ReGraph allows to create a hierarchies of graphs related by means of homomorphisms (or typing). In the context of a hierarchy, if there exists a homomorphism $G \rightarrow T$, we say that the graph $G$ is typed by the graph $T$. Graph hierarchy is a DAG, where nodes are graphs and edges are homomorphisms. A homomorphism maps every node of $G$ to some node in $T$ (a type) in such a way that:
In [1]:
from regraph import NXGraph, NXHierarchy, Rule
from regraph import plot_graph, plot_instance, plot_rule
In [2]:
%matplotlib inline
In [3]:
# Define graph G
g = NXGraph()
g.add_nodes_from(["protein", "binding", "region", "compound"])
g.add_edges_from([("region", "protein"), ("protein", "binding"), ("region", "binding"), ("compound", "binding")])
# Define graph T
t = NXGraph()
t.add_nodes_from(["action", "agent"])
t.add_edges_from([("agent", "agent"), ("agent", "action")])
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# Create a hierarchy
simple_hierarchy = NXHierarchy()
simple_hierarchy.add_graph("G", g, {"name": "Simple protein interaction"})
simple_hierarchy.add_graph("T", t, {"name": "Agent interaction"})
simple_hierarchy.add_typing(
"G", "T",
{"protein": "agent",
"region": "agent",
"compound": "agent",
"binding": "action",
}
)
print(simple_hierarchy)
The method get_graph returns the graph object corresponding to the provided graph id.
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type(simple_hierarchy.get_graph("T"))
Out[5]:
The method get_typing returns the dictionary object corresponding to the provided hierarchy edge and representing the associated graph homomorphism.
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simple_hierarchy.get_typing("G", "T")
Out[6]:
In [7]:
t_node_positions = plot_graph(simple_hierarchy.get_graph("T"))
g_node_positions = plot_graph(simple_hierarchy.get_graph("G"))
ReGraph implements the rewriting technique called sesqui-pushout rewriting that allows to transform graphs by applying rules through their instances (matchings). Rewriting an individual graphs in a hierarchy may require an update of other graphs and typings in this hierarchy, such updates are called propagation and are distinguished into two types: backward and forward propagation.
Backward propagation briefly:
Forward propagation briefly:
For more details, please see here.
ReGraph allows to rewrite individual graphs situated in the hierarchy using the method rewrite of NXHierarchy. The rewriting can be done in two modes:
Strict rewriting rewriting that does not allow propagation.
Not strict rewriting that allows propagation.
The rewrite takes as the input the following parameters:
graph_id, ID of the graph in the hierarchy to rewrite,rule, a rule object to apply,instance, a dictionary containing an instance of the lhs of the rule in the graph subject to rewriting, by default, tries to construct identity morphism of the nodes of the pattern,p_typing, a dictionary containing typing of graphs in the hierarchy by the interface of the rule, keys are IDs of hierarchy graphs, values are dictionaries containing the mapping of nodes from the hierarchy graphs to the inteface nodes (note that a node from a graph can be typed by a set of nodes in the interface of the rule, e.g. if we want to perform cloning of some types, etc).rhs_typing, a dictionary containing typing of the rhs by graphs of the hierarchy, keys are ids of hierarchy graphs, values are dictionaries containing the mapping of nodes from the lhs to the nodes of the typing graph given by the respective key of the value (note that a node from the rhs can be typed by a set of nodes of some graph, e.g. if we want to perform merging of some types, etc),strict, flag indicating if rewriting is strict, then any propagation is not allowed.Let us create a Rule object containing a rule we would like to apply.
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lhs = NXGraph()
lhs.add_nodes_from([1, 2])
lhs.add_edges_from([(1, 2)])
p = NXGraph()
p.add_nodes_from([1, 2])
p.add_edges_from([])
rhs = NXGraph()
rhs.add_nodes_from([1, 2, 3])
rhs.add_edges_from([(3, 1), (3, 2)])
# By default if `p_lhs` and `p_rhs` are not provided
# to a rule, it tries to construct this homomorphisms
# automatically by matching the names. In this case we
# have defined lhs, p and rhs in such a way that that
# the names of the matching nodes correspond
rule = Rule(p, lhs, rhs)
plot_rule(rule)
The created rule removes the edge 1->2, adds the new node 3 and two edges 3->1 and 3->2.
