an y t wo none mpty satura ted sets       󰇝     󰇞 for  󰇛 󰇜 must contain the ele ment    , hence condition ii i) of the Le mma is sati sfied . Consequently, according to the Lemma, the  follower s   󰇛  󰇜      󰇛  󰇜 converge to a consensus i n finite time, i.e., there exists   such that   󰇛  󰇜    󰇛󰇜 for all       and all     . To prove   󰇛  󰇜   󰇛 󰇜 for       and     , notice that for     ,   󰇛    󰇜   for the local reference scheme (11), and   󰇛    󰇜      󰇛 󰇜      for the leader ref erence scheme (1 5). Substituti ng these   󰇛    󰇜 into the dynamic equation (14) of   󰇛  󰇜 , we have for     that   󰇛    󰇜       󰇛  󰇜      󰇛󰇜 for the local reference sche me (11) , and   󰇛    󰇜       󰇛  󰇜         󰇛 󰇜        󰇛󰇜 for the leader reference scheme (1 5). Since the right hand sides of bo th ( A5) and ( A6) are independent of  , we have in bo th cases t hat   󰇛  󰇜  󰇛 󰇜 . This co mpletes the proof of (a) o f Theorem 1. (b) Sin ce   󰇛  󰇜    󰇛󰇜 for all       when     , w e have from (13) that   󰇛    󰇜        󰇛  󰇜          󰇛󰇜 or   󰇛    󰇜         󰇛   󰇛  󰇜    󰇜 󰇛󰇜 for     , and (16) follo ws fro m (A8). (c) The conclusion follo ws from (A5). (d) Substituting   󰇛  󰇜  󰇛 󰇜 into (A6), we have  󰇛  󰇜  󰇛 󰇜      󰇛  󰇜       󰇛󰇜 for     , and (17) follo ws fro m (A9 ) and (16).  Proof of Theore m 2 : (a) Consider an arbitrary leader-follower group in the HFON and let    󰇛󰇜 be the group 11 leader  s opinion and     󰇛 󰇜 (         ) be the followers  opinions. Since all the    followers are conn ected to each other through t he group lead er (the group lead er is a common element in an y saturated s et), we have from th e Lemma that the followers con verge to a consensus i n finite steps, i.e., there exists   such that     󰇛  󰇜      󰇛 󰇜 when     , so from (13) we have      󰇛    󰇜               󰇛  󰇜            󰇛󰇜 󰇛  󰇜 or      󰇛    󰇜                 󰇛  󰇜                󰇛  󰇜    󰇛  󰇜 for     . If     󰇛  󰇜 converges to   , then since          we have from (A11) that     󰇛  󰇜 converges to   , i.e., if the group leader  s opinion converges to the top lead er  s op inion   , then consensus of the follo wers i n the group also converges to   . Since the top leader and the agents i n Level L -1 form a basic leader-follower group, we have fro m T heorem 1 t hat the opinions of the agents in Level L-1 converge to the top leader  s opinion   . Hence, by induction, w e have that t he opinions of all the agents converge to the top lead er  s opinion    . 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