一入brain maping 到处是坑，从溯源 到连接 如今还得搞图论
别说了 让我静静 ！ 原文来自：Complex network measures of brain connectivity
The nature of nodes and links in individual brain networks is determined by combinations of brain mapping methods, anatomical parcellation schemes, and measures of connectivity
The choice of a given combination must be carefully motivated, as the nature of nodes and links largely determines the neurobiological interpretation of network topology
Nodes should ideally represent brain regions with coherent patterns of extrinsic anatomical or functional connections.
总而言之 nodes 是来及解剖分层和连接计算的结果，不同方法 得到的node可能不同的 看你用的是什么method
Binary links denote the presence or absence of connections (Fig. 2A), while weighted links also contain information about connection strengths
二值连接主要指node之间的连接存不存在，存在1 不存在0， 权重连接是node之间的连接强度，一般归一化后 都在 0<=w<=1
Weights in anatomical networks may represent the size, density, or coherence of anatomical tracts, while weights in functional and effective networks may represent respective magnitudes of correlational or causal interactions. Many recent studies discard link weights, as binary networks are in most cases simpler to characterize and have a more easily defined null model for statistical comparison (see below). On the other hand, weighted characterization usually focuses on somewhat different and complementary aspects of network organization
解剖网络里面权重有可能是尺寸 密度 相关性， 在功能和效率连接中 权重代表相关性的幅度 或者causal interactions (因果关系)， 很多研究表明不用weight 因为其麻烦，二值网络很好应用并且容易理解
Weak and non-significant links may represent spurious connections, particularly in functional or effective networks
Threshold values are often arbitrarily determined, and networks should ideally be characterized across a broad range of thresholds.
Independently, all self-connections or negative connections (such as functional anticorrelations) must currently be removed from the networks prior to analysis. Future network methods may be able to quantify
Measures of brain networks
Measurement values of all individual elements comprise a distribution, which provides a more global description of the network.
This distribution is most commonly characterized by its mean, although other features, such as distribution shape, may be more important if the distribution is nonhomogeneous.
In addition to these different representations, network measures also have binary and weighted, directed and undirected variants.
网络参数可以有二进制 权重 有向 无向变量
Weighted and directed variants of measures are typically generalizations of binary undirected variants and therefore reduce to the latter when computed on binary undirected networks.
we consider a basic and important measure known as the degree. The degree of an individual node is equal to the number of links connected to that node, which in practice is also equal to the number of neighbors of the node. Individual values of the degree therefore reflect importance of nodes in the network, as discussed below. The degrees of all nodes in the network comprise the degree distribution, which is an important marker of network development and resilience. The mean network degree is most commonly used as a measure of density, or the total “wiring cost” of the network. The directed variant of the degree distinguishes the number of inward links from the number of outward links, while the weighted variant of the degree, sometimes termed the strength, is defined as the sum of all neighboring link weights
这里引入degree 就是节点的对应的所有连接， 也等于它所有邻居的数量，
网络里面所有节点的degree 被称为 degree distribution
mean degree 是非常重要的，被称为 density 密度 或者 wiring cost
有向的连接可以分为income 和 output 两种
It is important to note that values of many network measures are greatly influenced by basic network characteristics, such as the number of nodes and links, and the degree distribution
Measures of functional segregation
Functional segregation in the brain is the ability for specialized processing to occur within densely interconnected groups of brain regions.
Measures of segregation primarily quantify the presence of such groups, known as clusters or modules,
比如 簇或者 模块
within the network. Measures of segregation have straightforward interpretations in anatomical and functional networks.
The presence of clusters in anatomical networks suggests the potential for functional segregation in these networks,
while the presence of clusters in functional networks suggests an organization of statistical dependencies indicative of segregated neural processing.
簇在功能连接意味着 处理信息 而产生的新的架构
Simple measures of segregation are based on the number of triangles in the network, with a high number of triangles implying segregation
如果有大量的三角形 意味着 隔离
Locally, the fraction of triangles around an individual node is known as the clustering coefficient and is equivalent to the fraction of the node’s neighbors that are also neighbors of each other (Watts and Strogatz, 1998).
