I am proposing a mathematical model of a scale free network based on the divisors of natural numbers (Probably somewhere in the definition, there can be made a "tuning parameter" which gives graphs with different properties):
To each number $n$ we construct a random undirected graph $G_n$:
We start with the number $1$ as a vertex in the graph $G_n$.
Then we iterate with $k$ from $2$ to $n$: Iterate over all previous nodes $d$ of $G_n$: If $d|k$ add with probability (independently of previous vertices and edges) :
$$p_d = \frac{k}{\sigma(k)d}$$ and undirected edge between $k$ and $d$, where $\sigma$ is the sum of divisors.
Here are some statistics computed for these random graphs empirically with Sagemath.
Question: Is it possible to analytically prove, some of these statistical properties? (I feel that the definition is "simple" enough to be manipulated mathematically.)
Here are some empirical statistics:
n,m,c,S,l,C,L
(3, 2, 1.33333333333333, 1.00000000000000, 1.33333333333333, 0.000000000000000, 1.33333333333333)
(4, 2, 1.00000000000000, 0.750000000000000, +Infinity, 0.000000000000000, 1.33333333333333)
(5, 3, 1.20000000000000, 0.800000000000000, +Infinity, 0.000000000000000, 1.50000000000000)
(6, 3, 1.00000000000000, 0.666666666666667, +Infinity, 0.000000000000000, 1.66666666666667)
(7, 3, 0.857142857142857, 0.571428571428571, +Infinity, 0.000000000000000, 1.50000000000000)
(8, 4, 1.00000000000000, 0.625000000000000, +Infinity, 0.000000000000000, 1.60000000000000)
(9, 10, 2.22222222222222, 1.00000000000000, 1.72222222222222, 0.304232804232804, 1.72222222222222)
(10, 6, 1.20000000000000, 0.700000000000000, +Infinity, 0.000000000000000, 1.90476190476190)
(11, 5, 0.909090909090909, 0.545454545454545, +Infinity, 0.000000000000000, 1.66666666666667)
(12, 9, 1.50000000000000, 0.833333333333333, +Infinity, 0.000000000000000, 2.22222222222222)
(13, 7, 1.07692307692308, 0.615384615384615, +Infinity, 0.000000000000000, 1.75000000000000)
(14, 10, 1.42857142857143, 0.785714285714286, +Infinity, 0.000000000000000, 2.07272727272727)
(15, 11, 1.46666666666667, 0.733333333333333, +Infinity, 0.134814814814815, 1.80000000000000)
(16, 16, 2.00000000000000, 0.812500000000000, +Infinity, 0.266666666666667, 1.96153846153846)
(17, 19, 2.23529411764706, 0.941176470588235, +Infinity, 0.247338935574230, 1.84166666666667)
(18, 15, 1.66666666666667, 0.833333333333333, +Infinity, 0.111952861952862, 2.18095238095238)
(19, 17, 1.78947368421053, 0.894736842105263, +Infinity, 0.000000000000000, 2.22058823529412)
(20, 17, 1.70000000000000, 0.850000000000000, +Infinity, 0.0671428571428571, 1.97058823529412)
(21, 17, 1.61904761904762, 0.857142857142857, +Infinity, 0.000000000000000, 2.73856209150327)
(22, 20, 1.81818181818182, 0.909090909090909, +Infinity, 0.0611888111888112, 2.50000000000000)
(23, 18, 1.56521739130435, 0.739130434782609, +Infinity, 0.116770186335404, 1.97058823529412)
(24, 23, 1.91666666666667, 0.916666666666667, +Infinity, 0.0742238562091503, 2.19047619047619)
(25, 27, 2.16000000000000, 0.840000000000000, +Infinity, 0.208235294117647, 1.99047619047619)
(26, 26, 2.00000000000000, 0.807692307692308, +Infinity, 0.139215686274510, 2.06190476190476)
(27, 25, 1.85185185185185, 0.851851851851852, +Infinity, 0.0743161462115711, 2.32015810276680)
(28, 21, 1.50000000000000, 0.678571428571429, +Infinity, 0.155549719887955, 1.95906432748538)
(29, 22, 1.51724137931034, 0.724137931034483, +Infinity, 0.0462643678160920, 2.29523809523810)
(30, 28, 1.86666666666667, 0.800000000000000, +Infinity, 0.178412698412698, 2.12318840579710)
(31, 32, 2.06451612903226, 0.903225806451613, +Infinity, 0.127287304282211, 2.38095238095238)
(32, 29, 1.81250000000000, 0.843750000000000, +Infinity, 0.108779761904762, 2.22222222222222)
(33, 23, 1.39393939393939, 0.666666666666667, +Infinity, 0.0608041196276490, 2.19047619047619)
(34, 29, 1.70588235294118, 0.823529411764706, +Infinity, 0.0982717197550957, 2.17989417989418)
(35, 33, 1.88571428571429, 0.885714285714286, +Infinity, 0.0668424908424908, 2.15053763440860)
(36, 36, 2.00000000000000, 0.888888888888889, +Infinity, 0.0863888888888889, 2.24596774193548)
(37, 31, 1.67567567567568, 0.783783783783784, +Infinity, 0.117387387387387, 2.11083743842365)
(38, 31, 1.63157894736842, 0.789473684210526, +Infinity, 0.0703508771929825, 2.28275862068966)
(39, 38, 1.94871794871795, 0.871794871794872, +Infinity, 0.0513498846832180, 2.22281639928699)
(40, 37, 1.85000000000000, 0.775000000000000, +Infinity, 0.187128205128205, 2.13763440860215)
(41, 37, 1.80487804878049, 0.780487804878049, +Infinity, 0.108269057049545, 2.18346774193548)
(42, 34, 1.61904761904762, 0.761904761904762, +Infinity, 0.0557616986188415, 2.71169354838710)
(43, 34, 1.58139534883721, 0.744186046511628, +Infinity, 0.0621586165772212, 2.27217741935484)
(44, 42, 1.90909090909091, 0.795454545454545, +Infinity, 0.175342130987292, 2.07731092436975)
(45, 40, 1.77777777777778, 0.755555555555556, +Infinity, 0.0296929408040519, 2.31729055258467)
(46, 46, 2.00000000000000, 0.826086956521739, +Infinity, 0.169875222816399, 2.10099573257468)
(47, 38, 1.61702127659574, 0.744680851063830, +Infinity, 0.000000000000000, 2.59159663865546)
(48, 37, 1.54166666666667, 0.729166666666667, +Infinity, 0.0417179802955665, 2.19663865546219)
where $n=$ number of nodes, $m=$ number of edges, $c =$ mean degree, $S=$ percentage of nodes in giant cluster, $l = $ average distance, $C=$ average local clustering coefficient, $L=$ average distance in giant component.
Originally asked at MO, but since it might not be research relevant, I decided to post it here.