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small fixes

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pull/1/head
Luca Lombardo 2 years ago
parent 6600f2ae1c
commit 74522d86d9
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3
.gitignore vendored
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# Makeindex log files
*.lpz
data/

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sources/1-s2.0-S009630032100881X-main.pdf
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# MAIL DI BINI
Buongiorno,
il progetto del corso di Calcolo Scientifico deve necessariamente intersecare gli argomenti sviluppati a lezione.
Le propongo allora questo progetto che tocca i metodi delle potenze, il processo di Arnoldi e il GMRES con restart e riguarda il page rank.
Nel lavoro che trova allegato (_autori Shen, Su, Carpentieri e Weng_) sono introdotti e confrontati alcuni metodi per effettuare il calcolo del vettore PageRank di una stessa rete ma con diversi valori del parametro di damping. Gli autori motivano questo interesse scrivendo che questo problema computazionale si incontra nel progetto di dispositivi anti-spam e citano per questo il lavoro _P.G. Constantine , D.F. Gleich , Random alpha PageRank, Internet Math. 6 (2009) 189–236._
Nel lavoro allegato vengono introdotti e confrontati alcuni metodi per effettuare questo calcolo. Precisamente:
`Algorithm 1 (pag. 6):` Metodo delle potenze adattato al calcolo del PageRank con diversi valori del parametro di damping
`Algorithm 4 (pag. 10):` Metodo del GMRES adattato al calcolo del PageRank con diversi valori del parametro di damping. Questo algoritmo utilizza il metodo di Arnoldi e il GMRES con restart che gli autori riportano per completezza rispettivamente nell'Algorithm 2 a pag. 7 e nell'Algorithm 3 a pag. 8.
Gli autori combinano poi `Algorithm 4` e `Algorithm 1` nell'`Algorithm 5` a pag. 10 per aumentare l'efficienza.
Il progetto riguarderebbe l'implementazione di `Algorithm 1` e `Algorithm 4` e un loro confronto per valutare l'efficienza in termini di numero di iterazioni e di tempo di cpu impiegato. Problemi test validi sono la matrice web-Stanford e web-BerkStan.
Se nella descrizione degli algoritmi ci fossero dei punti non chiari, bisogna andare a leggere la parte di testo che descrive gli algoritmi stessi. Ad esempio nella linea 14 dell'algoritmo 4 compare un vettore z che gli autori non definiscono dentro l'algoritmo. Questo vettore compare nella formula (40) e infatti viene definito nella riga subito dopo la (40).
Mi faccia sapere se incontra difficolta' di comprensione degli algoritmi (talvolta gli autori non sono molto precisi nel riportare i loro risultati). Ci risentiamo poi per mettere a punto i test da fare.
A presto,
Dario Bini

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{
"cells": [
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
"import networkx as nx\n",
"import scipy as sp\n",
"import scipy.sparse"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let's create two graphs from the list of edges downloaded from the Snap database. "
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"G1 = nx.read_edgelist('../data/web-Stanford.txt', create_using=nx.DiGraph(), nodetype=int)\n",
"\n",
"G2 = nx.read_edgelist('../data/web-BerkStan.txt', create_using=nx.DiGraph(), nodetype=int)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Creating the transition probability matrix"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [],
"source": [
"# square matrix of size n x n, where n is the number of nodes in the graph. The matrix is filled with zeros and the (i,j) element is x if the node i is connected to the node j. Where x is 1/(number of nodes connected to i).\n",
"\n",
"def create_matrix(G):\n",
" n = G.number_of_nodes()\n",
" P = sp.sparse.lil_matrix((n,n))\n",
" for i in G.nodes():\n",
" for j in G[i]: #G[i] is the list of nodes connected to i, it's neighbors\n",
" P[i-1,j-1] = 1/len(G[i])\n",
" return P"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"To ensure that the random process has a unique stationary distribution and it will not stagnate, the transition matrix P is usually modified to be an irreducible stochastic matrix A (called the Google matrix) as follows\n",
"\n",
"$$ A = \\alpha \\tilde{P} + (1-\\alpha)v e^T$$\n",
"\n",
"Where $\\tilde{P}$ is defined as \n",
"\n",
"$$ \\tilde{P} = P + v d^T$$\n",
"\n",
"Where $d \\in \\mathbb{N}^{n \\times 1}$ s a binary vector tracing the indices of dangling web-pages with no hyperlinks, i.e., $d(i ) = 1$ if the `ith` page has no hyperlink, $v \\in \\mathbb{R}^{n \\times 1}$ is a probability vector, $e = [1, 1, . . . , 1]^T$ , and $0 < \\alpha < 1$ is the so-called damping factor that represents the probability in the model that the surfer transfer by clicking a hyperlink rather than other ways"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [],
