plt1.set_title('Left edge')
plt2.set_xlabel(r'$\alpha$')
# plt2.set_ylabel(r'$x_{+}$')
plt2.set_title('Right edge')
plt2.legend()
fig.tight_layout()
fig.savefig('./figure/figure7_g_a_edge.png', dpi=300)
# pdf of space-sampled iid Gaussian theory, log scale
g = 0.9
nx = 40000
f_ls = [0.1, 0.2, 0.4]
fig = plt.figure(figsize=(8, 6))
x0_plot = 0.3
t0 = timeit.default_timer()
x, px = pdf_g(g, nx=nx)
x12_ls = []
x12c_ls = []
tf_plot = x >= x0_plot
line = plt.loglog(x[tf_plot],
px[tf_plot],
'k',
linewidth=1.5,
label='non-sampled')
px2 = sqrt(3) / (2 * np.pi) * x**(-5 / 3)
plt.loglog(x[tf_plot],
px2[tf_plot],
'--',
linewidth=1.5,
color=line[0].get_color())
x12_ls.append((x[0], x[-1]))
from numpy.random import randn from numpy.linalg import eigvals, eigvalsh from numpy import sqrt import timeit # created by Yu Hu ([email protected]), Aug 2020 plt.rcParams.update({'font.size': 18}) # different g g_ls = [0.6, 0.7, 0.8] nx_ls = [4000, 10000, 40000] ymax_ls = [1.5, 1.5, 1.7] for i, g in enumerate(g_ls): fig = plt.figure(figsize=(16, 6)) x, px = pdf_g(g, nx=nx_ls[i]) px2 = sqrt(3) / (2 * np.pi) * x**(-5 / 3) plt.subplot(121) plt.plot(x, px, color='b', label='g=' + str(g)) plt.plot([x[0], x[-1]], [0, 0], '.', markersize=10, color='b') plt.plot(x, px2, 'r--', label='power law') plt.ylim([0, ymax_ls[i]]) plt.xlabel('cov eigenvalue $x$') plt.ylabel('probability $p(x)$') plt.legend() plt.subplot(122) plt.loglog(x, px) plt.loglog(x, px2, 'r--', label='power law') plt.xlabel(r'$\log(x)$') plt.ylabel(r'$\log(p(x))$') plt.tight_layout()
# created by Yu Hu ([email protected]), Aug 2020 plt.rcParams.update({'font.size': 18}) # compare simulation and iid Gaussian theory g_ls = [0.5] N_ls = [100, 400, 800] ifig = 0 for g in g_ls: for N in N_ls: ifig += 1 J = g * randn(N, N) / sqrt(N) C = J2C(J) eig_C = eigvalsh(C) x, px = pdf_g(g, nx=1000) fig = plt.figure(figsize=(8, 6)) plt.plot(x, px, linewidth=1.5, label='theory') plt.hist(eig_C, 40, density=True, label='N=' + str(N)) plt.plot([x[0], x[-1]], [0, 0], '.', markersize=10) plt.xlabel('cov eigenvalues') plt.ylabel('probabilty') plt.legend() plt.title('g=' + str(g)) plt.tight_layout() fig.savefig('./figure/figure1a_' + str(ifig) + '.png', dpi=600) # different g g_ls = [0.3, 0.4, 0.5, 0.6, 0.7] x12_ls = [] x12c_ls = []
# r_quantile = 1.0238940717401241
print('0.995 quantile r:', r_quantile)
J = randn(N,N)/sqrt(N)*g
# kdivh = 0.25 # squared root of the kappa notations in text
kdivh = 0.45 # squared root of the kappa notations in text
kdiv = sqrt(kdivh**2/(1-kdivh**2)*(g**2/N)) # g is after removing div motifs
print('b entry var b/sqrt(N): ', kdiv*sqrt(N))
print('xuv, x=', N*kdiv)
b = randn(N) * kdiv # squared root of the kappa notations in text
print('check', np.linalg.norm(b)*np.linalg.norm(ones(N)))
print('outlier theory:', J_uv_outlier(g,N*kdiv))
print('outlier theory:', J_uv_outlier(g,np.linalg.norm(b)*np.linalg.norm(ones(N))))
Jm = J + np.outer(ones(N),b)
x,px = pdf_g(g)
C = J2C(J)
Cm = J2C(Jm)
eig_C = eigvalsh(C)
eig_Cm = eigvalsh(Cm)
fig = plt.figure(figsize=(8,4))
plt.hist(eig_C, 40, density=True, label='N='+str(N));
plt.plot(x,px, linewidth=1.5, label='g='+str(g))
x_lim = plt.xlim()
plt.xlabel('cov eigenvalues')
plt.ylabel('probability')
plt.legend()
plt.title(r'Cov spectrum with $J$')
plt.tight_layout()
fig.savefig('./figure/figure3a_1_cov_J.png', dpi=600)