Let us find instances of the created rule in the graph G.
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instances = simple_hierarchy.find_matching("G", rule.lhs)
print("Instances: ", instances)
for instance in instances:
plot_instance(
simple_hierarchy.get_graph("G"),
rule.lhs,
instance,
parent_pos=g_node_positions) #filename=("instance_example_%d.png" % i))
Let us fix the desired instance: we would like to remove the edge from protein to binding and add some new node connecting them.
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instance = {
1: "protein",
2: "binding"
}
Let us try to apply the rule to the selected instance as is in the strict rewriting mode.
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try:
rhs_instance = simple_hierarchy.rewrite("G", rule, instance, strict=True)
except Exception as e:
print("Error message: ", e)
print("Type: ", type(e))
We have failed to rewrite G, because we have not specified typing for the newly added node 3. Let us try again, but this time we will prove such typing.
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rhs_typing = {
"T": {3: "agent"}
}
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rhs_instance = simple_hierarchy.rewrite(
"G", rule, instance, rhs_typing=rhs_typing, strict=True)
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print("Instance of the RHS in G", rhs_instance)
plot_instance(
simple_hierarchy.get_graph("G"),
rule.rhs,
rhs_instance,
parent_pos=g_node_positions)
Out[14]:
We will now create a rule that applied to T and that clones the node agent into two nodes.
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lhs = NXGraph()
lhs.add_nodes_from(["agent"])
rule = Rule.from_transform(lhs)
_, rhs_clone = rule.inject_clone_node("agent")
plot_rule(rule)
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instance = {
"agent": "agent"
}
We try to apply the created rule to the graph T in the strict mode.
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try:
rhs_instance = simple_hierarchy.rewrite("T", rule, instance, strict=True)
except Exception as e:
print("Error message: ", e)
print("Type: ", type(e))
We have failed to rewrite T, because we have not specified typing for instances of agent in $p$. Let us try again, but this time we will prove such typing.
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p_typing = {
"G": {
'protein': 'agent',
'region': 'agent',
'compound': rhs_clone,
3: 'agent'
}
}
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rhs_instance = simple_hierarchy.rewrite("T", rule, instance, p_typing=p_typing, strict=True)
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print("Instance of the RHS in G", rhs_instance)
plot_instance(
simple_hierarchy.get_graph("T"),
rule.rhs,
rhs_instance,
parent_pos=t_node_positions)
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Let us relabel nodes in T.
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simple_hierarchy.relabel_graph_node('T', rhs_instance['agent'], 'organic_agent')
simple_hierarchy.relabel_graph_node('T', rhs_instance[rhs_clone], 'non_organic_agent')
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plot_graph(simple_hierarchy.get_graph('T'))
print(simple_hierarchy.get_typing("G", "T"))
To illustrate rewriting with propagation, let us consider a slighlty more sophisticated hierarchy.
