# Source code for networkx.algorithms.time_dependent

```
"""Time dependent algorithms."""
import networkx as nx
from networkx.utils import not_implemented_for
__all__ = ["cd_index"]
[docs]
@not_implemented_for("undirected")
@not_implemented_for("multigraph")
@nx._dispatchable(node_attrs={"time": None, "weight": 1})
def cd_index(G, node, time_delta, *, time="time", weight=None):
r"""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
----------
G : graph
A directed networkx graph whose nodes have `time` attributes and optionally
`weight` attributes (if a weight is not given, it is considered 1).
node : node
The node for which the CD index is calculated.
time_delta : numeric 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.
time : string (Optional, default is "time")
The name of the node attribute that will be used for the calculations.
weight : string (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.
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
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:
.. math::
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
"""
if not all(time in G.nodes[n] for n in G):
raise nx.NetworkXError("Not all nodes have a 'time' attribute.")
try:
# get target_date
target_date = G.nodes[node][time] + time_delta
# keep the predecessors that existed before the target date
pred = {i for i in G.pred[node] if G.nodes[i][time] <= target_date}
except:
raise nx.NetworkXError(
"Addition and comparison are not supported between 'time_delta' "
"and 'time' types."
)
# -1 if any edge between node's predecessors and node's successors, else 1
b = [-1 if any(j in G[i] for j in G[node]) else 1 for i in pred]
# n is size of the union of the focal node's predecessors and its successors' predecessors
n = len(pred.union(*(G.pred[s].keys() - {node} for s in G[node])))
if n == 0:
raise nx.NetworkXError("The cd index cannot be defined.")
# calculate cd index
if weight is None:
return round(sum(bi for bi in b) / n, 2)
else:
# If a node has the specified weight attribute, its weight is used in the calculation
# otherwise, a weight of 1 is assumed for that node
weights = [G.nodes[i].get(weight, 1) for i in pred]
return round(sum(bi / wt for bi, wt in zip(b, weights)) / n, 2)
```