In a homophilous network, nodes which are similar in some way are more likely to be linked together. The similarity may be a distance measure in some notional space.
Here, the graph at right is equivalent to an n × n adjacency matrix A, where the number of nodes
1 \le n = $n$ \le 15
10
inc("n");
figure[2].size(get("n"));
figure[1].load([0, matriculate(figure[2].data, get("m"))]);
dec("n");
figure[2].size(get("n"));
figure[1].load([0, matriculate(figure[2].data, get("m"))]);
, whose entries
A_{ij} = \begin{cases}
f(d_{ij}) &\text{if } f(d_{ij}) > m \\
0 &\text{otherwise}
\end{cases}
\text{ where }
f(d_{ij}) = $x$
[
"\\dfrac{1}{1 + e^{0.5(5 - d_{ij})}}",
"\\dfrac{1}{1 + e^{0.5(5 - d_{ij})}}",
"1 - \\dfrac{d_{ij}}{2}",
]0get("X")[get("Xi")]
var X = get("X");
inc("Xi");
set("x", X[get("Xi")]);
update("blurb_dij");
distanceFunction = distances[get("Xi")];
figure[1].load([0, matriculate(figure[2].data, get("m"))]);
var X = get("X");
dec("Xi");
var Xi = get("Xi");
set("x", X[get("Xi")]);
update("blurb_dij");
distanceFunction = distances[get("Xi")];
figure[1].load([0, matriculate(figure[2].data, get("m"))]);
,
$blurb$
[
"d_{ij} \\text{ is the } \\href{http://en.wikipedia.org/wiki/Euclidean_distance}{Euclidean} \\text{ distance between } \
i \\text{ and } j",
"d_{ij} \\text{ is the } \\href{http://en.wikipedia.org/wiki/Manhattan_distance}{Manhattan} \\text{ distance between } \
i \\text{ and } j",
"d_{ij} \
\\text{ is the } \\href{http://en.wikipedia.org/wiki/Hamming_distance}{Hamming} \\text{ distance between } \
\\text{“}x_{i}y_{i} \\text{\” and “} \
x_{j}y_{j} \\text{”}",
]get("blurbs")[get("Xi")]
set("blurb", get("blurbs")[get("Xi")]);
,
and
0 \le m = $m$ \le 1
0.35
inc("m");
figure[1].load([0, matriculate(figure[2].data, get("m"))]);
dec("m");
figure[1].load([0, matriculate(figure[2].data, get("m"))]);
is an arbitrary cutoff distance. If the edge weights are interpreted as probabilities of connections, then the matrix describes a distribution of possible graphs reflecting the same underlying social space.
Click and drag the points on the plot to change the network.