# NetworkX 1.9¶

Release date: 21 June 2014

Support for Python 3.1 is dropped in this release.

## Highlights¶

Completely rewritten maximum flow and flow-based connectivity algorithms with backwards incompatible interfaces

Community graph generators

Stoer–Wagner minimum cut algorithm

Linear-time Eulerian circuit algorithm

Linear algebra package changed to use SciPy sparse matrices

Algebraic connectivity, Fiedler vector, spectral ordering algorithms

Link prediction algorithms

Goldberg–Radzik shortest path algorithm

Semiconnected graph and tree recognition algorithms

## Flow package¶

The flow package (`networkx.algorithms.flow`

) is completely rewritten
with backward *incompatible* changes. It introduces a new interface to flow
algorithms. Existing code that uses the flow package will not work unmodified
with NetworkX 1.9.

# Main changes¶

We added two new maximum flow algorithms (

`preflow_push`

and`shortest_augmenting_path`

) and rewrote the Edmonds–Karp algorithm in`flow_fulkerson`

which is now in`edmonds_karp`

. @ysitu contributed implementations of all new maximum flow algorithms. The legacy Edmonds–Karp algorithm implementation in`ford_fulkerson`

is still available but will be removed in the next release.All maximum flow algorithm implementations (including the legacy

`ford_fulkerson`

) output now a residual network (i.e., a`DiGraph`

) after computing the maximum flow. See`maximum_flow`

documentation for the details on the conventions that NetworkX uses for defining a residual network.We removed the old

`max_flow`

and`min_cut`

functions. The main entry points to flow algorithms are now the functions`maximum_flow`

,`maximum_flow_value`

,`minimum_cut`

and`minimum_cut_value`

, which have new parameters that control maximum flow computation:`flow_func`

for specifying the algorithm that will do the actual computation (it accepts a function as argument that implements a maximum flow algorithm),`cutoff`

for suggesting a maximum flow value at which the algorithm stops,`value_only`

for stopping the computation as soon as we have the value of the flow, and`residual`

that accepts as argument a residual network to be reused in repeated maximum flow computation.All flow algorithms are required to accept arguments for these parameters but may selectively ignored the inapplicable ones. For instance,

`preflow_push`

algorithm can stop after the preflow phase without computing a maximum flow if we only need the flow value, but both`edmonds_karp`

and`shortest_augmenting_path`

always compute a maximum flow to obtain the flow value.The new function

`minimum_cut`

returns the cut value and a node partition that defines the minimum cut. The function`minimum_cut_value`

returns only the value of the cut, which is what the removed`min_cut`

function used to return before 1.9.The functions that implement flow algorithms (i.e.,

`preflow_push`

,`edmonds_karp`

,`shortest_augmenting_path`

and`ford_fulkerson`

) are not imported to the base NetworkX namespace. You have to explicitly import them from the flow package:

```
>>> from networkx.algorithms.flow import (ford_fulkerson, preflow_push,
... edmonds_karp, shortest_augmenting_path)
```

We also added a capacity-scaling minimum cost flow algorithm:

`capacity_scaling`

. It supports`MultiDiGraph`

and disconnected networks.

# Examples¶

Below are some small examples illustrating how to obtain the same output than in NetworkX 1.8.1 using the new interface to flow algorithms introduced in 1.9:

```
>>> import networkx as nx
>>> G = nx.icosahedral_graph()
>>> nx.set_edge_attributes(G, 'capacity', 1)
```

With NetworkX 1.8:

```
>>> flow_value = nx.max_flow(G, 0, 6)
>>> cut_value = nx.min_cut(G, 0, 6)
>>> flow_value == cut_value
True
>>> flow_value, flow_dict = nx.ford_fulkerson(G, 0, 6)
```

With NetworkX 1.9:

```
>>> from networkx.algorithms.flow import (ford_fulkerson, preflow_push,
... edmonds_karp, shortest_augmenting_path)
>>> flow_value = nx.maximum_flow_value(G, 0, 6)
>>> cut_value = nx.minimum_cut_value(G, 0, 6)
>>> flow_value == cut_value
True
>>> # Legacy: this returns the exact same output than ford_fulkerson in 1.8.1
>>> flow_value, flow_dict = nx.maximum_flow(G, 0, 6, flow_func=ford_fulkerson)
>>> # We strongly recommend to use the new algorithms:
>>> flow_value, flow_dict = nx.maximum_flow(G, 0, 6)
>>> # If no flow_func is passed as argument, the default flow_func
>>> # (preflow-push) is used. Therefore this is the same than:
>>> flow_value, flow_dict = nx.maximum_flow(G, 0, 6, flow_func=preflow_push)
>>> # You can also use alternative maximum flow algorithms:
>>> flow_value, flow_dict = nx.maximum_flow(G, 0, 6, flow_func=shortest_augmenting_path)
>>> flow_value, flow_dict = nx.maximum_flow(G, 0, 6, flow_func=edmonds_karp)
```

## Connectivity package¶

The flow-based connecitivity and cut algorithms from the connectivity
package (`networkx.algorithms.connectivity`

) are adapted to take
advantage of the new interface to flow algorithms. As a result, flow-based
connectivity algorithms are up to 10x faster than in NetworkX 1.8 for some
problems, such as sparse networks with highly skewed degree distributions.
A few backwards *incompatible* changes were introduced.

