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Dispersion (Backstrom-Kleinberg 2014)

Source: R/network-summary.R
dispersion.Rd

Per-pair measure of tie strength from the Facebook relationship-inference paper. For each pair \((u, v)\) where \(v\) is a neighbor of \(u\):

Usage

dispersion(x, u = NULL, v = NULL, normalized = TRUE, alpha = 1, b = 0, c = 0)

Arguments

x

Network input (matrix, igraph, network, cograph_network, tna object).

u

Optional source node (1-based index or node name). If NULL (default), compute for all sources.

v

Optional target node. If NULL, compute for all neighbors of u.

normalized

Logical. If TRUE (default), return the normalized form; otherwise the raw count.

alpha

Numeric normalization exponent. Default 1.

b

Numeric bias added to dispersion before exponentiation. Default 0.

c

Numeric bias added to embeddedness in the denominator. Default 0.

Value

  • Scalar if both u and v are specified.

  • Named numeric vector if exactly one of u, v is given, one element per neighbor of that node; the names are the neighbors' 1-based node indices as character strings, not their labels.

  • A data frame with columns from, to, dispersion when neither u nor v is given, one row per ordered (node, neighbor) pair, with from and to given as 1-based integer node indices.

  • numeric(0) for an empty graph.

Details

  1. Let \(S_T = N(u) \cap N(v)\) be their mutual friends (embeddedness).

  2. Count pairs \((s, t) \subset S_T\) such that:

    • \(s\) and \(t\) are not directly connected, AND

    • \(s\) and \(t\) share no common neighbor inside \(N(u)\) other than \(u\) and \(v\).

  3. The raw dispersion is this count. When normalized = TRUE, the result is \((\mathrm{dispersion} + b)^{\alpha} / (\mathrm{embeddedness} + c)\) (normalization is skipped when embeddedness + c == 0).

Matches networkx.dispersion bit-exact for all three call modes (single pair, single source, full matrix).

References

Backstrom, L., & Kleinberg, J. (2014). Romantic partnerships and the dispersion of social ties: A network analysis of relationship status on Facebook. In Proceedings of CSCW (pp. 831-841). ACM. https://arxiv.org/pdf/1310.6753v1.pdf

Examples

g <- igraph::make_graph("Zachary")
# Node 0 (R index 1) to node 33 (R index 34)
dispersion(g, u = 1, v = 34)
#> [1] 1
# All pairs from node 1
head(dispersion(g, u = 1))
#>        2        3        4        5        6        7 
#> 2.142857 0.800000 0.800000 0.000000 0.000000 0.000000