""" Networks is a module that implements a network of nodes connected by edges, and various algorithms and measurements implemented on networks. (This type of data structure is also known in mathematics as a graph. What we deal with specifically here are undirected graphs, where there is no direction or asymmetry between the nodes that delineate an edge.) This module contains the core code needed to define and manipulate networks, independently of how they are created or what their context is. We use the Networks module as part of other problem modules, such as SmallWorld and Percolation. """ # Import the necessary libraries. import NetGraphics # a separate module supporting network visualization import imp imp.reload(NetGraphics) # ----------------------------------------------------------------------- # # Defining undirected graph class: used by both Small World and Percolation # exercises. # # ----------------------------------------------------------------------- class UndirectedGraph: """An UndirectedGraph g contains a dictionary (g.connections) that maps a node identifier (key) to a list of nodes connected to (values). g.connections[node] returns a list [node2, node3, node4] of neighbors. Node identifiers can be any non-mutable Python type (e.g., integers, tuples, strings, but not lists). """ def __init__(self): """UndirectedGraph() creates an empty graph g. g.connections starts as an empty dictionary. When nodes are added, the corresponding values need to be inserted into lists. A method/function definition in a class must begin with an instance of the class in question; by convention, the name "self" is used for this instance.""" self.connections = {} def HasNode(self, node): """Returns True if the graph contains the specified node, and False otherwise. Check directly to see if the dictionary of connections contains the node, rather than (inefficiently) generating a list of all nodes and then searching for the specified node.""" # Hint: use the "in" operator to test if a key is in a dictionary pass def AddNode(self, node): """Uses HasNode(node) to determine if node has already been added.""" pass def AddEdge(self, node1, node2): """ Add node1 and node2 to network first Adds new edge (appends node2 to connections[node1] and vice-versa, since it's an undirected graph) Do so only if old edge does not already exist (node2 not in connections[node1]) """ pass def GetNodes(self): """g.GetNodes() returns all nodes (keys) in connections""" pass def GetNeighbors(self, node): """g.GetNeighbors(node) returns a copy of the list of neighbors of the specified node. A copy is returned (using the [:] operator) so that the user does not inadvertently change the neighbor list.""" pass # Simple test routines # def pentagraph(): """pentagraph() creates a simple 5-node UndirectedGraph, and then uses the imported NetGraphics module to layout and display the graph. The graph is returned from the function. """ g = UndirectedGraph() g.AddEdge(0,2) g.AddEdge(0,3) g.AddEdge(1,3) g.AddEdge(1,4) g.AddEdge(2,4) NetGraphics.DisplayCircleGraph(g) return g # def circle8(): """circle8() creates an 8-node UndirectedGraph, where the nodes are arranged in a circle (ring), and then uses the imported NetGraphics module to layout and display the graph. The graph is returned from the function. """ g = UndirectedGraph() g.AddEdge(0,1) g.AddEdge(1,2) g.AddEdge(2,3) g.AddEdge(3,4) g.AddEdge(4,5) g.AddEdge(5,6) g.AddEdge(6,7) g.AddEdge(7,0) NetGraphics.DisplayCircleGraph(g) return g # ***** After building and testing your network class, build some ***** # # ***** Small World or Percolation networks, making use of the ***** # # ***** corresponding hints file. Then return to analyze these ***** # # ***** networks using the general-purpose routines you write below. ***** # # ***** Small World exercise routines start here ***** # # ----------------------------------------------------------------------- # # Routines for finding path lengths on graphs: used by the Small World # Network exercise only # # ----------------------------------------------------------------------- def FindPathLengthsFromNode(graph, node): """Breadth--first search. See "Six degrees of separation" exercise from Sethna's book. Use a dictionary to store the distance to each node visited. Keys in the dictionary thus serve as markers of nodes that have already been visited, and should not be considered again.""" pass def FindAllPathLengths(graph): """ FindAllPathLengths returns a dictionary, indexed by node pairs, storing the shortest path length between those nodes, e.g. for small-world networks """ pass def FindAveragePathLength(graph): """Averages path length over all pairs of nodes""" pass # ----------------------------------------------------------------------- # # Routine for calculating the clustering coefficient of a graph. # This was a piece of the original Small World paper, but is not part of the # assigned exercise. # # ----------------------------------------------------------------------- def ComputeClusteringCoefficient(graph): """Computes clustering coefficient of graph""" pass # ----------------------------------------------------------------------- # # Routines for calculating "betweenness", which measures how many shortest # paths between pairs of nodes on a graph pass through a given node or edge. # Used in the Small Worlds exercise. # # References: (on the Web site) # Mark E. J. Newman, "Scientific collaboration networks. ii. shortest paths, # weighted networks, and criticality", Physical Review E 64: 016132, 2002. # Michelle Girvan and Mark E. J. Newman, "Community structure in social # and biological networks. Proceedings of the National Academy of Sciences # 12, 7821-7826, 2002. # # ----------------------------------------------------------------------- def EdgeAndNodeBetweennessFromNode(graph, node): """ Newman's edge and node betweenness inner loop Returns partial sum of edge, node betweenness """ pass def EdgeAndNodeBetweenness(graph): """Returns Newman's edge, node betweenness""" pass # ----------------------------------------------------------------------- # # Sample routines for reading in external files defining networks. Used # primarily for the later portions of the Small World exercise. # # ----------------------------------------------------------------------- def ReadGraphFromEdgeFile(filename, conversion=None): """Reads file with (node1,node2) for each edge in graph""" pass def ReadGraphFromNeighborFile(filename, conversion=None): """ Reads file with [node1,node2,node3] for completely interconnected group of nodes (as in actors in a movie) Should be read in as a bipartite graph! """ pass # ***** Percolation exercise routines start here ***** # # ----------------------------------------------------------------------- # # Routines for finding clusters in networks. Used in the Percolation exercise. # # ----------------------------------------------------------------------- def FindClusterFromNode(graph, node, visited=None): """Breadth--first search The dictionary "visited" should be initialized to False for all the nodes in the cluster you wish to find It's used in two different ways. (1) It's passed back to the calling program with all the nodes in the current cluster set to visited[nodeInCluster]=True, so that the calling program can skip nodes in this cluster in searching for other clusters. (2) It's used internally in this algorithm to keep track of the sites in the cluster that have already been found and incorporated See "Building a Percolation Network" in text for algorithm""" pass def FindAllClusters(graph): """For example, find percolation clusters Set up the dictionary "visited" for FindClusterFromNode Set up an empty list "clusters" Iterate over the nodes; if it haven't been visited, find the cluster containing it append it to the cluster list return clusters Check your answer using NetGraphics.DrawSquareNetworkBonds(g, cl) and NetGraphics.DrawSquareNetworkSites(g, cl) Optional: You may wish to sort your list of clusters according to their lengths, biggest to smallest For a list ell, the built-in method ell.sort() will sort the list from smallest to biggest; ell.sort(cmp) will sort the list according to the comparison function cmp(x, y) returns -1 if x < y, returns 0 if x==y, and returns 1 if x>y Define ReverseLengthCompare to compare two lists according to the unusual definition of inequality, l1 len(l2)! """ pass def GetSizeDistribution(clusters): """Given the clusters, makes up a dictionary giving the number of clusters of a given size. """ pass # Copyright (C) Cornell University # All rights reserved. # Apache License, Version 2.0