1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
|
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN"
"http://www.w3.org/TR/html4/loose.dtd">
<html >
<head><title>1.2.3.0 knn_q_from_degrees.py</title>
<meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1">
<meta name="generator" content="TeX4ht (http://www.cse.ohio-state.edu/~gurari/TeX4ht/)">
<meta name="originator" content="TeX4ht (http://www.cse.ohio-state.edu/~gurari/TeX4ht/)">
<!-- html,index=2,3,4,5,next -->
<meta name="src" content="mammult_doc.tex">
<meta name="date" content="2015-10-19 16:26:00">
<link rel="stylesheet" type="text/css" href="mammult_doc.css">
</head><body
>
<!--l. 3--><div class="crosslinks"><p class="noindent">[<a
href="mammult_docsu35.html" >next</a>] [<a
href="mammult_docsu33.html" >prev</a>] [<a
href="mammult_docsu33.html#tailmammult_docsu33.html" >prev-tail</a>] [<a
href="#tailmammult_docsu34.html">tail</a>] [<a
href="mammult_docsu31.html#mammult_docsu34.html" >up</a>] </p></div>
<h5 class="subsubsectionHead"><a
id="x38-370001.2.3"></a><span
class="cmtt-10x-x-109">knn</span><span
class="cmtt-10x-x-109">_q</span><span
class="cmtt-10x-x-109">_from</span><span
class="cmtt-10x-x-109">_degrees.py</span></h5>
<!--l. 3--><p class="noindent" ><span
class="cmbx-10x-x-109">NAME</span>
<!--l. 3--><p class="indent" > <span
class="cmbx-10x-x-109">knn</span><span
class="cmbx-10x-x-109">_q</span><span
class="cmbx-10x-x-109">_from</span><span
class="cmbx-10x-x-109">_degrees.py </span>- compute the inter-layer degree-degree
correlation function.
<!--l. 3--><p class="noindent" ><span
class="cmbx-10x-x-109">SYNOPSYS</span>
<!--l. 3--><p class="indent" > <span
class="cmbx-10x-x-109">knn</span><span
class="cmbx-10x-x-109">_q</span><span
class="cmbx-10x-x-109">_from</span><span
class="cmbx-10x-x-109">_degrees.py </span><span
class="cmmi-10x-x-109"><</span><span
class="cmitt-10x-x-109">filein</span><span
class="cmmi-10x-x-109">></span>
<!--l. 37--><p class="noindent" ><span
class="cmbx-10x-x-109">DESCRIPTION</span>
<!--l. 37--><p class="indent" > Compute the inter-layer degree correlation functions for two layers of a
multiplex, using the degrees of the nodes specified in the input file. The format of
the input file is as follows
<!--l. 37--><p class="indent" >   <span
class="cmti-10x-x-109">ki qi</span>
<!--l. 37--><p class="indent" > where <span
class="cmti-10x-x-109">ki </span>and <span
class="cmti-10x-x-109">qi </span>are, respectively, the degree at layer 1 and the degree at layer
2 of node <span
class="cmti-10x-x-109">i</span>.
<!--l. 37--><p class="indent" > If we consider two layers of a multiplex, and we denote by <span
class="cmmi-10x-x-109">k </span>the degree
of a node on the first layer and by <span
class="cmmi-10x-x-109">q </span>the degree of the same node on
the second layers, the inter-layer degree correlation function is defined
as
<table
class="equation-star"><tr><td>
<center class="math-display" >
<img
src="mammult_doc9x.png" alt="k(q) = -1-∑ k′P (k′|q)
Nk ′
k
" class="math-display" ></center></td></tr></table>
<!--l. 37--><p class="nopar" >
<!--l. 37--><p class="indent" > where <span
class="cmmi-10x-x-109">P</span>(<span
class="cmmi-10x-x-109">k</span><span
class="cmsy-10x-x-109">′|</span><span
class="cmmi-10x-x-109">q</span>) is the probability that a node with degree <span
class="cmmi-10x-x-109">q </span>on the second
layer has degree equal to <span
class="cmmi-10x-x-109">k</span><span
class="cmsy-10x-x-109">′ </span>on the first layer, and <span
class="cmmi-10x-x-109">N</span><sub><span
class="cmmi-8">k</span></sub> is the number of
nodes with degree <span
class="cmmi-10x-x-109">k </span>on the first layer. The quantity <span class="overline"><span
class="cmmi-10x-x-109">k</span></span>(<span
class="cmmi-10x-x-109">q</span>) is the expected
degree at layer 1 of node that have degree equal to <span
class="cmmi-10x-x-109">q </span>on layer 2. The dual
quantity:
<table
class="equation-star"><tr><td>
<center class="math-display" >
<img
src="mammult_doc10x.png" alt=" ∑
q(k) = -1- q′P (q′|k)
Nq q′
" class="math-display" ></center></td></tr></table>
<!--l. 37--><p class="nopar" >
<!--l. 37--><p class="indent" > is the average degree on layer 2 of nodes having degree <span
class="cmmi-10x-x-109">k </span>on layer
1.
