我有一组随机生成的形式图，我想计算每个图的熵。同一个问题，不同的话：我有几个网络，想计算每个网络的信息量。
这里有两个包含图熵正式定义的来源：
http://www.cs.washington.edu/homes/anuprao/pubs/CSE533Autumn2010/lecture4.pdf (PDF)
http://arxiv.org/abs/0711.4175v1
我正在寻找的代码将图形作为输入（作为边列表或邻接矩阵）并输出一些位或其他信息内容度量。
因为我在任何地方都找不到它的实现，所以我打算根据正式定义从头开始编写代码。如果有人已经解决了这个问题并愿意分享代码，将不胜感激。
【问题讨论】：
标签： graph social-networking entropy information-theory
我最终使用了不同的论文来定义图熵：
复杂网络的信息理论：关于进化和架构约束
房车Sole 和 S. Valverde (2004)
基于拓扑配置的网络熵及其对随机网络的计算
B.H.王W.X.王和周周
计算每个的代码如下。该代码假定您有一个无自环的无向、未加权图。它将邻接矩阵作为输入，并以位为单位返回熵的数量。它在 R 中实现并使用 sna package。
graphEntropy <- function(adj, type="SoleValverde") {
if (type == "SoleValverde") {
return(graphEntropySoleValverde(adj))
}
else {
return(graphEntropyWang(adj))
}
}
graphEntropySoleValverde <- function(adj) {
# Calculate Sole & Valverde, 2004 graph entropy
# Uses Equations 1 and 4
# First we need the denominator of q(k)
# To get it we need the probability of each degree
# First get the number of nodes with each degree
existingDegrees = degree(adj)/2
maxDegree = nrow(adj) - 1
allDegrees = 0:maxDegree
degreeDist = matrix(0, 3, length(allDegrees)+1) # Need an extra zero prob degree for later calculations
degreeDist[1,] = 0:(maxDegree+1)
for(aDegree in allDegrees) {
degreeDist[2,aDegree+1] = sum(existingDegrees == aDegree)
}
# Calculate probability of each degree
for(aDegree in allDegrees) {
degreeDist[3,aDegree+1] = degreeDist[2,aDegree+1]/sum(degreeDist[2,])
}
# Sum of all degrees mult by their probability
sumkPk = 0
for(aDegree in allDegrees) {
sumkPk = sumkPk + degreeDist[2,aDegree+1] * degreeDist[3,aDegree+1]
}
# Equivalent is sum(degreeDist[2,] * degreeDist[3,])
# Now we have all the pieces we need to calculate graph entropy
graphEntropy = 0
for(aDegree in 1:maxDegree) {
q.of.k = ((aDegree + 1)*degreeDist[3,aDegree+2])/sumkPk
# 0 log2(0) is defined as zero
if (q.of.k != 0) {
graphEntropy = graphEntropy + -1 * q.of.k * log2(q.of.k)
}
}
return(graphEntropy)
}
graphEntropyWang <- function(adj) {
# Calculate Wang, 2008 graph entropy
# Uses Equation 14
# bigN is simply the number of nodes
# littleP is the link probability. That is the same as graph density calculated by sna with gden().
bigN = nrow(adj)
littleP = gden(adj)
graphEntropy = 0
if (littleP != 1 && littleP != 0) {
graphEntropy = -1 * .5 * bigN * (bigN - 1) * (littleP * log2(littleP) + (1-littleP) * log2(1-littleP))
}
return(graphEntropy)
}
【讨论】：
如果您有一个加权图，一个好的开始是对所有权重进行排序和计数。然后您可以使用公式 -log(p)+log(2) (http://en.wikipedia.org/wiki/Binary_entropy_function) 来确定代码所需的位数。也许这不起作用，因为它是二元熵函数？
【讨论】：
您可以使用Koerner's entropy（= 香农熵应用于图形）。一个很好的参考文献是here。但是请注意，计算通常是 NP-hard（出于愚蠢的原因，您需要搜索所有顶点子集）。
【讨论】：