On a Co-evolving Opinion-Leadership Model in Social Networks

On a Co-ev olving Opinion-Leadership Mo del in So cial Net w orks Martina Alutto ∗ Lorenzo Zino ∗∗ Karl H. Joha nsson ∗ Angela F on tan ∗ ∗ Dep artment of De cision and Contr ol Systems, Scho ol o f Ele ctric al Engine ering and Computer Scienc e, KTH R oyal Institute of T e chnolo gy, Sto ckholm, Swe den (e-mails: { alutto; angfo n; kal lej } @kth.se) ∗∗ Dep artment of Ele ctr onics and T ele c ommunic ations, Polite cnic o di T orino, T urin, Italy (email: lor enzo.zino@p olito.i t) Abstract: Lea de r ship in so cial gro u ps is often a dynamic characteristic that emer g es from int eractions and opinion exch ange. E mpir ical evidence suggests that individuals with stro ng opinions tend to g ain influence, at the same time main taining a lignmen t with the so cial context is crucial for sustained leader s hip. Motiv ated by the so cial ps y c holog y lit erature that suppo rts these empirica l observ ations, we prop ose a novel dynamical system in whic h opinions and leade r ship c o -ev o lv e within a so cial net work. Our model extends the F riedkin–Johns e n framework by mak ing susce p tibilit y to peer influence time-dep enden t, tur ning it into the leadership v aria b le. Leadership strengthens when an a g en t holds strong yet so cially a ligned opinions, and dec line s when such a lignmen t is lo st, captur ing the trade-off betw e en conviction and so cial acceptance. After illustrating t he emerg en t b eha v ior of this co mplex system, we formally a nalyze the co upled dynamics, establishing sufficient conditions for convergence to a non-trivial equilibrium, and examining tw o time-sca le sepa ration regimes re fle c ting s c e n arios where opinion a n d leader ship ev olve at different speeds. Keywor ds: So cial net works and o pinion dynamics 1. INTR ODUCTION Leadership in so cial groups frequen tly emerges endoge- nously from in terpe r sonal interactions ra ther than being externally impos e d (Nak ay a ma et a l., 2019). Across so cial psychology and cognitive science, leadership is a so cial role that emerges when p eople’s views s tr ongly align with those of their p eers and when they are s een as impo rtan t and in- fluen tial in their relational en vironment (Surowiec ki, 2004 ; Isenberg, 198 6). People with strong or extreme o pinions frequently become more pro minen t in group discussions bec ause their arg u ment s are more confidently and coge ntly presented, whic h helps them influence gro up decis io ns and judgments (V an Swol, 20 0 9). Exp erimen tal evidence suppo rts this phenomenon, showing that individua ls with strong opinio ns a re t ypica lly less susceptible to so cial influ- ence, which manifests as per sisten t co m mitmen t to their views and increased per ceiv ed leader ship (Br a ndt et a l., 2015). How ev er, extr emit y alone do es no t guar an tee long- term influence. Recent res earc h sugg ests that leadership is a dyna m ic attribute sha ped by ong oing r elational a dapta- tion, as even strong ly p olarized individuals may gr adually conv er ge towards mo derate p ositions when exposed to per sisten t so cial pressure (Klein and Stavrov a, 2023). More generally , people tend to reduce the dissonance b et ween their o wn beliefs and thos e of their peers due to the ⋆ This wo rk was partiall y supp orted by the W allen berg AI, Au- tonomous Syste ms and S oft wa re Program ( W ASP) funded b y the Knu t and Alice W a llenberg F oundat ion. enduring h uman need for so cial v alidation, ro o ted in rela- tional co nsistency and epistemic certaint y (Hillman et al., 2023). In summary , a fundamen tal mechanism for co m- prehending the emerg ence and evolution o f leadership in complex s o cial systems is the co nflict b etw een convictions and conformity in so cial communities, within a n opinion formation and sha ring pro cess . Opinion dyna mics mo dels provide a mathematical frame- work for studying b elief up dating driven by so cial inter- actions. F oundational contributions, such as the F r ench– DeGro ot mo del (DeGro ot, 19 74) and the F riedkin–Johnsen mo del (F riedkin and Johnse n, 19 9 0), descr ib e b elief evolu- tion throug h weight ed av er aging pro cesses c a pturing con- sensus formation and stable disagr eement. More recen tly , the incorp ora tion of reflected apprais al mec ha nisms has enabled the endogeno us evolution of so cial p ower. Ac- cording to this theory , individuals adjust their influence based on how muc h their expr essed opinio ns affect those of their neighbors (F riedkin, 2011). This has fueled a growing research s tream on the co- evolution of opinions and influence in net worked s o cieties (Jia et al., 2015; Mirtabatabaei et al., 2014; Ohlin e t al., 202 2; Tian et al., 2021; W ang et al., 2 021, 202 2, 20 24; K ang and Li, 20 22; Liu et al., 2024). These c o ntributions show tha t influence is not static, but shap ed dynamically through feedba ck lo ops link ing susceptibility , opinion centrality , and netw ork structure. While these works deep en the unders ta nding of influence forma tion, the emergence o f lea dership has o ften bee n mo de le d by assuming exogeno us (and often constant) differences in stubb or nness or author it y . In contrast, em- pirical literature sugges ts that lea dership is intrinsically tied to and dyna mica lly shaped by the interpla