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Graph-tool
graph-tool is a Python (programming language), Python module for manipulation and statistical analysis of Graph (discrete mathematics), graphs (AKA Network theory, networks). The core data structures and algorithms of graph-tool are implemented in C++, making extensive use of metaprogramming, based heavily on the Boost Graph Library. Many algorithms are implemented in parallel using OpenMP, which provides increased performance on Multi-core processor, multi-core architectures. Features * Creation and manipulation of Directed graph, directed or Graph (discrete mathematics), undirected graphs. * Association of arbitrary information to the vertices, edges or even the graph itself, by means of property maps. * Filter vertices and/or edges "on the fly", such that they appear to have been removed. * Support for DOT language, dot, Graph Modelling Language and GraphML formats. * Convenient and powerful graph drawing based on cairo (graphics), cairo or Graphviz. * Support for typical st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Drawing
Graph drawing is an area of mathematics and computer science combining methods from geometric graph theory and information visualization to derive two-dimensional depictions of graph (discrete mathematics), graphs arising from applications such as social network analysis, cartography, linguistics, and bioinformatics. A drawing of a graph or network diagram is a pictorial representation of the vertex (graph theory), vertices and edge (graph theory), edges of a graph. This drawing should not be confused with the graph itself: very different layouts can correspond to the same graph., p. 6. In the abstract, all that matters is which pairs of vertices are connected by edges. In the concrete, however, the arrangement of these vertices and edges within a drawing affects its understandability, usability, fabrication cost, and aesthetics. The problem gets worse if the graph changes over time by adding and deleting edges (dynamic graph drawing) and the goal is to preserve the user's men ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cairo (graphics)
Cairo (stylized as cairo) is an Open-source software, open-source graphics library that provides a vector graphics-based, device-independent Application programming interface, API for software developers. It provides primitives for Plane (mathematics), two-dimensional drawing across a number of different front and back ends, backends. Cairo uses hardware acceleration when available. Software architecture Language bindings A library written in one programming language may be used in another language if language binding, bindings are written; Cairo has a range of bindings for various languages including C++, C Sharp (programming language), C# and other Common Language Infrastructure, CLI languages, Delphi (programming language), Delphi, Eiffel (programming language), Eiffel, Fortran, Factor (programming language), Factor, Harbour (programming language), Harbour, Haskell (programming language), Haskell, Julia (programming language), Julia, Lua (programming language), Lua, Perl, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Community Structure
In the study of complex networks, a network is said to have community structure if the nodes of the network can be easily grouped into (potentially overlapping) sets of nodes such that each set of nodes is densely connected internally. In the particular case of ''non-overlapping'' community finding, this implies that the network divides naturally into groups of nodes with dense connections internally and sparser connections between groups. But ''overlapping'' communities are also allowed. The more general definition is based on the principle that pairs of nodes are more likely to be connected if they are both members of the same community(ies), and less likely to be connected if they do not share communities. A related but different problem is community search, where the goal is to find a community that a certain vertex belongs to. Properties In the study of networks, such as computer and information networks, social networks and biological networks, a number of different chara ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Network Motif
Network motifs are recurrent and statistically significant subgraphs or patterns of a larger graph. All networks, including biological networks, social networks, technological networks (e.g., computer networks and electrical circuits) and more, can be represented as graphs, which include a wide variety of subgraphs. Network motifs are sub-graphs that repeat themselves in a specific network or even among various networks. Each of these sub-graphs, defined by a particular pattern of interactions between vertices, may reflect a framework in which particular functions are achieved efficiently. Indeed, motifs are of notable importance largely because they may reflect functional properties. They have recently gathered much attention as a useful concept to uncover structural design principles of complex networks. Although network motifs may provide a deep insight into the network's functional abilities, their detection is computationally challenging. Definitions Let and be two gr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clustering Coefficient
In graph theory, a clustering coefficient is a measure of the degree to which nodes in a graph tend to cluster together. Evidence suggests that in most real-world networks, and in particular social networks, nodes tend to create tightly knit groups characterised by a relatively high density of ties; this likelihood tends to be greater than the average probability of a tie randomly established between two nodes (Holland and Leinhardt, 1971; Watts and Strogatz, 1998). Two versions of this measure exist: the global and the local. The global version was designed to give an overall indication of Clustering_coefficient#Global_clustering_coefficient, the clustering in the network, whereas the local gives an indication of the Clustering_coefficient#Local_clustering_coefficient, extent of "clustering" of a single node. Local clustering coefficient The local clustering coefficient of a vertex (graph theory), vertex (node) in a Graph (discrete mathematics), graph quantifies how close its ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Centrality
