cd_index#

cd_index(G, node, time_delta, *, time='time', weight=None)[source]#

Compute the CD index for node within the graph G.

Calculates the CD index for the given node of the graph, considering only its predecessors who have the time attribute smaller than or equal to the time attribute of the node plus time_delta.

Parameters:

Ggraph: A directed networkx graph whose nodes have time attributes and optionally weight attributes (if a weight is not given, it is considered 1).
nodenode: The node for which the CD index is calculated.
time_deltanumeric or timedelta: Amount of time after the time attribute of the node. The value of time_delta must support comparison with the time node attribute. For example, if the time attribute of the nodes are datetime.datetime objects, then time_delta should be a datetime.timedelta object.
timestring (Optional, default is “time”): The name of the node attribute that will be used for the calculations.
weightstring (Optional, default is None): The name of the node attribute used as weight.

Returns:

float: The CD index calculated for the node node within the graph G.

Raises:

NetworkXError: If not all nodes have a time attribute or time_delta and time attribute types are not compatible or n equals 0.
NetworkXNotImplemented: If G is a non-directed graph or a multigraph.

Notes

This method implements the algorithm for calculating the CD index, as described in the paper by Funk and Owen-Smith [1]. The CD index is used in order to check how consolidating or destabilizing a patent is, hence the nodes of the graph represent patents and the edges show the citations between these patents. The mathematical model is given below:

\[CD_{t}=\frac{1}{n_{t}}\sum_{i=1}^{n}\frac{-2f_{it}b_{it}+f_{it}}{w_{it}},\]

where f_{it} equals 1 if i cites the focal patent else 0, b_{it} equals 1 if i cites any of the focal patents successors else 0, n_{t} is the number of forward citations in i and w_{it} is a matrix of weight for patent i at time t.

The datetime.timedelta package can lead to off-by-one issues when converting from years to days. In the example above timedelta(days=5 * 365) looks like 5 years, but it isn’t because of leap year days. So it gives the same result as timedelta(days=4 * 365). But using timedelta(days=5 * 365 + 1) gives a 5 year delta for this choice of years but may not if the 5 year gap has more than 1 leap year. To avoid these issues, use integers to represent years, or be very careful when you convert units of time.

References

[1]

Funk, Russell J., and Jason Owen-Smith. “A dynamic network measure of technological change.” Management science 63, no. 3 (2017): 791-817. http://russellfunk.org/cdindex/static/papers/funk_ms_2017.pdf

Examples

>>> from datetime import datetime, timedelta
>>> G = nx.DiGraph()
>>> nodes = {
...     1: {"time": datetime(2015, 1, 1)},
...     2: {"time": datetime(2012, 1, 1), "weight": 4},
...     3: {"time": datetime(2010, 1, 1)},
...     4: {"time": datetime(2008, 1, 1)},
...     5: {"time": datetime(2014, 1, 1)},
... }
>>> G.add_nodes_from([(n, nodes[n]) for n in nodes])
>>> edges = [(1, 3), (1, 4), (2, 3), (3, 4), (3, 5)]
>>> G.add_edges_from(edges)
>>> delta = timedelta(days=5 * 365)
>>> nx.cd_index(G, 3, time_delta=delta, time="time")
0.5
>>> nx.cd_index(G, 3, time_delta=delta, time="time", weight="weight")
0.12

Integers can also be used for the time values: >>> node_times = {1: 2015, 2: 2012, 3: 2010, 4: 2008, 5: 2014} >>> nx.set_node_attributes(G, node_times, “new_time”) >>> nx.cd_index(G, 3, time_delta=4, time=”new_time”) 0.5 >>> nx.cd_index(G, 3, time_delta=4, time=”new_time”, weight=”weight”) 0.12