y betw een opinion strength a nd so cial v alidation (V an Swol, 2009 ; Berglund and Gen tz, 2006). Mathematical mo dels capable of endogenously ca pturing such complex dynamics in a n analytically- tr actable formalism remain limited, c a lling for the developmen t of new mo deling appr oaches to a ccount for this imp or tant mechanism. T o address this ga p, we prop os e a nov el dynamical frame- work consisting in a coupled system of nonlinear ordinar y differential equations (ODEs), in which opinions and lead- ership co-evolve within a so cia l netw ork. In our formula- tion, an individua l’s leadership is sha p ed by tw o contrast- ing mechanisms, constan tan with the so cial psychology literature discussed in the above. In particular, leadership rises when the individual holds str ong opinio ns , and it de- clines when attitudes b eco me excessively misaligned with those of their reference g roup (e.g ., friends, colleag ues, or online communities). Our appro a ch can b e interpreted as a mo dification of the T aylor’s mo del (T aylor, 1968), the con- tin uous time counterpart of the F riedkin–Jo hns en mo del (F r iedkin and Johnsen, 19 90), in which the susceptibility to peer influence is no longer sta tic but ev o lves o ver time and b ecomes the v ariable repr esenting leadership. Indeed, unlike existing approaches that prescrib e influential indi- viduals a prio r i, we represent leader ship as a state v aria ble that s trengthens when an agent holds a strong opinio n that remains aligned with the prev alent social context, and weakens w hen suc h alig nment is lost. This mech anism captures the fundamental trade-o ff b etw een co nviction and so cial acceptance that governs infor ma l leader ship forma- tion. The main co ntribution of this pap er , b esides the for maliza- tion of suc h a mo deling fra mework, is threefold. First, w e present some ca s e studies o n sma ll netw or ks to illus tr ate the mo del and discuss its main prop er ties. Second, we provide a for mal characterization of the coupled dynamics and show s ufficient conditions ensuring conv ergence to a non-trivial equilibrium, using the center manifold metho d and the Banach fixed point theo rem. Third, we examine t wo time-sca le separatio n r egimes, whe r e opinio n up dating and leadership evolution o ccur at different ra tes, captur ing scenarios commonly obser ved in so cial systems. F or these scenarios , we firs t c har acterize the dynamics of the faster v ariable, and then w e establish sufficient conditions guar- anteeing convergence of the slower v aria ble to a unique equilibrium. Numerica l simulations illustrate the re s ults and sys tem behavior. The rest of the pape r is organized a s fo llows. Section 2 int ro duces the opinion-lea dership mo del, while Section 3 provides intuition o n its dynamics through illustrativ e examples. Sectio n 4 pr esents the analys is of equilibrium po ints and their stabilit y , while Section 5 discusses the time-scale s eparation analysis . Finally , Section 6 concludes the pap er a nd outlines dir ections for future research. 2. MODEL DESCRIPTION 2.1 Notation W e denote b y R and R + the sets o f real and nonnegative real num ber s , r esp ectively , while R n × n + indicates the set of real matrices with dimensio n n × n and no nnegative ent ries. The a ll-1 vector and the all-0 vector are denoted by 1 a nd 0 respectively . The identit y matrix and the all-0 matrix are deno ted by I and O , r esp ectively . The transp ose of a matrix A is deno ted by A T . F o r x in R n , let || x || 1 = P i | x i | and || x || ∞ = max i | x i | b e its l 1 - a nd l ∞ - norms, while diag( x ) denotes the diago nal matrix who se diagonal coincides with x . F or an irr educible matrix A in R n × n + , we let ρ ( A ) denote the sp ectral radius of A . Inequalities be tw een t wo vectors x and y in R n are meant to hold true entry-wise, i.e., x ≤ y mea ns that x i ≤ y i for every i , whereas x < y means that x i < y i for every i , and x  y means tha t x i ≤ y i for ev ery i and x j < y j for some j . Analogo us ho lds for the matr ix . Unless differe ntly sp ecified, we use the notation P j to denote P n j =1 . W e cons ider a so cia l netw o rk comp osed of n ag ents. The int eractions among agents are represented by a matrix W ∈ R + n × n , whic h is assumed to be row-sto chastic (i.e., P j W ij = 1, for all i ), so that W ij quantifies the weigh t that age nt i as signs to the opinion of a gent j . Ea ch agent i ∈ { 1 , . . . , n } is characterized by tw o dyna mica l v ariables: • An opinion v ar iable x i ( t ) ∈ R , representing the b elief of ag ent i o n a certain topic; • A leadership v ariable y i ( t ) ∈ [0 , 1 ], represe nting the degree of lea de r ship of a gent i , with lar ger v alues o f y i ( t ) denoting that agent i has a la rger tendency to lead the gr oup at time t . The join t evolution of opinions a nd leadership is governed by the follo wing system of O DEs:    ˙ x i = −  x i − X j W ij x j  (1 − y i ) − y i  x i − x i (0)  ˙ y i = α (1 − y i ) x 2 i − β y i  x i − X j W ij x j  2 , (1) for all i ∈ { 1 , . . . , n } , where α, β > 0 are sca lar parameters . Before analyzing the dynamics, we present a brief discus- sion to elucida te its mechanisms. F rom (1), we obser ve that the o pinion dynamics is driven by tw o main mechanisms. The first term, weight ed by (1 − y i ), captur