Introduction
Research on diffusion processes over networks has attracted considerable attention due to its applications ranging from epidemic spreading to the adoption of behaviors, technologies, and social norms [11 , 12 , 39 , 44 ] . Adoption is often driven not only by intrinsic value of
This work was supported in part by Digital Futures, Stockholm, Sweden, the Wallenberg AI, Autonomous Systems and Software Program (WASP) funded by the Knut and Alice Wallenberg Foundation, and by the European Union – Next Generation EU, Mission 4, Component 1, under the PRIN project TECHIE: A control and network-based approach for fostering the adoption of new technologies in the ecological transition , Cod. 2022KPHA24, CUP Master: D53D23001320006, CUP: B53D230027600006.
Email addresses: alutto@kth.se (Martina Alutto), fabriziodabbene@cnr.it (Fabrizio Dabbene), angfon@kth.se (Angela Fontan), kallej@kth.se (Karl H. Johansson), chiararavazzi@cnr.it (Chiara Ravazzi).
the product or behavior but also by social influence, making diffusion inherently a network-driven process emerging from local interactions. Empirical studies of product and service diffusion have further shown that adoption decisions depend on the local network structure and on the characteristics and position of previous adopters [25 ] . Several modeling frameworks have been proposed to model network diffusion, including the Bass model and threshold-based approaches [7 , 23 , 28 ] , as well as epidemic-inspired models that describe adoption as a contagion process [30 , 38 ] . In this context, SIR and SIS models, together with their extensions, have been widely used to represent spreading phenomena in social and technological systems, including reversible adoption, memory, and loss of interest [9 , 13 , 20 ] (Li et al. 2014). However, adoption is rarely determined by exposure alone. Individual willingness to adopt may also depend on beliefs, attitudes, and perceived social norms, which naturally motivates the integration of opinion dynamics into diffusion models. Classical models such as French–DeGroot (DeGroot [16 ] ) and Friedkin–Johnsen (Friedkin & Johnsen [19 ] ) describe how opinions evolve through social interactions and persistent individual predispositions. Building on these ideas, a growing body of work has coupled opinion evolution with an adoption dynamics resembling an epidemic one, recognizing a mutual interaction: opinions shape the willingness to adopt, while adoption itself may feed back into beliefs and attitudes ([5 , 10 , 15 , 22 , 33 , 41 , 42 , 45 , 47 ] (Xu & Ishii 2024)). A related direction is considered in ([40 ] ), where adoption is coupled with an opinion dynamics process evolving over a signed network, thus accounting for antagonistic interactions among agents. These models show that the joint evolution of opinions and adoption can generate behaviors that contagion-based models alone cannot capture, but they typically focus on a single alternative or on restricted forms of competition. When multiple alternatives compete for adoption, individual decisions depend not only on exposure and social influence, but also on the set of available alternatives and on the possibility of switching among them. Competition among technologies and products has long been studied in economics, where network externalities may generate strategic advantages, path dependence, and lock-in, while the introduction of new alternatives can reshape the competitive environment ([6 , 14 , 26 , 27 ] ). Complementary game-theoretic approaches, particularly network coordination games, describe technology adoption as a strategic process in which agents benefit from coordinating their choices with those of their neighbors ([17 , 18 , 37 ] ). Competition may lead to dominance of a single technology, but long-term coexistence can also emerge in the presence of heterogeneity, switching mechanisms, or competing contagions ([34 , 36 ] ). Related coexistence and competitive effects have also been observed in epidemic-inspired models of multi-virus, multi-contagion, and information spreading ([21 , 43 ] ). The relevance of accounting for more than two alternatives is also suggested by the evolution of real markets. Figure 1 shows that, during the expansion of iOS between 2007 and 2008, the relative ordering of Windows Mobile and BlackBerry reversed ([29 ] ). These aggregate data do not establish a causal effect of entry, but illustrate how competitive dynamics may evolve as the set of available alternatives changes. Intervention in spreading and adoption dynamics has also been widely studied. Empirical and model-based approaches have investigated how information provision, network targeting, and mechanisms affecting peer influence can alter diffusion, while the control literature has considered influence maximization, predictive control of technology
Figure 1. Global market shares of selected smartphone platforms, 2006–2009. Data from Kennedy & Pon (2011).
adoption, opinion shaping, and control of competing contagions ([3 , 4 , 10 , 28 , 34 ] (Alutto, Bellotti, Dabbene & Ravazzi 2026; Ruf et al. 2019)). Despite these developments, existing approaches typically capture only subsets of the mechanisms that govern competition among alternatives. In particular, competitive diffusion models generally do not account simultaneously for alternative-specific opinions, dissatisfaction with the currently adopted option, and switching, while coupled adoption-opinion control frameworks have mainly focused on a single alternative. A unified framework combining competition, endogenous opinion formation, dissatisfaction, switching, and dynamic intervention is still lacking. This leaves open the problem of understanding and controlling diffusion when several alternatives coexist and adoption decisions and opinions co-evolve over heterogeneous interaction networks. To address this gap, we introduce the Multi-technology Adoption-opinion co-Evolution (MATE) model, a discrete-time network framework for the simultaneous diffusion of m ≥ 2 m \ge2 m ≥ 2 [2 ] ) to an arbitrary number of competing alternatives. For m > 2 m>2 m > 2 m ≥ 2 m \ge2 m ≥ 2 adoption-diffused equilibria (Section 3), establishing existence under strictly positive dissatisfaction rates, and providing sufficient conditions for uniqueness and global convergence from non-adoption-free initial states. We further identify features specific to the multi-alternative setting: the cancellation underlying the two-alternative adoption ratio is lost, pairwise adoption depends on the broader competitive environment, and uniqueness of the adoption-diffused equilibrium is no longer guaranteed under the standing assumptions. Numerical examples additionally illustrate the coexistence of multiple equilibria and entry-induced reversals between incumbent alternatives. Third, we formulate two finite-horizon optimal opinion-shaping control problems, derive necessary conditions via the discrete-time Pontryagin maximum principle and compute numerical extremals using a forward-backward sweep algorithm (§ 4 ).
Notation
We denote by R \mathbb{R} R R + \mathbb{R}_{+} R + R + n × n \mathbb{R}_{+}^{n\times n} R + n × n n × n n\times n n × n 1 \mathbf{1} 1 0 \mathbf{0} 0 I I I A A A A T A^{\mathsf{T}} A T x ∈ R n x\in\mathbb{R}^{n} x ∈ R n ∥ x ∥ 1 = ∑ i ∣ x i ∣ \lVert x\rVert_{1}=\sum_i|x_i| ∥ x ∥ 1 = ∑ i ∣ x i ∣ ∥ x ∥ ∞ = max i ∣ x i ∣ \lVert x\rVert_{\infty}=\max_i|x_i| ∥ x ∥ ∞ = max i ∣ x i ∣ ℓ 1 \ell_1 ℓ 1 ℓ ∞ \ell_\infty ℓ ∞ diag ( x ) \operatorname{diag}(x) diag ( x ) x x x A ∈ R + n × n A\in\mathbb{R}_{+}^{n\times n} A ∈ R + n × n ρ ( A ) \rho(A) ρ ( A ) A A A x ≤ y x\le y x ≤ y x , y ∈ R n x,y\in\mathbb{R}^{n} x , y ∈ R n
Model Description
We consider a population divided into n n n m ≥ 2 m\ge2 m ≥ 2 K : = { 1 , … , m } \mathcal{K}:=\{1,\ldots,m\} K := { 1 , … , m } i i i V : = { 1 , … , n } \mathcal{V}:=\{1,\ldots,n\} V := { 1 , … , n } t ≥ 0 t\ge0 t ≥ 0
s i ( t ) s_i(t) s i ( t )
a i [ k ] ( t ) a_i^{[k]}(t) a i [ k ] ( t ) k ∈ K k\in\mathcal{K} k ∈ K
d i [ k ] ( t ) d_i^{[k]}(t) d i [ k ] ( t ) k ∈ K k\in\mathcal{K} k ∈ K
The compartmental structure describing the adoption process is illustrated in Figure 2 , where arrows indicate the possible transitions between the susceptible (S S S A [ k ] A^{[k]} A [ k ] D [ k ] D^{[k]} D [ k ] k k k G = ( V , E , W ) \mathcal{G}=(\mathcal{V},\mathcal{E},W) G = ( V , E , W ) E ⊆ V × V \mathcal{E}\subseteq\mathcal{V}\times\mathcal{V} E ⊆ V × V W ∈ R + n × n W\in\mathbb{R}_{+}^{n\times n} W ∈ R + n × n physical interaction matrix . Each entry W i j W_{ij} W ij j j j i i i
Figure 2. Schematic transition diagram of the multi-technology adoption-opinion model. Susceptible individuals may adopt technology k k k A [ k ] A^{[k]} A [ k ] D [ k ] D^{[k]} D [ k ] δ [ k ] \delta^{[k]} δ [ k ]
As mentioned above, in addition to physical interactions, adoption is driven by individual beliefs. To capture this effect, each community i i i opinion variable x i [ k ] ∈ [ 0 , 1 ] x_i^{[k]}\in[0,1] x i [ k ] ∈ [ 0 , 1 ] k k k p i [ k ] ∈ [ 0 , 1 ] p_i^{[k]}\in[0,1] p i [ k ] ∈ [ 0 , 1 ] i i i k k k G ~ = ( V , E ~ , W ~ ) \widetilde{\mathcal{G}}=(\mathcal{V},\widetilde{\mathcal{E}},\widetilde{W}) G = ( V , E , W ) E ~ ⊆ V × V \widetilde{\mathcal{E}}\subseteq\mathcal{V}\times\mathcal{V} E ⊆ V × V W ~ ∈ R + n × n \widetilde{W}\in\mathbb{R}_{+}^{n\times n} W ∈ R + n × n social interaction matrix . Each entry W ~ i j \widetilde{W}_{ij} W ij j j j i i i β i [ k ] x i [ k ] \beta_i^{[k]}x_i^{[k]} β i [ k ] x i [ k ] γ i [ k ] x i [ k ] \gamma_i^{[k]}x_i^{[k]} γ i [ k ] x i [ k ] β i [ k ] , γ i [ k ] ∈ ( 0 , 1 ) \beta_i^{[k]},\gamma_i^{[k]}\in(0,1) β i [ k ] , γ i [ k ] ∈ ( 0 , 1 ) δ i [ k ] ∈ [ 0 , 1 ] \delta_i^{[k]}\in[0,1] δ i [ k ] ∈ [ 0 , 1 ] i ∈ V i\in\mathcal{V} i ∈ V k ∈ K k\in\mathcal{K} k ∈ K
s i ( t + 1 ) = s i ( t ) − s i ( t ) ∑ k ∈ K β i [ k ] x i [ k ] ( t ) ∑ j ∈ V W i j a j [ k ] ( t ) , (1) s_i(t+1)=s_i(t)-s_i(t)\sum_{k\in\mathcal{K}}\beta_i^{[k]}x_i^{[k]}(t)\sum_{j\in\mathcal{V}}W_{ij}a_j^{[k]}(t), \tag*{(1)} s i ( t + 1 ) = s i ( t ) − s i ( t ) k ∈ K ∑ β i [ k ] x i [ k ] ( t ) j ∈ V ∑ W ij a j [ k ] ( t ) , (1) a i [ k ] ( t + 1 ) = a i [ k ] ( t ) + β i [ k ] x i [ k ] ( t ) s i ( t ) ∑ j ∈ V W i j a j [ k ] ( t ) − δ i [ k ] a i [ k ] ( t ) + γ i [ k ] x i [ k ] ( t ) ∑ ℓ ∈ K ∖ { k } d i [ ℓ ] ( t ) , (2) \begin{aligned}
a_i^{[k]}(t+1)&=a_i^{[k]}(t)+\beta_i^{[k]}x_i^{[k]}(t)s_i(t)\sum_{j\in\mathcal{V}}W_{ij}a_j^{[k]}(t)\\
&\quad-\delta_i^{[k]}a_i^{[k]}(t)+\gamma_i^{[k]}x_i^{[k]}(t)\sum_{\ell\in\mathcal{K}\setminus\{k\}}d_i^{[\ell]}(t), \tag*{(2)}
\end{aligned} a i [ k ] ( t + 1 ) = a i [ k ] ( t ) + β i [ k ] x i [ k ] ( t ) s i ( t ) j ∈ V ∑ W ij a j [ k ] ( t ) − δ i [ k ] a i [ k ] ( t ) + γ i [ k ] x i [ k ] ( t ) ℓ ∈ K ∖ { k } ∑ d i [ ℓ ] ( t ) , (2) d i [ k ] ( t + 1 ) = d i [ k ] ( t ) − d i [ k ] ( t ) ∑ ℓ ∈ K ∖ { k } γ i [ ℓ ] x i [ ℓ ] ( t ) + δ i [ k ] a i [ k ] ( t ) , (3) d_i^{[k]}(t+1)=d_i^{[k]}(t)-d_i^{[k]}(t)\sum_{\ell\in\mathcal{K}\setminus\{k\}}\gamma_i^{[\ell]}x_i^{[\ell]}(t)+\delta_i^{[k]}a_i^{[k]}(t), \tag*{(3)} d i [ k ] ( t + 1 ) = d i [ k ] ( t ) − d i [ k ] ( t ) ℓ ∈ K ∖ { k } ∑ γ i [ ℓ ] x i [ ℓ ] ( t ) + δ i [ k ] a i [ k ] ( t ) , (3) x i [ k ] ( t + 1 ) = ( 1 − λ i [ k ] − ξ i [ k ] ) p i [ k ] + λ i [ k ] ∑ j ∈ V W ~ i j x j [ k ] ( t ) + ξ i [ k ] ∑ j ∈ V W i j a j [ k ] ( t ) , (4) \begin{aligned}
x_i^{[k]}(t+1)&=(1-\lambda_i^{[k]}-\xi_i^{[k]})p_i^{[k]}\\
&\quad+\lambda_i^{[k]}\sum_{j\in\mathcal{V}}\widetilde{W}_{ij}x_j^{[k]}(t)+\xi_i^{[k]}\sum_{j\in\mathcal{V}}W_{ij}a_j^{[k]}(t), \tag*{(4)}
\end{aligned} x i [ k ] ( t + 1 ) = ( 1 − λ i [ k ] − ξ i [ k ] ) p i [ k ] + λ i [ k ] j ∈ V ∑ W ij x j [ k ] ( t ) + ξ i [ k ] j ∈ V ∑ W ij a j [ k ] ( t ) , (4) where λ i [ k ] , ξ i [ k ] ≥ 0 \lambda_i^{[k]},\xi_i^{[k]}\ge0 λ i [ k ] , ξ i [ k ] ≥ 0 λ i [ k ] + ξ i [ k ] < 1 \lambda_i^{[k]}+\xi_i^{[k]}<1 λ i [ k ] + ξ i [ k ] < 1 k ∈ K k\in\mathcal{K} k ∈ K [19 ] by incorporating a feedback term from the adoption layer. Specifically, opinions towards a technology are shaped by three factors: the fixed predisposition of each community, social interactions over the opinion network, and the observed adoption levels in the physical layer. Hence, λ i [ k ] \lambda_i^{[k]} λ i [ k ] ξ i [ k ] \xi_i^{[k]} ξ i [ k ] k k k i i i x i [ k ] x_i^{[k]} x i [ k ]
We refer to (1) as the Multi-technology AdopTion-opinion co-Evolution (MATE) model. This framework extends the adoption–opinion models proposed in (Alutto, Bellotti, Dabbene & Ravazzi 2026, Alutto et al. 2025), which focus on the diffusion of a single technology, and generalizes the preliminary two-technology setting studied in (Alutto, Dabbene, Fontan, Johanson & Ravazzi 2026) to the case of m ≥ 2 m\ge2 m ≥ 2
Preliminary assumptions and properties
We introduce the standing assumptions used throughout the analysis.
