Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/structuralholes.py @ 0:4f3585e2f14b draft default tip
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author | shellac |
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date | Mon, 22 Mar 2021 18:12:50 +0000 |
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1 """Functions for computing measures of structural holes.""" |
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2 |
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3 import networkx as nx |
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4 |
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5 __all__ = ["constraint", "local_constraint", "effective_size"] |
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6 |
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7 |
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8 def mutual_weight(G, u, v, weight=None): |
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9 """Returns the sum of the weights of the edge from `u` to `v` and |
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10 the edge from `v` to `u` in `G`. |
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11 |
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12 `weight` is the edge data key that represents the edge weight. If |
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13 the specified key is `None` or is not in the edge data for an edge, |
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14 that edge is assumed to have weight 1. |
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15 |
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16 Pre-conditions: `u` and `v` must both be in `G`. |
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17 |
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18 """ |
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19 try: |
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20 a_uv = G[u][v].get(weight, 1) |
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21 except KeyError: |
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22 a_uv = 0 |
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23 try: |
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24 a_vu = G[v][u].get(weight, 1) |
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25 except KeyError: |
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26 a_vu = 0 |
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27 return a_uv + a_vu |
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28 |
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29 |
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30 def normalized_mutual_weight(G, u, v, norm=sum, weight=None): |
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31 """Returns normalized mutual weight of the edges from `u` to `v` |
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32 with respect to the mutual weights of the neighbors of `u` in `G`. |
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33 |
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34 `norm` specifies how the normalization factor is computed. It must |
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35 be a function that takes a single argument and returns a number. |
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36 The argument will be an iterable of mutual weights |
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37 of pairs ``(u, w)``, where ``w`` ranges over each (in- and |
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38 out-)neighbor of ``u``. Commons values for `normalization` are |
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39 ``sum`` and ``max``. |
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40 |
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41 `weight` can be ``None`` or a string, if None, all edge weights |
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42 are considered equal. Otherwise holds the name of the edge |
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43 attribute used as weight. |
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44 |
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45 """ |
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46 scale = norm(mutual_weight(G, u, w, weight) for w in set(nx.all_neighbors(G, u))) |
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47 return 0 if scale == 0 else mutual_weight(G, u, v, weight) / scale |
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48 |
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49 |
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50 def effective_size(G, nodes=None, weight=None): |
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51 r"""Returns the effective size of all nodes in the graph ``G``. |
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52 |
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53 The *effective size* of a node's ego network is based on the concept |
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54 of redundancy. A person's ego network has redundancy to the extent |
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55 that her contacts are connected to each other as well. The |
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56 nonredundant part of a person's relationships it's the effective |
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57 size of her ego network [1]_. Formally, the effective size of a |
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58 node $u$, denoted $e(u)$, is defined by |
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59 |
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60 .. math:: |
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61 |
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62 e(u) = \sum_{v \in N(u) \setminus \{u\}} |
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63 \left(1 - \sum_{w \in N(v)} p_{uw} m_{vw}\right) |
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64 |
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65 where $N(u)$ is the set of neighbors of $u$ and $p_{uw}$ is the |
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66 normalized mutual weight of the (directed or undirected) edges |
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67 joining $u$ and $v$, for each vertex $u$ and $v$ [1]_. And $m_{vw}$ |
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68 is the mutual weight of $v$ and $w$ divided by $v$ highest mutual |
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69 weight with any of its neighbors. The *mutual weight* of $u$ and $v$ |
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70 is the sum of the weights of edges joining them (edge weights are |
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71 assumed to be one if the graph is unweighted). |
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72 |
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73 For the case of unweighted and undirected graphs, Borgatti proposed |
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74 a simplified formula to compute effective size [2]_ |
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75 |
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76 .. math:: |
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77 |
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78 e(u) = n - \frac{2t}{n} |
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79 |
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80 where `t` is the number of ties in the ego network (not including |
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81 ties to ego) and `n` is the number of nodes (excluding ego). |
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82 |
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83 Parameters |
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84 ---------- |
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85 G : NetworkX graph |
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86 The graph containing ``v``. Directed graphs are treated like |
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87 undirected graphs when computing neighbors of ``v``. |
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88 |
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89 nodes : container, optional |
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90 Container of nodes in the graph ``G`` to compute the effective size. |
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91 If None, the effective size of every node is computed. |
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92 |
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93 weight : None or string, optional |
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94 If None, all edge weights are considered equal. |
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95 Otherwise holds the name of the edge attribute used as weight. |
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96 |
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97 Returns |
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98 ------- |
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99 dict |
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100 Dictionary with nodes as keys and the effective size of the node as values. |
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101 |
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102 Notes |
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103 ----- |
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104 Burt also defined the related concept of *efficiency* of a node's ego |
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105 network, which is its effective size divided by the degree of that |
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106 node [1]_. So you can easily compute efficiency: |
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107 |
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108 >>> G = nx.DiGraph() |
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109 >>> G.add_edges_from([(0, 1), (0, 2), (1, 0), (2, 1)]) |
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110 >>> esize = nx.effective_size(G) |
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111 >>> efficiency = {n: v / G.degree(n) for n, v in esize.items()} |
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112 |
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113 See also |
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114 -------- |
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115 constraint |
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116 |
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117 References |
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118 ---------- |
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119 .. [1] Burt, Ronald S. |
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120 *Structural Holes: The Social Structure of Competition.* |
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121 Cambridge: Harvard University Press, 1995. |
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122 |
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123 .. [2] Borgatti, S. |
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124 "Structural Holes: Unpacking Burt's Redundancy Measures" |
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125 CONNECTIONS 20(1):35-38. |
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126 http://www.analytictech.com/connections/v20(1)/holes.htm |
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127 |
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128 """ |
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129 |
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130 def redundancy(G, u, v, weight=None): |
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131 nmw = normalized_mutual_weight |
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132 r = sum( |
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133 nmw(G, u, w, weight=weight) * nmw(G, v, w, norm=max, weight=weight) |