es the tendency of follo wers to conform to the average opinion of their neighbors, which is a well-kno wn phenomenon in socia l psychology (Asc h, 1952), and it is the key mechanism in the classica l F rench–DeGro ot o pinion dynamic mo del (De- Gro ot, 1974). In contrast, the second term reflects the stubbo rnness of leaders, who res is t c hanges and tend to remain consis tent with their initial opinion x i (0). In other words, for a fixed leadership v a lue y i , the first equation in (1 ) ca n b e int erpreted as a co nt in uous-time version of the classical F r iedkin–Johnse n mo del (F r iedkin and Johnsen, 19 90; T aylor, 1968 ). Instead, in this mo del, the agents’ leadership co-evolve with their opinion, shap ed by t w o opp osing mechanisms: the first term accounts for the tendency of agen ts with strong opinions to emer ge as leaders, as suppo rted b y the social psychology liter- ature (V an Swol, 20 09; Brandt et al., 20 15). H ere, the parameter α ca n be interpreted as a le adership gr owth r ate , co ntrolling how quickly str ong o pinions translate into increased leadership. Conv ersely , the se cond term repre- sents the los s of leadership due to so cial misalignmen t: agents whose opinions deviate significantly from tho se o f their neig hbo rs los e influence, as suggested b y the s o cial verification theory (Hillman et al., 20 23). The par ameter β therefore measur es the s ensitivity of le adership to disagr e e- ment , indicating how strongly misalig nment p ena lizes an agent’s leadership. W e refer to sy s tem (1) as the opinion- le adership mo del . If we conv enie ntly g ather the o pinion a nd leadership o f all agents in tw o vectors x = [ x 1 · · · x n ] ⊤ and y = [ y 1 · · · y n ] ⊤ , resp ectively , the system in (1) can b e r e w r itten compactly as:  ˙ x = − ( I − Y )( I − W ) x − Y ( x − x (0)) , ˙ y = α ( I − Y )diag ( x ) x − β Y diag (( I − W ) x ) ( I − W ) x, (2) where Y = diag ( y ) is the diagonal matrix who se diago nal ent ries are the leadership v ariables of the agents. The next result establishes well-posednes s of the mo del by showing that the set ∆ := [ − 1 , 1] n × [0 , 1] n is p o s itively inv ariant for the opinion-leader ship co evolu- tion mo del, i.e., if ( x (0) , y (0)) is in ∆, then ( x ( t ) , y ( t )) will be in ∆ for all t ≥ 0. L emma 1. F or any initial condition ( x (0) , y (0 )) ∈ ∆ there exists a uniq ue so lution ( x ( t ) , y ( t )) of (1) defined for all t ≥ 0. Mo reov er, ∆ is positively inv aria nt for (1 ). Pro of. Since the vector field of (1) is Lipsc hitz with resp ect to ( x, y ), lo cal existence a nd uniqueness for the Cauch y pr oblem are g uaranteed b y the P icard–L indel¨ of theorem (Hale , 20 0 9). The domain ∆ is compact and co n- vex, hence Nagumo’s theorem c a n be applied (Bla nchini, 1999, T heo rem 3 .1). T o v erify inv a r iance, we chec k the direction of the vector field at the b oundar ies of ∆. W e observe tha t, for all i = 1 , . . . , n , ˙ x i ≤ 0 if x i = 1 and ˙ x i ≥ 0 if x i = − 1. Similarly , ˙ y i ≤ 0 if y i = 1 and ˙ y i ≥ 0 if y i = 0. It follo ws that an y s olution ( x ( t ) , y ( t )) with ( x (0) , y (0 )) ∈ ∆ re mains in ∆ fo r a ll t ≥ 0, whic h implies that ∆ is a p ositively in v a riant set for (1). ✷ 3. ILLUSTRA TIVE E XAMPLES In this section, we analyze different scenario s to highlight the emerge nt phenomena that the mo del can captur e and repro duce. 3.1 Thr e e-no de network W e consider the o pinion-leaders hip mo del (1) in the c a se of a net work for med by 3 in teracting agents, starting with po larized opp os ite o pinions for agents 1 and 3 ( x 1 (0) = − 1 and x 3 (0) = 1), and neutr al opinion for agent 2 ( x 2 (0) = 0), which we also denote as the cent ral no de in what follows. W e ass ume a symmetric in teraction matrix of the form W ij = ( a if i = j, 1 − a 2 if i 6 = j, 0 20 40 60 80 100 0 0.2 0.4 0.6 0.8 1 node 1 node 2 node 3 (a) 0 20 40 60 80 100 0 0.2 0.4 0.6 0.8 1 node 1 node 2 node 3 (b) 0 500 1000 1500 2000 2500 0 0.2 0.4 0.6 0.8 1 node 1 node 2 node 3 (c) 0 50 100 150 200 0 0.2 0.4 0.6 0.8 1 node 1 node 2 node 3 (d) Fig. 1. Numerical simulation of the three-ag ent o pinion- leadership mo del with α = 0 . 6 a nd β = 1. (a) Symmetric case: agent 2 remains neutral while a gents 1 and 3 conv erge to mo derated polarized o pinions. (b)–(d) Perturbe d scenar ios showing how asy mmetry in the o pinion initializatio n or interaction s trengths breaks p ola rization symmetry and mo difies leader ship emergence. where the constant a ∈ (0 , 1 ) captures the r elative w e ight of the self-lo o p with res p e c t to the other en tries. Due to the symmetry of W , it can b e shown (pro of and nu merical simulations are here omitted for la ck o f space) that the influences of the t w o initially polarize d agent s acting on no de 2 ar e p erfectly balanced. As a res ult, the central agent r emains fixed at its initial neutral opinio n, i.e., x ∗ 2 = 0. Interestingly , we further o bs erve that the symmetry implies that the leadership o f the t wo p olar ized agents, even if initially different, even tually conv erges to the same v a lue (see Fig. 1a for a numerical s imulation). This v alue is lar ger than the one of the ne utr al a gent, due to the first term in the leadership dynamics that captures the real-world emerge nce of leaders among those with strong er opinions. W e now inv estigate p er turbations of this symmetric set- ting. First, we cons ider a weak e r mutual influence b etw een no des 1 and 3, i.e., we re duce the entries W 13 = W 31 = c < 1 − a 2 . In this cas e, illus trated in Fig. 1b, the leadership lev- els of agents 1 and 3 b eco me slightly smaller. With reduced m utual reinforcement, agents pola rized agents exp erience less opinion confirmation and thus exhibit r educed lea der- ship. Second, we assume that the ce nt ral no de has a small, but non-zero, initial co nditio n, i.e., x 2 (0) = 0 . 