三角形的占比 被称为 聚合系数， 等于相互是邻居节点的占比
The mean clustering coefficient for the network hence reflects, on average, the prevalence of clustered connectivity around individual nodes.
The mean clustering coefficient is normalized individually for each node (Table A1) and may therefore be disproportionately influenced by nodes with a low degree.
A classical variant of the clustering coefficient, known as the transitivity, is normalized collectively and consequently does not suffer from this problem (e.g., Newman, 2003). Both the clustering coefficient and the transitivity have been generalized for weighted (Onnela et al., 2005) and directed (Fagiolo, 2007) networks.
一个聚合系数的经典变量是 transitvity, 对聚合系数的归一化处理后的结果，
Measures of functional integration
Functional integration in the brain is the ability to rapidly combine specialized information from distributed brain regions.
Measures of integration characterize this concept by estimating the ease with which brain regions communicate and are commonly based on the concept of a path.
如何？ path !
Paths are sequences of distinct nodes and links and in anatomical networks represent potential routes of information flow between pairs of brain regions.
不同节点必然有信息流动 那就必然有流动渠道 那就是 path
Lengths of paths consequently estimate the potential for functional integration between brain regions, with shorter paths implying stronger potential for integration.
characteristic path length
The average shortest path length between all pairs of nodes in the network is known as the characteristic path length of the network (e.g., Watts and Strogatz, 1998) and is the most commonly used measure of functional integration. The average inverse shortest path length is a related measure known as the global efficiency
在所有节点中 平均最短路径被称为 CPL, 其 1/CPL 被称为 global efficiency
Unlike the characteristic path length, the global efficiency may be meaningfully computed on disconnected networks, as paths between disconnected nodes are defined to have infinite length, and correspondingly zero efficiency. More generally, the characteristic path length is primarily influenced by long paths (infinitely long paths are an illustrative extreme), while the global efficiency is primarily influenced by short paths
CPL一般都是连接网络， GE处理非网络连接的多，非连接节点之间的path 被视为 无限长，对应的GE就是0, 也可以说 CPL容易被长路径影响而 ＧＥ容易被段路径影响
Paths are easily generalized for directed and weighted networks (Fig. 3).
While a binary path length is equal to the number of links in the path,
a weighted path length is equal to the total sum of individual link lengths.
权重长度就是 link lengths.的总和
Link lengths are inversely related to link weights, as large weights typically represent strong associations and close proximity.
In comparison, functional networks have weaker connections between modules, and consequently a lower global efficiency
Small-world brain connectivity
Small-world networks are formally defined as networks that are significantly more clustered than random networks, yet have approximately the same characteristic path length as random networks
可以理解为不明功能的 一小撮 神经份子
Such local patterns are particularly diverse in directed networks. For instance, anatomical triangles may consist of feedforward loops, feedback loops, and bidirectional loops, with distinct frequencies of individual loops likely having specific functional implications. These patterns of local connectivity are known as (anatomical) motifs
关于 complex network analysis 的数学定义
这里面分了两个 第一个是 无向二值网络 第二个是 权重有向网络
我们先来考虑第一个 二值无向网络(binary and undirected )
N is the set of all nodes in the network, and n is the number of nodes. L is the set of all links in the network, and l is number of links. (i, j) is a link between nodes i and j, (i, j ∈ N). aij is the connection status between i and j: aij = 1 when link (i, j) exists (when i and j are neighbors); aij = 0 otherwise (aii = 0 for all i). We compute the number of links as l = ∑i,j∈N aij (to avoid ambiguity with directed links we count each undirected link twice, as aij and as aji).
l是(i j) i 节点 和 j节点之间的link
A(i j) 是 i j之间的连接状态 0 或者 1 (是否有链接)
那么多 所有的连接成为做I( i j)=sum(aij) 注意这里面所有连接被计算了两次
Degree :number of links connected to a node
Shortest path length: a basis for measuring integration
Number of triangles: a basis for measuring segregation
原理就是 这三个点之间的a都存在 由于双向的除以二 就是存在的三角形
Characteristic path length 个性最短路径
这个是比较重要的， 含义就是 每个节点和所有节点的平均路径
上面是triangles的总和 下面是 degree的总和平方（差1）