BCT Brain Conectivity Toolbox is a very useful and open software to do graph and network analysis in Brain imaging. which has lot kinds of measures to give the structure of brain adjancent matrix . something you need to know I will explain here :

Since for the EEG and MEG fMRI we are using the weight /binary undirectal networks

Preprocessing:

assump W as the connectivity matrix from Freq domain or Source domain.

- autofix the W to remove self connecity or NAN or other values which will effect the symmtric:Wad_fix = weight_conversion(W, ‘autofix’);
- then we need threshold the matrix to keep the potinetal weight value :adthreshold=threshold_proportional(Wad_fix, 0.3);
- we normarlize the matrix : Wad_nrm = weight_conversion(adthreshold, ‘normalize’);
- up to now , we have got a clean connecitvity adjancet matrix for later use .

Clustering Coffient

The clustering coefficient is the fraction of triangles around a node and is equivalent to the fraction of node’s neighbors that are neighbors of each other.值越大 聚合度越大

clustering_coef_wu.m this function give you the clustering coef value of every node .

**Community structure and modularity: **

The optimal community structure is a subdivision of the network into nonoverlapping groups of nodes in a way that maximizes the number of within-group edges, and minimizes the number of between-group edges. The modularity is a statistic that quantifies the degree to which the network may be subdivided into such clearly delineated groups. 最大化组之间的距离 最小化 组内距离 每个节点只使用一次

community_louvain.m (BU, WU, BD, WD, signed networks)

Louvain community detection algorithm with added finetuning.

**Rich club coefficient:**

The rich club coefficient at level k is the fraction of edges that connect nodes of degree k or higher out of the maximum number of edges that such nodes might share.

* K-core: *The k-core is the largest subnetwork comprising nodes of degree at least k. The k-core is computed by recursively peeling off nodes with degree lower than k, until no such nodes remain in the subnetwork. for binary network

**Distance and characteristic path length:**

The reachability matrix describes whether pairs of nodes are connected by paths (reachable). The distance matrix contains lengths of shortest paths between all pairs of nodes. The characteristic path length is the average shortest path length in the network. cpl是网络中最短路径的平均值

The global efficiency is the average inverse shortest path length in the network. 全局效能是平均最短路径的倒数，路径越短，全局效能越大

The local efficiency is the global efficiency computed on the neighborhood of the node, and is related to the clustering coefficient. 本地效能指的是全局效能子节点中的最短平均路径的倒数

**Functional motifs:**

Functional motifs are subsets of connection patterns embedded within structural motifs. Functional motif frequency is the frequency of functional motif occurence around a node. In weighted networks, the motif frequency may be supplemented by its weighted generalizations, the motif intensity and the motif coherence.

* Density:* Density is the fraction of present connections to possible connections. Connection weights are ignored in calculations. ；连接密度

* Strength: *Node strength is the sum of weights of links connected to the node. In directed networks, the in-strength is the sum of inward link weights and the out-strength is the sum of outward link weights.

节点强度 指的是weight sum