0 5. In this scenario, depicted in Fig. 1c, the lea dership of no des 1 and 3 still converge at a similar rate toward equilibr ium but no lo nger s ymmetrically . Hence, the agent with positive initial opinion (no de 3 ) even tually emerg es as a str onger leader than agent 1. Interestingly , we obser ve that a gent 2 exhibits a m uc h slower conv ergence b ecause their initial state is close to 0 and due to the symmetric of matrix W , the dis a greement term driving the leadership dynamics is small in magnitude. This leads to a g radual but steady leadership upda te, ev en tually b ecoming dominant in the so cial netw ork. Third, we consider a case where no de 2 has a nega tive initial o pinion as x 2 (0) = − 0 . 05 and a s tr onger influence from no de 1, i.e., le t W 12 > W 32 . In this case, as shown in Fig. 1d, the a bsence of symmetries leads to a more complex emergent b ehavior, whereby a gent 1 r e ceives a stro nger reinforcing influence from the central no de, resulting in a larger leader ship compare d to agent 3. 3.2 Imp act of mo del p ar ameters W e co nsider now a po pulation o f 8 in teracting age nt s, and we explo re the impact of the mo de l parameters . In particular, we fix α = 0 . 6 and we cons ider tw o scenar ios, with β = 0 . 1 a nd β = 1, re s p e ctively . A s imulation o f the o pinion-leaders hip mo del in (1) in the first scenario is illustrated in Figs. 2a–2b. In t his case, we observ e that the order of the opinions is pres erved throughout the evolution. In particular, agent 1 maintains the s trongest negative opinion, while agent 6 remains the one with the strongest p ositive opinio n. Agent 1 a chiev es a strong leadership that forces the p os itive opinion of a gents 4,6 a nd 8 to reduce, that is also the re a son why those are the o nes with less influence le vels at the equilibr ium. Because α = 0 . 6 is rela tively la rge compar ed to β = 0 . 1, the lea de r ship y i evolv es faster than the opinions, allowing agents to adjust their infl uence levels quickly while the opinions rema in la rgely stable. The second scenario is illustr ated in Figs. 2c–2d. Interest- ingly , we observe that here the order of opinio ns ch anges ov er time. F or instance, agent 1 starts with the most neg a- tive opinion but ends up less extreme, while agents initially with mo der ately p ositive opinions (e.g ., agents 2, 6, a nd 8) bec ome slightly more extreme and they emerg e as leaders in the so cia l netw ork. In this ca s e, due to the r elatively large v alue of β = 1 compared to α = 0 . 6, the second term in the leadership dynamics (which p enalizes disa greement) strongly affects y i , causing ra pid adjustments. As a r esult, agents with mo der ate o pinions a chiev e highe r levels of leadership. 4. EQUILIBRIA AND ST ABILITY ANAL YSIS In this section, we in vestigate the e xistence of eq uilibr ia for the opinion-lea dership mo del in (1) and their s ta bilit y . W e first es tablish that an y equilibrium point x ∗ , if it exists, is necessarily b ounded b y the initial condition of the o pinio ns. Mor e precise ly , given x = min j x j (0) a nd ¯ x = max j x j (0), x ∗ lies in the h yp er -rectang le X := [ x , ¯ x ] n . Pr op osition 2. Let x ∗ be an equilibrium of (1). T he n, x ∗ ∈ X . Pro of. First, observe that ( I − ( I − Y ) W ) 1 = 1 − W 1 − Y W 1 = Y 1 , so that 1 =  I − ( I − Y ( x )) W  − 1 Y 1 . Using this identit y together with (1 ), we es timate upp er and low er b ounds for x ∗ as x ∗ ≤ ¯ x [ I − ( I − Y ( x )) W ] − 1 Y 1 = ¯ x 1 , x ∗ ≥ x [ I − ( I − Y ( x )) W ] − 1 Y 1 = x 1 . Hence, each compo nent of x ∗ satisfies x ≤ x ∗ i ≤ ¯ x , for all i = 1 , . . . , n , whic h yields the claim. ✷ After having established this general r esult, we provide now a more precis e characteriz a tion of the equilibria o f the opinion-leader ship mo del, demonstra ting that the orig in is alwa y s an unstable equilibrium, and providing a sufficient condition for the con v ergence of the dyna mics. The or em 3. Consider the opinion- le a dership model in (1) with initial condition ( x (0) , y (0)). Then, the fo llowing hold: (i) ( 0 , 0 ) is an unsta ble equilibrium point; (ii) If x i (0) 6 = 0 for all i and, given ε > 0 and M := max {| x | , | ¯ x |} , it holds 3 β ( ¯ x − x ) 2 αε 2  2 M ε + 1  < 1 , (3) then the dyna mics conv erges to the unique equilib- rium point ( x ∗ , y ∗ ) tha t depends on the initial condi- tion and such that y ∗ > 0 and min i | x ∗ i | ≥ ε . Pro of. (i) Note that ( x ∗ , y ∗ ) ∈ ∆ is an equilibrium if a nd only if  0 = − x + ( I − Y ) W x + Y x (0) , 0 = ρ ( I − Y )dia g( x ) x − Y diag(( I − W ) x ) ( I − W ) x, where we define ρ := α/β . This implies that the eq uilib- rium p oints c onsist of the trivial p oint ( 0 , 0 ) and a ll p oints ( x ∗ , y ∗ ) with x ∗ i 6 = 0 and y ∗ i = ρx ∗ i 2 ρx ∗ i 2 + ( x ∗ i − P j W ij x ∗ j ) 