c02_graph_conc - Graph concepts Graphs are made up by...

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Graph concepts Graphs are made up by vertices (nodes) and edges (links) . An edge connects two vertices, or a vertex with itself – loop. AC, AC - multiple edges BB – loop The shape of the graph does not matter, only the way the nodes are onnected to each other connected to each other. imple graph - oes not have loops (self- dges) and does not have Simple graph does not have loops (self edges) and does not have multiple identical edges. Further reading: http://www.utm.edu/departments/math/graph/glossary.html
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Symmetrical and directed graphs Two distinct types of edges: symmetrical and directed (also called arcs). Two different graph frameworks: graph, digraph = directed graph. the digraph framework a symmetrical edge means the superposition In the digraph framework a symmetrical edge means the superposition of two opposite directed edges.
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Node degrees Node degree: the number of edges connected to the ode 4 k = node. directed networks we can define an egree i In directed networks we can define an in- degree and out-degree . The (total) degree is the sum of in- nd out egree out and out-degree. ource a node with in egree = 0 2 k in C = 1 k C = 3 = C k Source : a node with in-degree = 0. Sink : a node with out-degree = 0. g A F are sources B is a sink E.g. A, F are sources, B is a sink.
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Average degree N i k k 1 N – the number of nodes in the graph = i N 1 out in N out i out N in i in k k , k 1 k , k 1 k = : What is the relation between the number of 1 i 1 i N N = = Q: What is the relation between the number of edges in a (non-directed) graph and the sum of node degrees? How about in a directed graph?
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Statistics of node degrees Average degree N E 2 k N 1 k N 1 i i = = N E k k out in = = The degree distribution gives the fraction of nodes that have edges. ) k ( P k Similarly / gives the action of nodes that have in- egree / ) k ( P in in k ) k ( P out fraction of nodes that have in degree / out-degree . out k Ex. Calculate the degree distributions of the graphs in the left.
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Paths and circuits Adjacent nodes (vertices) – there is an edge joining them. In the digraph framework the adjacency is only defined in the direction of the arrow. Path: a sequence of nodes in which each node is adjacent to the next one. Edges can be part of a path only once. In the digraph framework a symmetrical edge can be used once in one direction and once in the opposite irection direction. Circuit : a path that starts and ends at the same ertex Ex. Give examples f circuits and cycles vertex . Cycle : a circuit that does not revisit any nodes. of circuits and cycles in the above graph
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Connectivity of undirected graphs Connected (undirected) graph: any two vertices can be joined by a path. A disconnected graph is made up by two or more connected components. Bridge: if we erase it, the graph becomes disconnected.
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Connectivity of directed graphs Strongly connected directed graph: has a path from each node to every other node and vice versa (e.g. AB path and BA path).
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c02_graph_conc - Graph concepts Graphs are made up by...

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