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134 for w in set(nx.all_neighbors(G, u)) |
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135 ) |
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136 return 1 - r |
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137 |
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138 effective_size = {} |
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139 if nodes is None: |
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140 nodes = G |
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141 # Use Borgatti's simplified formula for unweighted and undirected graphs |
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142 if not G.is_directed() and weight is None: |
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143 for v in nodes: |
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144 # Effective size is not defined for isolated nodes |
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145 if len(G[v]) == 0: |
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146 effective_size[v] = float("nan") |
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147 continue |
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148 E = nx.ego_graph(G, v, center=False, undirected=True) |
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149 effective_size[v] = len(E) - (2 * E.size()) / len(E) |
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150 else: |
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151 for v in nodes: |
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152 # Effective size is not defined for isolated nodes |
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153 if len(G[v]) == 0: |
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154 effective_size[v] = float("nan") |
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155 continue |
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156 effective_size[v] = sum( |
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157 redundancy(G, v, u, weight) for u in set(nx.all_neighbors(G, v)) |
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158 ) |
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159 return effective_size |
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160 |
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161 |
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162 def constraint(G, nodes=None, weight=None): |
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163 r"""Returns the constraint on all nodes in the graph ``G``. |
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164 |
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165 The *constraint* is a measure of the extent to which a node *v* is |
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166 invested in those nodes that are themselves invested in the |
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167 neighbors of *v*. Formally, the *constraint on v*, denoted `c(v)`, |
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168 is defined by |
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169 |
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170 .. math:: |
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171 |
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172 c(v) = \sum_{w \in N(v) \setminus \{v\}} \ell(v, w) |
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173 |
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174 where `N(v)` is the subset of the neighbors of `v` that are either |
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175 predecessors or successors of `v` and `\ell(v, w)` is the local |
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176 constraint on `v` with respect to `w` [1]_. For the definition of local |
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177 constraint, see :func:`local_constraint`. |
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178 |
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179 Parameters |
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180 ---------- |
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181 G : NetworkX graph |
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182 The graph containing ``v``. This can be either directed or undirected. |
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183 |
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184 nodes : container, optional |
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185 Container of nodes in the graph ``G`` to compute the constraint. If |
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186 None, the constraint of every node is computed. |
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187 |
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188 weight : None or string, optional |
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189 If None, all edge weights are considered equal. |
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190 Otherwise holds the name of the edge attribute used as weight. |
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191 |
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192 Returns |
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193 ------- |
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194 dict |
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195 Dictionary with nodes as keys and the constraint on the node as values. |
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196 |
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197 See also |
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198 -------- |
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199 local_constraint |
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200 |
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201 References |
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202 ---------- |
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203 .. [1] Burt, Ronald S. |
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204 "Structural holes and good ideas". |
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205 American Journal of Sociology (110): 349–399. |
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206 |
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207 """ |
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208 if nodes is None: |
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209 nodes = G |
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210 constraint = {} |
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211 for v in nodes: |
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212 # Constraint is not defined for isolated nodes |
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213 if len(G[v]) == 0: |
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214 constraint[v] = float("nan") |
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215 continue |
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216 constraint[v] = sum( |
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217 local_constraint(G, v, n, weight) for n in set(nx.all_neighbors(G, v)) |
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218 ) |
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219 return constraint |
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220 |
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221 |
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222 def local_constraint(G, u, v, weight=None): |
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223 r"""Returns the local constraint on the node ``u`` with respect to |
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224 the node ``v`` in the graph ``G``. |
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225 |
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226 Formally, the *local constraint on u with respect to v*, denoted |
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227 $\ell(v)$, is defined by |
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228 |
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229 .. math:: |
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230 |
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231 \ell(u, v) = \left(p_{uv} + \sum_{w \in N(v)} p_{uw} p{wv}\right)^2, |
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232 |
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233 where $N(v)$ is the set of neighbors of $v$ and $p_{uv}$ is the |
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234 normalized mutual weight of the (directed or undirected) edges |
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235 joining $u$ and $v$, for each vertex $u$ and $v$ [1]_. The *mutual |
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236 weight* of $u$ and $v$ is the sum of the weights of edges joining |
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237 them (edge weights are assumed to be one if the graph is |
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238 unweighted). |
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239 |
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240 Parameters |
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241 ---------- |
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242 G : NetworkX graph |
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243 The graph containing ``u`` and ``v``. This can be either |
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244 directed or undirected. |
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245 |
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246 u : node |
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247 A node in the graph ``G``. |
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248 |
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249 v : node |
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250 A node in the graph ``G``. |
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251 |
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252 weight : None or string, optional |
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253 If None, all edge weights are considered equal. |
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254 Otherwise holds the name of the edge attribute used as weight. |
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255 |
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256 Returns |
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257 ------- |
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258 float |
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259 The constraint of the node ``v`` in the graph ``G``. |
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260 |
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261 See also |
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262 -------- |
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263 constraint |
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264 |
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265 References |
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266 ---------- |
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267 .. [1] Burt, Ronald S. |
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268 "Structural holes and good ideas". |
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269 American Journal of Sociology (110): 349–399. |
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270 |
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271 """ |
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272 nmw = normalized_mutual_weight |
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273 direct = nmw(G, u, v, weight=weight) |
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274 indirect = sum( |
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275 nmw(G, u, w, weight=weight) * nmw(G, w, v, weight=weight) |
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276 for w in set(nx.all_neighbors(G, u)) |
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277 ) |
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278 return (direct + indirect) ** 2 |