02_connectance_degree.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Connectance & degree distribution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "https://poisotlab.github.io/EcologicalNetworks.jl/stable/"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "using EcologicalNetworks\n",
    "using EcologicalNetworksPlots\n",
    "using Plots\n",
    "using Random\n",
    "using Statistics\n",
    "using Distributions\n",
    "\n",
    "Random.seed!(42);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Connectance"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "B (generic function with 1 method)"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function B(n::Int64, p::Float64)\n",
    "    return simplify(BipartiteNetwork(rand(Float64, (n,n)) .<= p))\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "8×7 bipartite  ecological network (Bool, String) (L: 10)"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B(10, 0.1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "250"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n = 250"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "500-element Array{Float64,1}:\n",
       " 0.5263657094795438 \n",
       " 0.9446393474280155 \n",
       " 0.8191800594897347 \n",
       " 0.7826960426539047 \n",
       " 0.5725389356986399 \n",
       " 0.07122015013829008\n",
       " 0.7109143734757877 \n",
       " 0.6233409343040597 \n",
       " 0.8658557672748033 \n",
       " 0.46946476154174666\n",
       " 0.3496206541831892 \n",
       " 0.1585459539016172 \n",
       " 0.3591876615822245 \n",
       " ⋮                  \n",
       " 0.8908435953852336 \n",
       " 0.37406739570897063\n",
       " 0.6351651162674106 \n",
       " 0.7253448721187594 \n",
       " 0.17166376228217106\n",
       " 0.5382615147516283 \n",
       " 0.35423973149089094\n",
       " 0.6914516632906484 \n",
       " 0.9871142035105684 \n",
       " 0.4605677308345748 \n",
       " 0.9288062231898211 \n",
       " 0.14891595646883093"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P = rand(500)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "500-element Array{BipartiteNetwork{Bool,String},1}:\n",
       " 250×250 bipartite  ecological network (Bool, String) (L: 32973)\n",
       " 250×250 bipartite  ecological network (Bool, String) (L: 58995)\n",
       " 250×250 bipartite  ecological network (Bool, String) (L: 51305)\n",
       " 250×250 bipartite  ecological network (Bool, String) (L: 48755)\n",
       " 250×250 bipartite  ecological network (Bool, String) (L: 35826)\n",
       " 250×250 bipartite  ecological network (Bool, String) (L: 4457) \n",
       " 250×250 bipartite  ecological network (Bool, String) (L: 44264)\n",
       " 250×250 bipartite  ecological network (Bool, String) (L: 38931)\n",
       " 250×250 bipartite  ecological network (Bool, String) (L: 54130)\n",
       " 250×250 bipartite  ecological network (Bool, String) (L: 29179)\n",
       " 250×250 bipartite  ecological network (Bool, String) (L: 21965)\n",
       " 250×250 bipartite  ecological network (Bool, String) (L: 9873) \n",
       " 250×250 bipartite  ecological network (Bool, String) (L: 22360)\n",
       " ⋮                                                              \n",
       " 250×250 bipartite  ecological network (Bool, String) (L: 55627)\n",
       " 250×250 bipartite  ecological network (Bool, String) (L: 23315)\n",
       " 250×250 bipartite  ecological network (Bool, String) (L: 39636)\n",
       " 250×250 bipartite  ecological network (Bool, String) (L: 45379)\n",
       " 250×250 bipartite  ecological network (Bool, String) (L: 10704)\n",
       " 250×250 bipartite  ecological network (Bool, String) (L: 33633)\n",
       " 250×250 bipartite  ecological network (Bool, String) (L: 22059)\n",
       " 250×250 bipartite  ecological network (Bool, String) (L: 43115)\n",
       " 250×250 bipartite  ecological network (Bool, String) (L: 61765)\n",
       " 250×250 bipartite  ecological network (Bool, String) (L: 28669)\n",
       " 250×250 bipartite  ecological network (Bool, String) (L: 58078)\n",
       " 250×250 bipartite  ecological network (Bool, String) (L: 9438) "
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "networks = [B(n,p) for p in P]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
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     "execution_count": 30,
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   ],
   "source": [
    "histogram(connectance.(networks))"
   ]
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     "data": {
      "text/plain": [
       "K (generic function with 1 method)"
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     "execution_count": 32,
     "metadata": {},
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   ],
   "source": [
    "K(N) = collect(values(degree(N)))"
   ]
  },
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  {
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   "execution_count": 35,
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    "scatter(connectance.(networks), [var(K(n)) for n in networks])\n",
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    "yaxis!(\"Variance (degree)\")"
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    "scatter(connectance.(networks), [kurtosis(K(n)) for n in networks])\n",
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     "execution_count": 38,
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   ],
   "source": [
    "scatter(connectance.(networks), [skewness(K(n)) for n in networks])\n",
    "xaxis!(\"Connectance\")\n",
    "yaxis!(\"Skewness (degree)\")"
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  },
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  },
  {
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   "metadata": {},
   "source": [
    "## Degree"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "96×96 unipartite  ecological network (Bool, String) (L: 634)"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N = nz_stream_foodweb()[10]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
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       "</svg>\n"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I = initial(RandomInitialLayout, N)\n",
    "for step in 1:2000\n",
    "  position!(ForceDirectedLayout(0.8), I, N)\n",
    "end\n",
    "plot(I, N, aspectratio=1, size=(700,700))\n",
    "scatter!(I, N, nodefill=degree(N), nodesize=degree(N), mc=:YlGnBu)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "k = collect(values(degree(N)));\n",
    "histogram(k; leg=false, c=:lightgrey)\n",
    "xaxis!(\"Degree\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "degrees = collect(1:maximum(k))\n",
    "p_degrees = zeros(Float64, length(degrees))\n",
    "for (i, d) in enumerate(degrees)\n",
    "    p_degrees[i] = sum(k .== d)/richness(N)\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "scatter(degrees, p_degrees; leg=false)\n",
    "xaxis!(\"k\")\n",
    "yaxis!(\"P(k)\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Using the Lorenz curve"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "function lorenz(N::T) where {T <: AbstractEcologicalNetwork}\n",
    "    sp = collect(1:richness(N))./richness(N)\n",
    "    k = sort(collect(values(degree(N))))\n",
    "    lk = cumsum(k)./(2.0*links(N))\n",
    "    return sp, lk\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "N = web_of_life(\"M_PL_001\")\n",
    "plot(lorenz(N); leg=false, frame=:box, c=:black, aspectratio=1)\n",
    "xaxis!((0,1), \"Fraction of species\")\n",
    "yaxis!((0,1), \"Cumulative fraction of interactions\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": []
  }
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