Assumption 1 (i) The matrices W W W W ~ \widetilde{W} W W W W
(ii) For all k ∈ K k\in\mathcal{K} k ∈ K p [ k ] > 0 p^{[k]}>0 p [ k ] > 0
(iii) For each i ∈ V i\in\mathcal{V} i ∈ V ∑ k ∈ K β i [ k ] ≤ 1 \sum_{k\in\mathcal{K}}\beta_i^{[k]}\leq1 ∑ k ∈ K β i [ k ] ≤ 1 ∑ k ∈ K γ i [ k ] ≤ 1 \sum_{k\in\mathcal{K}}\gamma_i^{[k]}\leq1 ∑ k ∈ K γ i [ k ] ≤ 1
Assumption 1(i) implies that the interaction terms between communities are convex combinations of the fractions of adopters at neighboring communities. This guarantees that both the overall exposure term ∑ j W i j a j [ k ] \sum_j W_{ij}a_j^{[k]} ∑ j W ij a j [ k ] ∑ j W ~ i j x j [ k ] \sum_j\widetilde{W}_{ij}x_j^{[k]} ∑ j W ij x j [ k ] [ 0 , 1 ] [0,1] [ 0 , 1 ] W W W
Let a [ k ] ( t ) , d [ k ] ( t ) , x [ k ] ( t ) ∈ R n a^{[k]}(t),d^{[k]}(t),x^{[k]}(t)\in\mathbb{R}^n a [ k ] ( t ) , d [ k ] ( t ) , x [ k ] ( t ) ∈ R n k ∈ K k\in\mathcal{K} k ∈ K B [ k ] : = diag ( β [ k ] ) B^{[k]}:=\operatorname{diag}(\beta^{[k]}) B [ k ] := diag ( β [ k ] ) Γ [ k ] : = diag ( γ [ k ] ) \Gamma^{[k]}:=\operatorname{diag}(\gamma^{[k]}) Γ [ k ] := diag ( γ [ k ] ) Δ [ k ] : = diag ( δ [ k ] ) \Delta^{[k]}:=\operatorname{diag}(\delta^{[k]}) Δ [ k ] := diag ( δ [ k ] ) Λ [ k ] : = diag ( λ [ k ] ) \Lambda^{[k]}:=\operatorname{diag}(\lambda^{[k]}) Λ [ k ] := diag ( λ [ k ] ) Ξ [ k ] : = diag ( ξ [ k ] ) \Xi^{[k]}:=\operatorname{diag}(\xi^{[k]}) Ξ [ k ] := diag ( ξ [ k ] )
Y : = { ( s , ( a [ k ] ) k , ( d [ k ] ) k , ( x [ k ] ) k ) ∈ [ 0 , 1 ] ( 3 m + 1 ) n : s + ∑ k ∈ K ( a [ k ] + d [ k ] ) = 1 } . \mathcal{Y}:=\left\{(s,(a^{[k]})_{k},(d^{[k]})_{k},(x^{[k]})_{k})\in[0,1]^{(3m+1)n}:s+\sum_{k\in\mathcal{K}}(a^{[k]}+d^{[k]})=1\right\}. Y := { ( s , ( a [ k ] ) k , ( d [ k ] ) k , ( x [ k ] ) k ) ∈ [ 0 , 1 ] ( 3 m + 1 ) n : s + k ∈ K ∑ ( a [ k ] + d [ k ] ) = 1 } . For simplicity of notation, from now on, the state of the MATE model (1) at time t ≥ 0 t\geq0 t ≥ 0
y ( t ) : = ( s ( t ) , ( a [ k ] ) k ( t ) , ( d [ k ] ) k ( t ) , ( x [ k ] ) k ( t ) ) . (5) y(t):=(s(t),(a^{[k]})_k(t),(d^{[k]})_k(t),(x^{[k]})_k(t)). \tag*{(5)} y ( t ) := ( s ( t ) , ( a [ k ] ) k ( t ) , ( d [ k ] ) k ( t ) , ( x [ k ] ) k ( t )) . (5) Proposition 1 Consider the MATE model (1) under Assumption 1. If the initial condition belongs to Y \mathcal{Y} Y Y \mathcal{Y} Y
Corollary 1 Consider the MATE model (1) under Assumption 1. If the initial condition belongs to Y \mathcal{Y} Y
(i) s ( t + 1 ) ≤ s ( t ) s(t+1)\leq s(t) s ( t + 1 ) ≤ s ( t ) t ≥ 0 t\geq0 t ≥ 0
(ii) x [ k ] ( t ) > 0 x^{[k]}(t)>0 x [ k ] ( t ) > 0 t ≥ 1 t\geq1 t ≥ 1 k ∈ K k\in\mathcal{K} k ∈ K
In particular, Corollary 1 establishes nonincreasing monotonicity of the susceptibles. The proofs of Proposition 1 and Corollary 1 are in Appendix A.
Equilibrium points and stability analysis
In this section, we analyze the equilibrium points of the adoption–opinion model (1) and study their stability properties. We first analyze the adoption-free equilibrium (AFE) (§ 3.0.1 ), namely the equilibrium configuration in which no individuals adopt any technology and the whole population remains susceptible, and then the adoption-diffused equilibrium (ADE) (§ 3.0.2 ), characterized by a positive fraction of adopters and dissatisfied individuals across communities. Finally, we discuss the special case of m = 2 m=2 m = 2 § 3.0.3 ) and show that the within-community equilibrium adoption ratio depends only on dissatisfaction (user-experience) parameters. The proofs of all results in this section can be found in Appendix B.
Adoption-free equilibrium
Proposition 2 The MATE model (1) under Assumption 1 admits the AFE, defined as follows
y A F : = ( 1 , ( 0 ) k ∈ K , ( 0 ) k ∈ K , ( x A F [ k ] ) k ∈ K ) , (6) y_{\mathrm{AF}}:=(\mathbf{1},(\mathbf{0})_{k\in\mathcal{K}},(\mathbf{0})_{k\in\mathcal{K}},(x_{\mathrm{AF}}^{[k]})_{k\in\mathcal{K}}), \tag*{(6)} y AF := ( 1 , ( 0 ) k ∈ K , ( 0 ) k ∈ K , ( x AF [ k ] ) k ∈ K ) , (6) where, for each k ∈ K k\in\mathcal{K} k ∈ K
x A F [ k ] : = ( I − Λ [ k ] W ~ ) − 1 ( ( I − Λ [ k ] − Ξ [ k ] ) p [ k ] ) . (7) x_{\mathrm{AF}}^{[k]}:=(I-\Lambda^{[k]}\widetilde{W})^{-1}\bigl((I-\Lambda^{[k]}-\Xi^{[k]})p^{[k]}\bigr). \tag*{(7)} x AF [ k ] := ( I − Λ [ k ] W ) − 1 ( ( I − Λ [ k ] − Ξ [ k ] ) p [ k ] ) . (7) Moreover, y A F y_{\mathrm{AF}} y AF
Remark 1 Proposition 2 shows that y A F y_{\mathrm{AF}} y AF y A F y_{\mathrm{AF}} y AF y A F y_{\mathrm{AF}} y AF y A F y_{\mathrm{AF}} y AF y A F y_{\mathrm{AF}} y AF
Adoption-diffused equilibrium
This section characterizes the equilibrium structure of the MATE model beyond the adoption-free state. We first establish the existence of an ADE, in which the susceptible population is depleted throughout the network, and characterize its structure. We then rule out partial-adoption equilibria and provide sufficient conditions for the uniqueness and convergence to the ADE.
Before showing the first result, we denote any equilibrium with y ⋆ y^\star y ⋆ i i i
d ˉ i ⋆ : = ∑ k ∈ K d i [ k ] ⋆ , x ˉ i ⋆ : = ∑ k ∈ K γ i [ k ] x i [ k ] ⋆ . (8) \bar{d}_i^\star:=\sum_{k\in\mathcal{K}}d_i^{[k]\star},\qquad\bar{x}_i^\star:=\sum_{k\in\mathcal{K}}\gamma_i^{[k]}x_i^{[k]\star}. \tag*{(8)} d ˉ i ⋆ := k ∈ K ∑ d i [ k ] ⋆ , x ˉ i ⋆ := k ∈ K ∑ γ i [ k ] x i [ k ] ⋆ . (8) We start proving the existence of an equilibrium with s ⋆ = 0 s^\star=0 s ⋆ = 0
Theorem 1 Consider the MATE model (1) under Assumption 1. Assume δ i [ k ] > 0 \delta_i^{[k]}>0 δ i [ k ] > 0 i ∈ V i\in\mathcal{V} i ∈ V k ∈ K k\in\mathcal{K} k ∈ K
y ⋆ : = ( s ⋆ , ( a [ k ] ⋆ ) k ∈ K , ( d [ k ] ⋆ ) k ∈ K , ( x [ k ] ⋆ ) k ∈ K ) (9) y^\star:=(s^\star,(a^{[k]\star})_{k\in\mathcal{K}},(d^{[k]\star})_{k\in\mathcal{K}},(x^{[k]\star})_{k\in\mathcal{K}}) \tag*{(9)} y ⋆ := ( s ⋆ , ( a [ k ] ⋆ ) k ∈ K , ( d [ k ] ⋆ ) k ∈ K , ( x [ k ] ⋆ ) k ∈ K ) (9) such that s ⋆ = 0 s^\star=0 s ⋆ = 0 a i [ k ] ⋆ > 0 a_i^{[k]\star}>0 a i [ k ] ⋆ > 0 d i [ k ] ⋆ > 0 d_i^{[k]\star}>0 d i [ k ] ⋆ > 0 k ∈ K k\in\mathcal{K} k ∈ K k k k
x [ k ] ⋆ = ( I − Λ [ k ] W ) − 1 ( ( I − Λ [ k ] − Ξ [ k ] ) p [ k ] + Ξ [ k ] W a [ k ] ⋆ ) . (10) x^{[k]\star}=(I-\Lambda^{[k]}W)^{-1}\left((I-\Lambda^{[k]}-\Xi^{[k]})p^{[k]}+\Xi^{[k]}Wa^{[k]\star}\right). \tag*{(10)} x [ k ] ⋆ = ( I − Λ [ k ] W ) − 1 ( ( I − Λ [ k ] − Ξ [ k ] ) p [ k ] + Ξ [ k ] W a [ k ] ⋆ ) . (10) Moreover, the following structural identities hold:
d i [ k ] ⋆ = γ i [ k ] x i [ k ] ⋆ x ˉ i ⋆ d ˉ i ⋆ , a i [ k ] ⋆ = γ i [ k ] x i [ k ] ⋆ δ i [ k ] d ˉ i ⋆ ( 1 − γ i [ k ] x i [ k ] ⋆ x ˉ i ⋆ ) , (11) d_i^{[k]\star}=\frac{\gamma_i^{[k]}x_i^{[k]\star}}{\bar{x}_i^\star}\bar{d}_i^\star,\qquad
a_i^{[k]\star}=\frac{\gamma_i^{[k]}x_i^{[k]\star}}{\delta_i^{[k]}}\bar{d}_i^\star\left(1-\frac{\gamma_i^{[k]}x_i^{[k]\star}}{\bar{x}_i^\star}\right), \tag*{(11)} d i [ k ] ⋆ = x ˉ i ⋆ γ i [ k ] x i [ k ] ⋆ d ˉ i ⋆ , a i [ k ] ⋆ = δ i [ k ] γ i [ k ] x i [ k ] ⋆ d ˉ i ⋆ ( 1 − x ˉ i ⋆ γ i [ k ] x i [ k ] ⋆ ) , (11) where d ˉ i ⋆ \bar{d}_i^\star d ˉ i ⋆ x ˉ i ⋆ \bar{x}_i^\star x ˉ i ⋆
Remark 2. (11) shows an inverse dependence of the fraction of adopters a i [ k ] ⋆ a_i^{[k]\star} a i [ k ] ⋆ δ i [ k ] \delta_i^{[k]} δ i [ k ]
Remark 3. The assumption δ i [ k ] > 0 \delta_i^{[k]}>0 δ i [ k ] > 0 i ∈ V i\in\mathcal{V} i ∈ V k ∈ K k\in\mathcal{K} k ∈ K s i ⋆ = 0 s_i^\star=0 s i ⋆ = 0 i i i δ i [ k 0 ] = 0 \delta_i^{[k_0]}=0 δ i [ k 0 ] = 0 k 0 k_0 k 0 a i [ k 0 ] ⋆ = 1 a_i^{[k_0]\star}=1 a i [ k 0 ] ⋆ = 1
The previous theorem characterizes equilibria with no susceptibles. The following proposition shows that, under Assumption 1, an equilibrium cannot exhibit adoption in only a subset of communities.
Proposition 3 Consider the MATE model (1) under Assumption 1. Let y ⋆ y^\star y ⋆ i ∈ V i\in\mathcal{V} i ∈ V s i ⋆ = 0 s_i^\star=0 s i ⋆ = 0 s ⋆ = 0 s^\star=0 s ⋆ = 0
Corollary 2 Consider the MATE model (1) under Assumption 1, and assume that δ i [ k ] > 0 \delta_i^{[k]}>0 δ i [ k ] > 0 i ∈ V i\in\mathcal{V} i ∈ V k ∈ K k\in\mathcal{K} k ∈ K
The next results establish sufficient conditions for the uniqueness and convergence to the ADE.
Theorem 2 Consider the MATE model (1) under Assumption 1. Assume that δ i [ k ] > 0 \delta_i^{[k]}>0 δ i [ k ] > 0 i ∈ V i\in\mathcal{V} i ∈ V k ∈ K k\in\mathcal{K} k ∈ K
max k ∈ K ( max i ξ i [ k ] 1 − max i λ i [ k ] max i 1 δ i [ k ] ( 1 + 1 2 min r δ i [ r ] ) ) < 1. (12) \max_{k\in\mathcal{K}}\left(\frac{\max_i \xi_i^{[k]}}{1-\max_i\lambda_i^{[k]}}\max_i\frac{1}{\delta_i^{[k]}}\left(1+\frac{1}{2\min_{r}\delta_i^{[r]}}\right)\right)<1. \tag*{(12)} k ∈ K max ( 1 − max i λ i [ k ] max i ξ i [ k ] i max δ i [ k ] 1 ( 1 + 2 min r δ i [ r ] 1 ) ) < 1. (12) then the ADE is unique.
Remark 4. The sufficient condition in (9) ensures that the feedback between opinions and adoption is sufficiently weak. The first factor bounds the sensitivity of the opinions at equilibrium to adoption, while the second bounds the sensitivity of the adoption at equilibrium to opinions.
Before the next result, define the following for all i i i
δ i , min : = min k δ i [ k ] , γ i , min : = min k γ i [ k ] , (13) \delta_{i,\min}:=\min_k\delta_i^{[k]},\qquad\gamma_{i,\min}:=\min_k\gamma_i^{[k]}, \tag*{(13)} δ i , m i n := k min δ i [ k ] , γ i , m i n := k min γ i [ k ] , (13) x ‾ i : = min k ( 1 − λ i [ k ] − ξ i [ k ] ) p i [ k ] > 0 , \underline{x}_i:=\min_k(1-\lambda_i^{[k]}-\xi_i^{[k]})p_i^{[k]}>0, x i := k min ( 1 − λ i [ k ] − ξ i [ k ] ) p i [ k ] > 0 , and
ε i : = x ‾ i δ i , min 2 γ i , min 3 > 0. (14) \varepsilon_i:=\underline{x}_i\delta_{i,\min}^2\gamma_{i,\min}^3>0. \tag*{(14)} ε i := x i δ i , m i n 2 γ i , m i n 3 > 0. (14) Theorem 3 Consider the MATE model (1) under Assumption 1. Assume that δ i [ k ] > 0 \delta_i^{[k]}>0 δ i [ k ] > 0 i ∈ V i\in\mathcal{V} i ∈ V k ∈ K k\in\mathcal{K} k ∈ K
6 min i ε i max k ∈ K max i ξ i [ k ] 1 − max i λ i [ k ] < 1 , (15) \frac{6}{\min_i\varepsilon_i}\max_{k\in\mathcal{K}}\frac{\max_i\xi_i^{[k]}}{1-\max_i\lambda_i^{[k]}}<1, \tag*{(15)} min i ε i 6 k ∈ K max 1 − max i λ i [ k ] max i ξ i [ k ] < 1 , (15) then every trajectory with s ( 0 ) ≠ 1 s(0)\ne1 s ( 0 ) = 1 y ⋆ y^\star y ⋆ y ( t ) → y ⋆ y(t)\to y^\star y ( t ) → y ⋆ t → ∞ t\to\infty t → ∞
Remark 5. The sufficient condition (11) requires the feedback between opinion variations and adopter-dissatisfied dynamics to be sufficiently weak.
Example 1. To illustrate the uniqueness result of Theorem 2, we consider a network with n = 30 n=30 n = 30 m = 5 m=5 m = 5 A [ k ] ( t ) : = 1 n 1 T a [ k ] ( t ) A^{[k]}(t):=\frac{1}{n}\mathbf{1}^{\mathsf{T}}a^{[k]}(t) A [ k ] ( t ) := n 1 1 T a [ k ] ( t ) k k k Figure 3 (a) considers a parameter regime satisfying condition (9). Trajectories initialized from different adoption profiles approach the same equilibrium, consistently with the uniqueness of the ADE. In contrast, Figure 3 (b) considers a parameter regime in which the uniqueness condition (9) is violated. In this case, two different initial adoption profiles converge numerically to distinct equilibria, showing that multiple ADEs and path-dependent long-term outcomes may arise when the sufficient uniqueness condition is not satisfied.
Figure 3. Time evolution of the average population fractions with m = 5 m=5 m = 5
Although convergence is theoretically guaranteed only under the sufficient condition (11), numerical experiments suggest that convergence may also occur beyond this regime.
Special Case: Two Technologies
We now specialize the analysis to K = { 1 , 2 } \mathcal{K}=\{1,2\} K = { 1 , 2 } m = 2 m=2 m = 2 [2 ] , and highlight analogies and differences with the general multi-technology case, focus of the present work.
Proposition 4 Consider the MATE model (1) with m = 2 m=2 m = 2 δ i [ k ] > 0 \delta_i^{[k]}>0 δ i [ k ] > 0 i ∈ V i\in\mathcal{V} i ∈ V k ∈ { 1 , 2 } k\in\{1,2\} k ∈ { 1 , 2 } y ⋆ y^\star y ⋆ s ⋆ = 0 s^\star=0 s ⋆ = 0 i ∈ V i\in\mathcal{V} i ∈ V
a i [ 2 ] ⋆ a i [ 1 ] ⋆ = δ i [ 1 ] δ i [ 2 ] , d i [ 2 ] ⋆ d i [ 1 ] ⋆ = γ i [ 2 ] χ i [ 2 ] ⋆ γ i [ 1 ] χ i [ 1 ] ⋆ . (16) \frac{a_i^{[2]\star}}{a_i^{[1]\star}}=\frac{\delta_i^{[1]}}{\delta_i^{[2]}}, \qquad
\frac{d_i^{[2]\star}}{d_i^{[1]\star}}=\frac{\gamma_i^{[2]}\chi_i^{[2]\star}}{\gamma_i^{[1]}\chi_i^{[1]\star}}. \tag*{(16)} a i [ 1 ] ⋆ a i [ 2 ] ⋆ = δ i [ 2 ] δ i [ 1 ] , d i [ 1 ] ⋆ d i [ 2 ] ⋆ = γ i [ 1 ] χ i [ 1 ] ⋆ γ i [ 2 ] χ i [ 2 ] ⋆ . (16) The result follows from Theorem 1 specialized to m = 2 m=2 m = 2 [2 ] ; hence, the proof is omitted.
