The hipster effect: When anti-conformists all look the same

Figure 1.  The binary hipster model: (A) each individual has a state $ s_i = \pm 1 $, and switches to the other state depending on the trend felt $ m_{i} $ and its hipster nature ($ \varepsilon_{i} = -1 $) or mainstream ($ \varepsilon_{i} = 1 $) nature. The random transitions occur at a rate $ \varphi(- \varepsilon_{i}s_{i}m_{i}) $ where $ \varphi(x) $ is a sigmoidal function (B) depending on a sharpness parameter $ \beta $ that account for the level of determinism in the transition: the larger $ \beta $, the sharper the transition, and the less probable non-preferred transitions. We depicted distinct possible transition functions with color indicating the value of $ \beta $ (darker lines correspond to larger $ \beta $). The table in (C) summarizes the 4 possible situations for the two types of individuals and 2 types of majoritary trend

Figure 2.  Non-delayed system with $ \bar J = 1 $: (A) the critical noise level $ \beta_c $ associated with the pitchfork bifurcation of the mean-field system. (B-C) simulations of the stochastic hipster model with $ n = 1\,000 $ agents and no delay. (B): $ q = 1/3 $: subcritical system (left, $ \beta = 1 $) shows a disordered regime, at the critical regime the system switches between distinct disordered states where hipsters and mainstream align transiently (middle, $ \beta = \beta_{c} = 2.5 $), and super-critical system (right, $ \beta = 5 $): the mainstream find a consensus and hipsters oppose to it. (C) $ q = 1/2 $: no bifurcation occurs. At relatively low noise ($ \beta = 15 $), while the average remains close to $ 0 $, the system switches between different states where hipsters and mainstreams are aligned, with switching times exponentially distributed (right)

Figure 3.  (A) Delay-induce Hopf bifurcation in the plane $ (\lambda,\tau) $. (B-C) simulations of the discrete system for $ n = 1\,000 $, $ \bar{J} = 1 $, and $ \lambda = 2 $, with distinct parameter combinations: (B) $ q = 1 $ and $ \beta = 2 $, and (C): $ q = 2/3 $ and $ \beta = 6 $. In both cases, the Hopf bifurcation arises at $ \tau\approx 0.604 $. We depict the behavior of the system for $ \tau = 0.5 $ (B1, C1), $ 0.7 $ (B2, C2) and $ 1.5 $ (B3, C3). Top row: time evolution of all states as a function of time, bottom row: empirical mean for the hipster (blue) or mainstream (red) state

Figure 4.  Space-dependent delays and connectivity in a hipster-only situation (see Supplementary Fig. 12 for the dependence in $ q $).: bifurcations as a function of the length $ a $ of the interval on which hipsters communicate. Parameters $ \beta = 4 $, $ \gamma = 0.3 $. Left: Computation of the Hopf bifurcation line as a function of $ \tau_{0} $ and $ a $ (orange line, bottom left) as the intersection of the two surfaces $ S_{1} $ (orange) and $ S_{2} $ (green surface). For $ \tau_0 = 0.18 $, the system shows no synchronization for small intervals (top right, $ a = 0.1 $), synchronization for intermediate intervals (middle right, $ a = 1 $), and no synchronization for large intervals (bottom right, $ a = 3 $), as visible in the simulation of the Markov chain with $ N = 3\,000 $ individuals, together with the computed trend below

Figure 5.  Analytical surfaces associated with transitions between the different regimes theoretically derived in section 3.1 (A, B), and bifurcation surfaces (C). (A-B): orange plane (independent of $ q $) shows the parameters associated with $ D = 0 $, and corresponds to the switch from regimes associated with one pitchfork bifurcation to regimes associated with two pitchfork bifurcations (see proposition 1). The 5-branch pitchfork bifurcation at $ T = 2\sqrt{-D} $ (green surface) separates pitchforks and Hopf bifurcations regimes. Hopf bifurcation type switches at a Bautin bifurcation (blue plane independent of $ q $). (A) $ J_{11} = 1 $, $ J_{21} = 3 $; the Bautin and 5-branch pitchfork intersect along a line projected on the $ (q,J_{21}) $ plane in (A'), where the system may present a codimension-three bifurcation. (B) $ J_{11} = 1.2 $ and $ J_{21} = 1 $; Bautin plane does not intersect the Hopf bifurcation surface within the range of parameters studied, and the type of Hopf bifurcation is always supercritical. (C) Bifurcation surfaces as a function of $ q $ and $ \mu = tJ_{11}J_{22} $; surface plotted: $ \beta_{c} $, $ \beta_{p1} $, $ \beta_{p2} $, $ \beta_{h} $ and the 5-branch pitchfork bifurcation (orange line) for $ J_{11} = 0.9 $ and $ J_{22} = 0.3 $

