"Among the types of centrality that have been considered in the literature, many have to do with distances between nodes. Take, for instance, a node in an undirected connected network: if the sum of distances to all other nodes is large, the node under consideration is peripheral; this is the starting point to define Bavelas's closeness centrality \\cite{closeness}, which is the reciprocal of peripherality (i.e., the reciprocal of the sum of distances to all other nodes). \n",
"Among the types of centrality that have been considered in the literature, many have to do with distances between nodes. Take, for instance, a node in an undirected connected network: if the sum of distances to all other nodes is large, the node under consideration is peripheral; this is the starting point to define Bavelas's closeness centrality \\cite{closeness}, which is the reciprocal of peripherality (i.e., the reciprocal of the sum of distances to all other nodes). \n",
"\n",
"\n",
"The role played by shortest paths is justified by one of the most well-known features of complex networks, the so-called small-world phenomenon. A small-world network is a graph where the average distance between nodes is logarithmic in the size of the network, whereas the clustering coefficient is larger (that is, neighborhoods tend to be denser) than in a random Erdős-Rényi graph with the same size and average distance. The fact that social networks (whether electronically mediated or not) exhibit the small-world property is known at least since Milgram's famous experiment \\cite{} and is arguably the most popular of all features of complex networks. For instance, the average distance of the Facebook graph was recently established to be just $4.74$.\n"
"The role played by shortest paths is justified by one of the most well-known features of complex networks, the so-called small-world phenomenon. A small-world network is a graph where the average distance between nodes is logarithmic in the size of the network, whereas the clustering coefficient is larger (that is, neighborhoods tend to be denser) than in a random Erdős-Rényi graph with the same size and average distance. The fact that social networks (whether electronically mediated or not) exhibit the small-world property is known at least since Milgram's famous experiment and is arguably the most popular of all features of complex networks. For instance, the average distance of the Facebook graph was recently established to be just $4.74$.\n"
]
]
},
},
{
{
@ -473,8 +473,7 @@
"\n",
"\n",
"The degree distribution, $P(k)$, is the fraction of sites having degree $k$. We know from the literature that many real networks do not exhibit a Poisson degree distribution, as predicted in the ER model. In fact, many of them exhibit a distribution with a long, power-law, tail, $P(k) \\sim k^{-\\gamma}$ with some $γ$, usually between $2$ and 3$.\n",
"The degree distribution, $P(k)$, is the fraction of sites having degree $k$. We know from the literature that many real networks do not exhibit a Poisson degree distribution, as predicted in the ER model. In fact, many of them exhibit a distribution with a long, power-law, tail, $P(k) \\sim k^{-\\gamma}$ with some $γ$, usually between $2$ and 3$.\n",
"\n",
"\n",
"For know, we will just compute the average degree of our networks and add it to the dataframe.\n",
"For know, we will just compute the average degree of our networks and add it to the dataframe."
"\n"
]
]
},
},
{
{
@ -981,7 +980,7 @@
"outputs": [],
"outputs": [],
"source": [
"source": [
"for G in checkins_graphs:\n",
"for G in checkins_graphs:\n",
" degree_distribution(G)"
" degree_distribution(G, log=True)"
]
]
},
},
{
{
@ -1020,7 +1019,7 @@
" print(G.name)\n",
" print(G.name)\n",
" print(\"Number of nodes: \", G.number_of_nodes())\n",
" print(\"Number of nodes: \", G.number_of_nodes())\n",
" print(\"Number of edges: \", G.number_of_edges())\n",
" print(\"Number of edges: \", G.number_of_edges())\n",
" degree_distribution(G)"
" degree_distribution(G, log=False)"
]
]
},
},
{
{
@ -1035,7 +1034,7 @@
" print(G.name)\n",
" print(G.name)\n",
" print(\"Number of nodes: \", G.number_of_nodes())\n",
" print(\"Number of nodes: \", G.number_of_nodes())\n",
" print(\"Number of edges: \", G.number_of_edges())\n",
" print(\"Number of edges: \", G.number_of_edges())\n",
" degree_distribution(G)"
" degree_distribution(G, log=False)"
]
]
},
},
{
{
@ -1051,13 +1050,17 @@
"cell_type": "markdown",
"cell_type": "markdown",
"metadata": {},
"metadata": {},
"source": [
"source": [
"The degree distribution alone is not enough to characterize the network. There are many other quantities, such as the degree-degree correlation (between connected nodes), the spatial correlations, the clustering coefficient, the betweenness or central-ity distribution, and the self-similarity exponents."
