fluctuability

Calculates fluctuatbility

fluctuability(netin, calc='overtime')

Fluctuability of temporal networks.

This is the variation of the network’s edges over time. [fluct-1] This is the unique number of edges through time divided by the overall number of edges.

Parameters:

• netin (array or dict) – Temporal network input (graphlet or contact) (nettype: ‘bd’, ‘bu’, ‘wu’, ‘wd’)
• calc (str) – Version of fluctuabiility to calcualte. ‘overtime’

Returns:

fluct – Fluctuability

Return type:

array

Notes

Fluctuability quantifies the variability of edges. Given x number of edges, F is higher when those are repeated edges among a smaller set of edges and lower when there are distributed across more edges.

\[F = {{\sum_{i,j} H_{i,j}} \over {\sum_{i,j,t} G_{i,j,t}}}\]

where \(H_{i,j}\) is a binary matrix where it is 1 if there is at least one t such that G_{i,j,t} = 1 (i.e. at least one temporal edge exists).

F is not normalized which makes comparisions of F across very different networks difficult (could be added).

Examples

This example compares the fluctability of two different networks with the same number of edges. Below two temporal networks, both with 3 nodes and 3 time-points. Both get 3 connections.

>>> import teneto
>>> import numpy as np
>>> # Manually specify node (i,j) and temporal (t) indicies.
>>> ind_highF_i = [0,0,1]
>>> ind_highF_j = [1,2,2]
>>> ind_highF_t = [1,2,2]
>>> ind_lowF_i = [0,0,0]
>>> ind_lowF_j = [1,1,1]
>>> ind_lowF_t = [0,1,2]
>>> # Define 2 networks below and set above edges to 1
>>> G_highF = np.zeros([3,3,3])
>>> G_lowF = np.zeros([3,3,3])
>>> G_highF[ind_highF_i,ind_highF_j,ind_highF_t] = 1
>>> G_lowF[ind_lowF_i,ind_lowF_j,ind_lowF_t] = 1

The two different networks look like this:

(Source code)

Now calculate the fluctability of the two networks above.

>>> F_high = teneto.networkmeasures.fluctuability(G_highF)
>>> F_high
1.0
>>> F_low = teneto.networkmeasures.fluctuability(G_lowF)
>>> F_low
0.3333333333333333

Here we see that the network with more unique connections has the higher fluctuability.

[fluct-1]

Thompson et al (2017) “From static to temporal network theory applications to functional brain connectivity.” Network Neuroscience, 2: 1. p.69-99 [Link]