Figure 6.  Pitchfork bifurcations with $ D<0 $ and $ T>2\sqrt{-D} $. (A) $ q = 0.6 $, $ J11 = 3.25 $, $ J_{12} = 1 $, $ J_{21} = 2 $ and $ J_{22} = 0.25 $. Both bifurcations are supercritical (respectively, $ \gamma\simeq -0.54 $ and $ \gamma\simeq-0.38 $). Blue solid lines: stable fixed points, dashed lines: unstable fixed points. (B) $ q = 0.5 $, $ J_{11} = 3 $, $ J_{12} = 4.5 $, $ J_{21} = 0.5 $ and $ J_{22} = 0.25 $; pitchfork at $ \beta_{p1} $ is supercritical ($ \gamma\simeq -8.91 $) and at $ \beta_{p2} $ subcritical ($ \gamma\simeq 0.66 $): the two branches of non-disordered equilibria collide, and a subcritical Hopf bifurcation arises (dashed pink line) undergoing a fold-double-homoclinic bifurcation (Dh), associated with the presence of a stable periodic orbit (pink solid line) persisting for larger values of $ \beta $ tested. (C) $ q = 0.1 $, $ J_{11} = 1.2 $, $ J_{12} = 1 $, $ J_{21} = 5 $, $ J_{22} = 3.5 $. The pitchfork at $ \beta_{p1} $ is subcritical ($ \gamma\simeq 0.08 $), and at $ \beta_{p2} $ supercritical ($ \gamma\simeq 1.56 $). A saddle-node bifurcation is numerically identified on the branch of the unstable fixed points emerging from $ \beta_{p1} $. Phase planes are hand-drawn below the diagrams in 7 typical situations identified (matching the colors on the diagrams), and responses of a stochastic network with $ n = 4\,000 $ agents confirm these dynamics for values of $ \beta $ in each regime: (A): $ \beta = 0.5,\,2,\,4 $, (B): $ \beta = 0.5,\, 1.5,\, 2.3,\, 3 $, (C): $ \beta = 1,\,1.57, 4 $. Type of each individual depicted in color (+1: white, -1: black), mainstream are on top, and average type for hipsters (blue) or mainstreams (red) are added on top of this diagram. Bifurcation diagrams generated with XPP Aut [15], simulations performed on custom code on Matlab.

Figure 7.  Super- and Sub-critical Hopf bifurcations in the asymmetric Hipster model. (A, subpercritical) $ q = 0.4 $, $ J11 = 3.1 $, $ J_{12} = 3.5 $, $ J_{21} = 3 $ and $ J_{22} = 2 $. (B, subcritical) $ q = 0.1 $, $ J11 = 0.8 $, $ J_{12} = 5 $, $ J_{21} = 3 $ and $ J_{22} = 5 $. Top: bifurcation diagrams. Bottom panels: simulations of the network model, with $ n = 4\,000 $, and various values of the noise parameter $ \beta $: A1: $ \beta = 1 $, A2: $ \beta = 4 $, A3: $ \beta = 5.5 $, B1: $ \beta = 3.5 $, B2: $ \beta = 6 $, B3: $ \beta = 11 $

Figure 8.  Codimension-two diagrams. Top: Two codimension two diagrams, (A) $ J_{11} = 0.9 $, $ J_{12} = 0.9 $, $ J_{21} = 0.6 $ and $ J_{22} = 0.3 $, where the pitchfork bifurcation $ \beta_{p2} $ is always supercritical, or (B) parameters associated with Fig. 6, center. Blue: $ \beta_{p1} $, Red: $ \beta_{p2} $ (solid: supercritical, dashed: subcritical), yellow: $ \beta_{h} $ depicted analytically. Dotted blue line is the saddle-node bifurcation of non-disordered equilibria and dotted yellow line is the subcritical Hopf bifurcation on non-disordered states, both computed using Matlab Matcont package [13,14]. BT: Bogdanov-Takens, P: transition between sub- and super-critical pitchfork bifurcations. (A1) $ q = 0.3 $, (A2): $ q = 0.4 $ (grey lines in (A)). (A3): idealized unfolding of the 5-branch pitchfork bifurcation (black: fixed points, gray: cycles, circle color as in (A-B), except saddle-node represented in pink). (B1) $ q = 0.42 $, (B2): $ q = 0.38 $, and the case $ q = 0.5 $ is Fig. 6. (B3) idealized unfolding of the 5-branch pitchfork