Remark 6 For m = 2 m=2 m = 2 m > 2 m>2 m > 2 ( 1 − γ i [ k ] χ i [ k ] ⋆ / x ˉ i ⋆ ) (1-\gamma_i^{[k]}\chi_i^{[k]\star}/\bar{x}_i^\star) ( 1 − γ i [ k ] χ i [ k ] ⋆ / x ˉ i ⋆ ) γ i [ k ] χ i [ k ] ⋆ \gamma_i^{[k]}\chi_i^{[k]\star} γ i [ k ] χ i [ k ] ⋆
The extension from two to multiple competing technologies substantially changes the equilibrium properties. For m = 2 m=2 m = 2
Example 2 We illustrate a competitive effect specific to m > 2 m>2 m > 2 t e t_e t e m = 2 m=2 m = 2 m = 3 m=3 m = 3 a [ 3 ] ( t e ) = d [ 3 ] ( t e ) = 0 a^{[3]}(t_e)=d^{[3]}(t_e)=0 a [ 3 ] ( t e ) = d [ 3 ] ( t e ) = 0 x [ 3 ] ( t e ) = p [ 3 ] x^{[3]}(t_e)=p^{[3]} x [ 3 ] ( t e ) = p [ 3 ] Figure 4 , technology 1 initially has larger adoption than technology 2, whereas after entry their ordering is reversed, although technology 3 remains non-dominant. Thus, the entrant reshapes the competition through opinion-dependent switching.
Figure 4. Effect of technology entry on the adoption dynamics. The introduction of technology 3 changes the relative ordering of technologies 1 and 2, while the entrant remains non-dominant.
Opinion shaping control
Having characterized the equilibrium and convergence properties of the MATE dynamics (1), we now investigate how external interventions can steer technology adoption through the opinion layer. We consider interventions that modify the baseline predisposition toward
a technology without directly altering the adoption, dissatisfaction, or switching rates. Such interventions may represent information campaigns, educational initiatives, advertising, or targeted communication aimed at improving individuals’ perceptions of a technology rather than directly enforcing adoption. Empirical studies suggest that these interventions can significantly affect attitudes and adoption intentions (Li et al. 2020). We model the control as a shift of the fixed predisposition p [ k ] p^{[k]} p [ k ]
Optimal control problem definition
For each community i ∈ V i\in\mathcal{V} i ∈ V k ∈ K k\in\mathcal{K} k ∈ K p u , i [ k ] ( t ) : = p i [ k ] ( 1 − u i [ k ] ( t ) ) + u i [ k ] ( t ) p_{u,i}^{[k]}(t):=p_i^{[k]}(1-u_i^{[k]}(t))+u_i^{[k]}(t) p u , i [ k ] ( t ) := p i [ k ] ( 1 − u i [ k ] ( t )) + u i [ k ] ( t ) u i [ k ] ( t ) ∈ [ 0 , 1 ] u_i^{[k]}(t)\in[0,1] u i [ k ] ( t ) ∈ [ 0 , 1 ] u i [ k ] ( t ) = 0 u_i^{[k]}(t)=0 u i [ k ] ( t ) = 0 u i [ k ] ( t ) = 1 u_i^{[k]}(t)=1 u i [ k ] ( t ) = 1
x i [ k ] ( t + 1 ) = ( 1 − λ i [ k ] − ξ i [ k ] ) p u , i [ k ] ( t ) + λ i [ k ] ∑ j ∈ V W ~ i j x j [ k ] ( t ) + ξ i [ k ] ∑ j ∈ V W i j a j [ k ] ( t ) , (17) x_i^{[k]}(t+1)=(1-\lambda_i^{[k]}-\xi_i^{[k]})p_{u,i}^{[k]}(t)
+\lambda_i^{[k]}\sum_{j\in\mathcal{V}}\widetilde{W}_{ij}x_j^{[k]}(t)+\xi_i^{[k]}\sum_{j\in\mathcal{V}}W_{ij}a_j^{[k]}(t), \tag*{(17)} x i [ k ] ( t + 1 ) = ( 1 − λ i [ k ] − ξ i [ k ] ) p u , i [ k ] ( t ) + λ i [ k ] j ∈ V ∑ W ij x j [ k ] ( t ) + ξ i [ k ] j ∈ V ∑ W ij a j [ k ] ( t ) , (17) while the variables s i ( t ) s_i(t) s i ( t ) a i [ k ] ( t ) a_i^{[k]}(t) a i [ k ] ( t ) d i [ k ] ( t ) d_i^{[k]}(t) d i [ k ] ( t ) U : = [ 0 , 1 ] n ∣ K ∣ ⊂ R n ∣ K ∣ U:=[0,1]^{n|\mathcal{K}|}\subset\mathbb{R}^{n|\mathcal{K}|} U := [ 0 , 1 ] n ∣ K ∣ ⊂ R n ∣ K ∣ U : = { u ( ⋅ ) = ( u ( 0 ) , … , u ( T − 1 ) ) : u ( t ) ∈ U , t ∈ [ 0 , T − 1 ] } \mathcal{U}:=\{u(\cdot)=(u(0),\ldots,u(T-1)):u(t)\in U,t\in[0,T-1]\} U := { u ( ⋅ ) = ( u ( 0 ) , … , u ( T − 1 )) : u ( t ) ∈ U , t ∈ [ 0 , T − 1 ]} u ( t ) u(t) u ( t ) K \mathcal{K} K V \mathcal{V} V
y ( t + 1 ) = f ( y ( t ) , u ( t ) ) , y ( 0 ) = y 0 , (18) y(t+1)=f(y(t),u(t)), \qquad y(0)=y_0, \tag*{(18)} y ( t + 1 ) = f ( y ( t ) , u ( t )) , y ( 0 ) = y 0 , (18) where y ( t ) y(t) y ( t ) s i ( t ) s_i(t) s i ( t ) a i [ k ] ( t ) a_i^{[k]}(t) a i [ k ] ( t ) d i [ k ] ( t ) d_i^{[k]}(t) d i [ k ] ( t ) x i [ k ] ( t ) x_i^{[k]}(t) x i [ k ] ( t ) p u , i [ k ] ( t ) ∈ [ p i [ k ] , 1 ] p_{u,i}^{[k]}(t)\in[p_i^{[k]},1] p u , i [ k ] ( t ) ∈ [ p i [ k ] , 1 ]
To quantify the intervention effort, we assign a positive cost to each control component. Let Q = diag ( q i [ k ] ) ∈ R n m × n m Q=\operatorname{diag}(q_i^{[k]})\in\mathbb{R}^{nm\times nm} Q = diag ( q i [ k ] ) ∈ R nm × nm u ( t ) ⊤ Q u ( t ) = ∑ k ∈ K ∑ i ∈ V q i [ k ] ( u i [ k ] ( t ) ) 2 u(t)^\top Q u(t)=\sum_{k\in\mathcal{K}}\sum_{i\in\mathcal{V}}q_i^{[k]}(u_i^{[k]}(t))^2 u ( t ) ⊤ Q u ( t ) = ∑ k ∈ K ∑ i ∈ V q i [ k ] ( u i [ k ] ( t ) ) 2 q i [ k ] > 0 q_i^{[k]}>0 q i [ k ] > 0 i i i k k k k 0 k_0 k 0 U k 0 : = { u ∈ [ 0 , 1 ] n m : u i [ k ] = 0 ∀ i ∈ V , k ≠ k 0 } U_{k_0}:=\{u\in[0,1]^{nm}:u_i^{[k]}=0\ \forall i\in\mathcal{V},\ k\ne k_0\} U k 0 := { u ∈ [ 0 , 1 ] nm : u i [ k ] = 0 ∀ i ∈ V , k = k 0 } U k 0 U_{k_0} U k 0
Problem 1 (Cumulative adoption maximization) Given weights r 1 [ k ] > 0 r_1^{[k]}>0 r 1 [ k ] > 0 k k k u ∗ ∈ U u^*\in U u ∗ ∈ U
J 1 ( u ) = ∑ t = 0 T − 1 ( ∑ k ∈ K r 1 [ k ] 1 ⊤ a [ k ] ( t ) − u ( t ) ⊤ Q u ( t ) ) , J_1(u)=\sum_{t=0}^{T-1}\left(\sum_{k\in\mathcal{K}}r_1^{[k]}\mathbf{1}^\top a^{[k]}(t)-u(t)^\top Q u(t)\right), J 1 ( u ) = t = 0 ∑ T − 1 ( k ∈ K ∑ r 1 [ k ] 1 ⊤ a [ k ] ( t ) − u ( t ) ⊤ Q u ( t ) ) , subject to the controlled dynamics (14).
Problem 2 (Targeted adoption maximization) Given a target technology k 0 ∈ K k_0\in\mathcal{K} k 0 ∈ K B > 0 B>0 B > 0 u ∗ ∈ U k 0 u^*\in U_{k_0} u ∗ ∈ U k 0
J 2 ( u ) = 1 ⊤ a [ k 0 ] ( T ) , J_2(u)=\mathbf{1}^\top a^{[k_0]}(T), J 2 ( u ) = 1 ⊤ a [ k 0 ] ( T ) , subject to the controlled dynamics (14) and the budget constraint
∑ t = 0 T − 1 u ( t ) ⊤ Q u ( t ) ≤ B . (19) \sum_{t=0}^{T-1}u(t)^\top Q u(t)\le B. \tag*{(19)} t = 0 ∑ T − 1 u ( t ) ⊤ Q u ( t ) ≤ B . (19) Both problems can be written in the common form J ( u ) = ∑ t = 0 T − 1 ℓ ( y ( t ) , u ( t ) ) + Φ ( y ( T ) ) J(u)=\sum_{t=0}^{T-1}\ell(y(t),u(t))+\Phi(y(T)) J ( u ) = ∑ t = 0 T − 1 ℓ ( y ( t ) , u ( t )) + Φ ( y ( T )) ℓ 1 ( y ( t ) , u ( t ) ) : = ∑ k ∈ K r 1 [ k ] 1 ⊤ a [ k ] ( t ) − u ( t ) ⊤ Q u ( t ) \ell_1(y(t),u(t)):=\sum_{k\in\mathcal{K}}r_1^{[k]}\mathbf{1}^\top a^{[k]}(t)-u(t)^\top Q u(t) ℓ 1 ( y ( t ) , u ( t )) := ∑ k ∈ K r 1 [ k ] 1 ⊤ a [ k ] ( t ) − u ( t ) ⊤ Q u ( t ) Φ 1 ( y ( T ) ) : = 0 \Phi_1(y(T)):=0 Φ 1 ( y ( T )) := 0 ℓ 2 ( y ( t ) , u ( t ) ) : = 0 \ell_2(y(t),u(t)):=0 ℓ 2 ( y ( t ) , u ( t )) := 0 Φ 2 ( y ( T ) ) : = 1 ⊤ a [ k 0 ] ( T ) \Phi_2(y(T)):=\mathbf{1}^\top a^{[k_0]}(T) Φ 2 ( y ( T )) := 1 ⊤ a [ k 0 ] ( T )
Remark 7 The control affects adoption with a two-step delay, since u ( t ) u(t) u ( t ) x ( t + 1 ) x(t+1) x ( t + 1 ) a ( t + 2 ) a(t+2) a ( t + 2 )
Existence of optimal controls
We first show that both finite-horizon formulations admit an optimal solution.
Theorem 4 Consider the controlled adoption-opinion model (14), with initial condition y ( 0 ) = y 0 y(0)=y_0 y ( 0 ) = y 0 u 1 ∗ ∈ U u_1^*\in U u 1 ∗ ∈ U u 2 ∗ ∈ U k 0 u_2^*\in U_{k_0} u 2 ∗ ∈ U k 0 u 1 ∗ u_1^* u 1 ∗ u 2 ∗ u_2^* u 2 ∗
J 1 ( u 1 ∗ ) = max u ∈ U J 1 ( u ) , J 2 ( u 2 ∗ ) = max u ∈ U k 0 ∑ t = 0 T − 1 u ( t ) ⊤ Q u ( t ) ≤ B J 2 ( u ) . J_1(u_1^*)=\max_{u\in U}J_1(u),\qquad J_2(u_2^*)=\max_{\substack{u\in U_{k_0}\\ \sum_{t=0}^{T-1}u(t)^\top Q u(t)\le B}}J_2(u). J 1 ( u 1 ∗ ) = u ∈ U max J 1 ( u ) , J 2 ( u 2 ∗ ) = u ∈ U k 0 ∑ t = 0 T − 1 u ( t ) ⊤ Q u ( t ) ≤ B max J 2 ( u ) . The proof can be found in Appendix C.1.