In graph theory and network analysis, indicators of centrality assign numbers or rankings to nodes within a graph corresponding to their network position. Applications include identifying the most influential person(s) in a social network, key infrastructure nodes in the Internet or urban networks, super-spreaders of disease, and brain networks. Centrality concepts were first developed in social network analysis, and many of the terms used to measure centrality reflect their sociological origin.Newman, M.E.J. 2010. ''Networks: An Introduction.'' Oxford, UK: Oxford University Press. Definition and characterization of centrality indices Centrality indices are answers to the question "What characterizes an important vertex?" The answer is given in terms of a real-valued function on the vertices of a graph, where the values produced are expected to provide a ranking which identifies the most important nodes. The word "importance" has a wide number of meanings, leading to many d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximum Flow
In optimization theory, maximum flow problems involve finding a feasible flow through a flow network that obtains the maximum possible flow rate. The maximum flow problem can be seen as a special case of more complex network flow problems, such as the circulation problem. The maximum value of an s-t flow (i.e., flow from source s to sink t) is equal to the minimum capacity of an s-t cut (i.e., cut severing s from t) in the network, as stated in the max-flow min-cut theorem. History The maximum flow problem was first formulated in 1954 by T. E. Harris and F. S. Ross as a simplified model of Soviet railway traffic flow. In 1955, Lester R. Ford, Jr. and Delbert R. Fulkerson created the first known algorithm, the Ford–Fulkerson algorithm.Ford, L.R., Jr.; Fulkerson, D.R., ''Flows in Networks'', Princeton University Press (1962). In their 1955 paper, Ford and Fulkerson wrote that the problem of Harris and Ross is formulated as follows (see p. 5):Consider a rail network con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dominator (graph Theory)
In computer science, a node of a control-flow graph dominates a node if every path from the ''entry node'' to must go through . Notationally, this is written as (or sometimes ). By definition, every node dominates itself. There are a number of related concepts: * A node ''strictly dominates'' a node if dominates and does not equal . * The ''immediate dominator'' or idom of a node is the unique node that strictly dominates but does not strictly dominate any other node that strictly dominates . Every node reachable from the entry node has an immediate dominator (except the entry node). * The ''dominance frontier'' of a node is the set of all nodes such that dominates an immediate predecessor of , but does not strictly dominate . It is the set of nodes where 's dominance stops. * A ''dominator tree'' is a tree where each node's children are those nodes it immediately dominates. The start node is the root of the tree. History Dominance was first introduced by Re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connected Component (graph Theory)
In graph theory, a component of an undirected graph is a connected subgraph that is not part of any larger connected subgraph. The components of any graph partition its vertices into disjoint sets, and are the induced subgraphs of those sets. A graph that is itself connected has exactly one component, consisting of the whole graph. Components are sometimes called connected components. The number of components in a given graph is an important graph invariant, and is closely related to invariants of matroids, topological spaces, and matrices. In random graphs, a frequently occurring phenomenon is the incidence of a giant component, one component that is significantly larger than the others; and of a percolation threshold, an edge probability above which a giant component exists and below which it does not. The components of a graph can be constructed in linear time, and a special case of the problem, connected-component labeling, is a basic technique in image analysis. Dy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimum Spanning Tree
A minimum spanning tree (MST) or minimum weight spanning tree is a subset of the edges of a connected, edge-weighted undirected graph that connects all the vertices together, without any cycles and with the minimum possible total edge weight. That is, it is a spanning tree whose sum of edge weights is as small as possible. More generally, any edge-weighted undirected graph (not necessarily connected) has a minimum spanning forest, which is a union of the minimum spanning trees for its connected components. There are many use cases for minimum spanning trees. One example is a telecommunications company trying to lay cable in a new neighborhood. If it is constrained to bury the cable only along certain paths (e.g. roads), then there would be a graph containing the points (e.g. houses) connected by those paths. Some of the paths might be more expensive, because they are longer, or require the cable to be buried deeper; these paths would be represented by edges with larger weight ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subgraph Isomorphism Problem
In theoretical computer science, the subgraph isomorphism problem is a computational task in which two graphs G and H are given as input, and one must determine whether G contains a subgraph that is isomorphic to H. Subgraph isomorphism is a generalization of both the maximum clique problem and the problem of testing whether a graph contains a Hamiltonian cycle, and is therefore NP-complete. However certain other cases of subgraph isomorphism may be solved in polynomial time. Sometimes the name subgraph matching is also used for the same problem. This name puts emphasis on finding such a subgraph as opposed to the bare decision problem. Decision problem and computational complexity To prove subgraph isomorphism is NP-complete, it must be formulated as a decision problem. The input to the decision problem is a pair of graphs G and ''H''. The answer to the problem is positive if ''H'' is isomorphic to a subgraph of ''G'', and negative otherwise. Formal question: Let G=(V,E), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Isomorphism
In graph theory, an isomorphism of graphs ''G'' and ''H'' is a bijection between the vertex sets of ''G'' and ''H'' : f \colon V(G) \to V(H) such that any two vertices ''u'' and ''v'' of ''G'' are adjacent in ''G'' if and only if f(u) and f(v) are adjacent in ''H''. This kind of bijection is commonly described as "edge-preserving bijection", in accordance with the general notion of isomorphism being a structure-preserving bijection. If an isomorphism exists between two graphs, then the graphs are called isomorphic, often denoted by G\simeq H. In the case when the isomorphism is a mapping of a graph onto itself, i.e., when ''G'' and ''H'' are one and the same graph, the isomorphism is called an automorphism of ''G''. Graph isomorphism is an equivalence relation on graphs and as such it partitions the class of all graphs into equivalence classes. A set of graphs isomorphic to each other is called an isomorphism class of graphs. The question of whether graph isomorphism can be dete ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
