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        "author": "Bruno Messias; Thomas K. Peron",
        "author_email": "messias.physics@gmail.com",
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        "classifiers": [
            "Development Status :: 4 - Beta",
            "Intended Audience :: Science/Research",
            "License :: OSI Approved :: MIT License",
            "Programming Language :: Python :: 3",
            "Topic :: Scientific/Engineering :: Information Analysis",
            "Topic :: Scientific/Engineering :: Mathematics",
            "Topic :: Scientific/Engineering :: Physics"
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        "description": "# ![](docs/source/imgs/stdog.png) \n# Structure and Dynamics on Graphs (Beta)\n\nThe main goal of StDoG is to provide a package which can be used to study\ndynamical and structural properties (like spectra) on graphs with a large\nnumber of vertices. The modules of StDoG are being built by\ncombining codes written in *Tensorflow* + *CUDA* and *C++*.\n\n## 1 - Install\n\n```\npip install stdog\n```\n\n## 2 - Examples\n\n### 2.1 - Dynamics\n\n#### 2.1.1 - Kuramoto\n\n##### Tensorflow\n```python\nimport numpy as np\nimport igraph as ig\nfrom stdog.utils.misc import ig2sparse  #Function to convert igraph format to sparse matrix\n\n\nnum_couplings = 40\nN = 20480\n\nG = ig.Graph.Erdos_Renyi(N, 3/N)\nadj = ig2sparse(G)\n\nomegas = np.random.normal(size= N).astype(\"float32\")\ncouplings = np.linspace(0.0,4.,num_couplings)\nphases =  np.array([\n    np.random.uniform(-np.pi,np.pi,N)\n    for i_l in range(num_couplings)\n\n],dtype=np.float32)\n\n\nprecision =32\n\ndt = 0.01\nnum_temps = 50000\ntotal_time = dt*num_temps\ntotal_time_transient = total_time\ntransient = False\n```\n\n```python\nfrom stdog.dynamics.kuramoto import Heuns\n\nheuns_0 = Heuns(adj, phases, omegas, couplings, total_time, dt,         \n    device=\"/gpu:0\", # or /cpu:\n    precision=precision, transient=transient)\n\nheuns_0.run()\nheuns_0.transient = True\nheuns_0.total_time = total_time_transient\nheuns_0.run()\norder_parameter_list = heuns_0.order_parameter_list # (num_couplings, total_time//dt)\n```\n```python\nimport matplotlib.pyplot as plt\n\nr = np.mean(order_parameter_list, axis=1)\nstdr = np.std(order_parameter_list, axis=1)\n\nplt.ion()\nfig, ax1 = plt.subplots()\nax1.plot(couplings,r,'.-')\nax2 = ax1.twinx()\nax2.plot(couplings,stdr,'r.-')\nplt.show()\n```\n![](docs/imgs/heuns_tf.png)\n\n#### CUDA - Faster than Tensorflow implementation\n\nIf CUDA is available. You can install our another package,\n[stdogpkg/cukuramoto](https://github.com/stdogpkg/cukuramoto) (C)\n```\npip install cukuramoto\n```\n\n```python\nfrom stdog.dynamics.kuramoto.cuheuns import CUHeuns as cuHeuns\n\nheuns_0 = cuHeuns(adj, phases, omegas,  couplings,\n    total_time, dt, block_size = 1024, transient = False)\n\nheuns_0.run()\n\nheuns_0.transient = True\nheuns_0.total_time = total_time_transient\nheuns_0.run()\norder_parameter_list = heuns_0.order_parameter_list #\n```\n### 2.2 Spectral \n\n#### Spectral Density\n\nThe Kernel Polynomial Method [1] can\u00a0estimate the spectral density of large sparse Hermitan matrices with a computational cost almost linear. This method combines three key ingredients: the Chebyshev expansion + the stochastic trace estimator [2] + kernel smoothing.