2 , ∀ i = 1 , . . . , n. (4) W e can now study the lo ca l stability of ( 0 , 0 ) using the linearization theo rem. Linearizing (2) around ( 0 , 0 ), w e find the J acobian matrix J ( 0 , 0 ) =  W − I diag ( x (0)) O O  . Since W is row-stochastic, the sp ectr um of J ( 0 , 0 ) contains n − 1 eigenv alues with negative rea l part and n + 1 ze ro eigenv alues. Therefore, the linear iz ation theorem alone do es no t a llow us to conclude on stability and w e rely on the cen ter manifold method. The stable subspace is generated by the eig env ector s asso ciated to the negativ e eigenv alues of ( W − I ). The central subspace is instea d generated by the eigenv ector s as s o ciated to the null eigen- v alues, which means V C = { ( x, y ) : ( W − I ) x + diag ( x (0)) y = 0 } , and the cen ter ma nifold is lo c a lly approximated with the function y = h ( x ) where h ( 0 ) = 0 , ˙ h ( 0 ) = 0 and h ( x ) = o ( | x | ). Ther efore, the reduced equation on the center ma nifo ld is ˙ y = α (1 − h ( x )) x 2 + o ( | x | 2 ) , for which the equilibrium ( 0 , 0 ) is unstable. Therefor e, the origin is unsta ble also for the original system (2). (ii) W e pro ceed with the characterization of the equilib- rium p oints given in ( 4). Substituting (4) int o the firs t equation of (1) at the equilibrium yields 0 20 40 60 80 100 -1 -0.5 0 0.5 1 Agent 1 Agent 2 Agent 3 Agent 4 Agent 5 Agent 6 Agent 7 Agent 8 (a) Opinions, β = 0 . 1 0 20 40 60 80 100 0 0.5 1 Agent 1 Agent 2 Agent 3 Agent 4 Agent 5 Agent 6 Agent 7 Agent 8 (b) Leadership, β = 0 . 1 0 50 100 150 -1 -0.5 0 0.5 1 Agent 1 Agent 2 Agent 3 Agent 4 Agent 5 Agent 6 Agent 7 Agent 8 (c) Opinions, β = 1 0 50 100 150 0 0.5 1 Agent 1 Agent 2 Agent 3 Agent 4 Agent 5 Agent 6 Agent 7 Agent 8 (d) Leadership, β = 1 Fig. 2. Simulation of the opinion-leadership mo del (1 ) with 8 agent s with α = 0 . 6 and (a,b) β = 0 . 1 or (c,d) β = 1 .  x ∗ i − X j W ij x ∗ j  3 + ρx ∗ i 2 ( x ∗ i − x i (0)) = 0 , ∀ i ∈ 1 , . . . , n. T o characterize the eq uilibria, we s tudy the cor resp onding fixed-p oint equation x = − 1 ρ ( X − 1 ) 2 diag(( I − W ) x ) 2 ( I − W ) x + x (0) , (5) where X = diag( x 1 , . . . , x n ). F or a given ε > 0, de fine X ε := { x ∈ X : | x i | ≥ ε > 0 , ∀ i } , a nd let φ : X ε → X ε denote the right-hand side of (5). The n, fixed p o ints of φ corres p o nd exactly to the equilibria of the sy s tem. T o analyze existence and uniqueness, we co mpute the Jacobian of φ . Its entries are ∂ φ i ( x ) ∂ x i = − 3 x i ( x i − P j W ij x j ) 2 (1 − W ii ) − 2 ( x i − P j W ij x j ) 3 ρx 3 i = − ( x i − P j W ij x j ) 2 ( x i (1 − 3 W ii ) + 2 P j W ij x j ) ρx 3 i , ∂ φ i ( x ) ∂ x k = − 3( x i − P j W ij x j ) 2 ( − W ik ) ρx 2 i , ∀ k 6 = i. Hence, the infinit y nor m of the J acobian satisfies the following inequalit y:     ∂ φ ( x ) ∂ x     ∞ = max i        ∂ φ i ( x ) ∂ x i     + X k 6 = i     ∂ φ i ( x ) ∂ x k        ≤ 1 ρ  6 M ( ¯ x − x ) 2 ε 3 + 3( ¯ x − x ) 2 ε 2  = 3( ¯ x − x ) 2 ρε 2  2 M ε + 1  , where the inequality follows from Pr o p osition 2 and the definitions of M a nd X ε . Under condition (3), φ is therefo r e a con tr action m ap. By the Banac h fixed-p oint theor em (Kirk and Khamsi, 2001), it admits a unique fix ed p oint x ∗ on the complete metric space X ε . Moreover, for a ny initial c ondition x (0) satisfying (3 ), the sequence of iterates x n +1 = φ ( x n ) con v erges to x ∗ ( x (0)). Since x ∗ ( x (0)) is a fixed point of φ , it corres p o nds to a n equilibrium of the dynamics (2). Therefore , under co ndition (3), the system conv erges to the eq uilibr ium ( x ∗ ( x (0)) , y ∗ ( x (0))), with y ∗ ( x (0)) given by (4). ✷ R emark 4. It is worth highlighting that condition (3) is o nly sufficient and do e s no t characterize the e xact stability boundar y . The admissible r egion should thus be int erpreted as a conserv ative inner approximation of the true stability region. Fig. 3. Reg ion o f admissible pa rameter v alues ( ε, ρ ) for which condition (3) holds true. The r esults o f Theore m 3 a re illus trated in Fig. 3. In particular, we illustra te the set of admissible pair s ( ε, ρ ) satisfying (3) for the case in which ¯ x = 0 . 1 and x = − 0 . 1, so that M = 0 . 1. The region ensuring existence and con vergence (depicted in cy a n) incr eases for larger v alues of ρ = α/β , mea ning that a weaker sensitivity to disagreement enables equilibrium conv ergence ev en when opinions rema in r elatively clo se to the o rigin (small ε ). While Theorem 3 provides a sufficient co ndition for co n- vergence to an equilibrium, it do es not allow, in general, to establish a closed-form expression for such an equilibrium. Now we der ive a corolla r y s howing that, in the simple scenario in which agents shar e the same initial opinion, a complete characterization of the b ehavior of the opinion- leadership mo del is pos sible. Cor ol lary 5. Consider the opinion-lea dership mo del in (1). If x (0) = x 0 1 with x 0 6 = 0, then the sy s tem conv erges to the unique equilibrium ( x ∗ , 1 ) = ( x 0 1 , 1 ). Pro of. Obser ve that for each agent i , X j W ij x j (0) = x 0 X j W ij = x 0 , since W is row-sto chastic. Hence, x i (0) − P j W ij x j (0) = 0, and the dy na mics for x i bec omes ˙ x i (0) = 0. But then, this can b e applied fo r a ll t ≥ 0 since x i ( t ) = x i (0) = x 0 and also x i ( t ) − P j W ij x j ( t ) = 0 for all t ≥ 0. Similarly , the