Detecting phase transitions in collective behavior using manifold's curvature

Figure 1.  An abrupt phase change of the crowd behavior where a walking crowd suddenly starts running [2]. (a) The first phase of the motion (walking) is embedded onto the blue colored manifold, while the second phase (running) is embedded onto the red colored manifold. Two snapshots showing walking and running at time steps $t_1$ and $t_2$ are embedded onto points $\boldsymbol{p}^{(t_1)}$ and $\boldsymbol{p}^{(t_2)}$, respectively, in the corresponding manifolds. The locus of singularities ($\mathcal{L}$) is represented by orange color. (b) The scaled residual variance versus the dimensionality, that gives the dimensionality of the underlying manifold by an elbow, is obtained by running Isomap upon frames in each phase with 6 nearest neighbors. Embedding dimensionalities of sub-manifolds representing walking (blue circle) and running (red square) of the crowd are three and four, respectively

Figure 2.  (a) Superimposing a neighborhood of the curve $C$ at point $\boldsymbol{p}$ with an arc $\boldsymbol{p}_1\boldsymbol{p}\boldsymbol{p}_2$ which subtend an small angle of $2T$ at the origin of a translating circle. (b) Zoomed and rotated secular sector $\boldsymbol{p}_1\boldsymbol{p}_2\boldsymbol{O}$ such that the blue arrow is horizontal

Figure 3.  Local distribution of data around the point $\boldsymbol{p}$ on a two dimensional manifold ($\mathcal{M}^2$). Principal sections $\Pi^{(1)}_{\boldsymbol{p}}$ and $\Pi^{(2)}_{\boldsymbol{p}}$ are created by using the shape operator at $\boldsymbol{p}$, and curves $C^{(1)}$ and $C^{(2)}$ are produced by intersecting $\Pi^{(1)}_{\boldsymbol{p}}$ and $\Pi^{(2)}_{\boldsymbol{p}}$ with $\mathcal{M}^2$, respectively

Figure 4.  (a) A three dimensional sombrero-hat of 2000 points consisting two sub-manifolds (blue and green) and locus of singularities (red) is intersected with the plane $\{\beta_1\hat{\boldsymbol{i}}+\beta_2\hat{\boldsymbol{k}} \vert \beta_1, \beta_2 \in \mathbb{R}\}$ to produce (b) a curve in $\mathbb{R}^3$. (c) Isomap residual plots those show embedding dimensionalities by elbows reveal that the dimensionalities of two sub-manifolds are two while the dimensionality of the locus is three

Figure 5.  Detecting phase transitions in a particle swarm simulated using the Vicsek model with alternating noise levels. (a) The distribution of $(\sigma_4/\sigma_1)_{n}$ versus frame numbers. Therein, the range of frames for each sub-manifold is represented by a left-right arrow and the frame number at each phase transition is represented by a red circle along with the frame number associated. (b) The plot of 20 largest phase changes including frame numbers of three phase transitions. (c) Isomap residual variance versus dimensionality of each sub-manifold

Figure 6.  Detecting a phase transition between phases of walking and running in a human crowd [3]. (a) The distribution of $(\sigma_3/\sigma_1)_{n}$ versus frame numbers. Therein, while the snapshots show instances of the crowd in each phase, left-right arrows and the red circle represent ranges of frames in each sub-manifold and the frame at the phase transition, respectively. (b) The plot of 20 largest phase changes representing the phase transition in red along with its frame number

Figure 7.  Detecting a transition in a bird flock between phases sitting and flying [1]. (a) The distribution of $(\sigma_3/\sigma_1)_{n}$ shows ranges of frames representing two sub-manifolds by left-right arrows and instances phases of the flock by snapshots. (b) The plot of 20 largest phase changes. The frame at the phase transition is represented by red in Figures (a) and (b)

Figure 8.  Detecting phase transitions in a fish school. (a) The distribution of $(\sigma_6/\sigma_1)_{n}$ with left-right arrows showing ranges of frames in sub-manifolds and red dots showing frames at phase transitions. (b) Snapshots of the school before (left) and after (right) each phase transition. (c) The plot of 20 largest phase changes consisting four phase transitions marked in red with their frame numbers

Figure 9.  Two dimensional saddle surface $\mathcal{M}^2$, described by the Equation (28) for $x_1, x_2 \in \mathbb{U} [-2,2]$