Network Diffusion of Innovations

Diffusion networks

  • Tries to explain how new ideas and practices (innovations) spread within and between communities.

  • While a lot of factors have been shown to influence diffusion (Spatial, Economic, Cultural, Biological, etc.), Social Networks is a prominent one.

  • More complex than contagion \(\implies\) a single tie is no longer enough for an innovation to spread across a social system.

  • We think of this in terms of adoption thresholds and social exposure.

Thresholds

  • Network thresholds (Valente, 1995), \(\tau\), are defined as the required proportion or number of neighbors that leads you to adopt a particular behavior (innovation), \(a=1\). In (very) general terms

    \[ a_i = \left\{\begin{array}{ll} 1 &\mbox{if } \tau_i\leq E_i \\ 0 & \mbox{Otherwise} \end{array}\right. \qquad E_i \equiv \frac{\sum_{j\neq i}\mathbf{X}_{ij}a_j}{\sum_{j\neq i}\mathbf{X}_{ij}} \]

    Where \(E_i\) is i’s exposure to the innovation and \(\mathbf{X}\) is the adjacency matrix (the network).

  • This can be generalized and extended to include covariates and other weighting schemes (that’s what netdiffuseR is all about).

netdiffuseR

Overview

netdiffuseR is an R package that:

  • Is designed for Visualizing, Analyzing and Simulating network diffusion data (in general).

  • Depends on some pretty popular packages:

    • RcppArmadillo: So it’s fast,
    • Matrix: So it’s big,
    • statnet and igraph: So it’s not from scratch

  • Can handle big graphs, more than 4 billion elements adjacency matrix (PR for RcppArmadillo)

  • Already on CRAN with ~4,000 downloads since its first version, Feb 2016,

  • A lot of features to make it easy to read network (dynamic) data, making it a nice companion of other net packages.

Datasets

  • Among netdiffuseR features, we find three classical Diffusion Network Datasets:

    • brfarmersDiffNet Brazilian farmers and the innovation of Hybrid Corn Seed (1966).
    • medInnovationsDiffNet Doctors and the innovation of Tetracycline (1955).
    • kfamilyDiffNet Korean women and Family Planning methods (1973).
    brfarmersDiffNet
    # Dynamic network of class -diffnet-
#  Name               : Brazilian Farmers
#  Behavior           : Adoption of Hybrid Corn Seeds
#  # of nodes         : 692 (1001, 1002, 1004, 1005, 1007, 1009, 1010, 1020, ...)
#  # of time periods  : 21 (1946 - 1966)
#  Type               : directed
#  Final prevalence   : 1.00
#  Static attributes  : village, idold, age, liveout, visits, contact, coo... (146)
#  Dynamic attributes : -
    medInnovationsDiffNet
    # Dynamic network of class -diffnet-
#  Name               : Medical Innovation
#  Behavior           : Adoption of Tetracycline
#  # of nodes         : 125 (1001, 1002, 1003, 1004, 1005, 1006, 1007, 1008, ...)
#  # of time periods  : 18 (1 - 18)
#  Type               : directed
#  Final prevalence   : 1.00
#  Static attributes  : city, detail, meet, coll, attend, proage, length, ... (58)
#  Dynamic attributes : -
    kfamilyDiffNet
    # Dynamic network of class -diffnet-
#  Name               : Korean Family Planning
#  Behavior           : Family Planning Methods
#  # of nodes         : 1047 (10002, 10003, 10005, 10007, 10010, 10011, 10012, 10014, ...)
#  # of time periods  : 11 (1 - 11)
#  Type               : directed
#  Final prevalence   : 1.00
#  Static attributes  : village, recno1, studno1, area1, id1, nmage1, nmag... (430)
#  Dynamic attributes : -

Visualization methods

set.seed(12315)
x <- rdiffnet(
  300, t = 4, rgraph.args = list(k=8, p=.3),
  seed.graph = "small-world",
  seed.nodes = "central"
  )

plot(x)

plot_diffnet(x)

plot_diffnet2(x)

plot_diffnet2(x, add.map = "last", diffmap.alpha = .75, include.white = NULL)

plot_adopters(x)

plot_threshold(x)

plot_infectsuscep(x, K=2, logscale = FALSE)

plot_hazard(x)

Problems

  1. Using the diffnet object in data_intro.rmd, use the function plot_threshold specifying shapes and colors according to the variables ItrustMyFriends and Age. Do you see any pattern?
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

University of Southern California
Center for Applied Network Analysis (CANA)