dynamics for y i reduces to ˙ y i = ρ (1 − y i ) x 2 0 , whic h v anishes only for y i = 1 (since x 0 6 = 0). Therefore, ( x ∗ , y ∗ ) = ( x 0 1 , 1 ) is the unique equilibrium. No te that the condition in (3) is satisfied in this case, since x coincides with ¯ x . The n the system will c onv er ge to the eq uilibr ium ( x 0 1 , 1 ). ✷ In pla in w ords, Corollary 5, all agen ts maintain the same opinion over time, and every agent reaches the ma ximum level o f leader ship, reflecting the complete alignment of opinions within the net work. 5. TIME-SCALE SEP ARA TION In many so cial and or ganizationa l contexts, the dynamics of opinions and leadership may evolv e a t a different pace. F or instance, in offline so cia l c o mmunit ies, individuals may quickly adapt their expressed o pinions in resp onse to p eer pressure, while the proces s of g aining or losing le a dership status typically dev elops more slowly , dep ending o n repu- tation, credibility , o r long -term vis ibility (Nak ayama et al., 2019). Conversely , in s ettings such as o nline pla tfor ms, the rise o f lea dership o r influence (e.g., b eco ming a trend-setter or influencer) ma y o ccur v ery r apidly , while opinions on sp ecific iss ue s remain relatively stable (Onnela a nd Reed- Tso chas, 2 010). T o ca pture these t w o scenario s , we in tro duce a time-sc ale sep ar ation betw een the opinion and leadership dynamics. W e rewrite the o pinion-leader s hip dynamics as    ˙ x i = −  x i − X j W ij x j  (1 − y i ) − y i  x i − x i (0)  τ ˙ y i = α (1 − y i ) x 2 i − β y i  x i − X j W ij x j  2 , (6) so tha t the parameter τ > 0 acts as a timescale se pa ra- tion v a riable (Berglund a nd Gen tz, 20 0 6), co ntrolling the relative sp eed of the leadership dynamics with resp ect to opinion adaptatio n. Dep ending on the v alue of τ , the sys- tem exhibits tw o distinct limiting reg imes . When τ → 0, leadership adjusts r apidly relative to opinions, leading to the fast-le adership r e gime , whic h is representativ e, e.g., of the online socia l netw or k scenario describ ed in the ab ove. When τ → ∞ , instead, lea dership evolves m uch more slowly than opinions, giving rise to the slow-le adershi p r e gime , representative o f the offline context discussed in the previo us parag raph. This distinction allows the mo del to capture a wide range of dynamical behaviors ac ross differe nt so cial or or ganiza- tional settings. Moreover, as we shall e x tensively discuss in the rest of this s e ction, suc h a time-scale separ ation a llows us to per fo rm a deeper analytical study of the system, gaining analytical ins ig hts in to the emergent lea dership behavior of the popula tion. 5.1 F ast-le adership r e gime In the fast-lea dership reg ime (i.e., in the limit τ → 0), the leadership v ariable y evolv es on a m uch faster time scale than the opinions x , whic h can be initially assumed to b e constant, y ielding the following result. Pr op osition 6. Consider the opinion-leadership in (6) un- der the fast-leaders hip r egime ( τ → 0 ). Giv en x constant, the leader s hip of each node y i will converge to y ∗ i ( x ) =      0 if x i = 0 , ρx 2 i ρx 2 i + ( x i − P j W ij x j ) 2 if x i 6 = 0 . (7) for all i ∈ 1 , . . . , n . Pro of. Over the time scale on which y evolves, we can assume x to b e quasi-constant. In this approximation, 0 1000 2000 3000 -1 -0.5 0 0.5 1 (a) Opini ons 0 1000 2000 3000 0 0.2 0.4 0.6 0.8 1 (b) Leadership Fig. 4. Simulation of the mo del in (6 ) under the fast- leadership re g ime ( τ → 0). the second e q uation of (6) is linear in y i and solving for the equilibrium ˙ y i = 0, w e obtain (7). With similar consideratio ns as in the pro of of Theo rem 3, we get that the equilibrium ( 0 , 0 ) is unstable. F o cusing now on the equilibrium po ints w her e y ∗ i ( x ) > 0 , stability can be assessed by examining th e eigenv a lues of the Jaco bian matrix, that is J y = − diag  ρx 2 i +  x i − X j W ij x j  2  . Since J y is diagona l with str ictly negative entries, all its eigenv alues are neg ative. By standa rd results from linear system theory , this implies that ea ch y i conv er ges to its equilibrium y ∗ i for fixed x . ✷ Once the lea de r ship v ariable y has co nverged to its equi- librium manifold co mputed in P rop ositio n 6, according to the time-scale separatio n ar gument (Berglund and Gentz, 2006), the opinion ev olves acco rding to the reduced dy- namics ˙ x i = − ( x i − P j W ij x j ) 3 + ρx 2 i ( x i − x i (0)) ρx 2 i + ( x i − P j W ij x j ) 2 . (8) Due to lack of space, w e do not rep ort here the full pro of on the characterizatio n of the e q uilibria, but we pr ovide a brief outline. T o characterize the equilibria, we can study the corres po nding fixed-p oint eq uation, as defined in (5). Then we can prov e existence a nd stabilit y of an equilibrium ( x ∗ , y ∗ ) as do ne in Prop ositio n 3(ii). Figure 4 illustrates the b ehavior of the opinion-leader s hip mo del (6) under a fast-leaders hip time- s cale r egime. In this case, for all i , the le a dership v ar iables y i quickly co nv er ge to the manifold defined b y (7), after which the opinio n dynamics x i ( t ) evolve more slowly , eventu ally r eaching an equilibrium. 