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hierarchy = NXHierarchy()
colors = NXGraph()
colors.add_nodes_from([
"green", "red"
])
colors.add_edges_from([
("red", "green"),
("red", "red"),
("green", "green")
])
hierarchy.add_graph("colors", colors)
shapes = NXGraph()
shapes.add_nodes_from(["circle", "square"])
shapes.add_edges_from([
("circle", "square"),
("square", "circle"),
("circle", "circle")
])
hierarchy.add_graph("shapes", shapes)
quality = NXGraph()
quality.add_nodes_from(["good", "bad"])
quality.add_edges_from([
("bad", "bad"),
("bad", "good"),
("good", "good")
])
hierarchy.add_graph("quality", quality)
g1 = NXGraph()
g1.add_nodes_from([
"red_circle",
"red_square",
])
g1.add_edges_from([
("red_circle", "red_square"),
("red_circle", "red_circle"),
("red_square", "red_circle")
])
g1_colors = {
"red_circle": "red",
"red_square": "red",
}
g1_shapes = {
"red_circle": "circle",
"red_square": "square",
}
hierarchy.add_graph("g1", g1)
hierarchy.add_typing("g1", "colors", g1_colors)
hierarchy.add_typing("g1", "shapes", g1_shapes)
g2 = NXGraph()
g2.add_nodes_from([
"good_circle",
"good_square",
"bad_circle",
])
g2.add_edges_from([
("good_circle", "good_square"),
("good_square", "good_circle"),
("bad_circle", "good_circle"),
("bad_circle", "bad_circle"),
])
g2_shapes = {
"good_circle": "circle",
"good_square": "square",
"bad_circle": "circle"
}
g2_quality = {
"good_circle": "good",
"good_square": "good",
"bad_circle": "bad",
}
hierarchy.add_graph("g2", g2)
hierarchy.add_typing("g2", "shapes", g2_shapes)
hierarchy.add_typing("g2", "quality", g2_quality)
g3 = NXGraph()
g3.add_nodes_from([
"good_red_circle",
"bad_red_circle",
"good_red_square",
])
g3.add_edges_from([
("bad_red_circle", "good_red_circle"),
("good_red_square", "good_red_circle"),
("good_red_circle", "good_red_square")
])
g3_g1 = {
"good_red_circle": "red_circle",
"bad_red_circle": "red_circle",
"good_red_square": "red_square"
}
g3_g2 = {
"good_red_circle": "good_circle",
"bad_red_circle": "bad_circle",
"good_red_square": "good_square",
}
hierarchy.add_graph("g3", g3)
hierarchy.add_typing("g3", "g1", g3_g1)
hierarchy.add_typing("g3", "g2", g3_g2)
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for graph in hierarchy.graphs():
print("Graph ", graph)
plot_graph(hierarchy.get_graph(graph))
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print(hierarchy)
Some of the graphs in the hierarchy are now typed by multiple graphs, which is reflected in the types of nodes, as in the example below:
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print("Node types in G3:\n")
for node in hierarchy.get_graph("g3").nodes():
print(node, hierarchy.node_type("g3", node))
Having constructed a sophisticated rewriting rule typed by some nodes in the hierarchy one may want to store this rule and to be able to propagate any changes that happen in the hierarchy to the rule as well.
ReGraph's NXHierarchy allows to add graph rewriting rules as nodes in the hierarchy. Rules in the hierarchy can be typed by graphs, but rule nodes are not allowed to have incoming edges, i.e. nothing can be typed by a rule.
In the example below, a rule is added to the previously constructed hierarchy and typed by graphs g1 and g2:
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lhs = NXGraph()
lhs.add_nodes_from([1, 2])
lhs.add_edges_from([(1, 2)])
p = NXGraph()
p.add_nodes_from([1, 11, 2])
p.add_edges_from([(1, 2)])
rhs = NXGraph.copy(p)
rhs.add_nodes_from([3])
p_lhs = {1: 1, 11: 1, 2: 2}
p_rhs = {1: 1, 11: 11, 2: 2}
r1 = Rule(p, lhs, rhs, p_lhs, p_rhs)
hierarchy.add_rule("r1", r1, {"desc": "Rule 1: typed by two graphs"})
lhs_typing1 = {1: "red_circle", 2: "red_square"}
rhs_typing1 = {3: "red_circle"}
lhs_typing2 = {1: "good_circle", 2: "good_square"}
rhs_typing2 = {3: "bad_circle"}
hierarchy.add_rule_typing("r1", "g1", lhs_typing1, rhs_typing1)
hierarchy.add_rule_typing("r1", "g2", lhs_typing2, rhs_typing2)
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plot_rule(hierarchy.get_rule('r1'))
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g1_lhs_typing, g1_rhs_typing = hierarchy.get_typing('r1', 'g1')
g2_lhs_typing, g2_rhs_typing = hierarchy.get_typing('r1', 'g2')
print("Typing of R1 by G1: ")
print("\tLHS", g1_lhs_typing)
print("\tP (is implicit)")
print("\tRHS", g1_rhs_typing)
print("Typing of R1 by G2: ")
print("\tLHS", g2_lhs_typing)
print("\tP (is implicit)")
print("\tRHS", g2_rhs_typing)
We now show how graph rewriting can be performed in such an hierarchy. In the previous example we perfromed strict rewriting in a hierarchy, where no propagation was performed.
