local_variation

Networkmeasure: local_variation

local_variation(data)

Calculates the local variaiont of inter-contact times. [LV-1], [LV-2]

Parameters:data (array, dict) – This is either (1) temporal network input (graphlet or contact) with nettype: ‘bu’, ‘bd’. (2) dictionary of ICTs (output of intercontacttimes). Returns:LV – Local variation per edge. Return type:array

Notes

The local variation is like the bursty coefficient and quantifies if a series of inter-contact times are periodic, random or Poisson distributed or bursty.

It is defined as:

\[LV = {3 \over {n-1}}\sum_{i=1}^{n-1}{{{\iota_i - \iota_{i+1}} \over {\iota_i + \iota_{i+1}}}^2}\]

Where \(\iota\) are inter-contact times and i is the index of the inter-contact time (not a node index). n is the number of events, making n-1 the number of inter-contact times.

The possible range is: \(0 \geq LV \gt 3\).

When periodic, LV=0, Poisson, LV=1 Larger LVs indicate bursty process.

Examples

First import all necessary packages

>>> import teneto
>>> import numpy as np

Now create 2 temporal network of 2 nodes and 60 time points. The first has periodict edges, repeating every other time-point:

>>> G_periodic = np.zeros([2, 2, 60])
>>> ts_periodic = np.arange(0, 60, 2)
>>> G_periodic[:,:,ts_periodic] = 1

The second has a more bursty pattern of edges:

>>> ts_bursty = [1, 8, 9, 32, 33, 34, 39, 40, 50, 51, 52, 55]
>>> G_bursty = np.zeros([2, 2, 60])
>>> G_bursty[:,:,ts_bursty] = 1

Now we call local variation for each edge.

>>> LV_periodic = teneto.networkmeasures.local_variation(G_periodic)
>>> LV_periodic
array([[nan,  0.],
       [ 0., nan]])

Above we can see that between node 0 and 1, LV=0 (the diagonal is nan). This is indicative of a periodic contacts (which is what we defined). Doing the same for the second example:

>>> LV_bursty = teneto.networkmeasures.local_variation(G_bursty)
>>> LV_bursty
array([[       nan, 1.28748748],
       [1.28748748,        nan]])

When the value is greater than 1, it indicates a bursty process.

nans are returned if there are no intercontacttimes

References

[LV-1]

Shinomoto et al (2003) Differences in spiking patterns among cortical neurons. Neural Computation 15.12 [Link]

[LV-2]

Followed eq., 4.34 in Masuda N & Lambiotte (2016) A guide to temporal networks. World Scientific. Series on Complex Networks. Vol 4 [Link]