Figure 9.  Role of the timescales in synchronization: a symmetrically interacting hipster model with distinct transition rates function: $ \varphi_{1}(x) = 2(1+\tanh(x)) $ for mainstreams, and $ \varphi_{2} = \alpha \varphi_{1} $ for hipsters. Top: codimension two bifurcation diagrams as a function of $ \alpha $ and $ \beta $ for (A) $ q = 0.3 $, (B) $ q = 0.6 $. Pink lines: Hopf bifurcations (solid: supercritical, dashed: subcritical), blue: supercritical pitchfork bifurcation (dashed: yielding unstable fixed points, solid: yielding stable fixed points) (computed with Matcont). Purple: fold of limit cycles, hand-drawn. (A1-B1) codimension-one diagram for $ \alpha = 0.2 $: blue lines are fixed points, pink lines cycles, solid / dashes: stability. (A1): the disordered state loses stability at a Hopf bifurcation, then undergoing a pitchfork bifurcation yielding consensus equilibria. These equilibria gain stability through a Hopf bifurcation. Cycles collide at a double-homoclinic fold of cycle bifurcation, similar to the one observed in Fig. \ref{fig:pitchforks}, middle. (B1): the disordered state loses stability in favor of a cycle that persists for larger $ \beta $. (A2-A4) network simulations with $ n = 4\,000 $ and $ \beta = 1,\, 2.5 $ or $ 3.5 $ respectively. (B2-B3) $ \beta = 3 $ or $ 6 $

Figure 10.  Synchronization in models with $ P>2 $ choices. (A) Extreme scenario: jump occur, similar to the binary model, according to conformity to the trend, and switches depend on occupation levels: hipsters (mainstreams) switch to the least (most) occupied state. $ P = 10 $ choices, $ \beta = 2 $, $ q = 0.2 $ and fixed delay $ \tau = 10 $. (B) Random scenario: Pott's model with mainstreams and hipsters and delay: when an individual switches state, it chooses uniformly at random among other states. $ P = 4 $, $ \beta = 15 $, $ q = 0.7 $ and $ \tau = 15 $

Figure 11.  Delays advance the emergence of oscillations in the hipster model with asymmetric interactions. Parameters as in (A) Fig. 6(B) with $ \beta = 2 $ and $ \tau = 10 $, (B) Fig. 7 (A) with $ \beta = 1.5 $ and $ \tau = 4 $ and (C) Fig. 7 (B) with $ \beta = 3.5 $ and $ \tau = 4 $. In all cases, the parameters correspond to the absence of oscillations in the non-delayed system, and delays induce synchrony

Figure 12.  Dependence of the Hopf bifurcation curve upon variation of $ q $ in the spatially extended case (section 2.2). Decreasing $ q $ tends to stabilize fixed points, and larger delays are necessary to synchronize the system. Upper-left diagram: orange surfaces correspond to $ S_{1} $, for $ q = 1 $ (lower surface) or $ q = 0.8 $, and green surface is $ S_{2} $, independent of $ q $. The intersection of these curves provide the locus of Hopf bifurcations depicted below, and we note on the right the absence of oscillations at $ q = 0.8 $ where a system with $ q = 1 $ oscillates (top), and larger delays reveal those oscillations

Figure 13.  Modifications of the codimension-two bifurcation diagram (analytical curves) for parameters associated with Fig. 8 (A) for various values of $ J_{22} $ or $ J_{12} $. We observe in both cases the disappearance of Hopf bifurcations either by increasing $ J_{22} $ or decreasing $ J_{12} $, leaving the system with a unique pitchfork bifurcation. In all diagrams, the abscissa is $ q $ and the ordinate is $ \beta $; blue curve is $ \beta_{p1} $ or $ \beta_{c} $, red: $ \beta_{p2} $, yellow: $ \beta_{h} $