In such different domains as statistical physics, neurosciences, spin glasses, social science, economics and finance, large ensemble of interacting individuals evolving following (mainstream) or against (hipsters) the majority are ubiquitous. Moreover, in a variety of applications, interactions between agents occur after specific delays that depends on the time needed to transport, transmit or take into account information. This paper focuses on the role of opposition to majority and delays in the emerging dynamics in a population composed of mainstream and anti-conformist individuals. To this purpose, we introduce a class of simple statistical system of interacting agents taking into account (ⅰ) the presence of mainstream and anti-conformist individuals and (ⅱ) delays, possibly heterogeneous, in the transmission of information. In this simple model, each agent can be in one of two states, and can change state in continuous time with a rate depending on the state of others in the past. We express the thermodynamic limit of these systems as the number of agents diverge, and investigate the solutions of the limit equation, with a particular focus on synchronized oscillations induced by delayed interactions. We show that when hipsters are too slow in detecting the trends, they will consistently make the same choice, and realizing this too late, they will switch, all together to another state where they remain alike. Another modality synchronizing hipsters are asymmetric interactions, particularly when the cross-interaction between hipsters and mainstreams aree prominent, i.e. when hipsters radically oppose to mainstream and mainstreams wish to follow the majority, even when led by hipsters. We demonstrate this phenomenon analytically using bifurcation theory and reduction to normal form. We find that, in the case of asymmetric interactions, the level of randomness in the decisions themselves also leads to synchronization of the hipsters. Beyond the choice of the best suit to wear this winter, this study may have important implications in understanding synchronization of nerve cells, investment strategies in finance, or emergent dynamics in social science, domains in which delays of communication and the geometry of information accessibility are prominent.
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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} $
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The binary hipster model: (A) each individual has a state
Non-delayed system with
(A) Delay-induce Hopf bifurcation in the plane
Space-dependent delays and connectivity in a hipster-only situation (see Supplementary Fig. 12 for the dependence in
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
Pitchfork bifurcations with
Super- and Sub-critical Hopf bifurcations in the asymmetric Hipster model. (A, subpercritical)
Codimension-two diagrams. Top: Two codimension two diagrams, (A)
Role of the timescales in synchronization: a symmetrically interacting hipster model with distinct transition rates function:
Synchronization in models with
Delays advance the emergence of oscillations in the hipster model with asymmetric interactions. Parameters as in (A) Fig. 6(B) with
Dependence of the Hopf bifurcation curve upon variation of
Modifications of the codimension-two bifurcation diagram (analytical curves) for parameters associated with Fig. 8 (A) for various values of