Necessary optimality conditions
We now derive necessary optimality conditions for Problems 1 and 2 by applying the discrete-time version of the Pontryagin maximum principle [8 ] . For both problems, we introduce a costate sequence μ ( t ) \mu(t) μ ( t ) y ( t ) y(t) y ( t ) μ ( t ) = ( μ s ( t ) , { μ a [ k ] ( t ) } k , { μ d [ k ] ( t ) } k , { μ x [ k ] ( t ) } k ) \mu(t)=(\mu_s(t),\{\mu_{a^{[k]}}(t)\}_k,\{\mu_{d^{[k]}}(t)\}_k,\{\mu_{x^{[k]}}(t)\}_k) μ ( t ) = ( μ s ( t ) , { μ a [ k ] ( t ) } k , { μ d [ k ] ( t ) } k , { μ x [ k ] ( t ) } k ) μ x i [ k ] ( t ) \mu_{x_i^{[k]}}(t) μ x i [ k ] ( t ) x i [ k ] ( t ) x_i^{[k]}(t) x i [ k ] ( t ) u i [ k ] ( t ) u_i^{[k]}(t) u i [ k ] ( t ) x i [ k ] ( t + 1 ) x_i^{[k]}(t+1) x i [ k ] ( t + 1 )
∂ x i [ k ] ( t + 1 ) ∂ u i [ k ] ( t ) = ( 1 − λ i [ k ] − ξ i [ k ] ) ( 1 − p i [ k ] ) , (20) \frac{\partial x_i^{[k]}(t+1)}{\partial u_i^{[k]}(t)}=(1-\lambda_i^{[k]}-\xi_i^{[k]})(1-p_i^{[k]}), \tag*{(20)} ∂ u i [ k ] ( t ) ∂ x i [ k ] ( t + 1 ) = ( 1 − λ i [ k ] − ξ i [ k ] ) ( 1 − p i [ k ] ) , (20) which is nonnegative for all i ∈ V i\in\mathcal{V} i ∈ V k ∈ K k\in\mathcal{K} k ∈ K
∂ ∂ u i [ k ] ( − u ( t ) ⊤ Q u ( t ) ) = − 2 q i [ k ] u i [ k ] ( t ) . (21) \frac{\partial}{\partial u_i^{[k]}}\left(-u(t)^\top Q u(t)\right)=-2q_i^{[k]}u_i^{[k]}(t). \tag*{(21)} ∂ u i [ k ] ∂ ( − u ( t ) ⊤ Q u ( t ) ) = − 2 q i [ k ] u i [ k ] ( t ) . (21) Problem 1 The Hamiltonian is H 1 , t ( y ( t ) , u ( t ) , μ ( t + 1 ) ) : = ℓ 1 ( y ( t ) , u ( t ) ) + μ ( t + 1 ) ⊤ f ( y ( t ) , u ( t ) ) \mathcal{H}_{1,t}(y(t),u(t),\mu(t+1)):=\ell_1(y(t),u(t))+\mu(t+1)^\top f(y(t),u(t)) H 1 , t ( y ( t ) , u ( t ) , μ ( t + 1 )) := ℓ 1 ( y ( t ) , u ( t )) + μ ( t + 1 ) ⊤ f ( y ( t ) , u ( t )) u 1 ∗ u_1^* u 1 ∗ y ∗ y^* y ∗ μ ( t ) \mu(t) μ ( t ) μ ( T ) = 0 \mu(T)=0 μ ( T ) = 0 μ ( t ) = ∇ y H 1 , t ( y ∗ ( t ) , u 1 ∗ ( t ) , μ ( t + 1 ) ) \mu(t)=\nabla_y\mathcal{H}_{1,t}(y^*(t),u_1^*(t),\mu(t+1)) μ ( t ) = ∇ y H 1 , t ( y ∗ ( t ) , u 1 ∗ ( t ) , μ ( t + 1 )) t = T − 1 , … , 0 t=T-1,\ldots,0 t = T − 1 , … , 0
u 1 ∗ ( t ) ∈ arg max u ∈ U H 1 , t ( y ∗ ( t ) , u , μ ( t + 1 ) ) , t = 0 , … , T − 1. u_1^*(t)\in\operatorname*{arg\,max}_{u\in U}\mathcal{H}_{1,t}(y^*(t),u,\mu(t+1)),\qquad t=0,\ldots,T-1. u 1 ∗ ( t ) ∈ u ∈ U arg max H 1 , t ( y ∗ ( t ) , u , μ ( t + 1 )) , t = 0 , … , T − 1. Differentiating H 1 , t \mathcal{H}_{1,t} H 1 , t u i [ k ] ( t ) u_i^{[k]}(t) u i [ k ] ( t )
∂ H 1 , t ∂ u i [ k ] = − 2 q i [ k ] u i [ k ] ( t ) + ( 1 − λ i [ k ] − ξ i [ k ] ) ( 1 − p i [ k ] ) μ x i [ k ] ( t + 1 ) . \frac{\partial\mathcal{H}_{1,t}}{\partial u_i^{[k]}}=-2q_i^{[k]}u_i^{[k]}(t)+(1-\lambda_i^{[k]}-\xi_i^{[k]})(1-p_i^{[k]})\mu_{x_i^{[k]}}(t+1). ∂ u i [ k ] ∂ H 1 , t = − 2 q i [ k ] u i [ k ] ( t ) + ( 1 − λ i [ k ] − ξ i [ k ] ) ( 1 − p i [ k ] ) μ x i [ k ] ( t + 1 ) . For a fixed state-costate pair, maximizing the Hamiltonian with respect to u i [ k ] ( t ) u_i^{[k]}(t) u i [ k ] ( t )
u ^ 1 , i [ k ] ( t ) = 1 2 q i [ k ] ( 1 − λ i [ k ] − ξ i [ k ] ) ( 1 − p i [ k ] ) μ x i [ k ] ( t + 1 ) . (22) \widehat{u}_{1,i}^{[k]}(t)=\frac{1}{2q_i^{[k]}}(1-\lambda_i^{[k]}-\xi_i^{[k]})(1-p_i^{[k]})\mu_{x_i^{[k]}}(t+1). \tag*{(22)} u 1 , i [ k ] ( t ) = 2 q i [ k ] 1 ( 1 − λ i [ k ] − ξ i [ k ] ) ( 1 − p i [ k ] ) μ x i [ k ] ( t + 1 ) . (22) Hence, the pointwise maximizer entering the PMP conditions is
u 1 , i [ k ] ∗ ( t ) = min { 1 , max { 0 , u ^ 1 , i [ k ] ( t ) } } , i ∈ V , k ∈ K . (23) u_{1,i}^{[k]*}(t)=\min\left\{1,\max\left\{0,\widehat{u}_{1,i}^{[k]}(t)\right\}\right\},\qquad i\in\mathcal{V},\ k\in\mathcal{K}. \tag*{(23)} u 1 , i [ k ] ∗ ( t ) = min { 1 , max { 0 , u 1 , i [ k ] ( t ) } } , i ∈ V , k ∈ K . (23) For a fixed state-costate pair, H 1 , t \mathcal{H}_{1,t} H 1 , t u ( t ) u(t) u ( t ) − u ( t ) ⊤ Q u ( t ) -u(t)^\top Q u(t) − u ( t ) ⊤ Q u ( t )
Problem 2 Since the horizon is finite, Problem 2 can be reduced to a finite-dimensional nonlinear program in the control sequence. For the budget constraint g ( u ) : = ∑ t = 0 T − 1 u ( t ) ⊤ Q u ( t ) − B ≤ 0 g(u):=\sum_{t=0}^{T-1}u(t)^\top Q u(t)-B\le0 g ( u ) := ∑ t = 0 T − 1 u ( t ) ⊤ Q u ( t ) − B ≤ 0 B > 0 B>0 B > 0 u ≠ 0 u\ne0 u = 0 ∇ g ( u ) ≠ 0 \nabla g(u)\ne0 ∇ g ( u ) = 0 η ≥ 0 \eta\ge0 η ≥ 0 (19) , then the augmented Hamiltonian is H 2 , t ( y ( t ) , u ( t ) , μ ( t + 1 ) ) : = μ ( t + 1 ) ⊤ f ( y ( t ) , u ( t ) ) − η u ( t ) ⊤ Q u ( t ) \mathcal{H}_{2,t}(y(t),u(t),\mu(t+1)):=\mu(t+1)^\top f(y(t),u(t))-\eta u(t)^\top Q u(t) H 2 , t ( y ( t ) , u ( t ) , μ ( t + 1 )) := μ ( t + 1 ) ⊤ f ( y ( t ) , u ( t )) − η u ( t ) ⊤ Q u ( t ) μ x i [ k 0 ] ( T ) = 1 \mu_{x_i^{[k_0]}}(T)=1 μ x i [ k 0 ] ( T ) = 1 i ∈ V i\in\mathcal{V} i ∈ V μ ( t ) = ∇ y H 2 , t ( y ∗ ( t ) , u ∗ ( t ) , μ ( t + 1 ) ) \mu(t)=\nabla_y\mathcal{H}_{2,t}(y^*(t),u^*(t),\mu(t+1)) μ ( t ) = ∇ y H 2 , t ( y ∗ ( t ) , u ∗ ( t ) , μ ( t + 1 )) t = T − 1 , … , 0 t=T-1,\ldots,0 t = T − 1 , … , 0
u 2 ∗ ( t ) ∈ arg max u ∈ U k 0 H 2 , t ( y ∗ ( t ) , u , μ ( t + 1 ) ) , t = 0 , … , T − 1 , u_2^*(t)\in\arg\max_{u\in\mathcal{U}_{k_0}}\mathcal{H}_{2,t}(y^*(t),u,\mu(t+1)),\qquad t=0,\ldots,T-1, u 2 ∗ ( t ) ∈ arg u ∈ U k 0 max H 2 , t ( y ∗ ( t ) , u , μ ( t + 1 )) , t = 0 , … , T − 1 , together with the complementarity conditions
η ≥ 0 , ∑ t = 0 T − 1 u 2 ∗ ( t ) ⊤ Q u 2 ∗ ( t ) ≤ B , \eta\ge0,\qquad\sum_{t=0}^{T-1}u_2^*(t)^\top Q u_2^*(t)\le B, η ≥ 0 , t = 0 ∑ T − 1 u 2 ∗ ( t ) ⊤ Q u 2 ∗ ( t ) ≤ B , η ( ∑ t = 0 T − 1 u 2 ∗ ( t ) ⊤ Q u 2 ∗ ( t ) − B ) = 0. (24) \eta\left(\sum_{t=0}^{T-1}u_2^*(t)^\top Q u_2^*(t)-B\right)=0. \tag*{(24)} η ( t = 0 ∑ T − 1 u 2 ∗ ( t ) ⊤ Q u 2 ∗ ( t ) − B ) = 0. (24) Differentiating H 2 , t \mathcal{H}_{2,t} H 2 , t u i [ k 0 ] ( t ) u_i^{[k_0]}(t) u i [ k 0 ] ( t ) (20) –(21) gives
∂ H 2 , t ∂ u i [ k 0 ] = − 2 η q i [ k 0 ] u i [ k 0 ] ( t ) + ( 1 − λ i [ k 0 ] − ξ i [ k 0 ] ) ( 1 − p i [ k 0 ] ) μ x i [ k 0 ] ( t + 1 ) . \frac{\partial\mathcal{H}_{2,t}}{\partial u_i^{[k_0]}}=-2\eta q_i^{[k_0]}u_i^{[k_0]}(t)+(1-\lambda_i^{[k_0]}-\xi_i^{[k_0]})(1-p_i^{[k_0]})\mu_{x_i^{[k_0]}}(t+1). ∂ u i [ k 0 ] ∂ H 2 , t = − 2 η q i [ k 0 ] u i [ k 0 ] ( t ) + ( 1 − λ i [ k 0 ] − ξ i [ k 0 ] ) ( 1 − p i [ k 0 ] ) μ x i [ k 0 ] ( t + 1 ) . Since the control is restricted to the target technology, the maximization is performed only over the components u i [ k 0 ] ( t ) u_i^{[k_0]}(t) u i [ k 0 ] ( t ) η > 0 \eta>0 η > 0
u ^ 2 , i [ k 0 ] ( t ) = 1 2 η q i [ k 0 ] ( 1 − λ i [ k 0 ] − ξ i [ k 0 ] ) ( 1 − p i [ k 0 ] ) μ x i [ k 0 ] ( t + 1 ) , \widehat{u}_{2,i}^{[k_0]}(t)=\frac{1}{2\eta q_i^{[k_0]}}(1-\lambda_i^{[k_0]}-\xi_i^{[k_0]})(1-p_i^{[k_0]})\mu_{x_i^{[k_0]}}(t+1), u 2 , i [ k 0 ] ( t ) = 2 η q i [ k 0 ] 1 ( 1 − λ i [ k 0 ] − ξ i [ k 0 ] ) ( 1 − p i [ k 0 ] ) μ x i [ k 0 ] ( t + 1 ) , and the corresponding pointwise maximizer is
u 2 , i [ k 0 ] ∗ ( t ) = min { 1 , max { 0 , u ^ 2 , i [ k 0 ] ( t ) } } , i ∈ V , (25) u_{2,i}^{[k_0]*}(t)=\min\left\{1,\max\left\{0,\widehat{u}_{2,i}^{[k_0]}(t)\right\}\right\},\qquad i\in\mathcal{V}, \tag*{(25)} u 2 , i [ k 0 ] ∗ ( t ) = min { 1 , max { 0 , u 2 , i [ k 0 ] ( t ) } } , i ∈ V , (25) while u 2 , i [ k ] ∗ ( t ) = 0 u_{2,i}^{[k]*}(t)=0 u 2 , i [ k ] ∗ ( t ) = 0 i ∈ V i\in\mathcal{V} i ∈ V k ≠ k 0 k\ne k_0 k = k 0 η \eta η (24) . If the budget constraint is active, then ∑ t = 0 T − 1 u ∗ ( t ) ⊤ Q u ∗ ( t ) = B \sum_{t=0}^{T-1}u^*(t)^\top Q u^*(t)=B ∑ t = 0 T − 1 u ∗ ( t ) ⊤ Q u ∗ ( t ) = B η = 0 \eta=0 η = 0 η = 0 \eta=0 η = 0 (23) and (25) . This is because the control does not act directly on adoption, but only on the controlled baseline opinions. The effect of adoption on the computed intervention is encoded in the costate variables. In particular, the component μ x i [ k ] ( t + 1 ) \mu_{x_i^{[k]}}(t+1) μ x i [ k ] ( t + 1 ) i i i k k k
Numerical computation of extremals
The necessary optimality conditions derived above define a coupled forward-backward system: the state dynamics is initialized at y ( 0 ) = y 0 y(0)=y_0 y ( 0 ) = y 0 [35 ] . Starting from an admissible control sequence u ( 0 ) u^{(0)} u ( 0 ) h h h (23) for Problem 1, (25) for Problem 2. To improve numerical stability, we use the relaxed update
u ( h + 1 ) ( t ) = ϖ u ~ ( h + 1 ) ( t ) + ( 1 − ϖ ) u ( h ) ( t ) , ϖ ∈ ( 0 , 1 ) , (26) u^{(h+1)}(t)=\varpi\widetilde{u}^{(h+1)}(t)+(1-\varpi)u^{(h)}(t),\qquad\varpi\in(0,1), \tag*{(26)} u ( h + 1 ) ( t ) = ϖ u ( h + 1 ) ( t ) + ( 1 − ϖ ) u ( h ) ( t ) , ϖ ∈ ( 0 , 1 ) , (26) where u ~ ( h + 1 ) ( t ) \widetilde{u}^{(h+1)}(t) u ( h + 1 ) ( t ) η > 0 \eta>0 η > 0 ∑ t = 0 T − 1 u ( t ) ⊤ Q u ( t ) = B \sum_{t=0}^{T-1}u(t)^\top Q u(t)=B ∑ t = 0 T − 1 u ( t ) ⊤ Q u ( t ) = B
We now illustrate the two optimal control problems on a synthetic network of n = 10 n=10 n = 10 m = 5 m=5 m = 5 (12) is verified numerically. The time horizon is T = 300 T=300 T = 300 Q Q Q q i [ k ] = 0.8 q_i^{[k]}=0.8 q i [ k ] = 0.8 i , k i,k i , k (26) . Figure 5 shows, for each technology k k k 1 n ∑ i a i [ k ] ( t ) \frac{1}{n}\sum_i a_i^{[k]}(t) n 1 ∑ i a i [ k ] ( t ) 1 n ∑ i d i [ k ] ( t ) \frac{1}{n}\sum_i d_i^{[k]}(t) n 1 ∑ i d i [ k ] ( t ) Figure 6 displays the average control effort over time and its allocation across alternatives. The controller initially concentrates resources on the most responsive alternatives, then progressively rebalances as adoption levels saturate. This time-varying reallocation reflects the backward propagation of the costate: as adoption of a technology approaches its controlled equilibrium, the marginal value μ x i [ k ] \mu_{x_i^{[k]}} μ x i [ k ] Figure 7 shows the mean adoption and dissatisfied fraction for the target technology k 0 = 1 k_0=1 k 0 = 1 T T T k 0 k_0 k 0 a [ k 0 ] a^{[k_0]} a [ k 0 ] δ [ k 0 ] a [ k 0 ] \delta^{[k_0]}a^{[k_0]} δ [ k 0 ] a [ k 0 ] D [ k 0 ] D^{[k_0]} D [ k 0 ]
Figure 5. Problem 1 (m = 5 m=5 m = 5 T = 300 T=300 T = 300
Figure 6. Average control effort for Problems 1 and 2 (top), and per-alternative allocation for Problem 1 (bottom).
Figure 7. Problem 2 with target k 0 = 1 k_0=1 k 0 = 1 T = 300 T=300 T = 300
Code availability. All numerical experiments can be reproduced using the code and parameter sets available at https://github.com/martinalutto/competing-technologies-simulations.
Conclusion
We proposed a networked adoption-opinion framework for multiple competing alternatives, jointly capturing social influence, adoption, dissatisfaction, and opinion-dependent switching. We established existence of an adoption-diffused equilibrium and provided sufficient conditions for uniqueness and convergence. The comparison between m = 2 m=2 m = 2 m > 2 m>2 m > 2
Two main limitations remain. First, the model has not yet been identified or validated against empirical data, so the results are structural rather than predictive. Second, the current compartmental structure does not capture permanent exit or true extinction of an alternative. Future work will address these aspects, less conservative uniqueness and convergence conditions, and time-varying interaction networks.