"The degree distribution alone is not enough to characterize the network. There are many other quantities, such as the degree-degree correlation (between connected nodes), the spatial correlations, the clustering coefficient, the betweenness or central-ity distribution, and the self-similarity exponents.\n",
"\n",
"--- \n",
"\n",
"Now let's try to compute the same analysis made before for this random models"
]
]
}
}
],
],
"metadata": {
"metadata": {
"kernelspec": {
"kernelspec": {
"display_name": "Python 3.10.6 64-bit",
"display_name": "Python 3.10.8 64-bit",
"language": "python",
"language": "python",
"name": "python3"
"name": "python3"
},
},
@ -1071,12 +1074,12 @@
"name": "python",
"name": "python",
"nbconvert_exporter": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"pygments_lexer": "ipython3",
"version": "3.10.6 (main, Nov 14 2022, 16:10:14) [GCC 11.3.0]"
"Cell \u001b[0;32mIn[6], line 10\u001b[0m\n\u001b[1;32m 6\u001b[0m G \u001b[38;5;241m=\u001b[39m create_random_graphs(graph, \u001b[38;5;124m\"\u001b[39m\u001b[38;5;124merods\u001b[39m\u001b[38;5;124m\"\u001b[39m)\n\u001b[1;32m 8\u001b[0m \u001b[38;5;66;03m# add the basic information to the dataframe\u001b[39;00m\n\u001b[1;32m 9\u001b[0m analysis_results_erods \u001b[38;5;241m=\u001b[39m analysis_results_erods\u001b[38;5;241m.\u001b[39mappend({\n\u001b[0;32m---> 10\u001b[0m \u001b[38;5;124m'\u001b[39m\u001b[38;5;124mGraph\u001b[39m\u001b[38;5;124m'\u001b[39m: \u001b[43mG\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mname\u001b[49m,\n\u001b[1;32m 11\u001b[0m \u001b[38;5;124m'\u001b[39m\u001b[38;5;124mNumber of Nodes\u001b[39m\u001b[38;5;124m'\u001b[39m: G\u001b[38;5;241m.\u001b[39mnumber_of_nodes(),\n\u001b[1;32m 12\u001b[0m \u001b[38;5;124m'\u001b[39m\u001b[38;5;124mNumber of Edges\u001b[39m\u001b[38;5;124m'\u001b[39m: G\u001b[38;5;241m.\u001b[39mnumber_of_edges(),\n\u001b[1;32m 13\u001b[0m \u001b[38;5;124m'\u001b[39m\u001b[38;5;124mlog N\u001b[39m\u001b[38;5;124m'\u001b[39m: np\u001b[38;5;241m.\u001b[39mlog(G\u001b[38;5;241m.\u001b[39mnumber_of_nodes())\n\u001b[1;32m 14\u001b[0m }, ignore_index\u001b[38;5;241m=\u001b[39m\u001b[38;5;28;01mTrue\u001b[39;00m)\n\u001b[1;32m 16\u001b[0m \u001b[38;5;66;03m# compute the average degree and add it to the dataframe\u001b[39;00m\n\u001b[1;32m 17\u001b[0m avg_deg \u001b[38;5;241m=\u001b[39m np\u001b[38;5;241m.\u001b[39mmean([d \u001b[38;5;28;01mfor\u001b[39;00m n, d \u001b[38;5;129;01min\u001b[39;00m G\u001b[38;5;241m.\u001b[39mdegree()])\n",
"\u001b[0;31mAttributeError\u001b[0m: 'NoneType' object has no attribute 'name'"
"Cell \u001b[0;32mIn[7], line 25\u001b[0m\n\u001b[1;32m 22\u001b[0m analysis_results_ws\u001b[38;5;241m.\u001b[39mloc[analysis_results_ws[\u001b[38;5;124m'\u001b[39m\u001b[38;5;124mGraph\u001b[39m\u001b[38;5;124m'\u001b[39m] \u001b[38;5;241m==\u001b[39m G\u001b[38;5;241m.\u001b[39mname, \u001b[38;5;124m'\u001b[39m\u001b[38;5;124mAverage Clustering Coefficient\u001b[39m\u001b[38;5;124m'\u001b[39m] \u001b[38;5;241m=\u001b[39m avg_clustering\n\u001b[1;32m 24\u001b[0m \u001b[38;5;66;03m# compute the average shortest path length and add it to the dataframe\u001b[39;00m\n\u001b[0;32m---> 25\u001b[0m average_shortest_path_length \u001b[38;5;241m=\u001b[39m \u001b[43maverage_shortest_path\u001b[49m\u001b[43m(\u001b[49m\u001b[43mG\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 26\u001b[0m analysis_results_ws\u001b[38;5;241m.