Table 2.

Graph theoretical metrics used for analysis.

DOI: http://dx.doi.org/10.7554/eLife.12366.021

Metric	Brief interpretation	Definition
Degree	Number of edges connected to a node. This generalizes to in- or out-degree in directed graphs, describing the number of incoming or outgoing connections for a node, respectively.	${\displaystyle k_i=\sum _{j\in n}{a}_{ij}}$ $N=\mathrm{set} \mathrm{of} \mathrm{all} \mathrm{nodes}$ $a_{ij}=\left\{\begin{array}{lc}1,& \mathrm{if} \mathrm{edge} \mathrm{from} \mathrm{node} i \mathrm{to} j \mathrm{exists}\\ 0,& \mathrm{otherwise}\end{array}\right.$ Directed versions: $k_i^{in}=\sum _{j\in N}{a}_{ji}$‍ ‍ ‍ ‍ ‍‍ ‍ ‍ ‍ ‍ ‍ $k_i^{out}=\sum _{j\in N}{a}_{ij}$ $a_{ij}=\left\{\begin{array}{lc}1,& \mathrm{if} \mathrm{directed} \mathrm{edge} \mathrm{from} \mathrm{node} i \mathrm{to} j \mathrm{exists}\\ 0,& \mathrm{otherwise}\end{array}\right.$
Clustering coefficient (Watts and Strogatz, 1998)	Level of connectivity among nearest neighbors of node $i$	${\displaystyle \frac{2t_i}{k_i(k_i-1)}}$ $t_i = \mathrm{number} \mathrm{of} \mathrm{triangles} \mathrm{that} \mathrm{include} \mathrm{node} i$
Characteristic path length (Watts and Strogatz, 1998)	Mean shortest undirected path length over all pairs of nodes	${\displaystyle \frac{1}{n(n-1)}\sum _{i,j\in N,j\ne i}{d}_{ij}}$ $n = \mathrm{number} \mathrm{of} \mathrm{nodes}$ $d_{ij} = \mathrm{shortest} \mathrm{undirected} \mathrm{path} \mathrm{from} i \mathrm{to} j$
Global efficiency (Latora and Marchiori, 2001)	Mean inverse shortest undirected path length over all pairs of nodes	${\displaystyle \frac{1}{n(n-1)}\sum _{i,j \in N,j\ne i}\frac{1}{d_{ij}}}$ $d_{ij}= \mathrm{shortest} \mathrm{undirected} \mathrm{path} \mathrm{from} i \mathrm{to} j$
Nodal efficiency (generalized from [Achard and Bullmore, 2007])	Mean inverse shortest directed path length from a single node to all other nodes	${\displaystyle \frac{1}{(n-1)}\sum _{j\in N,j\ne i}\frac{1}{d_{ij}}}$ $d_{ij} = \mathrm{shortest} \mathrm{directed} \mathrm{path} \mathrm{from} i \mathrm{to} j$
Reciprocity coefficient	Proportion of edges from node $i$ to node $j$ that have a reciprocal connection from node $j$ to node $i$ (when $i\ne j$)	${\displaystyle \frac{1}{N_e}\sum _{i,j\in N,j\ne i}{a}_{ij}{a}_{ji}}$ $a_{ij} =\left\{\begin{array}{lc}1,& \mathrm{if} \mathrm{directed} \mathrm{edge} \mathrm{from} i \mathrm{to} j \mathrm{exists}\\ 0,& \mathrm{otherwise}\end{array}\right.$ $N_e = \mathrm{total} \mathrm{number} \mathrm{of} \mathrm{undirected} \mathrm{edges}$