"source": [
"n = G1.number_of_nodes()\n",
"P = create_matrix(G1) "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"the vector `d` solves the dangling nodes problem"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [],
"source": [
"# define d as a nx1 sparse matrix, where n is the number of nodes in the graph. The vector is filled with d(i) = 1 if the i row of the matrix P is filled with zeros, other wise is 0\n",
"d = sp.sparse.lil_matrix((n,1))\n",
"for i in range(n):\n",
" if P[i].sum() == 0:\n",
" d[i] = 1"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The vector v is a probability vector, the sum of its elements bust be one"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [],
"source": [
"# define v as the probability vector of size n x 1, where n is the number of nodes in the graph. The vector is filled with 1/n\n",
"# https://en.wikipedia.org/wiki/Probability_vector\n",
"\n",
"v = sp.sparse.lil_matrix((n,1))\n",
"for i in range(n):\n",
" v[i] = 1/n "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now we can compute the transition matrix\n"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [],
"source": [
"Pt = P + v.dot(d.T)\n",
"\n",
"# Pt is a sparse matrix too"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [],
"source": [
"# e is a nx1 sparse matrix filled with ones\n",
"e = sp.sparse.lil_matrix((1,n))\n",
"for i in range(n):\n",
" e[0,i] = 1"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [],
"source": [
"# # v*eT is a nxn sparse matrix filled all with 1/n, let's call it B\n",
"\n",
"# B = sp.sparse.lil_matrix((n,n))\n",
"# for i in range(n):\n",
"# for j in range(n):\n",
"# B[i,j] = 1/n\n",
"\n",
"# A = alpha*Pt + (1-alpha)*B"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Algorithm 1 Shifted-Power method for PageRank with multiple damping factors:"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [],
"source": [
"a = [0.85, 0.9, 0.95, 0.99]\n",
"tau = 10**-8\n",
"max_mv = 1000\n",
"\n",
"# this should return mv (the number of iteration needed for the convergence), and two vector of lenght len(a) called x and r. Where x is the vector of the pagerank and r is the residual vector\n",
"\n",
"\n",
"\n",
"def Algorithm1(Pt, v, tau, max_mv, a: list):\n",
" u = Pt.dot(v) - v\n",
" mv = 1\n",
" for i in range(len(a)):\n",
" r = sp.sparse.lil_matrix((len(a),1))\n",
" r[i] = a[i]*u\n",
" Res = sp.sparse.lil_matrix((len(a),1))\n",
" Res[i] = np.linalg.norm(r[i])\n",
"\n",
" if Res[i] > tau:\n",
" x = sp.sparse.lil_matrix((len(a),1))\n",
" x[i] = r[i] + v\n",
" \n",
" while max(Res) > tau and mv < max_mv:\n",
" u = Pt.dot(u)\n",
" mv += 1\n",
"\n",
" for i in range(len(a)):\n",
" if Res[i] >= tau:\n",
" r = sp.sparse.lil_matrix((len(a),1))\n",
" r[i] = a[i]*u\n",
" Res = sp.sparse.lil_matrix((len(a),1))\n",
" Res[i] = np.linalg.norm(r[i])\n",
"\n",
" if Res[i] > tau:\n",
" x = sp.sparse.lil_matrix((len(a),1))\n",
" x[i] = r[i] + x[i]\n",
" return mv, x, r "
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [
{
"ename": "ValueError",
"evalue": "shape mismatch in assignment",
"output_type": "error",
"traceback": [
"\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
"\u001b[0;31mValueError\u001b[0m Traceback (most recent call last)",
"\u001b[0;32m/tmp/ipykernel_128897/741213869.py\u001b[0m in \u001b[0;36m<module>\u001b[0;34m\u001b[0m\n\u001b[1;32m 1\u001b[0m \u001b[0;31m# launch the algorithm\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 2\u001b[0;31m \u001b[0mmv\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mr\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mAlgorithm1\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mPt\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mv\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mtau\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mmax_mv\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0ma\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
"\u001b[0;32m/tmp/ipykernel_128897/1942976890.py\u001b[0m in \u001b[0;36mAlgorithm1\u001b[0;34m(Pt, v, tau, max_mv, a)\u001b[0m\n\u001b[1;32m 12\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mi\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mrange\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mlen\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 13\u001b[0m \u001b[0mr\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0msp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msparse\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mlil_matrix\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mlen\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 14\u001b[0;31m \u001b[0mr\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mi\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0ma\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mi\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mu\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 15\u001b[0m \u001b[0mRes\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0msp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msparse\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mlil_matrix\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mlen\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 16\u001b[0m \u001b[0mRes\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mi\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mnp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mlinalg\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mnorm\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mr\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mi\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
"\u001b[0;32m/usr/lib/python3/dist-packages/scipy/sparse/_lil.py\u001b[0m in \u001b[0;36m__setitem__\u001b[0;34m(self, key, x)\u001b[0m\n\u001b[1;32m 330\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_set_intXint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mkey\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mkey\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 331\u001b[0m \u001b[0;31m# Everything else takes the normal path.\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 332\u001b[0;31m \u001b[0mIndexMixin\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m__setitem__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mkey\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 333\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 334\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0m_mul_scalar\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mother\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
"\u001b[0;32m/usr/lib/python3/dist-packages/scipy/sparse/_index.py\u001b[0m in \u001b[0;36m__setitem__\u001b[0;34m(self, key, x)\u001b[0m\n\u001b[1;32m 130\u001b[0m if not ((broadcast_row or x.shape[0] == i.shape[0]) and\n\u001b[1;32m 131\u001b[0m (broadcast_col or x.shape[1] == i.shape[1])):\n\u001b[0;32m--> 132\u001b[0;31m \u001b[0;32mraise\u001b[0m \u001b[0mValueError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'shape mismatch in assignment'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 133\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mshape\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m==\u001b[0m \u001b[0;36m0\u001b[0m \u001b[0;32mor\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mshape\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m==\u001b[0m \u001b[0;36m0\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 134\u001b[0m \u001b[0;32mreturn\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
"\u001b[0;31mValueError\u001b[0m: shape mismatch in assignment"
]
}
],
"source": [
"# launch the algorithm\n",
"mv, x, r = Algorithm1(Pt, v, tau, max_mv, a)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Algorithm 2 Arnoldi process"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"def Algorithm2():\n",
" pass"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Algorithm 4 shifted-GMRES method for PageRank with multiple damping factors: "
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"def Algorithm4(Pt, v, m , a: list, tau , maxit, x: list):\n",
" iter = 1\n",
" \n",
" e1 = sp.sparse.lil_matrix((len(a),1))\n",
" e1[0] = 1\n",
"\n",
" # identity matrix sparse\n",
" I = sp.sparse.lil_matrix((len(a),len(a)))\n",
" for i in range(n):\n",
" I[i,i] = 1\n",
"\n",
" # create the page rank vector x\n",
" x = sp.sparse.lil_matrix((n,1))\n",
" for i in range(n):\n",
" x[i] = 1/n\n",
"\n",
" # create the vector r \n",
" r = sp.sparse.lil_matrix((len(a),1))\n",
" for i in range(len(a)):\n",
" r[i] = ((1-a[i])/a[i]).dot(v) - ((1/a[i]).dot(I) - Pt).dot(x[i]).dot(e1)\n",
" \n",
" # create the vector Res\n",
" Res = sp.sparse.lil_matrix((len(a),1))\n",
" for i in range(len(a)):\n",
" Res[i] = a[i] * np.linalg.norm(r[i])\n",
"\n",
" mv = 0\n",
"\n",
" while max(Res) > tau and mv < maxit:\n",
" # find the k that satisfy the condition res[k] = max(res[i])\n",
" k = 0\n",
" for i in range(len(a)):\n",
" if Res[i] == max(Res):\n",
" k = i\n",
" break\n",
" \n",
" # compute a new vector called delta where delta[i] = (res[i]*a[k])/(res[k]*a[i])\n",
" delta = sp.sparse.lil_matrix((len(a),1))\n",
" for i in range(len(a)) and i != k:\n",
" delta[i] = (Res[i]*a[k])/(Res[k]*a[i])\n",
"\n",
" # run algorithm 2\n",
" # TO DO\n",
"\n",
" # j depends on the algorithm 2\n",
" mv = mv + j\n",
"\n",
" \n",
" # .............\n",
"\n",
" \n",
" "
]
}
],
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