Proofs of the results in Section 2
Proof of Proposition 1
Assume that y ( t ) ∈ Y y(t)\in\mathcal{Y} y ( t ) ∈ Y t ≥ 0 t\geq0 t ≥ 0 y ( t + 1 ) y(t+1) y ( t + 1 ) Y \mathcal{Y} Y i ∈ V i\in\mathcal{V} i ∈ V
s i ( t + 1 ) = s i ( t ) ( 1 − ∑ k ∈ K β i [ k ] x i [ k ] ( t ) ∑ j ∈ V W i j [ k ] ( t ) ) ≥ 0. s_i(t+1)=s_i(t)\left(1-\sum_{k\in\mathcal{K}}\beta_i^{[k]}x_i^{[k]}(t)\sum_{j\in\mathcal{V}}W_{ij}^{[k]}(t)\right)\geq0. s i ( t + 1 ) = s i ( t ) 1 − k ∈ K ∑ β i [ k ] x i [ k ] ( t ) j ∈ V ∑ W ij [ k ] ( t ) ≥ 0. For every i ∈ V i\in\mathcal{V} i ∈ V k ∈ K k\in\mathcal{K} k ∈ K a i [ k ] ( t + 1 ) ≥ ( 1 − δ i [ k ] ) a i [ k ] ( t ) a_i^{[k]}(t+1)\geq(1-\delta_i^{[k]})a_i^{[k]}(t) a i [ k ] ( t + 1 ) ≥ ( 1 − δ i [ k ] ) a i [ k ] ( t ) δ i [ k ] ∈ [ 0 , 1 ] \delta_i^{[k]}\in[0,1] δ i [ k ] ∈ [ 0 , 1 ] a i [ k ] ( t + 1 ) ≥ 0 a_i^{[k]}(t+1)\geq0 a i [ k ] ( t + 1 ) ≥ 0 x i [ k ] ( t ) ∈ [ 0 , 1 ] x_i^{[k]}(t)\in[0,1] x i [ k ] ( t ) ∈ [ 0 , 1 ] d i [ k ] ( t + 1 ) ≥ 0 d_i^{[k]}(t+1)\geq0 d i [ k ] ( t + 1 ) ≥ 0 k ∈ K k\in\mathcal{K} k ∈ K s i ( t + 1 ) + ∑ k ∈ K ( a i [ k ] ( t + 1 ) + d i [ k ] ( t + 1 ) ) = s i ( t ) + ∑ k ∈ K ( a i [ k ] ( t ) + d i [ k ] ( t ) ) = 1 s_i(t+1)+\sum_{k\in\mathcal{K}}\left(a_i^{[k]}(t+1)+d_i^{[k]}(t+1)\right)=s_i(t)+\sum_{k\in\mathcal{K}}\left(a_i^{[k]}(t)+d_i^{[k]}(t)\right)=1 s i ( t + 1 ) + ∑ k ∈ K ( a i [ k ] ( t + 1 ) + d i [ k ] ( t + 1 ) ) = s i ( t ) + ∑ k ∈ K ( a i [ k ] ( t ) + d i [ k ] ( t ) ) = 1 i ∈ V i\in\mathcal{V} i ∈ V
s i ( t + 1 ) + ∑ k ∈ K ( a i [ k ] ( t + 1 ) + d i [ k ] ( t + 1 ) ) = 1. s_i(t+1)+\sum_{k\in\mathcal{K}}\left(a_i^{[k]}(t+1)+d_i^{[k]}(t+1)\right)=1. s i ( t + 1 ) + k ∈ K ∑ ( a i [ k ] ( t + 1 ) + d i [ k ] ( t + 1 ) ) = 1. from (1d) and row-stochasticity of W ~ \widetilde{W} W W W W x [ k ] ( t + 1 ) ∈ [ 0 , 1 ] n x^{[k]}(t+1)\in[0,1]^n x [ k ] ( t + 1 ) ∈ [ 0 , 1 ] n x [ k ] ( t ) ∈ [ 0 , 1 ] n x^{[k]}(t)\in[0,1]^n x [ k ] ( t ) ∈ [ 0 , 1 ] n a [ k ] ( t ) ∈ [ 0 , 1 ] n a^{[k]}(t)\in[0,1]^n a [ k ] ( t ) ∈ [ 0 , 1 ] n y ( t + 1 ) y(t+1) y ( t + 1 ) Y \mathcal{Y} Y Y \mathcal{Y} Y t ≥ 0 t\geq0 t ≥ 0
Proof of Corollary 1
(i) By Proposition 1, the subtracted term in (1a) is nonnegative for all i ∈ V i\in\mathcal{V} i ∈ V s i ( t + 1 ) ≤ s i ( t ) s_i(t+1)\leq s_i(t) s i ( t + 1 ) ≤ s i ( t ) i ∈ V i\in\mathcal{V} i ∈ V x [ k ] ( t + 1 ) ≥ ( I − Λ [ k ] − Ξ [ k ] ) p [ k ] > 0 x^{[k]}(t+1)\geq(I-\Lambda^{[k]}-\Xi^{[k]})p^{[k]}>0 x [ k ] ( t + 1 ) ≥ ( I − Λ [ k ] − Ξ [ k ] ) p [ k ] > 0 t ≥ 0 t\geq0 t ≥ 0 k ∈ K k\in\mathcal{K} k ∈ K
Proofs of the results in Section 3
Proof of Proposition 2
With a [ k ] = 0 a^{[k]}=0 a [ k ] = 0 d [ k ] = 0 d^{[k]}=0 d [ k ] = 0 W ~ \widetilde{W} W λ i [ k ] + ξ i [ k ] < 1 \lambda_i^{[k]}+\xi_i^{[k]}<1 λ i [ k ] + ξ i [ k ] < 1 i i i ρ ( Λ [ k ] W ~ ) ≤ ∥ Λ [ k ] W ~ ∥ ∞ = max i λ i [ k ] < 1 \rho(\Lambda^{[k]}\widetilde{W})\leq\|\Lambda^{[k]}\widetilde{W}\|_\infty=\max_i\lambda_i^{[k]}<1 ρ ( Λ [ k ] W ) ≤ ∥ Λ [ k ] W ∥ ∞ = max i λ i [ k ] < 1 I − Λ [ k ] W ~ I-\Lambda^{[k]}\widetilde{W} I − Λ [ k ] W x A F [ k ] > 0 x_{AF}^{[k]}>0 x A F [ k ] > 0 ( I − Λ [ k ] W ~ ) − 1 ≥ 0 (I-\Lambda^{[k]}\widetilde{W})^{-1}\geq0 ( I − Λ [ k ] W ) − 1 ≥ 0 ( I − Λ [ k ] − Ξ [ k ] ) p [ k ] > 0 (I-\Lambda^{[k]}-\Xi^{[k]})p^{[k]}>0 ( I − Λ [ k ] − Ξ [ k ] ) p [ k ] > 0 p [ k ] ≤ 1 p^{[k]}\leq1 p [ k ] ≤ 1
( I − Λ [ k ] − Ξ [ k ] ) p [ k ] ≤ ( I − Λ [ k ] − Ξ [ k ] ) 1 ≤ ( I − Λ [ k ] ) 1 = ( I − Λ [ k ] W ~ ) 1 , (27) (I-\Lambda^{[k]}-\Xi^{[k]})p^{[k]}\leq(I-\Lambda^{[k]}-\Xi^{[k]})1
\leq(I-\Lambda^{[k]})1=(I-\Lambda^{[k]}\widetilde{W})1, \tag*{(27)} ( I − Λ [ k ] − Ξ [ k ] ) p [ k ] ≤ ( I − Λ [ k ] − Ξ [ k ] ) 1 ≤ ( I − Λ [ k ] ) 1 = ( I − Λ [ k ] W ) 1 , (27) where the last equality follows from W ~ 1 = 1 \widetilde{W}1=1 W 1 = 1 ( I − Λ [ k ] W ~ ) − 1 (I-\Lambda^{[k]}\widetilde{W})^{-1} ( I − Λ [ k ] W ) − 1 x A F [ k ] ≤ 1 x_{AF}^{[k]}\leq1 x A F [ k ] ≤ 1
Now fix any ε ∈ ( 0 , 1 ) \varepsilon\in(0,1) ε ∈ ( 0 , 1 ) y A F y_{AF} y A F s ( 0 ) = ( 1 − ε ) 1 s(0)=(1-\varepsilon)1 s ( 0 ) = ( 1 − ε ) 1 a [ k 0 ] ( 0 ) = ε 1 a^{[k_0]}(0)=\varepsilon1 a [ k 0 ] ( 0 ) = ε 1 k 0 ∈ K k_0\in\mathcal{K} k 0 ∈ K x [ k ] ( 0 ) = x A F [ k ] x^{[k]}(0)=x_{AF}^{[k]} x [ k ] ( 0 ) = x A F [ k ] k ∈ K k\in\mathcal{K} k ∈ K s ( t ) ≤ s ( 0 ) s(t)\leq s(0) s ( t ) ≤ s ( 0 ) t t t ∥ s ( t ) − 1 ∥ ∞ ≥ ε \|s(t)-1\|_\infty\geq\varepsilon ∥ s ( t ) − 1 ∥ ∞ ≥ ε t ≥ 0 t\geq0 t ≥ 0 ∥ y ( t ) − y A F ∥ ∞ ≥ ∥ s ( t ) − 1 ∥ ∞ ≥ ε ∀ t ≥ 0 \|y(t)-y_{AF}\|_\infty\geq\|s(t)-1\|_\infty\geq\varepsilon\ \forall t\geq0 ∥ y ( t ) − y A F ∥ ∞ ≥ ∥ s ( t ) − 1 ∥ ∞ ≥ ε ∀ t ≥ 0 ε > 0 \varepsilon>0 ε > 0 y A F y_{AF} y A F y A F y_{AF} y A F y A F y_{AF} y A F
Proof of Theorem 1
The proof proceeds in three steps. First, we identify a compact set containing the equilibrium opinion profiles. Second, for any fixed opinion profile, we uniquely determine the corresponding adoption and dissatisfaction variables. Finally, we formulate the equilibrium conditions as a continuous map and apply Brouwer’s fixed-point theorem.
At equilibrium, (1d) yields (7). Since ( I − Λ [ k ] W ~ ) − 1 (I-\Lambda^{[k]}\widetilde{W})^{-1} ( I − Λ [ k ] W ) − 1 a [ k ] ⋆ ≥ 0 a^{[k]\star}\geq0 a [ k ] ⋆ ≥ 0 x [ k ] ⋆ ≥ x A F [ k ] > 0 x^{[k]\star}\geq x_{AF}^{[k]}>0 x [ k ] ⋆ ≥ x A F [ k ] > 0 x [ k ] ⋆ ≤ 1 x^{[k]\star}\leq1 x [ k ] ⋆ ≤ 1 x ⋆ x^\star x ⋆ C : = ∏ k ∈ K [ x A F [ k ] , 1 ] = { x ∈ [ 0 , 1 ] n m : x A F [ k ] ≤ x [ k ] ≤ 1 } \mathcal{C}:=\prod_{k\in\mathcal{K}}[x_{AF}^{[k]},1]=\left\{x\in[0,1]^{nm}:x_{AF}^{[k]}\leq x^{[k]}\leq1\right\} C := ∏ k ∈ K [ x A F [ k ] , 1 ] = { x ∈ [ 0 , 1 ] nm : x A F [ k ] ≤ x [ k ] ≤ 1 } x ⋆ ∈ C x^\star\in\mathcal{C} x ⋆ ∈ C a ⋆ a^\star a ⋆ d ⋆ d^\star d ⋆ s i ⋆ = 0 s_i^\star=0 s i ⋆ = 0
d i [ k ] ⋆ a i [ k ] ⋆ = γ i [ k ] x i [ k ] ⋆ ∑ ℓ ∈ K ∖ { k } d i [ ℓ ] ⋆ , (28) d_i^{[k]\star}a_i^{[k]\star}=\gamma_i^{[k]}x_i^{[k]\star}\sum_{\ell\in\mathcal{K}\setminus\{k\}}d_i^{[\ell]\star}, \tag*{(28)} d i [ k ] ⋆ a i [ k ] ⋆ = γ i [ k ] x i [ k ] ⋆ ℓ ∈ K ∖ { k } ∑ d i [ ℓ ] ⋆ , (28) d i [ k ] ⋆ a i [ k ] ⋆ = ( ∑ ℓ ∈ K ∖ { k } γ i [ ℓ ] x i [ ℓ ] ⋆ ) d i [ k ] ⋆ , (29) d_i^{[k]\star}a_i^{[k]\star}=\left(\sum_{\ell\in\mathcal{K}\setminus\{k\}}\gamma_i^{[\ell]}x_i^{[\ell]\star}\right)d_i^{[k]\star}, \tag*{(29)} d i [ k ] ⋆ a i [ k ] ⋆ = ℓ ∈ K ∖ { k } ∑ γ i [ ℓ ] x i [ ℓ ] ⋆ d i [ k ] ⋆ , (29) for every i ∈ V i\in\mathcal{V} i ∈ V k ∈ K k\in\mathcal{K} k ∈ K γ i [ k ] x i [ k ] ⋆ ( d ˉ i ⋆ − d i [ k ] ⋆ ) = ( x ˉ i ⋆ − γ i [ k ] x i [ k ] ⋆ ) d i [ k ] ⋆ \gamma_i^{[k]}x_i^{[k]\star}(\bar d_i^\star-d_i^{[k]\star})=(\bar x_i^\star-\gamma_i^{[k]}x_i^{[k]\star})d_i^{[k]\star} γ i [ k ] x i [ k ] ⋆ ( d ˉ i ⋆ − d i [ k ] ⋆ ) = ( x ˉ i ⋆ − γ i [ k ] x i [ k ] ⋆ ) d i [ k ] ⋆
d i [ k ] ⋆ = γ i [ k ] x i [ k ] ⋆ x ˉ i ⋆ d ˉ i ⋆ . (30) d_i^{[k]\star}=\frac{\gamma_i^{[k]}x_i^{[k]\star}}{\bar x_i^\star} \bar d_i^\star. \tag*{(30)} d i [ k ] ⋆ = x ˉ i ⋆ γ i [ k ] x i [ k ] ⋆ d ˉ i ⋆ . (30) Substituting (B.3) into (B.2b), we obtain
a i [ k ] ⋆ = γ i [ k ] x i [ k ] ⋆ δ i [ k ] ( 1 − γ i [ k ] x i [ k ] ⋆ x ˉ i ⋆ ) d ˉ i ⋆ . (31) a_i^{[k]\star}=\frac{\gamma_i^{[k]}x_i^{[k]\star}}{\delta_i^{[k]}}\left(1-\frac{\gamma_i^{[k]}x_i^{[k]\star}}{\bar x_i^\star}\right)\bar d_i^\star. \tag*{(31)} a i [ k ] ⋆ = δ i [ k ] γ i [ k ] x i [ k ] ⋆ ( 1 − x ˉ i ⋆ γ i [ k ] x i [ k ] ⋆ ) d ˉ i ⋆ . (31) Since s i ⋆ = 0 s_i^\star=0 s i ⋆ = 0 Y \mathcal{Y} Y ∑ k ∈ K ( a i [ k ] ⋆ + d i [ k ] ⋆ ) = 1 \sum_{k\in\mathcal{K}}(a_i^{[k]\star}+d_i^{[k]\star})=1 ∑ k ∈ K ( a i [ k ] ⋆ + d i [ k ] ⋆ ) = 1 i ∈ V i\in\mathcal{V} i ∈ V
d ˉ i ⋆ ∑ k ∈ K ( γ i [ k ] x i [ k ] ⋆ δ i [ k ] ( 1 − γ i [ k ] x i [ k ] ⋆ x ˉ i ⋆ ) + γ i [ k ] x i [ k ] ⋆ x ˉ i ⋆ ) = 1 , \bar d_i^\star\sum_{k\in\mathcal{K}}\left(\frac{\gamma_i^{[k]}x_i^{[k]\star}}{\delta_i^{[k]}}\left(1-\frac{\gamma_i^{[k]}x_i^{[k]\star}}{\bar x_i^\star}\right)+\frac{\gamma_i^{[k]}x_i^{[k]\star}}{\bar x_i^\star}\right)=1, d ˉ i ⋆ k ∈ K ∑ ( δ i [ k ] γ i [ k ] x i [ k ] ⋆ ( 1 − x ˉ i ⋆ γ i [ k ] x i [ k ] ⋆ ) + x ˉ i ⋆ γ i [ k ] x i [ k ] ⋆ ) = 1 , that is, using (5),
d ˉ i ⋆ = ( ∑ k ∈ K γ i [ k ] x i [ k ] ⋆ δ i [ k ] ( 1 − γ i [ k ] x i [ k ] ⋆ x ˉ i ⋆ ) + 1 ) − 1 . (32) \bar d_i^\star=\left(\sum_{k\in\mathcal{K}}\frac{\gamma_i^{[k]}x_i^{[k]\star}}{\delta_i^{[k]}}\left(1-\frac{\gamma_i^{[k]}x_i^{[k]\star}}{\bar x_i^\star}\right)+1\right)^{-1}. \tag*{(32)} d ˉ i ⋆ = ( k ∈ K ∑ δ i [ k ] γ i [ k ] x i [ k ] ⋆ ( 1 − x ˉ i ⋆ γ i [ k ] x i [ k ] ⋆ ) + 1 ) − 1 . (32) Thus, for every x ∈ C x\in\mathcal{C} x ∈ C a ( x ) a(x) a ( x ) d ( x ) d(x) d ( x ) x x x a [ k ] ( x ) , d [ k ] ( x ) ∈ [ 0 , 1 ] n a^{[k]}(x),d^{[k]}(x)\in[0,1]^n a [ k ] ( x ) , d [ k ] ( x ) ∈ [ 0 , 1 ] n L : C → [ 0 , 1 ] n m L:\mathcal{C}\to[0,1]^{nm} L : C → [ 0 , 1 ] nm
