# Erdos

Erdos is a graph library written in Julia. Installation is straightforward:


julia> Pkg.add("Erdos")



Erdos defines two abstract graph types, AGraph and ADiGraph, from which all concrete undirected and directed graph types are derived.

Two concrete graph types, Graph and Network, and two digraph types, DiGraph and DiNetwork are implemented. In all these types the graph topology is represented internally as an adjacency list for each vertex. (Di)Networks come with some additional features over (Di)Graphs:

• each edge has a unique index;
• vertex/edge maps (also called properties) can be stored internally.

All graphs in Erdos have 1:n indexed vertices, where n is the number of vertices. Multi-edges are not allowed. Self-edges are experimentally supported. Provided this constraints, new graph types can be easily introduced just defining a few basic methods.

## Basic examples

### Constructors

Build your first graph using the basic constructors.


julia> using Erdos

julia> g = CompleteGraph(100)

Graph{Int64}(100, 4950)

julia> g = Network(10, 30) # a random graph with 10 vertex and 30 edges

Network(10, 30)

julia> g = Graph{Int32}(100)

Graph{Int32}(100, 0)

julia> g = DiGraph()

DiGraph{Int64}(0, 0)

julia> g = erdos_renyi(10,30)

Graph{Int64}(10, 30)

julia> g = random_regular_graph(10,3,seed=17)

Graph{Int64}(10, 15).



The default graph and digraph types are Graph{Int} and DiGraph{Int}.

Use the Graph{Int32} type to save memory if working with very large graphs.

### Queries


julia> g = CompleteBipartiteGraph(5,10)

Graph{Int64}(15, 50)

julia> is_bipartite(g)

true

julia> nv(g) # number of vertices

15

julia> ne(g) #number of edges

50

julia> has_edge(g,1,2)

false

julia> has_edge(g,1,6)

true

julia> degree(g,1)

10



### Modifiers:


julia> g=DiGraph(10)

DiGraph{Int64}(10, 0)

julia> add_edge!(g,1,2)

true

julia> g

DiGraph{Int64}(10, 1)

# trying to add an edge between non-existen vertices results in a

# (silent) failure

julia> add_edge!(g,1,11)

false

# has the addition of an already existent edge

julia> add_edge!(g,1,2)

false

julia> add_edge!(g,2,1)

true

julia> rem_edge!(g,2,1)

true

# returns the index of the vertex

julia> add_vertex!(g)

11

# vertex removal will cause the switch of the last index

# with the removed one, to keep the indexes continuity.

julia> rem_vertex!(g,1)

true

julia> rem_vertex!(g,12)

false



### Iterators


julia> g = Graph(10,20) #erdos renyi random graph with 10 vertices and 20 edges

Graph{Int64}(10, 20)

julia> numedges = 0;

# iterate over all edges

julia> for e in edges(g)

i, j = src(e), dst(e) # source and destination of an edge

@assert i <= j  # default for undirected graphs

numedge += 1

end

julia> ne(g) == numedge

true

julia> k=0;

julia> for i in neighbors(g,1)

@assert has_edge(g, 1, i);

k += 1

end

julia> degree(g, 1) == k

true



### I/O

Erdos supports many standard graph formats. Here is an example with Pajek's .net

format:


julia> g = DiGraph(10,20)

DiGraph{Int64}(10, 20)

julia> writegraph("test.net", g)

1

julia> h = readgraph("test.net")

DiGraph{Int64}(10, 20)

julia> g == h

true



### Datasets

A collection of real world graphs is available through readgraph:


julia> g = readgraph(:karate)

Graph{Int64}(34, 78)

julia> g = readgraph(:condmat,Graph{UInt32})

Graph{UInt32}(16726, 47594)



## Ready to explore

Refer to the documentation to explore all the features of Erdos.

Here is a comprehensive list of the implemente algorithms. (EE) denotes algorithms in the companion package ErdosExtras.

• core functions: vertices and edges addition and removal, degree (in/out/all), neighbors (in/out/all)

• maps dictionary like types to store properties associated to vertices and edges

• networks store vertex/edge/graph properties (maps) inside the graph itself

• connectivity: strongly- and weakly-connected components, bipartite checks, condensation, attracting components, neighborhood, k-core

• operators: complement, reverse, reverse!, union, join, intersect, difference, symmetric difference, blkdiag, induced subgraphs, products (cartesian/scalar)

• shortest paths: Dijkstra, Dijkstra with predecessors, Bellman-Ford, Floyd-Warshall, A*

• graph datasets: A collection of real world graphs (e.g. Zachary's karate club)

• graph generators: notorious graphs, euclidean graphs and random graphs (Erdős–Rényi, Watts-Strogatz, random regular, arbitrary degree sequence, stochastic block model)

• I/O formats: graphml, gml, gexf, dot, net, gt. For some of these formats vertex/edge/graph properties can be read and written.

• centrality: betweenness, closeness, degree, pagerank, Katz

• traversal operations: cycle detection, BFS and DFS DAGs, BFS and DFS traversals with visitors, DFS topological sort, maximum adjacency / minimum cut, multiple random walks

• flow operations: maximum flow, minimum s-t cut

• matching: minimum weight matching on arbitrary graphs (EE), minimum b-matching (EE)

• travelling salesman problem: a TSP solver based on linear programming (EE)

• dismantling: collective influencer heuristic

• clique enumeration: maximal cliques

• linear algebra / spectral graph theory: adjacency matrix, Laplacian matrix, non-backtracking matrix

• community: modularity, community detection, core-periphery, clustering coefficients

• distance within graphs: eccentricity, diameter, periphery, radius, center

• distance between graphs: spectral_distance, edit_distance