<!--l. 57--><p class="noindent" ><span
class="cmbx-10x-x-109">OUTPUT</span>
<!--l. 57--><p class="indent" > The program prints on <span
class="cmtt-10x-x-109">stdout </span>a list of lines in the format:
<!--l. 57--><p class="indent" >   <span
class="cmti-10x-x-109">k </span><span class="overline"><span
class="cmmi-10x-x-109">q</span></span>(<span
class="cmmi-10x-x-109">k</span>)
<!--l. 57--><p class="indent" > where <span
class="cmti-10x-x-109">k </span>is the degree on layer 1 and <span class="overline"><span
class="cmmi-10x-x-109">q</span></span>(<span
class="cmmi-10x-x-109">k</span>) is the average degree on layer 2 of
nodes having degree equal to <span
class="cmmi-10x-x-109">k </span>on layer 1.
<!--l. 57--><p class="indent" > The program also prints on <span
class="cmtt-10x-x-109">stderr </span>a list of lines in the format:
<!--l. 57--><p class="indent" >   <span
class="cmti-10x-x-109">q </span><span class="overline"><span
class="cmmi-10x-x-109">k</span></span>(<span
class="cmmi-10x-x-109">q</span>)
<!--l. 57--><p class="indent" > where <span
class="cmti-10x-x-109">q </span>is the degree on layer 2 and <span class="overline"><span
class="cmmi-10x-x-109">k</span></span>(<span
class="cmmi-10x-x-109">q</span>) is the average degree on layer 1 of
nodes having degree equal to <span
class="cmmi-10x-x-109">q </span>on layer 2.
<!--l. 64--><p class="noindent" ><span
class="cmbx-10x-x-109">REFERENCE</span>
<!--l. 64--><p class="indent" > V. Nicosia, V. Latora, “Measuring and modeling correlations in multiplex
networks”, <span
class="cmti-10x-x-109">Phys. Rev. E </span><span
class="cmbx-10x-x-109">92</span>, 032805 (2015).
<!--l. 64--><p class="indent" > Link to paper: <a
href="http://journals.aps.org/pre/abstract/10.1103/PhysRevE.92.032805" class="url" ><span
class="cmtt-10x-x-109">http://journals.aps.org/pre/abstract/10.1103/PhysRevE.92.032805</span></a>
<!--l. 64--><p class="indent" > V. Nicosia, G. Bianconi, V. Latora, M. Barthelemy, “Growing multiplex
networks”, <span
class="cmti-10x-x-109">Phys. Rev. Lett. </span><span
class="cmbx-10x-x-109">111</span>, 058701 (2013).
<!--l. 64--><p class="indent" > Link to paper: <a
href="http://prl.aps.org/abstract/PRL/v111/i5/e058701" class="url" ><span
class="cmtt-10x-x-109">http://prl.aps.org/abstract/PRL/v111/i5/e058701</span></a>
<!--l. 64--><p class="indent" > V. Nicosia, G. Bianconi, V. Latora, M. Barthelemy, “Non-linear growth and
condensation in multiplex networks”, <span
class="cmti-10x-x-109">Phys. Rev. E </span><span
class="cmbx-10x-x-109">90</span>, 042807 (2014).
<!--l. 64--><p class="indent" > Link to paper: <a
href="http://journals.aps.org/pre/abstract/10.1103/PhysRevE.90.042807" class="url" ><span
class="cmtt-10x-x-109">http://journals.aps.org/pre/abstract/10.1103/PhysRevE.90.042807</span></a>
<!--l. 4--><div class="crosslinks"><p class="noindent">[<a
href="mammult_docsu35.html" >next</a>] [<a
href="mammult_docsu33.html" >prev</a>] [<a
href="mammult_docsu33.html#tailmammult_docsu33.html" >prev-tail</a>] [<a
href="mammult_docsu34.html" >front</a>] [<a
href="mammult_docsu31.html#mammult_docsu34.html" >up</a>] </p></div>
<!--l. 4--><p class="indent" > <a
id="tailmammult_docsu34.html"></a>
</body></html>
|