5.2 Slow-le adership r e gime In the slow-leadership r egime (i.e., in the limit τ → ∞ ), the evolution of the opinion v ariable x o ccurs o n a muc h faster time scale than that of leadership y , which can b e initially assumed as constant, obtaining the following r esult. Firs t, define the L aplacian matrix L := I − W . Pr op osition 7. Consider the opinion-leadership in (6 ) un- der the s low-leadership regime ( τ → ∞ ). Given y consta nt , the opinion will con verge to x ∗ ( y ) =  L + Y W  − 1 Y x (0) , (9) where Y = diag ( y 1 , . . . , y n ). 0 2 4 6 8 10 -1 -0.5 0 0.5 1 (a) Opinions 0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1 (b) Leadership Fig. 5. Simulation of the model in (6) under the slow- leadership re g ime ( τ → ∞ ). Pro of. Over the time sca le on which x evolv es, we can treat y as quasi-co nstant. This allows us to a nalyze the opinion dy na mics separately , considering y fixed. In this limit, the equilibrium of the opinion dynamics ˙ x = 0 is given by (9). The conv er g ence to x ∗ ( y ) follows from the fact that, for a fixed y , the opinion dynamics defines a linear system a nd equilibrium stability ca n b e ass essed by examining the e igenv alues o f the Jaco bian matrix, that is J x = −  ( I − Y ) L + Y  . Since W is r ow-sto chastic, all eigenv alues of J x are nega- tive. By standar d results from linear system theory , this implies that x converges exp onentially to the unique equi- librium x ∗ ( y ) for fixed y . ✷ R emark 8. The expression from Prop osition 7 ca n b e int erpreted as a modificatio n of the equilibr ium of a F riedkin–Jo hns e n opinio n dynamics mo del (F r iedkin a nd Johnsen, 1990), where eac h agent’s final opinion is a con- vex combination o f their initial opinio n and the weight ed av er age of their neighbors’ opinions, with weigh ts mo du- lated by their curr ent leadership y i . According to the time-scale sepa ration principle (Berglund and Gentz, 2006), once the fast v aria ble x has rea ched its equilibrium manifold x ∗ ( y ), the s low lea dership v ar iable y evolv es a ccording to the reduced dynamics ˙ y = α ( I − Y )diag ( x ∗ ( y )) x ∗ ( y ) − β Y diag ( Lx ∗ ( y )) Lx ∗ ( y ) . (10) This is illustrated in Fig. 5, which shows how opinions x i ( t ) rapidly settle in to a quasi-s teady co nfiguration, while the leadership v aria bles y i adjust on a muc h slower timescale. In this scenario , the opinion dynamics effectively react to a nearly c o nstant lea dership. Finally , in the cont ext o f a slo w-leadership time-scale, the following result pro vides a sufficien t condition fo r the existence and uniqueness of an equilibrium p o int for the system (10 ). Pr op osition 9. Consider the reduced dynamics in (10) in the slow-leadership regime ( τ → ∞ ), and let x i (0) 6 = 0 for all i . If 4 M 3 x 2 ρ ¯ x 2 + ( ¯ x − x ) 2 ρx 2 + ( ¯ x − x ) 2  M (2 + ρ ) ρx 2 + ( ¯ x − x ) 2 + 1  < 1 , (11) then there exists a unique equilibrium y ∗ and the dynamics will converge to y ∗ for any initial condition y (0). Pro of. Define the implicit map ϕ : [0 , 1] → [0 , 1] such that at the equilibrium w e can study the following fixed-po int equation y = ϕ ( y ) :=  ρX 2 +  diag( Lx ∗ ( y ))  2  − 1 ρX 2 , (12) where X = diag ( x ∗ ( y )). Note that from Prop ositio n 2 and (9), it fo llows that, giv en the set X ϕ := " ρx 2 ρ ¯ x 2 + ( ¯ x − x ) 2 , 1 # n , φ ( X ϕ ) ⊆ X ϕ . As done in Theor em 3, in or der to analy ze existence and uniquenes s of fix ed p oints of ϕ , whic h corres p o nd exactly to equilibr ia o f the system (10), we compute the Ja c obian matrix of ϕ . Defining the following diagonal matrix A ( y ) := ρX 2 +  diag( L x ∗ ( y ))  2 , we get ∂ ϕ ∂ y ( y ) = − ρA − 1 ( y ) ∂ A ∂ y ( y ) A − 1 ( y ) X 2 +2 ρA − 1 ( y ) X ∂ x ∗ ∂ y ( y ) , where ∂ A ∂ y ( y ) = 2  ρX +diag( Lx ∗ ( y )) L  ∂ x ∗ ∂ y ( y ) , and ∂ x ∗ ∂ y ( y ) =  L + Y W  − 1  − W  L + Y W  − 1 Y X (0)+ X (0)  =  L + Y W  − 1 diag  x (0) − W x ∗ ( y )  . (13) Recalling that W is r ow-stochastic, it follows that the infinit y norm o f  L + Y W  − 1 satisfies   ( L + Y W ) − 1   ∞ ≤ 1 1 − k ( I − Y ) W k ∞ = 1 1 − (1 − min i y i ) = ρ ¯ x 2 + ( ¯ x − x ) 2 ρx 2 . (14) W e can now b ound the infinity norm of the Jacobian matrix of ϕ as follo ws, wher e for brevity , w e omit some int ermediate steps,     ∂ ϕ ( y ) ∂ y     ∞ ≤ 4 M 3 x 2 ρ ¯ x 2 + ( ¯ x − x ) 2 ρx 2 + ( ¯ x − x ) 2  M (2 + ρ ) ρx 2 + ( ¯ x − x ) 2 + 1  . Since (11) holds true, ϕ is a contraction map. By the Banach fixed-p oint theorem (Kirk and Khamsi, 2 001), the contraction mapping ϕ admits a unique fixed point y ∗ in X φ , and the sequence of iterates y n +1 = ϕ ( y n ) conv er ges to y ∗ for any initial condition y (0). Since y ∗ is a fixed p oint of ϕ , it co rresp o nds to an equilibrium o f the reduced dynamics (10). Therefore, under condition (11), the reduced system converges to the equilibrium p oint. ✷ The as s umption in Pr op osition 9 is illustrated in Fig. 6, where we show the set o f admissible pairs ( ¯ x, ρ ) satisfy- ing (11) for the case in whic h x = − 1. The plot illustrates how the (conserv a tive) r egion ensuring existence and con- vergence of the dynamics in the slow-leadership regime increases as the ra tio ρ = α/ β incr eases, sugg esting that conv er gence is eas ie r to a chieve when strong- opinionated individuals are mo re likely to emerge as leaders. 