{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Arc Diagram of Star Wars Characters that Interacted in The Force Awakens ##"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"[Arc diagram](https://en.wikipedia.org/wiki/Arc_diagram) is a special layout for graphs. It places the nodes on a line and the links are arcs of circles. Here we\n",
"plot the arc diagram associated to characters in Stars Wars: The Force Awakens. The nodes are characters. They are colored according to the number of times the corresponding character spoke across the 7th episode of Star Wars.\n",
" "
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"url=\"https://raw.githubusercontent.com/evelinag/StarWars-social-network/master/\"+\\\n",
"\"networks/starwars-episode-7-interactions-allCharacters.json\""
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"import requests\n",
"import json\n",
"r = requests.get(url)\n",
"data = json.loads(r.content)"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"dict_keys(['nodes', 'links'])\n"
]
}
],
"source": [
"print (data.keys()) "
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [],
"source": [
"import numpy as np"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"{'name': 'LUKE', 'value': 2, 'colour': '#3881e5'}"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"L = len(data['nodes'])# number of nodes (characters)\n",
"data['nodes'][0]"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [],
"source": [
"labels = [item['name'].lower().title() for item in data['nodes']]\n",
"values = [item['value'] for item in data['nodes']]\n",
"hover_text = [f'{labels[k]}, {values[k]} scenes' for k in range(L)]"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"{'source': 2, 'target': 1, 'value': 1}\n"
]
}
],
"source": [
"print(data['links'][0])"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Define the graph edges, i.e. the pairs of nodes that are linked:"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [],
"source": [
"edges = [(item['source'], item['target']) for item in data['links']] #Edges"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [],
"source": [
"interact_strength = [item['value'] for item in data['links']]\n",
"keys = sorted(set(interact_strength)) "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Associate to each link a width value for the corresponding arc, depending on the number \n",
"of interactions between the nodes connected by that arc: \n"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [],
"source": [
"widths = [0.5+k*0.25 for k in range(5)] + [2+k*0.25 for k in range(4)]+[3, 3.25, 3.75, 4.25, 5, 5.25, 7]\n",
"d = dict(zip(keys, widths)) \n",
"nwidths = [d[val] for val in interact_strength] "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Below we define a few functions that compute points on arcs of circles connecting two nodes/characters. These arcs are defined as quadratic \n",
"rational Bezier curves ([[Gallier]](http://www.cis.upenn.edu/~jean/gbooks/geom1.html)), of control points $b_0, b_1, b_2$.\n",
"\n",
"$b_0$ and $b_2$ are the points on the x-axis, representing two connected nodes. $b_1$ is chosen such that the triangle $\\Delta b_0b_1b_2$ is equilateral. The weights of the control points are respectively (1, 0.5, 1)."
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [],
"source": [
"def get_b1(b0, b2):\n",
" # b0, b1 list of x, y coordinates\n",
" if len(b0) != len(b2) != 2:\n",
" raise ValueError('b0, b1 must be lists of two elements')\n",
" b1 = 0.5 * (np.asarray(b0)+np.asarray(b2))+\\\n",
" 0.5 * np.array([0,1.0]) * np.sqrt(3) * np.linalg.norm(np.array(b2)-np.array(b0))\n",
" return b1.tolist() "
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {},
"outputs": [],
"source": [
"def dim_plus_1(b, w):#lift the points b0, b1, b2 to 3D points a0, a1, a2 (see Gallier book)\n",
" #b is a list of 3 lists of 2D points, i.e. a list of three 2-lists \n",
" #w is a list of numbers (weights) of len equal to the len of b\n",
" if not isinstance(b, list) or not isinstance(b[0], list):\n",
" raise ValueError('b must be a list of three 2-lists')\n",
" if len(b) != len(w) != 3:\n",
" raise ValueError('the number of weights must be equal to the nr of points')\n",
" else:\n",
" a = np.array([point + [w[i]] for (i, point) in enumerate(b)])\n",
" a[1, :2] *= w[1]\n",
" return a"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {},
"outputs": [],
"source": [
"def Bezier_curve(bz, nr): #the control point coordinates are passed in a list bz=[bz0, bz1, bz2] \n",
" # bz is a list of three 2-lists \n",
" # nr is the number of points to be computed on each arc\n",
" t = np.linspace(0, 1, nr)\n",
" #for each parameter t[i] evaluate a point on the Bezier curve with the de Casteljau algorithm\n",
" N = len(bz) \n",
" points = [] # the list of points to be computed on the Bezier curve\n",
" for i in range(nr):\n",
" aa = np.copy(bz) \n",
" for r in range(1, N):\n",
" aa[:N-r,:] = (1-t[i]) * aa[:N-r,:] + t[i] * aa[1:N-r+1,:] # convex combination of points\n",
" points.append(aa[0,:]) \n",