\n\n```python\nimport igraph as ig\nimport numpy as np\n\nN = 3000\nG = ig.Graph.Erdos_Renyi(N, 3/N)\n\nW = np.array(G.get_adjacency().data, dtype=np.float64)\nvals = np.linalg.eigvalsh(W).real\n```\n\n```python\nimport stdog.spectra as spectra\nfrom stdog.utils.misc import ig2sparse \n\nW = ig2sparse(G)\nnum_moments = 300\nnum_vecs = 200\nextra_points = 10\nek, rho = spectra.dos.kpm(W, num_moments, num_vecs, extra_points, device=\"/gpu:0\")\n```\n\n```python\nimport matplotlib.pyplot as plt\nplt.hist(vals, density=True, bins=100, alpha=.9, color=\"steelblue\")\nplt.scatter(ek, rho, c=\"tomato\", zorder=999, alpha=0.9, marker=\"d\")\nplt.ylim(0, 1)\nplt.show()\n```\n![kpm](docs/imgs/kpm_dos.png)\n#### Trace Functions through Stochastic Lanczos Quadrature (SLQ)[3]\n\n\n##### Computing custom trace functions\n\n```python\nfrom stdog.spectra.trace_function import slq\nimport tensorflow as tf\n\ndef trace_function(eig_vals):\n    return tf.exp(eig_vals)\n\nnum_vecs = 100\nnum_steps = 50\napproximated_estrada_index, _ = slq(L_sparse, num_vecs, num_steps,  trace_function, device=\"/gpu:0\")\nexact_estrada_index =  np.sum(np.exp(vals_laplacian))\napproximated_estrada_index, exact_estrada_index\n```\nThe above code returns\n\n```\n(3058.012, 3063.16457163222)\n```\n##### Entropy\n```python\nimport scipy\nimport scipy.sparse\nfrom stdog.spectra.trace_function import entropy as slq_entropy\n\ndef entropy(eig_vals):\n  s = 0.\n  for val in eig_vals:\n    if val > 0:\n      s += -val*np.log(val)\n  return s\n\nL = np.array(G.laplacian(normalized=True), dtype=np.float64)\nvals_laplacian = np.linalg.eigvalsh(L).real\n\nexact_entropy =  entropy(vals_laplacian)\n\nL_sparse = scipy.sparse.coo_matrix(L)\n    \nnum_vecs = 100\nnum_steps = 50\napproximated_entropy = slq_entropy(\n    L_sparse, num_vecs, num_steps, device=\"/cpu:0\")\n\napproximated_entropy, exact_entropy\n```\n```\n(-509.46283, -512.5283224633046)\n```\n\n## References\n\n1 -  Wang, L.W., 1994. Calculating the density of states and\noptical-absorption spectra of large quantum systems by the plane-wave moments\nmethod. Physical Review B, 49(15), p.10154.\n\n2 - Hutchinson, M.F., 1990. A stochastic estimator of the trace of the\ninfluence matrix for laplacian smoothing splines. Communications in\nStatistics-Simulation and Computation, 19(2), pp.433-450.\n\n3 - Ubaru, S., Chen, J., & Saad, Y. (2017). Fast Estimation of tr(f(A)) via Stochastic Lanczos Quadrature. \nSIAM Journal on Matrix Analysis and Applications, 38(4), 1075-1099.\n\n\n## 3 - How to cite\n\n[Thomas Peron](https://tkdmperon.github.io/), [Bruno Messias](http://brunomessias.com/), Ang\u00e9lica S. Mata, [Francisco A. Rodrigues](http://conteudo.icmc.usp.br/pessoas/francisco/), and [Yamir Moreno](http://cosnet.bifi.es/people/yamir-moreno/). On the onset of synchronization of Kuramoto oscillators in scale-free networks. [arXiv:1905.02256](https://arxiv.org/abs/1905.02256) (2019).\n\n## 4 - Acknowledgements\n\nThis work has been supported also by FAPESP grants  11/50761-2  and  2015/22308-2.   Research  carriedout using the computational resources of the Center forMathematical  Sciences  Applied  to  Industry  (CeMEAI)funded by FAPESP (grant 2013/07375-0).\n \n### Responsible authors\n\n[@devmessias](https://github.com/devmessias), [@tkdmperon](https://github.com/tkdmperon)",
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