The functions for local connectivity and cuts accept now arguments for the new parameters defined for the flow interface:

`flow_func`

for defining the algorithm that will perform the underlying maximum flow computations,`residual`

that accepts as argument a residual network to be reused in repeated maximum flow computations, and`cutoff`

for defining a maximum flow value at which the underlying maximum flow algorithm stops. The big speed improvement with respect to 1.8 comes mainly from the reuse of the residual network and the use of`cutoff`

.We removed the flow-based local connectivity and cut functions from the base namespace. Now they have to be explicitly imported from the connectivity package. The main entry point to flow-based connectivity and cut functions are the functions

`edge_connectivity`

,`node_connectivity`

,`minimum_edge_cut`

, and`minimum_node_cut`

. All these functions accept a couple of nodes as optional arguments for computing local connectivity and cuts.We improved the auxiliary network for connectivity functions: The node mapping dict needed for node connectivity and minimum node cuts is now a graph attribute of the auxiliary network. Thus we removed the

`mapping`

parameter from the local versions of connectivity and cut functions. We also changed the parameter name for the auxuliary digraph from`aux_digraph`

to`auxiliary`

.We changed the name of the function

`all_pairs_node_connectiviy_matrix`

to`all_pairs_node_connectivity`

. This function now returns a dictionary instead of a NumPy 2D array. We added a new parameter`nbunch`

for computing node connectivity only among pairs of nodes in`nbunch`

.A

`stoer_wagner`

function is added to the connectivity package for computing the weighted minimum cuts of undirected graphs using the Stoer–Wagner algorithm. This algorithm is not based on maximum flows. Several heap implementations are also added in the utility package (`networkx.utils`

) for use in this function.`BinaryHeap`

is recommended over`PairingHeap`

for Python implementations without optimized attribute accesses (e.g., CPython) despite a slower asymptotic running time. For Python implementations with optimized attribute accesses (e.g., PyPy),`PairingHeap`

provides better performance.

## Other new functionalities¶

A

`disperson`

function is added in the centrality package (`networkx.algorithms.centrality`

) for computing the dispersion of graphs.A community package (

`networkx.generators.community`

) is added for generating community graphs.An

`is_semiconnected`

function is added in the connectivity package (`networkx.algorithms.connectivity`

) for recognizing semiconnected graphs.The

`eulerian_circuit`

function in the Euler package (`networkx.algorithm.euler`

) is changed to use a linear-time algorithm.A

`non_edges`

function in added in the function package (`networkx.functions`

) for enumerating nonexistent edges between existing nodes of graphs.The linear algebra package (

`networkx.linalg`

) is changed to use SciPy sparse matrices.Functions

`algebraic_connectivity`

,`fiedler_vector`

and`spectral_ordering`

are added in the linear algebra package (`networkx.linalg`

) for computing the algebraic connectivity, Fiedler vectors and spectral orderings of undirected graphs.A link prediction package (

`networkx.algorithms.link_prediction`

) is added to provide link prediction-related functionalities.Write Support for the graph6 and sparse6 formats is added in the read/write package (

`networkx.readwrite`

).A

`goldberg_radzik`

function is added in the shortest path package (`networkx.algorithms.shortest_paths`

) for computing shortest paths using the Goldberg–Radzik algorithm.A tree package (

`networkx.tree`

) is added to provide tree recognition functionalities.A context manager

`reversed`

is added in the utility package (`networkx.utils`

) for temporary in-place reversal of graphs.

## Miscellaneous changes¶

The functions in the components package (

`networkx.algorithms.components`

) such as`connected_components`

,`connected_components_subgraph`

now return generators instead of lists. To recover the earlier behavior, use`list(connected_components(G))`

.JSON helpers in the JSON graph package (

`networkx.readwrite.json_graph`

) are removed. Use functions from the standard library (e.g.,`json.dumps`

) instead.Support for Python 3.1 is dropped. Basic support is added for Jython 2.7 and IronPython 2.7, although they remain not officially supported.

Numerous reported issues are fixed.