" return np.array(points) "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The points of the 3D-Bezier curve computed from the lifted control points $b_0, b_1, b_2$\n",
"are projected through a central projection of center $O(0,0,0)$ onto the plane $z=1$, to get the \n",
"2D rational Bezier curve, i.e. an arc of circle\n",
"whose tangents at the nodes $b_0, b_2$, form an angle of $\\pi/3$ with the line $b_0b_2$:"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {},
"outputs": [],
"source": [
"def Rational_Bezier_curve(a, nr):\n",
" discrete_curve = Bezier_curve(a, nr ) \n",
" return [p[:2]/p[2] for p in discrete_curve]"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {},
"outputs": [],
"source": [
"import plotly.plotly as py\n",
"import plotly.graph_objs as go"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Define the trace for nodes placed on the x-axis. The nodes are colored with the colorscale `pl_density`: "
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {},
"outputs": [],
"source": [
"pl_density = [[0.0, 'rgb(230,240,240)'],\n",
" [0.1, 'rgb(187,220,228)'],\n",
" [0.2, 'rgb(149,197,226)'],\n",
" [0.3, 'rgb(123,173,227)'],\n",
" [0.4, 'rgb(115,144,227)'],\n",
" [0.5, 'rgb(119,113,213)'],\n",
" [0.6, 'rgb(120,84,186)'],\n",
" [0.7, 'rgb(115,57,151)'],\n",
" [0.8, 'rgb(103,35,112)'],\n",
" [0.9, 'rgb(82,20,69)'],\n",
" [1.0, 'rgb(54,14,36)']]"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {},
"outputs": [],
"source": [
"node_trace = dict(type='scatter',\n",
" x=list(range(L)),\n",
" y=[0]*L,\n",
" mode='markers',\n",
" marker=dict(size=12, \n",
" color=values, \n",
" colorscale=pl_density,\n",
" showscale=False,\n",
" line=dict(color='rgb(50,50,50)', width=0.75)),\n",
" text=hover_text,\n",
" hoverinfo='text')"
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {},
"outputs": [],
"source": [
"data = []\n",
"tooltips = [] #list of strings to be displayed when hovering the mouse over the middle of the circle arcs\n",
"xx = []\n",
"yy = []"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Generate the arcs of circles as rational Bezier curves and append each arc to a list:"
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {},
"outputs": [],
"source": [
"X = list(range(L)) # node x-coordinates\n",
"nr = 75 \n",
"for i, (j, k) in enumerate(edges):\n",
" if j < k:\n",
" tooltips.append(f'interactions({labels[j]}, {labels[k]})={interact_strength[i]}')\n",
" else:\n",
" tooltips.append(f'interactions({labels[k]}, {labels[j]})={interact_strength[i]}')\n",
" b0 = [X[j], 0.0]\n",
" b2 = [X[k], 0.0]\n",
" b1 = get_b1(b0, b2)\n",
" a = dim_plus_1([b0, b1, b2], [1, 0.5, 1])\n",
" pts = Rational_Bezier_curve(a, nr)\n",
" xx.append(pts[nr//2][0]) #abscissa of the middle point on the computed arc\n",
" yy.append(pts[nr//2][1]) #ordinate of the same point\n",
" x,y = zip(*pts)\n",
" \n",
" data.append(dict(type='scatter',\n",
" x=x, \n",
" y=y, \n",
" name='',\n",
" mode='lines', \n",
" line=dict(width=nwidths[i], color='#6b8aca', shape='spline'),\n",
" hoverinfo='none'\n",
" )\n",
" )"
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {},
"outputs": [],
"source": [
"data.append(dict(type='scatter',\n",
" x=xx,\n",
" y=yy,\n",
" name='',\n",
" mode='markers',\n",
" marker=dict(size=0.5, color='#6b8aca'),\n",
" text=tooltips,\n",
" hoverinfo='text'))\n",
"data.append(node_trace)"
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {},
"outputs": [],
"source": [
"title = \"Arc Diagram of Star Wars Characters that Interacted in The Force Awakens\"\n",
"\n",
"anno_text = \"Data source: \"+\\\n",
" \" [1]\""
]
},
{
"cell_type": "code",
"execution_count": 28,
"metadata": {},
"outputs": [],
"source": [
"layout = dict(\n",
" title=title, \n",
" font=dict(size=10), \n",
" width=900,\n",
" height=460,\n",
" showlegend=False,\n",
" xaxis=dict(anchor='y',\n",
" showline=False, \n",
" zeroline=False,\n",
" showgrid=False,\n",
" tickvals=list(range(27)), \n",
" ticktext=labels,\n",
" tickangle=50,\n",
" ),\n",
" yaxis=dict(visible=False), \n",
" hovermode='closest',\n",
" margin=dict(t=80, b=110, l=10, r=10),\n",
" annotations=[dict(showarrow=False, \n",
" text=anno_text,\n",
" xref='paper', \n",
" yref='paper', \n",
" x=0.05, \n",
" y=-0.3, \n",
" xanchor='left', \n",
" yanchor='bottom', \n",
" font=dict(size=11 ))\n",
" ]\n",
" \n",
" \n",
" )\n",
"fig = go.FigureWidget(data=data, layout=layout)\n",
"#py.plot(fig, filename='arc-diagram-FA') # plot online"
]
},
{
"cell_type": "code",
"execution_count": 30,
"metadata": {},
"outputs": [
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"FigureWidget({\n",
" 'data': [{'hoverinfo': 'none',\n",
" 'line': {'color': '#6b8aca', 'shape': 'spline'…"
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},
"metadata": {},
"output_type": "display_data"
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"source": [
"fig"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The online plot is posted at the URL [https://plot.ly/~empet/13574]( https://plot.ly/~empet/13574)."
]
},
{
"cell_type": "code",
"execution_count": null,
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