The following example illustrates how the ReGraph propagates the changes made by rewriting on any level to all the graphs (as well as the rules) typed by the one target of rewriting.
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lhs = NXGraph()
lhs.add_nodes_from(["a", "b"])
lhs.add_edges_from([
("a", "b"),
("b", "a")
])
p = NXGraph()
p.add_nodes_from(["a", "a1", "b"])
p.add_edges_from([
("a", "b"),
("a1", "b")
])
rhs = NXGraph.copy(p)
rule = Rule(
p, lhs, rhs,
{"a": "a", "a1": "a", "b": "b"},
{"a": "a", "a1": "a1", "b": "b"},
)
plot_rule(rule)
We have created a rule that clones the node a and reconnects the edges between a and b.
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instances = hierarchy.find_matching("shapes", lhs)
print("Instances:")
for instance in instances:
print(instance)
plot_instance(hierarchy.get_graph("shapes"), rule.lhs, instance)
We rewrite the graph shapes with the fixed instances (so, the node circle is cloned).
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rhs_instances = hierarchy.rewrite("shapes", rule, {"a": "circle", "b": "square"})
Observe the following plots, the cloning of circle was propagated to all the ancestors of shapes, because we didn't specify how to retype intances of circle for these ancestors using the p_typing parameter. This is an example of previously mentioned backward propagation.
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for graph in hierarchy.graphs():
print("Graph ", graph)
plot_graph(hierarchy.get_graph(graph))
Even the rule r1 was affected as the result of propagation, all its circle nodes were cloned.
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plot_rule(hierarchy.get_rule('r1'))
Let us now consider a small example of forward propagation. We will create a rule that performs some additions and merges of nodes.
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pattern = NXGraph()
pattern.add_nodes_from(["a", "b"])
rule = Rule.from_transform(pattern)
rhs_node = rule.inject_merge_nodes(["a", "b"])
rule.inject_add_node("c")
rule.inject_add_edge("c", rhs_node)
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instance = {
"a": "good_circle",
"b": "bad_circle",
}
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old_position = plot_instance(hierarchy.get_graph("g2"), rule.lhs, instance)
plot_rule(rule)
Application of this rule will merge nodes bad_circle and good_circle in the graph g2. It with then add a new node and connect it with an edge to the merged node. Let us specify some typings of the new node in the RHS: we will set the new node to be typed as circle in the graph shapes.
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rhs_typing = {
"shapes": {
"c": "circle"
}
}
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rhs_instance = hierarchy.rewrite("g2", rule, instance, rhs_typing=rhs_typing)
Observe the following graphs, as the resule of forward propagation nodes good and bad were merged in the graph qualities. In addition, a new node typing the node c in the rule was added to the graph qualities.
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for graph in hierarchy.graphs():
print("Graph ", graph)
plot_graph(hierarchy.get_graph(graph))
Because NetworkX graphs are in-memory objects, they are destroyed as soon as the Python application is no longer running. ReGraph provides some utils for serialization of NXHierarchy objects and implements the following methods for loading and exporting your hierarchy in JSON-format:
NXHierarchy.to_json creates a json representations of the hierarchy;
NXHierarchy.from_json loads an hierarchy from json representation (returns new Hierarchy object);
NXHierarchy.export exports the hierarchy to a file (json format);NXHierarchy.load loads an hierarchy from a .json file (returns new object as well).
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hierarchy_json = hierarchy.to_json()
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import json
print(json.dumps(hierarchy_json, indent=" "))
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new_hierarchy = NXHierarchy.from_json(hierarchy_json)
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new_hierarchy == hierarchy
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