Network Diffusion of Innovations

Diffusion networks

  • Explains 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.

  • There are many components in the diffusion network model including network exposures, thresholds, infectiousness, susceptibility, hazard rates, diffusion rates (bass model), clustering (Moran’s I), and so on.

Thresholds

  • One of the cannonical concepts is the network threshold. Network thresholds (Valente, 1995; 1996), \(\tau\), 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 network 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, e.g., an adjacency matrix with more than 4 billion elements (PR for RcppArmadillo)

  • Already on CRAN with ~6,000 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

  • netdiffuseR has the three classic Diffusion Network Datasets:

    • medInnovationsDiffNet Doctors and the innovation of Tetracycline (1955).
    • brfarmersDiffNet Brazilian farmers and the innovation of Hybrid Corn Seed (1966).
    • 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(
  400, t = 6, rgraph.args = list(k=6, p=.3),
  seed.graph = "small-world",
  seed.nodes = "central", rewire = FALSE, threshold.dist = 1/4
  )

plot(x)

plot_diffnet(x)

plot_diffnet2(x)

plot_adopters(x)

plot_threshold(x)

plot_infectsuscep(x, K=2)
# Warning in plot_infectsuscep.list(graph$graph, graph$toa, t0, normalize, :
# When applying logscale some observations are missing.

plot_hazard(x)

Problems

  1. Using the diffnet object in intro.rda, use the function plot_threshold specifying shapes and colors according to the variables ItrustMyFriends and Age. Do you see any pattern? (solution script and solution plot)
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

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