sid

sid(tnet, communities, axis=0, calc='overtime', decay=0)[source]

Segregation integration difference (SID). An estimation of each community or global difference of within versus between community strength.[sid-1]_

Parameters:
  • tnet (array, dict) – Temporal network input (graphlet or contact). Allowerd nettype: ‘bu’, ‘bd’, ‘wu’, ‘wd’
  • communities (array) – a Nx1 vector or NxT array of community assignment.
  • axis (int) – Dimension that is returned 0 or 1 (default 0). Note, only relevant for directed networks. i.e. if 0, node i has Aijt summed over j and t. and if 1, node j has Aijt summed over i and t.
  • calc (str) –

    ‘overtime’ returns SID over time (a 1 x community vector) (default);

    ’community_pairs’ returns a community x community x time matrix, which is the SID for each community pairing;

    ’community_avg’ (returns a community x time matrix). Which is the normalized average of each community to all other communities.

    ’community_pairs_norm’ (returns a community x time matrix). Which is the normalized average of each community pair. Each pair is normalized to the average of both communities in the pair.

  • decay (int) – if calc = ‘community_pairs’ or ‘community_avg’, then decay is possible where the centrality of the previous time point is carried over to the next time point but decays at a value of $e^decay$ such that the temporal centrality measure becomes: $D(t+1) = e^{-decay}D(t) + D(t+1)$.
Returns:

sid – segregation-integration difference. Format: 2d or 3d numpy array (depending on calc) representing (community,community,time) or (community,time)
Return type:

array

Notes

SID tries to quantify if there is more segergation or intgration compared to other time-points. If SID > 0, then there is more segregation than usual. If SID < 0, then there is more integration than usual.

There are three different variants of SID, one is a global measure (calc=’overtime’), the second is a value per community (calc=’community_avg’), the third is a value for each community-community pairing (calc=’community_pairs’).

First we calculate the temporal strength for each edge. This is calculate by

\[S_{i,t} = \sum_j G_{i,j,t}\]

The pairwise SID, when the network is undirected, is calculated by

\[SID_{A,B,t} = ({2 \over {N_A (N_A - 1)}}) S_{A,t} - ({{1} \over {N_A * N_B}}) S_{A,B,t})\]

Where \(S_{A,t}\) is the average temporal strength at time-point t for community A. \(N_A\) is the number of nodes in community A.

When calculating the SID for a community, it is calculated byL

\[SID_{A,t} = \sum_b^C({2 \over {N_A (N_A - 1)}}) S_{A,t} - ({{1} \over {N_A * N_b}}) S_{A,b,t})\]

Where C is the number of communities.

When calculating the SID globally, it is calculated byL

\[SID_{t} = \sum_a^C\sum_b^C({2 \over {N_a (N_a - 1)}}) S_{A,t} - ({{1} \over {N_a * N_b}}) S_{a,b,t})\]

References

[sid-1]Fransson et al (2018) Brain network segregation and integration during an epoch-related working memory fMRI experiment. Neuroimage. 178. [Link]