Dynamics of a minimal consumer network with uni-directional influence

Fashion cycle dynamics induced by agents' heterogeneity for generic bell-shaped attractiveness functions

“…We notice that the statement of the previous result coincides with Proposition II.1 in [40], since the more general formulation we are here considering does affect neither the (19) with ∈ (0, 0.9). We denote in orange a = 0, Q3 in cyan a = 1, in red a = a * , in green the attractor surrounding it, in blue a = a * 1 and a = a * 2 .…”

“…where σ , as explained after Definition 2.3, is a positive parameter describing the sensitivity of the attractiveness to the distance between the visibility level V i,t induced by the preference structure i ∈ {α, β} and the threshold visibility level V, which allows maximizing the attractiveness degree. We notice that A i,t in (19) and A i,t in (20) satisfy all the assumptions in Definition 2.3. In regard to the parameter values, we let free to vary, while we fix the remaining parameters as follows:…”

“…Solid (dashed) curves refer to stable (unstable) equilibria and cycles. In (B) the bifurcation diagram of g for f 1 in (19) with ∈ (0.514, 0.520). The magenta points are generated by the initial condition a 0 = 0.55, while the green points are generated by the initial condition a 0 = 0.657.…”

“…(14) and (15)). Namely, even if V i , i ∈ {α, β}, do not depend on σ , such parameter is usually present in the expressions of (∂A α (a)/∂(σ d 2 α ))| a=a * and of (∂A α (a)/∂(σ d 2 α ))| a=a * j , as it is easy to check, for instance, with the functional form for attractiveness in (19). In fact, with the Gaussian map in (19), for suitable parameter configurations, there may not exist threshold stability values analogous to those in (14) and (15) with respect to σ , since e.g.…”

“…Finally, we stress that, rather than considering stable preferences as we do, it is possible to deal with preference adjusting consumers, like in [22][23][24][25][26]33], where cycles are found in a model with agents having globally interdependent preferences. The setting proposed in those works has been extended in [19,20], to deal with the case in which the interdependence among preferences is local in nature, and in [32], where the authors propose a stochastic version of the original framework. We recall that consumption choices of agents disposed on a network and having locally interdependent preferences have been considered in [10], too.…”