\u001b[39mloc[analysis_results_ws[\u001b[38;5;124m'\u001b[39m\u001b[38;5;124mGraph\u001b[39m\u001b[38;5;124m'\u001b[39m] \u001b[38;5;241m==\u001b[39m G\u001b[38;5;241m.\u001b[39mname, \u001b[38;5;124m'\u001b[39m\u001b[38;5;124mAverage Shortest Path Length\u001b[39m\u001b[38;5;124m'\u001b[39m] \u001b[38;5;241m=\u001b[39m average_shortest_path_length\n\u001b[1;32m 28\u001b[0m \u001b[38;5;66;03m# compute the betweenness centrality and add it to the dataframe\u001b[39;00m\n",
"File \u001b[0;32m~/github/small-worlds/utils.py:497\u001b[0m, in \u001b[0;36maverage_shortest_path\u001b[0;34m(G, k)\u001b[0m\n\u001b[1;32m 494\u001b[0m \u001b[39mprint\u001b[39m(\u001b[39m\"\u001b[39m\u001b[39m\\t\u001b[39;00m\u001b[39mNumber of edges after removing \u001b[39m\u001b[39m{}\u001b[39;00m\u001b[39m% o\u001b[39;00m\u001b[39mf nodes: \u001b[39m\u001b[39m{}\u001b[39;00m\u001b[39m\"\u001b[39m \u001b[39m.\u001b[39mformat((k)\u001b[39m*\u001b[39m\u001b[39m100\u001b[39m, G_copy\u001b[39m.\u001b[39mnumber_of_edges()))\n\u001b[1;32m 496\u001b[0m tmp \u001b[39m=\u001b[39m \u001b[39m0\u001b[39m\n\u001b[0;32m--> 497\u001b[0m connected_components \u001b[39m=\u001b[39m \u001b[39mlist\u001b[39m(nx\u001b[39m.\u001b[39mconnected_components(G_copy))\n\u001b[1;32m 498\u001b[0m \u001b[39m# remove all the connected components with less than 10 nodes\u001b[39;00m\n\u001b[1;32m 499\u001b[0m connected_components \u001b[39m=\u001b[39m [c \u001b[39mfor\u001b[39;00m c \u001b[39min\u001b[39;00m connected_components \u001b[39mif\u001b[39;00m \u001b[39mlen\u001b[39m(c) \u001b[39m>\u001b[39m \u001b[39m10\u001b[39m]\n",
"\u001b[0;31mUnboundLocalError\u001b[0m: local variable 'G_copy' referenced before assignment"
" # use numpy and scipy to speed up the process\n",
" adj = sp.csr_matrix(adj)\n",
" n, m = adj.shape \n",
" assert n == m # make sure the adjacency matrix is square\n",
" adj_triu = sp.triu(adj) # only consider the upper triangular part of the adjacency matrix\n",
" adj_tuple = sp.find(adj_triu) # get the indices and values of the non-zero elements\n",
" adj_edges = np.array(list(zip(adj_tuple[0], adj_tuple[1]))) # get the edges\n",
" adj_data = adj_tuple[2] # get the edge weights\n",
" nnz = adj_edges.shape[0] # number of non-zero elements\n",
" assert nnz == adj_data.shape[0] # make sure the number of edges and edge weights are the same\n",
" for _ in range(nswap): # repeat nswap times\n",
" # choose random edges to swap\n",
" edge_idx = np.random.choice(nnz, size=2, replace=False) # choose two random edges\n",
" edge1 = adj_edges[edge_idx[0]] # get the first edge\n",
" edge2 = adj_edges[edge_idx[1]] # get the second edge\n",
" # make sure the edges are not self-loops and not already connected\n",
" if edge1[0] == edge2[0] or edge1[0] == edge2[1] or edge1[1] == edge2[0] or edge1[1] == edge2[1] or adj[edge1[0], edge2[1]] or adj[edge2[0], edge1[1]]: \n",
" continue # if the edges are self-loops or already connected, try again\n",
Luca Lombardo
2 years ago