L [ k ] ( x ) : = ( I − Λ [ k ] W ~ ) − 1 ( ( I − Λ [ k ] − Ξ [ k ] ) p [ k ] + Ξ [ k ] W ~ a [ k ] ( x ) ) . (33) L^{[k]}(x):=(I-\Lambda^{[k]}\widetilde{W})^{-1}\left((I-\Lambda^{[k]}-\Xi^{[k]})p^{[k]}+\Xi^{[k]}\widetilde{W}a^{[k]}(x)\right). \tag*{(33)} L [ k ] ( x ) := ( I − Λ [ k ] W ) − 1 ( ( I − Λ [ k ] − Ξ [ k ] ) p [ k ] + Ξ [ k ] W a [ k ] ( x ) ) . (33) By continuity of a [ k ] ( x ) a^{[k]}(x) a [ k ] ( x ) L L L C \mathcal{C} C L ( C ) ⊆ C L(\mathcal{C})\subseteq\mathcal{C} L ( C ) ⊆ C x ∈ C x\in\mathcal{C} x ∈ C a [ k ] ( x ) ≥ 0 a^{[k]}(x)\geq0 a [ k ] ( x ) ≥ 0 a [ k ] ( x ) ≤ 1 a^{[k]}(x)\leq1 a [ k ] ( x ) ≤ 1 W W W W a [ k ] ( x ) ≤ 1 Wa^{[k]}(x)\leq1 W a [ k ] ( x ) ≤ 1 x A F [ k ] ≤ L [ k ] ( x ) ≤ 1 x_{AF}^{[k]}\leq L^{[k]}(x)\leq1 x A F [ k ] ≤ L [ k ] ( x ) ≤ 1 k ∈ K k\in\mathcal{K} k ∈ K L ( x ) ∈ C L(x)\in\mathcal{C} L ( x ) ∈ C x ∈ C x\in\mathcal{C} x ∈ C C \mathcal{C} C x ⋆ ∈ C x^\star\in\mathcal{C} x ⋆ ∈ C L ( x ⋆ ) = x ⋆ L(x^\star)=x^\star L ( x ⋆ ) = x ⋆ a ⋆ : = a ( x ⋆ ) a^\star:=a(x^\star) a ⋆ := a ( x ⋆ ) d ⋆ : = d ( x ⋆ ) d^\star:=d(x^\star) d ⋆ := d ( x ⋆ ) y ⋆ y^\star y ⋆ x ⋆ ∈ C x^\star\in\mathcal{C} x ⋆ ∈ C x [ k ] ⋆ > 0 x^{[k]\star}>0 x [ k ] ⋆ > 0 k k k x ˉ i ⋆ > 0 \bar x_i^\star>0 x ˉ i ⋆ > 0 x ˉ i ⋆ − γ i [ k ] x i [ k ] ⋆ = ∑ ℓ ∈ K ∖ { k } γ i [ ℓ ] x i [ ℓ ] ⋆ > 0 \bar x_i^\star-\gamma_i^{[k]}x_i^{[k]\star}=\sum_{\ell\in\mathcal{K}\setminus\{k\}}\gamma_i^{[\ell]}x_i^{[\ell]\star}>0 x ˉ i ⋆ − γ i [ k ] x i [ k ] ⋆ = ∑ ℓ ∈ K ∖ { k } γ i [ ℓ ] x i [ ℓ ] ⋆ > 0 d i [ k ] ⋆ > 0 d_i^{[k]\star}>0 d i [ k ] ⋆ > 0 a i [ k ] ⋆ > 0 a_i^{[k]\star}>0 a i [ k ] ⋆ > 0 i ∈ V i\in\mathcal{V} i ∈ V k ∈ K k\in\mathcal{K} k ∈ K
Proof of Proposition 3
Let J : = { i ∈ V : s i ⋆ = 0 } \mathcal{J}:=\{i\in\mathcal{V}:s_i^\star=0\} J := { i ∈ V : s i ⋆ = 0 } J ≠ ∅ \mathcal{J}\neq\varnothing J = ∅ J ≠ V \mathcal{J}\neq\mathcal{V} J = V i ∈ J i\in\mathcal{J} i ∈ J s i ⋆ = 0 s_i^\star=0 s i ⋆ = 0 ∑ k ∈ K ( a i [ k ] ∗ + d i [ k ] ∗ ) = 1 \sum_{k\in\mathcal{K}}(a_i^{[k]*}+d_i^{[k]*})=1 ∑ k ∈ K ( a i [ k ] ∗ + d i [ k ] ∗ ) = 1 d i [ k ] ∗ > 0 d_i^{[k]*}>0 d i [ k ] ∗ > 0 i ∈ V i\in\mathcal{V} i ∈ V k ∈ K k\in\mathcal{K} k ∈ K γ i [ k ] > 0 \gamma_i^{[k]}>0 γ i [ k ] > 0 γ i [ k ] x i [ k ] ∗ > 0 \gamma_i^{[k]}x_i^{[k]*}>0 γ i [ k ] x i [ k ] ∗ > 0 a i [ k ] = 0 a_i^{[k]}=0 a i [ k ] = 0 k k k ∑ ℓ ∈ K ∖ { k } d i [ ℓ ] ∗ = 0 \sum_{\ell\in\mathcal{K}\setminus\{k\}}d_i^{[\ell]*}=0 ∑ ℓ ∈ K ∖ { k } d i [ ℓ ] ∗ = 0 k k k d i [ k ] ∗ = 0 d_i^{[k]*}=0 d i [ k ] ∗ = 0 k k k k 0 ∈ K k_0\in\mathcal{K} k 0 ∈ K a i [ k 0 ] ∗ > 0 a_i^{[k_0]*}>0 a i [ k 0 ] ∗ > 0 W W W J \mathcal{J} J V \mathcal{V} V p ∈ J p\in\mathcal{J} p ∈ J q ∉ J q\notin\mathcal{J} q ∈ / J W q p > 0 W_{qp}>0 W q p > 0 k 0 ∈ K k_0\in\mathcal{K} k 0 ∈ K a p [ k 0 ] ∗ > 0 a_p^{[k_0]*}>0 a p [ k 0 ] ∗ > 0 q ∉ J q\notin\mathcal{J} q ∈ / J s q ∗ > 0 s_q^*>0 s q ∗ > 0
0 = s q ∗ ∑ k ∈ K β q [ k ] x q [ k ] ∗ ∑ r ∈ V W q r a r [ k ] ∗ . 0=s_q^*\sum_{k\in\mathcal{K}}\beta_q^{[k]}x_q^{[k]*}\sum_{r\in\mathcal{V}}W_{qr}a_r^{[k]*}. 0 = s q ∗ k ∈ K ∑ β q [ k ] x q [ k ] ∗ r ∈ V ∑ W q r a r [ k ] ∗ . However, the term corresponding to k = k 0 k=k_0 k = k 0 s q ∗ β q [ k 0 ] x q [ k 0 ] ∗ W q p a p [ k 0 ] ∗ > 0 s_q^*\beta_q^{[k_0]}x_q^{[k_0]*}W_{qp}a_p^{[k_0]*}>0 s q ∗ β q [ k 0 ] x q [ k 0 ] ∗ W q p a p [ k 0 ] ∗ > 0 J = V \mathcal{J}=\mathcal{V} J = V s ∗ = 0 s^*=0 s ∗ = 0
Proof of Corollary 2
Let y ∗ y^* y ∗ s ∗ = 0 s^*=0 s ∗ = 0 i ∈ V i\in\mathcal{V} i ∈ V s ∗ = 0 s^*=0 s ∗ = 0 x i [ k ] ∗ > 0 x_i^{[k]*}>0 x i [ k ] ∗ > 0 d i [ k ] > 0 d_i^{[k]}>0 d i [ k ] > 0 a i [ k ] ∗ , d i [ k ] ∗ > 0 a_i^{[k]*},d_i^{[k]*}>0 a i [ k ] ∗ , d i [ k ] ∗ > 0 k k k y ∗ y^* y ∗ s i ∗ > 0 s_i^*>0 s i ∗ > 0 i i i β i [ k ] x i [ k ] ∗ \beta_i^{[k]}x_i^{[k]*} β i [ k ] x i [ k ] ∗ W a [ k ] = 0 Wa^{[k]}=0 W a [ k ] = 0 k k k W W W a [ k ] ≥ 0 a^{[k]}\ge0 a [ k ] ≥ 0 a [ k ] = 0 a^{[k]}=0 a [ k ] = 0 d [ k ] = 0 d^{[k]}=0 d [ k ] = 0 k k k s ∗ = 1 s^*=1 s ∗ = 1 x [ k ] ∗ = x A F x^{[k]*}=x_{\mathrm{AF}} x [ k ] ∗ = x AF y ∗ = y A F y^*=y_{\mathrm{AF}} y ∗ = y AF
Proof of Theorem 2
We establish uniqueness of the ADE by showing that the equilibrium map L L L
We endow C ⊂ R n m C\subset\mathbb{R}^{nm} C ⊂ R nm ℓ ∞ \ell_\infty ℓ ∞ ∥ x ∥ ∞ : = max k ∈ K ∥ x [ k ] ∥ ∞ \|x\|_\infty:=\max_{k\in\mathcal{K}}\|x^{[k]}\|_\infty ∥ x ∥ ∞ := max k ∈ K ∥ x [ k ] ∥ ∞ i ∈ V i\in\mathcal{V} i ∈ V k ∈ K k\in\mathcal{K} k ∈ K
g i [ k ] ( x ) : = γ i [ k ] x i [ k ] δ i [ k ] ( 1 − γ i [ k ] x i [ k ] x ˉ i ) , x ˉ i : = ∑ r ∈ K γ i [ r ] x i [ r ] . g_i^{[k]}(x):=\frac{\gamma_i^{[k]}x_i^{[k]}}{\delta_i^{[k]}}\left(1-\frac{\gamma_i^{[k]}x_i^{[k]}}{\bar{x}_i}\right),\qquad
\bar{x}_i:=\sum_{r\in\mathcal{K}}\gamma_i^{[r]}x_i^{[r]}. g i [ k ] ( x ) := δ i [ k ] γ i [ k ] x i [ k ] ( 1 − x ˉ i γ i [ k ] x i [ k ] ) , x ˉ i := r ∈ K ∑ γ i [ r ] x i [ r ] . Then, from (B.5)–(B.4), d ˉ i ( x ) = ( 1 + ∑ r ∈ K g i [ r ] ( x ) ) − 1 \bar{d}_i(x)=\left(1+\sum_{r\in\mathcal{K}}g_i^{[r]}(x)\right)^{-1} d ˉ i ( x ) = ( 1 + ∑ r ∈ K g i [ r ] ( x ) ) − 1 a i [ k ] ( x ) = g i [ k ] ( x ) d ˉ i ( x ) a_i^{[k]}(x)=g_i^{[k]}(x)\bar{d}_i(x) a i [ k ] ( x ) = g i [ k ] ( x ) d ˉ i ( x ) a [ k ] a^{[k]} a [ k ] i ∈ V i\in\mathcal{V} i ∈ V a i [ k ] ( x ) a_i^{[k]}(x) a i [ k ] ( x ) ( x i [ ℓ ] ) ℓ ∈ K (x_i^{[\ell]})_{\ell\in\mathcal{K}} ( x i [ ℓ ] ) ℓ ∈ K ∂ a i [ k ] ( x ) / ∂ x i [ ℓ ] \partial a_i^{[k]}(x)/\partial x_i^{[\ell]} ∂ a i [ k ] ( x ) / ∂ x i [ ℓ ] ℓ ∈ K \ell\in\mathcal{K} ℓ ∈ K
∂ g i [ k ] ∂ x i [ k ] = γ i [ k ] δ i [ k ] ( 1 − γ i [ k ] x i [ k ] x ˉ i ) 2 , ℓ = k . (34) \frac{\partial g_i^{[k]}}{\partial x_i^{[k]}}=\frac{\gamma_i^{[k]}}{\delta_i^{[k]}}\left(1-\frac{\gamma_i^{[k]}x_i^{[k]}}{\bar{x}_i}\right)^2,\qquad\ell=k. \tag*{(34)} ∂ x i [ k ] ∂ g i [ k ] = δ i [ k ] γ i [ k ] ( 1 − x ˉ i γ i [ k ] x i [ k ] ) 2 , ℓ = k . (34) ∂ g i [ k ] ∂ x i [ ℓ ] = γ i [ ℓ ] δ i [ k ] ( γ i [ k ] x i [ k ] x ˉ i ) 2 , ℓ ≠ k . (35) \frac{\partial g_i^{[k]}}{\partial x_i^{[\ell]}}=\frac{\gamma_i^{[\ell]}}{\delta_i^{[k]}}\left(\frac{\gamma_i^{[k]}x_i^{[k]}}{\bar{x}_i}\right)^2,\qquad\ell\ne k. \tag*{(35)} ∂ x i [ ℓ ] ∂ g i [ k ] = δ i [ k ] γ i [ ℓ ] ( x ˉ i γ i [ k ] x i [ k ] ) 2 , ℓ = k . (35) Since 0 ≤ γ i [ r ] x i [ r ] / x ˉ i ≤ 1 0\le\gamma_i^{[r]}x_i^{[r]}/\bar{x}_i\le1 0 ≤ γ i [ r ] x i [ r ] / x ˉ i ≤ 1
∣ ∂ g i [ k ] ∂ x i [ ℓ ] ∣ = γ i [ ℓ ] δ i [ k ] ≤ γ i [ ℓ ] δ i [ k ] , (36) \left|\frac{\partial g_i^{[k]}}{\partial x_i^{[\ell]}}\right|=\frac{\gamma_i^{[\ell]}}{\delta_i^{[k]}}\le\frac{\gamma_i^{[\ell]}}{\delta_i^{[k]}}, \tag*{(36)} ∂ x i [ ℓ ] ∂ g i [ k ] = δ i [ k ] γ i [ ℓ ] ≤ δ i [ k ] γ i [ ℓ ] , (36) for every k , ℓ ∈ K k,\ell\in\mathcal{K} k , ℓ ∈ K d ˉ i \bar{d}_i d ˉ i
∂ ∂ x i [ ℓ ] ( 1 + ∑ r ∈ K g ( x i [ r ] ) ) = γ i [ ℓ ] δ i [ ℓ ] − 2 ( γ i [ ℓ ] ) 2 x i [ ℓ ] δ i [ ℓ ] x ˉ i + γ i [ ℓ ] x ˉ i 2 ∑ r ∈ K ( γ i [ r ] x i [ r ] ) 2 δ i [ r ] . (37) \frac{\partial}{\partial x_i^{[\ell]}}\left(1+\sum_{r\in\mathcal{K}}g(x_i^{[r]})\right)=\frac{\gamma_i^{[\ell]}}{\delta_i^{[\ell]}}-\frac{2(\gamma_i^{[\ell]})^2x_i^{[\ell]}}{\delta_i^{[\ell]}\bar{x}_i}+\frac{\gamma_i^{[\ell]}}{\bar{x}_i^2}\sum_{r\in\mathcal{K}}\frac{(\gamma_i^{[r]}x_i^{[r]})^2}{\delta_i^{[r]}}. \tag*{(37)} ∂ x i [ ℓ ] ∂ ( 1 + r ∈ K ∑ g ( x i [ r ] ) ) = δ i [ ℓ ] γ i [ ℓ ] − δ i [ ℓ ] x ˉ i 2 ( γ i [ ℓ ] ) 2 x i [ ℓ ] + x ˉ i 2 γ i [ ℓ ] r ∈ K ∑ δ i [ r ] ( γ i [ r ] x i [ r ] ) 2 . (37) From γ i [ r ] x i [ r ] ≤ x ˉ i \gamma_i^{[r]}x_i^{[r]}\le\bar{x}_i γ i [ r ] x i [ r ] ≤ x ˉ i
∣ γ i [ ℓ ] δ i [ ℓ ] − 2 ( γ i [ ℓ ] ) 2 x i [ ℓ ] δ i [ ℓ ] x ˉ i − γ i [ ℓ ] x ˉ i ∣ = γ i [ ℓ ] δ i [ ℓ ] ∣ 1 − 2 γ i [ ℓ ] x i [ ℓ ] x ˉ i ∣ ≤ γ i [ ℓ ] δ i , min , \left|\frac{\gamma_i^{[\ell]}}{\delta_i^{[\ell]}}-\frac{2(\gamma_i^{[\ell]})^2x_i^{[\ell]}}{\delta_i^{[\ell]}\bar{x}_i}-\frac{\gamma_i^{[\ell]}}{\bar{x}_i}\right|
=\frac{\gamma_i^{[\ell]}}{\delta_i^{[\ell]}}\left|1-\frac{2\gamma_i^{[\ell]}x_i^{[\ell]}}{\bar{x}_i}\right|
\le\frac{\gamma_i^{[\ell]}}{\delta_{i,\min}}, δ i [ ℓ ] γ i [ ℓ ] − δ i [ ℓ ] x ˉ i 2 ( γ i [ ℓ ] ) 2 x i [ ℓ ] − x ˉ i γ i [ ℓ ] = δ i [ ℓ ] γ i [ ℓ ] 1 − x ˉ i 2 γ i [ ℓ ] x i [ ℓ ] ≤ δ i , m i n γ i [ ℓ ] , using the first definition in (10a). Moreover, regarding the third factor in (B.9),
γ i [ ℓ ] x ˉ i 2 ∑ r ∈ K ( γ i [ r ] x i [ r ] ) 2 δ i [ r ] ≤ γ i [ ℓ ] δ i , min ∑ r ∈ K ( γ i [ r ] x i [ r ] x ˉ i ) 2 ≤ γ i [ ℓ ] δ i , min . (38) \frac{\gamma_i^{[\ell]}}{\bar{x}_i^2}\sum_{r\in\mathcal{K}}\frac{(\gamma_i^{[r]}x_i^{[r]})^2}{\delta_i^{[r]}}\le\frac{\gamma_i^{[\ell]}}{\delta_{i,\min}}\sum_{r\in\mathcal{K}}\left(\frac{\gamma_i^{[r]}x_i^{[r]}}{\bar{x}_i}\right)^2\le\frac{\gamma_i^{[\ell]}}{\delta_{i,\min}}. \tag*{(38)} x ˉ i 2 γ i [ ℓ ] r ∈ K ∑ δ i [ r ] ( γ i [ r ] x i [ r ] ) 2 ≤ δ i , m i n γ i [ ℓ ] r ∈ K ∑ ( x ˉ i γ i [ r ] x i [ r ] ) 2 ≤ δ i , m i n γ i [ ℓ ] . (38) Hence, from (B.9),
∣ ∂ ∂ x i [ ℓ ] ( 1 + ∑ r ∈ K g ( x i [ r ] ) ) ∣ ≤ 2 γ i [ ℓ ] δ i , min . (39) \left|\frac{\partial}{\partial x_i^{[\ell]}}\left(1+\sum_{r\in\mathcal{K}}g(x_i^{[r]})\right)\right|\le\frac{2\gamma_i^{[\ell]}}{\delta_{i,\min}}. \tag*{(39)} ∂ x i [ ℓ ] ∂ ( 1 + r ∈ K ∑ g ( x i [ r ] ) ) ≤ δ i , m i n 2 γ i [ ℓ ] . (39) Observe also that since, for each r ∈ K r\in\mathcal{K} r ∈ K ( γ i [ r ] x i [ r ] ) 2 ≤ ( γ i [ r ] x i [ r ] ) ∑ h ∈ K γ i [ h ] x i [ h ] (\gamma_i^{[r]}x_i^{[r]})^2\le(\gamma_i^{[r]}x_i^{[r]})\sum_{h\in\mathcal{K}}\gamma_i^{[h]}x_i^{[h]} ( γ i [ r ] x i [ r ] ) 2 ≤ ( γ i [ r ] x i [ r ] ) ∑ h ∈ K γ i [ h ] x i [ h ]
1 x ˉ i ∑ r ∈ K ( γ i [ r ] x i [ r ] ) 2 δ i [ r ] ≤ ∑ r ∈ K γ i [ r ] x i [ r ] δ i [ r ] . \frac{1}{\bar{x}_i}\sum_{r\in\mathcal{K}}\frac{(\gamma_i^{[r]}x_i^{[r]})^2}{\delta_i^{[r]}}\le\sum_{r\in\mathcal{K}}\frac{\gamma_i^{[r]}x_i^{[r]}}{\delta_i^{[r]}}. x ˉ i 1 r ∈ K ∑ δ i [ r ] ( γ i [ r ] x i [ r ] ) 2 ≤ r ∈ K ∑ δ i [ r ] γ i [ r ] x i [ r ] . Therefore,