6. CONCLUSION This w or k introduces a nov el mo deling framework that captures the endogenous emergence of leaders hip in so- cial netw orks thro ugh the coupled evolution of opinions. By embedding a dynamic leadership mec hanism within a -1 -0.5 0 0.5 1 5 10 15 20 Fig. 6. Region of admiss ible par a meter v alues ( ¯ x, ρ ) for which condition (11) holds true. F riedkin-–Jo hnsen inspired opinio n mo del, we emphasized a fundamental trade- o ff observed in so cial systems: strong and confidently expr essed convictions can promo te lea der- ship emer gence, yet e xcessive deviation from the prev a iling so cial co ntext tends to w eaken influence due to reduced v alidation fro m peer s . W e formally analyzed the resulting nonlinea r dynamics and established sufficien t conditions ensuring c onv er gence to a nontrivial equilibr ium p oint. In addition, we exa mined t wo r elev ant time-sca le sepa ration regimes, showing how the relative sp eed of opinion and leadership ev olution affects the long-term so cial configur ation. The pr o p osed fra mework o ffers several opp ortunities for further rese arch. F rom a modeling pers p ec tive, one may consider alter native functional forms for the leader s hip dynamics, or simplifying as sumptions s uch as restricting individual opinions to a p o sitive domain, to derive s tronger analytical g uarantees. On the a pplied side , incor p o rating heterogeneo us interaction patterns e xtracted from empir- ical data would a llow v alida ting the mo del a nd testing its predictive p ower in real so cial sy stems. REFERENCES Asch, S.E. (1952). S o cial psycholo gy. Prentice-Hall, Inc. Berglund, N. and Gen tz, B. (2006). Noise-induc e d phe- nomena in slow-fast dynamic al systems: A sample-p aths appr o ach . Spr inger, Lo ndo n. Blanchini, F. (1999). Set inv ar iance in control. Automat- ic a , 3 5(11), 1747 –176 7. Brandt, M.J., Ev ans, A.M., and Crawford, J.T. (2015). The unthinking or confident extremist? P o litica l ex- tremists are more likely than mo der ates to reject exp erimenter-generated a nchors. Psychol. Sci. , 2 6(2), 189–2 02. DeGro ot, M.H. (1974). Rea ching a consensus . J. Am. St at. Asso c. , 6 9 (345), 118– 121. F riedkin, N.E. (2011). A formal theo ry of reflected a p- praisals in the e volution o f p ower. A dm. Sci. Q. , 56(4), 501–5 29. F riedkin, N.E. and Johnse n, E.C. (19 90). So c ial influence and opinio ns. J. Math. So ciol. , 15(3 -4), 193–2 06. Hale, J.K. (2009). Or dinary differ ential e qu ations . Courier Corp ora tion. Hillman, J.G., F owlie, D.I., and MacDonald, T.K. (202 3). So cial verification theor y: A new wa y to conceptualize v alidation, dissonance , and b elonging . Pers. So c. Psy- chol. R ev. , 2 7 (3), 309–33 1. Isenberg, D.J . (1986 ). Group p o larization: A critica l review and meta-ana lysis. J. Pers. So c. Psychol. , 50(6), 1141. Jia, P ., Mirtabatabaei, A., F riedkin, N.E., and Bullo, F. (2015). O pinion dynamics and the evolution of so cial power in influence networks. SIAM R ev. , 5 7(3), 36 7– 397. Kang, R. a nd Li, X. (20 2 2). Co evolution of o pinion dynam- ics on ev olving signed a ppraisal net w orks. Automatic a , 137, 1101 38. Kirk, W.A. and Khamsi, M.A. (20 0 1). An Intr o du ct ion to Metric Sp ac es and Fixe d Point The ory . J o hn Wiley & Sons, New Y or k. Klein, N. and Stavrov a, O. (2023 ). Resp ondents with more extreme views show moder ation of opinio ns in multi- year surveys in the USA and the Netherlands. Commun. Psychol. , 1(1), 3 7. Liu, F., Cui, S., Chen, G., Mei, W., and Gao , H. (202 4). Mo deling, a nalysis, and manipulation o f co-e volution betw een appraisa l dynamics and opinion dynamics . Au- tomatic a , 16 7, 111 7 97. Mirtabatabaei, A., Jia, P ., F riedkin, N.E., and Bullo, F. (2014). On the r eflected appr aisals dynamics of influence net works with stubbor n agents. In Pr o c. 2014 IEE E Am. Contr ol Conf. , 3 978– 3983 . Nak ay ama, S., Kr asner, E., Zino, L., and Porfiri, M. (2019). So cia l information and sp ontaneous e mer gence of lea ders in human groups. J . R. So c. Int erfac e , 16(151 ), 2 0180 938. Ohlin, D., Bencherki, F., and T egling, E. (20 22). Achieving consensus in netw ork s o f increa singly stubb or n voters. In Pr o c. IEEE 61st Conf. De cis. Contr ol , 3531–35 37. Onnela, J.P . and Ree d- Tso chas, F. (201 0). Spontaneous emergence of so cial influence in online sys tems. Pr o c. Natl. A c ad. Sci. USA , 107(4 3), 18375– 1838 0. Surowiec ki, J. (2004 ). The Wisdom of Cr owds . DoubleDay , New Y ork . T aylor, M. (1968 ). T owards a mathematical theory of influence and a ttitude change. Hum . Rela t. , 21(2), 1 21– 139. Tian, Y., Jia, P ., Mirtabataba ei, A., W a ng, L., F riedkin, N.E., and Bullo, F. (202 1). So cial p ow er ev o lution in influence netw o rks with s tubbo rn individuals . IE EE T r ans. Autom. Contr ol , 67(2), 57 4–58 8. V a n Swol, L.M. (2009 ). Extreme members and g roup po larizatio n. So c. Influ. , 4(3), 1 85–1 99. W a ng, L., Be rnardo, C., Hong, Y., V a sca, F., Shi, G., and Altafini, C. (2021). Achieving consensus in spite of stubb ornness : time-v ar y ing concatenated F riedkin- Johnsen mo dels. In Pr o c. IEEE 60th Conf. D e cis. Contr ol , 49 64–4 969. W a ng, L., Chen, G., Bernar do , C., Hong, Y., Shi, G., and Altafini, C. (202 4). So cial p ow er g ames in conca tenated opinion dynamics. IEEE T ra ns. Autom. Contr ol , 6 9(11), 7614– 7629 . W a ng, L., Chen, G., Hong, Y., Shi, G., and Altafini, C. (2022 ). A soc ia l power game for the co ncatenated F riedkin-Johns e n model. In Pr o c. IEEE 61st Conf. De cis. Contr ol , 35 13–35 18.