1 + ∑ r ∈ K γ i [ r ] x i [ r ] δ i [ r ] − 1 x ˉ i ∑ r ∈ K ( γ i [ r ] x i [ r ] ) 2 δ i [ r ] ≥ 1. (40) 1+\sum_{r\in\mathcal{K}}\frac{\gamma_i^{[r]}x_i^{[r]}}{\delta_i^{[r]}}-\frac{1}{\bar{x}_i}\sum_{r\in\mathcal{K}}\frac{(\gamma_i^{[r]}x_i^{[r]})^2}{\delta_i^{[r]}}\ge1. \tag*{(40)} 1 + r ∈ K ∑ δ i [ r ] γ i [ r ] x i [ r ] − x ˉ i 1 r ∈ K ∑ δ i [ r ] ( γ i [ r ] x i [ r ] ) 2 ≥ 1. (40) Then d ˉ i ≤ 1 \bar{d}_i\le1 d ˉ i ≤ 1
∣ ∂ d ˉ i ∂ x i [ ℓ ] ∣ = ∣ ∂ ∂ x i [ ℓ ] ( 1 + ∑ r ∈ K g ( x i [ r ] ) ) ( 1 + ∑ r ∈ K g ( x i [ r ] ) ) 2 ∣ ≤ 2 γ i [ ℓ ] δ i , min . (41) \left|\frac{\partial\bar{d}_i}{\partial x_i^{[\ell]}}\right|=\left|\frac{\frac{\partial}{\partial x_i^{[\ell]}}\left(1+\sum_{r\in\mathcal{K}}g(x_i^{[r]})\right)}{\left(1+\sum_{r\in\mathcal{K}}g(x_i^{[r]})\right)^2}\right|\le\frac{2\gamma_i^{[\ell]}}{\delta_{i,\min}}. \tag*{(41)} ∂ x i [ ℓ ] ∂ d ˉ i = ( 1 + ∑ r ∈ K g ( x i [ r ] ) ) 2 ∂ x i [ ℓ ] ∂ ( 1 + ∑ r ∈ K g ( x i [ r ] ) ) ≤ δ i , m i n 2 γ i [ ℓ ] . (41) While from 0 ≤ γ i [ r ] x i [ r ] / x ˉ i ≤ 1 0\le\gamma_i^{[r]}x_i^{[r]}/\bar{x}_i\le1 0 ≤ γ i [ r ] x i [ r ] / x ˉ i ≤ 1 θ ( 1 − θ ) ≤ 1 4 \theta(1-\theta)\le\frac14 θ ( 1 − θ ) ≤ 4 1 θ ∈ [ 0 , 1 ] \theta\in[0,1] θ ∈ [ 0 , 1 ]
0 ≤ g i [ k ] ( x ) ≤ x ˉ i 4 δ i [ k ] ≤ 1 4 δ i [ k ] , (42) 0\le g_i^{[k]}(x)\le\frac{\bar{x}_i}{4\delta_i^{[k]}}\le\frac{1}{4\delta_i^{[k]}}, \tag*{(42)} 0 ≤ g i [ k ] ( x ) ≤ 4 δ i [ k ] x ˉ i ≤ 4 δ i [ k ] 1 , (42) where the last inequality follows from Assumption 1(iii). By (B.8), (B.12), (B.13) and (B.14), for all ℓ ∈ K \ell\in\mathcal{K} ℓ ∈ K
∣ ∂ a i [ k ] ∂ x i [ ℓ ] ( x ) ∣ ≤ ∣ ∂ g i [ k ] ∂ x i [ ℓ ] ∣ ∣ d ˉ i ∣ + ∣ ∂ d ˉ i ∂ x i [ ℓ ] g i [ k ] ( x ) ∣ ≤ γ i [ ℓ ] δ i [ k ] + 1 4 δ i [ k ] 2 γ i [ ℓ ] δ i , min = γ i [ ℓ ] δ i [ k ] ( 1 + 1 2 δ i , min ) . (43) \begin{aligned}
\left|\frac{\partial a_i^{[k]}}{\partial x_i^{[\ell]}}(x)\right|\le\left|\frac{\partial g_i^{[k]}}{\partial x_i^{[\ell]}}\right||\bar{d}_i|+\left|\frac{\partial\bar{d}_i}{\partial x_i^{[\ell]}}g_i^{[k]}(x)\right| \\
&\le\frac{\gamma_i^{[\ell]}}{\delta_i^{[k]}}+\frac{1}{4\delta_i^{[k]}}\frac{2\gamma_i^{[\ell]}}{\delta_{i,\min}}=\frac{\gamma_i^{[\ell]}}{\delta_i^{[k]}}\left(1+\frac{1}{2\delta_{i,\min}}\right). \tag*{(43)}
\end{aligned} ∂ x i [ ℓ ] ∂ a i [ k ] ( x ) ≤ ∂ x i [ ℓ ] ∂ g i [ k ] ∣ d ˉ i ∣ + ∂ x i [ ℓ ] ∂ d ˉ i g i [ k ] ( x ) ≤ δ i [ k ] γ i [ ℓ ] + 4 δ i [ k ] 1 δ i , m i n 2 γ i [ ℓ ] = δ i [ k ] γ i [ ℓ ] ( 1 + 2 δ i , m i n 1 ) . (43) Since a i [ k ] ( x ) a_i^{[k]}(x) a i [ k ] ( x ) ( x i [ ℓ ] ) ℓ ∈ K (x_i^{[\ell]})_{\ell\in\mathcal{K}} ( x i [ ℓ ] ) ℓ ∈ K x , y ∈ C x,y\in\mathcal{C} x , y ∈ C
∥ a [ k ] ( x ) − a [ k ] ( y ) ∥ ∞ ≤ max i ∈ V ∑ ℓ ∈ K sup z ∈ C ∥ ∂ a i [ k ] ( z ) ∂ x i [ ℓ ] ∥ ∥ x − y ∥ ∞ . \lVert a^{[k]}(x)-a^{[k]}(y)\rVert_{\infty}\leq\max_{i\in\mathcal{V}}\sum_{\ell\in\mathcal{K}}\sup_{z\in\mathcal{C}}\left\lVert\frac{\partial a_i^{[k]}(z)}{\partial x_i^{[\ell]}}\right\rVert\lVert x-y\rVert_{\infty}. ∥ a [ k ] ( x ) − a [ k ] ( y ) ∥ ∞ ≤ i ∈ V max ℓ ∈ K ∑ z ∈ C sup ∂ x i [ ℓ ] ∂ a i [ k ] ( z ) ∥ x − y ∥ ∞ . Using (B.15) and Assumption 1(iii), this yields
∥ a [ k ] ( x ) − a [ k ] ( y ) ∥ ∞ ≤ max i ∈ V ∑ ℓ ∈ K γ i [ ℓ ] δ i [ k ] ( 1 + 1 2 δ i , min ) ∥ x − y ∥ ∞ ≤ max i ∈ V 1 δ i [ k ] ( 1 + 1 2 δ i , min ) ∥ x − y ∥ ∞ . (44) \begin{aligned}
\lVert a^{[k]}(x)-a^{[k]}(y)\rVert_{\infty}\leq\max_{i\in\mathcal{V}}\sum_{\ell\in\mathcal{K}}\frac{\gamma_i^{[\ell]}}{\delta_i^{[k]}}\left(1+\frac{1}{2\delta_{i,\min}}\right)\lVert x-y\rVert_{\infty} \\
&\leq\max_{i\in\mathcal{V}}\frac{1}{\delta_i^{[k]}}\left(1+\frac{1}{2\delta_{i,\min}}\right)\lVert x-y\rVert_{\infty}. \tag*{(44)}
\end{aligned} ∥ a [ k ] ( x ) − a [ k ] ( y ) ∥ ∞ ≤ i ∈ V max ℓ ∈ K ∑ δ i [ k ] γ i [ ℓ ] ( 1 + 2 δ i , m i n 1 ) ∥ x − y ∥ ∞ ≤ i ∈ V max δ i [ k ] 1 ( 1 + 2 δ i , m i n 1 ) ∥ x − y ∥ ∞ . (44) The constant in (B.16) is uniform over x , y ∈ C x,y\in\mathcal{C} x , y ∈ C L L L L ( C ) ⊆ C L(\mathcal{C})\subseteq\mathcal{C} L ( C ) ⊆ C x , y ∈ C x,y\in\mathcal{C} x , y ∈ C k ∈ K k\in\mathcal{K} k ∈ K L [ k ] ( x ) − L [ k ] ( y ) = ( I − Λ [ k ] W ~ ) − 1 Ξ [ k ] W ( a [ k ] ( x ) − a [ k ] ( y ) ) L^{[k]}(x)-L^{[k]}(y)=(I-\Lambda^{[k]}\widetilde{W})^{-1}\Xi^{[k]}W(a^{[k]}(x)-a^{[k]}(y)) L [ k ] ( x ) − L [ k ] ( y ) = ( I − Λ [ k ] W ) − 1 Ξ [ k ] W ( a [ k ] ( x ) − a [ k ] ( y )) ∞ \infty ∞
∥ L [ k ] ( x ) − L [ k ] ( y ) ∥ ∞ ≤ ∥ ( I − Λ [ k ] W ~ ) − 1 Ξ [ k ] W ∥ ∞ ∥ a [ k ] ( x ) − a [ k ] ( y ) ∥ ∞ ≤ max i ξ i [ k ] 1 − max i λ i [ k ] max i ∈ V 1 δ i [ k ] ( 1 + 1 2 δ i , min ) ∥ x − y ∥ ∞ . \begin{aligned}
\lVert L^{[k]}(x)-L^{[k]}(y)\rVert_{\infty}
&\leq\left\lVert(I-\Lambda^{[k]}\widetilde{W})^{-1}\Xi^{[k]}W\right\rVert_{\infty}\lVert a^{[k]}(x)-a^{[k]}(y)\rVert_{\infty}\\
&\leq\frac{\max_i\xi_i^{[k]}}{1-\max_i\lambda_i^{[k]}}\max_{i\in\mathcal{V}}\frac{1}{\delta_i^{[k]}}\left(1+\frac{1}{2\delta_{i,\min}}\right)\lVert x-y\rVert_{\infty}.
\end{aligned} ∥ L [ k ] ( x ) − L [ k ] ( y ) ∥ ∞ ≤ ( I − Λ [ k ] W ) − 1 Ξ [ k ] W ∞ ∥ a [ k ] ( x ) − a [ k ] ( y ) ∥ ∞ ≤ 1 − max i λ i [ k ] max i ξ i [ k ] i ∈ V max δ i [ k ] 1 ( 1 + 2 δ i , m i n 1 ) ∥ x − y ∥ ∞ . since W ~ \widetilde{W} W λ i [ k ] < 1 \lambda_i^{[k]}<1 λ i [ k ] < 1 i i i k ∈ K k\in\mathcal{K} k ∈ K c < 1 c<1 c < 1 ∥ L ( x ) − L ( y ) ∥ ∞ ≤ c ∥ x − y ∥ ∞ \lVert L(x)-L(y)\rVert_{\infty}\leq c\lVert x-y\rVert_{\infty} ∥ L ( x ) − L ( y ) ∥ ∞ ≤ c ∥ x − y ∥ ∞ L L L C \mathcal{C} C L L L x ⋆ ∈ C x^\star\in\mathcal{C} x ⋆ ∈ C x ⋆ x^\star x ⋆ d ⋆ d^\star d ⋆ a ⋆ a^\star a ⋆ s ⋆ = 0 s^\star=0 s ⋆ = 0
Proof of Theorem 3
We first observe that condition (11) implies the uniqueness condition (9). Indeed, for every k ∈ K k\in\mathcal{K} k ∈ K
max i 1 δ i [ k ] ( 1 + 1 2 δ i , min ) ≤ max i 1 δ i , min ( 1 + 1 2 δ i , min ) ≤ 3 2 min i δ i , min 2 ≤ 6 min i ε i , \begin{aligned}
\max_i\frac{1}{\delta_i^{[k]}}\left(1+\frac{1}{2\delta_{i,\min}}\right)\leq\max_i\frac{1}{\delta_{i,\min}}\left(1+\frac{1}{2\delta_{i,\min}}\right) \\
&\leq\frac{3}{2\min_i\delta_{i,\min}^{2}}\leq\frac{6}{\min_i\varepsilon_i},
\end{aligned} i max δ i [ k ] 1 ( 1 + 2 δ i , m i n 1 ) ≤ i max δ i , m i n 1 ( 1 + 2 δ i , m i n 1 ) ≤ 2 min i δ i , m i n 2 3 ≤ min i ε i 6 , where the second inequality uses δ i , min ≤ 1 \delta_{i,\min}\leq1 δ i , m i n ≤ 1 ε i = x i δ i , min 2 γ i , min 2 ≤ δ i , min 2 \varepsilon_i=x_i\delta_{i,\min}^{2}\gamma_{i,\min}^{2}\leq\delta_{i,\min}^{2} ε i = x i δ i , m i n 2 γ i , m i n 2 ≤ δ i , m i n 2 max k max i ξ i [ k ] 1 − max i λ i [ k ] \max_k\frac{\max_i\xi_i^{[k]}}{1-\max_i\lambda_i^{[k]}} max k 1 − m a x i λ i [ k ] m a x i ξ i [ k ] y ( t + 1 ) − y ( t ) → 0 y(t+1)-y(t)\to0 y ( t + 1 ) − y ( t ) → 0
By Proposition 1, the trajectory remains in the compact set Y \mathcal{Y} Y s ( t ) s(t) s ( t ) s ˉ : = lim t → ∞ s ( t ) \bar{s}:=\lim_{t\to\infty}s(t) s ˉ := lim t → ∞ s ( t ) 0 ≤ s ˉ ≤ s ( 0 ) 0\leq\bar{s}\leq s(0) 0 ≤ s ˉ ≤ s ( 0 ) s i ( t ) − s i ( t + 1 ) → 0 s_i(t)-s_i(t+1)\to0 s i ( t ) − s i ( t + 1 ) → 0 i i i w i ( t ) : = [ ( a i [ k ] ( t ) ) k ∈ K , ( d i [ k ] ( t ) ) k ∈ K ] ⊤ ∈ R 2 m w_i(t):=[(a_i^{[k]}(t))_{k\in\mathcal{K}},(d_i^{[k]}(t))_{k\in\mathcal{K}}]^\top\in\mathbb{R}^{2m} w i ( t ) := [( a i [ k ] ( t ) ) k ∈ K , ( d i [ k ] ( t ) ) k ∈ K ] ⊤ ∈ R 2 m
(1b)–(1c) can be written as w i ( t + 1 ) : = P i ( t ) w i ( t ) + b i ( t ) w_i(t+1):=P_i(t)w_i(t)+b_i(t) w i ( t + 1 ) := P i ( t ) w i ( t ) + b i ( t )
P i ( t ) : = [ diag ( 1 − δ i [ k ] ) k T i ( t ) diag ( δ i [ k ] ) k diag ( 1 − ∑ ℓ ≠ k γ i [ ℓ ] x i [ ℓ ] ( t ) ) k ] , P_i(t):=\begin{bmatrix}
\operatorname{diag}(1-\delta_i^{[k]})_{k} & T_i(t)\\
\operatorname{diag}(\delta_i^{[k]})_{k} & \operatorname{diag}\left(1-\sum_{\ell\neq k}\gamma_i^{[\ell]}x_i^{[\ell]}(t)\right)_{k}
\end{bmatrix}, P i ( t ) := [ diag ( 1 − δ i [ k ] ) k diag ( δ i [ k ] ) k T i ( t ) diag ( 1 − ∑ ℓ = k γ i [ ℓ ] x i [ ℓ ] ( t ) ) k ] , with
[ T i ( t ) ] k ℓ : = { γ i [ k ] x i [ k ] ( t ) , k ≠ ℓ , 0 , k = ℓ , [T_i(t)]_{k\ell}:=
\begin{cases}
\gamma_i^{[k]}x_i^{[k]}(t), & k\neq\ell,\\
0, & k=\ell,
\end{cases} [ T i ( t ) ] k ℓ := { γ i [ k ] x i [ k ] ( t ) , 0 , k = ℓ , k = ℓ , and b i ( t ) : = [ ( β i [ k ] x i [ k ] ( t ) s i ( t ) ( W a [ k ] ( t ) ) i ) k , 0 ] ⊤ b_i(t):=[(\beta_i^{[k]}x_i^{[k]}(t)s_i(t)(Wa^{[k]}(t))_i)_{k},0]^\top b i ( t ) := [( β i [ k ] x i [ k ] ( t ) s i ( t ) ( W a [ k ] ( t ) ) i ) k , 0 ] ⊤ P i ( t ) P_i(t) P i ( t ) 1 ⊤ b i ( t ) = s i ( t ) − s i ( t + 1 ) → 0 \mathbf{1}^{\top}b_i(t)=s_i(t)-s_i(t+1)\to0 1 ⊤ b i ( t ) = s i ( t ) − s i ( t + 1 ) → 0 b i ( t ) ≥ 0 b_i(t)\geq0 b i ( t ) ≥ 0 b i ( t ) → 0 b_i(t)\to0 b i ( t ) → 0 b i ( t + 1 ) − b i ( t ) → 0 b_i(t+1)-b_i(t)\to0 b i ( t + 1 ) − b i ( t ) → 0
We now establish a uniform contraction property for products of three consecutive matrices P i ( t ) P_i(t) P i ( t ) x i [ k ] ( t ) ≥ x i x_i^{[k]}(t)\geq x_i x i [ k ] ( t ) ≥ x i t ≥ 1 t\geq1 t ≥ 1 i , k i,k i , k x i [ ℓ ] ( t ) ≤ 1 x_i^{[\ell]}(t)\leq1 x i [ ℓ ] ( t ) ≤ 1 γ i [ ℓ ] ≤ 1 \gamma_i^{[\ell]}\leq1 γ i [ ℓ ] ≤ 1 t ≥ 1 t\geq1 t ≥ 1
γ i [ k ] x i [ k ] ( t ) ≥ γ i , min x i , (45) \gamma_i^{[k]}x_i^{[k]}(t)\geq\gamma_{i,\min}x_i, \tag*{(45)} γ i [ k ] x i [ k ] ( t ) ≥ γ i , m i n x i , (45) 1 − ∑ ℓ ≠ r γ i [ ℓ ] x i [ ℓ ] ( t ) ≥ 1 − ∑ ℓ ≠ r γ i [ ℓ ] ≥ γ i [ r ] ≥ γ i , min . (46) 1-\sum_{\ell\neq r}\gamma_i^{[\ell]}x_i^{[\ell]}(t)\geq1-\sum_{\ell\neq r}\gamma_i^{[\ell]}\geq\gamma_i^{[r]}\geq\gamma_{i,\min}. \tag*{(46)} 1 − ℓ = r ∑ γ i [ ℓ ] x i [ ℓ ] ( t ) ≥ 1 − ℓ = r ∑ γ i [ ℓ ] ≥ γ i [ r ] ≥ γ i , m i n . (46) Fix any r ∈ K r\in\mathcal{K} r ∈ K A k A_k A k D k D_k D k D r D_r D r k = r k=r k = r A r → D r → D r → D r A_r\to D_r\to D_r\to D_r A r → D r → D r → D r D r → D r → D r → D r D_r\to D_r\to D_r\to D_r D r → D r → D r → D r k ≠ r k\neq r k = r A k → D k → A r → D r A_k\to D_k\to A_r\to D_r A k → D k → A r → D r D k → A r → D r → D r D_k\to A_r\to D_r\to D_r D k → A r → D r → D r δ i , min γ i , min 2 \delta_{i,\min}\gamma_{i,\min}^{2} δ i , m i n γ i , m i n 2 γ i , min x i δ i , min γ i , min \gamma_{i,\min}x_i\delta_{i,\min}\gamma_{i,\min} γ i , m i n x i δ i , m i n γ i , m i n x i δ i , min γ i , min 2 x_i\delta_{i,\min}\gamma_{i,\min}^{2} x i δ i , m i n γ i , m i n 2 x i , δ i , min , γ i , min ∈ ( 0 , 1 ] x_i,\delta_{i,\min},\gamma_{i,\min}\in(0,1] x i , δ i , m i n , γ i , m i n ∈ ( 0 , 1 ] ε i : = x i δ i , min 2 γ i , min 3 \varepsilon_i:=x_i\delta_{i,\min}^{2}\gamma_{i,\min}^{3} ε i := x i δ i , m i n 2 γ i , m i n 3 Q i ( t ) : = P i ( t + 3 ) P i ( t + 2 ) P i ( t + 1 ) Q_i(t):=P_i(t+3)P_i(t+2)P_i(t+1) Q i ( t ) := P i ( t + 3 ) P i ( t + 2 ) P i ( t + 1 ) D r D_r D r [ Q i ( t ) ] D r j ≥ ε i [Q_i(t)]_{D_rj}\geq\varepsilon_i [ Q i ( t ) ] D r j ≥ ε i j j j R i ( t ) : = Q i ( t ) − ε i e D r 1 ⊤ R_i(t):=Q_i(t)-\varepsilon_i\mathbf{e}_{D_r}\mathbf{1}^{\top} R i ( t ) := Q i ( t ) − ε i e D r 1 ⊤ R i ( t ) ≥ 0 R_i(t)\geq0 R i ( t ) ≥ 0 ∥ R i ( t ) ∥ 1 = 1 − ε i \lVert R_i(t)\rVert_1=1-\varepsilon_i ∥ R i ( t ) ∥ 1 = 1 − ε i q ∈ R 2 m q\in\mathbb{R}^{2m} q ∈ R 2 m
∥ Q i ( t ) q ∥ 1 ≤ ε i ∥ 1 ⊤ q ∥ + ( 1 − ε i ) ∥ q ∥ 1 . (47) \lVert Q_i(t)q\rVert_1\leq\varepsilon_i\lVert\mathbf{1}^{\top}q\rVert+(1-\varepsilon_i)\lVert q\rVert_1. \tag*{(47)} ∥ Q i ( t ) q ∥ 1 ≤ ε i ∥ 1 ⊤ q ∥ + ( 1 − ε i ) ∥ q ∥ 1 . (47) This is the standard contraction mechanism associated with ergodicity coefficients of stochastic matrices [24 Sec. 3] . Now define z i ( t ) : = w i ( t + 1 ) − w i ( t ) z_i(t):=w_i(t+1)-w_i(t) z i ( t ) := w i ( t + 1 ) − w i ( t ) v [ k ] ( t ) : = x [ k ] ( t + 1 ) − x [ k ] ( t ) v^{[k]}(t):=x^{[k]}(t+1)-x^{[k]}(t) v [ k ] ( t ) := x [ k ] ( t + 1 ) − x [ k ] ( t ) w i w_i w i
z i ( t + 1 ) = P i ( t + 1 ) z i ( t ) + ( P i ( t + 1 ) − P i ( t ) ) w i ( t ) + b i ( t + 1 ) − b i ( t ) . z_i(t+1)=P_i(t+1)z_i(t)+(P_i(t+1)-P_i(t))w_i(t)+b_i(t+1)-b_i(t). z i ( t + 1 ) = P i ( t + 1 ) z i ( t ) + ( P i ( t + 1 ) − P i ( t )) w i ( t ) + b i ( t + 1 ) − b i ( t ) . Iterating it over three steps yields
z i ( t + 3 ) = Q i ( t ) z i ( t ) + P i ( t + 3 ) P i ( t + 2 ) ( ( P i ( t + 1 ) − P i ( t ) ) w i ( t ) + b i ( t + 1 ) − b i ( t ) ) + P i ( t + 3 ) ( ( P i ( t + 2 ) − P i ( t + 1 ) ) w i ( t + 1 ) + b i ( t + 2 ) − b i ( t + 1 ) ) + ( ( P i ( t + 3 ) − P i ( t + 2 ) ) w i ( t + 2 ) + b i ( t + 3 ) − b i ( t + 2 ) ) . (48) \begin{aligned}
z_i(t+3)={}&Q_i(t)z_i(t)\\
&+P_i(t+3)P_i(t+2)\big((P_i(t+1)-P_i(t))w_i(t)+b_i(t+1)-b_i(t)\big)\\
&+P_i(t+3)\big((P_i(t+2)-P_i(t+1))w_i(t+1)+b_i(t+2)-b_i(t+1)\big)\\
&+\big((P_i(t+3)-P_i(t+2))w_i(t+2)+b_i(t+3)-b_i(t+2)\big). \tag*{(48)}
\end{aligned} z i ( t + 3 ) = Q i ( t ) z i ( t ) + P i ( t + 3 ) P i ( t + 2 ) ( ( P i ( t + 1 ) − P i ( t )) w i ( t ) + b i ( t + 1 ) − b i ( t ) ) + P i ( t + 3 ) ( ( P i ( t + 2 ) − P i ( t + 1 )) w i ( t + 1 ) + b i ( t + 2 ) − b i ( t + 1 ) ) + ( ( P i ( t + 3 ) − P i ( t + 2 )) w i ( t + 2 ) + b i ( t + 3 ) − b i ( t + 2 ) ) . (48) Only the D k D_k D k P i ( t ) P_i(t) P i ( t )
∥ P i ( t + 1 ) − P i ( t ) ∥ 1 ≤ 2 ∑ ℓ ≠ k γ i [ ℓ ] ∣ v i [ ℓ ] ( t ) ∣ ≤ 2 ( ∑ k γ i [ k ] ) max k ∣ v i [ k ] ( t ) ∣ ≤ 2 max k ∣ v i [ k ] ( t ) ∣ \lVert P_i(t+1)-P_i(t)\rVert_1\leq2\sum_{\ell\ne k}\gamma_i^{[\ell]}|v_i^{[\ell]}(t)|
\leq2\left(\sum_k\gamma_i^{[k]}\right)\max_k|v_i^{[k]}(t)|\leq2\max_k|v_i^{[k]}(t)| ∥ P i ( t + 1 ) − P i ( t ) ∥ 1 ≤ 2 ℓ = k ∑ γ i [ ℓ ] ∣ v i [ ℓ ] ( t ) ∣ ≤ 2 ( k ∑ γ i [ k ] ) k max ∣ v i [ k ] ( t ) ∣ ≤ 2 k max ∣ v i [ k ] ( t ) ∣ Since w i ( t ) ≥ 0 w_i(t)\geq0 w i ( t ) ≥ 0 ∥ w i ( t ) ∥ 1 = 1 − s i ( t ) ≤ 1 \lVert w_i(t)\rVert_1=1-s_i(t)\leq1 ∥ w i ( t ) ∥ 1 = 1 − s i ( t ) ≤ 1 ∑ k γ i [ k ] ≤ 1 \sum_k\gamma_i^{[k]}\leq1 ∑ k γ i [ k ] ≤ 1
∥ ( P i ( t + 1 ) − P i ( t ) ) w i ( t ) ∥ 1 ≤ 2 max k ∣ v i [ k ] ( t ) ∣ . (49) \lVert(P_i(t+1)-P_i(t))w_i(t)\rVert_1\leq2\max_k|v_i^{[k]}(t)|. \tag*{(49)} ∥( P i ( t + 1 ) − P i ( t )) w i ( t ) ∥ 1 ≤ 2 k max ∣ v i [ k ] ( t ) ∣. (49) Moreover,
1 ⊤ z i ( t ) = 1 ⊤ w i ( t + 1 ) − 1 ⊤ w i ( t ) = s i ( t ) − s i ( t + 1 ) → 0. (50) \mathbf{1}^{\top}z_i(t)=\mathbf{1}^{\top}w_i(t+1)-\mathbf{1}^{\top}w_i(t)=s_i(t)-s_i(t+1)\to0. \tag*{(50)} 1 ⊤ z i ( t ) = 1 ⊤ w i ( t + 1 ) − 1 ⊤ w i ( t ) = s i ( t ) − s i ( t + 1 ) → 0. (50) Since P i ( t ) P_i(t) P i ( t ) ∥ P i ( t ) ∥ 1 = 1 \lVert P_i(t)\rVert_1=1 ∥ P i ( t ) ∥ 1 = 1 b i ( t + 1 ) − b i ( t ) → 0 b_i(t+1)-b_i(t)\to0 b i ( t + 1 ) − b i ( t ) → 0
lim sup t → ∞ ∥ z i ( t ) ∥ 1 ≤ ( 1 − ϵ i ) lim sup t → ∞ ∥ z i ( t ) ∥ 1 + 6 max k lim sup t → ∞ ∥ v [ k ] ( t ) ∥ ∞ . \limsup_{t\to\infty}\lVert z_i(t)\rVert_1\leq(1-\epsilon_i)\limsup_{t\to\infty}\lVert z_i(t)\rVert_1+6\max_k\limsup_{t\to\infty}\lVert v^{[k]}(t)\rVert_\infty. t → ∞ lim sup ∥ z i ( t ) ∥ 1 ≤ ( 1 − ϵ i ) t → ∞ lim sup ∥ z i ( t ) ∥ 1 + 6 k max t → ∞ lim sup ∥ v [ k ] ( t ) ∥ ∞ . Therefore
lim sup t → ∞ ∥ z i ( t ) ∥ 1 ≤ 6 ϵ i max k lim sup t → ∞ ∥ v [ k ] ( t ) ∥ ∞ . \limsup_{t\to\infty}\lVert z_i(t)\rVert_1\leq\frac{6}{\epsilon_i}\max_k\limsup_{t\to\infty}\lVert v^{[k]}(t)\rVert_\infty. t → ∞ lim sup ∥ z i ( t ) ∥ 1 ≤ ϵ i 6 k max t → ∞ lim sup ∥ v [ k ] ( t ) ∥ ∞ . Taking the max over i i i
max i lim sup t → ∞ ∥ z i ( t ) ∥ 1 ≤ 6 min i ϵ i max k lim sup t → ∞ ∥ v [ k ] ( t ) ∥ ∞ . (51) \max_i\limsup_{t\to\infty}\lVert z_i(t)\rVert_1\leq\frac{6}{\min_i\epsilon_i}\max_k\limsup_{t\to\infty}\lVert v^{[k]}(t)\rVert_\infty. \tag*{(51)} i max t → ∞ lim sup ∥ z i ( t ) ∥ 1 ≤ min i ϵ i 6 k max t → ∞ lim sup ∥ v [ k ] ( t ) ∥ ∞ . (51) Conversely, subtracting the opinion dynamics at two consecutive times gives
v [ k ] ( t + 1 ) = Λ [ k ] W ~ v [ k ] ( t ) + Ξ [ k ] W ( a [ k ] ( t + 1 ) − a [ k ] ( t ) ) . v^{[k]}(t+1)=\Lambda^{[k]}\widetilde{W}v^{[k]}(t)+\Xi^{[k]}W\left(a^{[k]}(t+1)-a^{[k]}(t)\right). v [ k ] ( t + 1 ) = Λ [ k ] W v [ k ] ( t ) + Ξ [ k ] W ( a [ k ] ( t + 1 ) − a [ k ] ( t ) ) . Since W W W W ~ \widetilde{W} W ∥ a [ k ] ( t + 1 ) − a [ k ] ( t ) ∥ ∞ ≤ max i ∥ z i ( t ) ∥ 1 \lVert a^{[k]}(t+1)-a^{[k]}(t)\rVert_\infty\leq\max_i\lVert z_i(t)\rVert_1 ∥ a [ k ] ( t + 1 ) − a [ k ] ( t ) ∥ ∞ ≤ max i ∥ z i ( t ) ∥ 1
lim sup t → ∞ ∥ v [ k ] ( t ) ∥ ∞ ≤ max i ξ i [ k ] 1 − max i λ i [ k ] max i lim sup t → ∞ ∥ z i ( t ) ∥ 1 . \limsup_{t\to\infty}\lVert v^{[k]}(t)\rVert_\infty\leq\frac{\max_i\xi_i^{[k]}}{1-\max_i\lambda_i^{[k]}}\max_i\limsup_{t\to\infty}\lVert z_i(t)\rVert_1. t → ∞ lim sup ∥ v [ k ] ( t ) ∥ ∞ ≤ 1 − max i λ i [ k ] max i ξ i [ k ] i max t → ∞ lim sup ∥ z i ( t ) ∥ 1 . Taking the maximum over k k k
max k lim sup t → ∞ ∥ v [ k ] ( t ) ∥ ∞ ≤ max k max i ξ i [ k ] 1 − max i λ i [ k ] max i lim sup t → ∞ ∥ z i ( t ) ∥ 1 . (52) \max_k\limsup_{t\to\infty}\lVert v^{[k]}(t)\rVert_\infty\leq\max_k\frac{\max_i\xi_i^{[k]}}{1-\max_i\lambda_i^{[k]}}\max_i\limsup_{t\to\infty}\lVert z_i(t)\rVert_1. \tag*{(52)} k max t → ∞ lim sup ∥ v [ k ] ( t ) ∥ ∞ ≤ k max 1 − max i λ i [ k ] max i ξ i [ k ] i max t → ∞ lim sup ∥ z i ( t ) ∥ 1 . (52) Combining (B.22) and (B.23), condition (11) implies max k lim sup t → ∞ ∥ v [ k ] ( t ) ∥ ∞ = 0 \max_k\limsup_{t\to\infty}\lVert v^{[k]}(t)\rVert_\infty=0 max k lim sup t → ∞ ∥ v [ k ] ( t ) ∥ ∞ = 0 lim sup t → ∞ ∥ z i ( t ) ∥ 1 = 0 \limsup_{t\to\infty}\lVert z_i(t)\rVert_1=0 lim sup t → ∞ ∥ z i ( t ) ∥ 1 = 0 i i i k k k x i [ k ] ( t + 1 ) − x i [ k ] ( t ) → 0 x_i^{[k]}(t+1)-x_i^{[k]}(t)\to0 x i [ k ] ( t + 1 ) − x i [ k ] ( t ) → 0 a i [ k ] ( t + 1 ) − a i [ k ] ( t ) → 0 a_i^{[k]}(t+1)-a_i^{[k]}(t)\to0 a i [ k ] ( t + 1 ) − a i [ k ] ( t ) → 0 d i [ k ] ( t + 1 ) − d i [ k ] ( t ) → 0 d_i^{[k]}(t+1)-d_i^{[k]}(t)\to0 d i [ k ] ( t + 1 ) − d i [ k ] ( t ) → 0 s i ( t + 1 ) − s i ( t ) → 0 s_i(t+1)-s_i(t)\to0 s i ( t + 1 ) − s i ( t ) → 0 y ( t + 1 ) − y ( t ) → 0 y(t+1)-y(t)\to0 y ( t + 1 ) − y ( t ) → 0
It remains to characterize the limit points. Since Y \mathcal{Y} Y y ˉ \bar{y} y ˉ y ( t + 1 ) − y ( t ) → 0 y(t+1)-y(t)\to0 y ( t + 1 ) − y ( t ) → 0 s ( 0 ) ≠ 1 s(0)\ne1 s ( 0 ) = 1 s ( t ) s(t) s ( t ) y ∗ y^* y ∗ y ( t ) → y ∗ y(t)\to y^* y ( t ) → y ∗
Proofs of the results in Section 4
Proof of Theorem 4
The proof reduces the finite-horizon control problems to the maximization of continuous functions over compact feasible sets. Since the horizon is finite, U \mathcal{U} U U U U U k 0 U_{k_0} U k 0 U \mathcal{U} U
{ u ∈ U k 0 : ∑ t = 0 T − 1 u ( t ) ⊤ Q u ( t ) ≤ B } \left\{u\in U_{k_0}:\sum_{t=0}^{T-1}u(t)^{\top}Qu(t)\leq B\right\} { u ∈ U k 0 : t = 0 ∑ T − 1 u ( t ) ⊤ Q u ( t ) ≤ B } is a nonempty closed subset of U k 0 U_{k_0} U k 0 u ( ⋅ ) ∈ U u(\cdot)\in\mathcal{U} u ( ⋅ ) ∈ U y ( t + 1 ) = f ( y ( t ) , u ( t ) ) y(t+1)=f(y(t),u(t)) y ( t + 1 ) = f ( y ( t ) , u ( t )) y ( 0 ) = y 0 y(0)=y_0 y ( 0 ) = y 0 f f f y ( 0 ) , … , y ( T ) y(0),\ldots,y(T) y ( 0 ) , … , y ( T ) u ( ⋅ ) u(\cdot) u ( ⋅ ) J 1 J_1 J 1 J 2 J_2 J 2
