Undirected Graphs¶

class netin.UnDiGraph(n: int, k: int, f_m: float, seed: object | None = None)¶

Undirected graph base model.

Parameters¶

n: int: number of nodes (minimum=2)
k: int: minimum degree of nodes (minimum=1)
f_m: float: fraction of minorities (minimum=1/n, maximum=(n-1)/n)
seed: object: seed for random number generator

Notes¶

The initialization is an undirected graph with n nodes and no edges. Then, everytime a node is selected as source, it gets connected to k target nodes. Target nodes are selected depending on the chosen mechanism of edge formation:

PA: Preferential attachment (in-degree), see netin.PA [BarabasiAlbert1999]
PAH: Preferential attachment (in-degree) with homophily, see netin.PAH [Karimi2018]
PATC: Preferential attachment (in-degree) with triadic closure, see netin.PATC [HolmeKim2002]
PATCH: Preferential attachment (in-degree) with homophily and triadic closure, see netin.PATCH

References¶

[BarabasiAlbert1999] (1,2,3,4)

1. Barabasi and R. Albert “Emergence of scaling in random networks”, Science 286, pp 509-512, 1999.

[Karimi2018] (1,2,3,4,5)

Karimi, M. Génois, C. Wagner, P. Singer, & M. Strohmaier, M “Homophily influences ranking of minorities in social networks”, Scientific reports 8(1), 11077, 2018.

[HolmeKim2002] (1,2,3,4)

Holme and B. J. Kim “Growing scale-free networks with tunable clustering” Phys. Rev. E 2002.

calculate_degree_powerlaw_exponents() → tuple[float, float]¶

Returns the powerlaw exponents for the majority and minority class.

Returns¶

pl_Mfloat: Powerlaw exponent for the majority class
pl_m: float: Powerlaw exponent for the minority class

fit_degree_powerlaw() → tuple[Fit, Fit]¶

Returns the powerlaw fit of the degree distribution to a powerlaw for the majority and minority class.

Returns¶

fit_Mpowerlaw.Fit: Powerlaw fit for the majority class
fit_m: powerlaw.Fit: Powerlaw fit for the minority class

generate()¶

An undirected graph of n nodes is grown by attaching new nodes each with k edges. Each edge is either drawn by preferential attachment, homophily, and/or triadic closure.

For triadic closure, a candidate is chosen uniformly at random from all triad-closing edges (of the new node). Otherwise, or if there are no triads to close, edges are connected via preferential attachment and/or homophily.

Homophily varies ranges from 0 (heterophilic) to 1 (homophilic), where 0.5 is neutral. Similarly, triadic closure varies from 0 (no triadic closure) to 1 (full triadic closure).

PA: An undirected graph with h_mm = h_MM in [0.5, None] and tc = 0 is a BA preferential attachment model.
PAH: An undirected graph with h_mm not in [0.5, None] and h_MM not in [0.5, None] and tc = 0 is a PA model with homophily.
PATC: An undirected graph with h_mm = h_MM in [0.5, None] and tc > 0 is a PA model with triadic closure.
PATCH: An undirected graph with h_mm not in [0.5, None] and h_MM not in [0.5, None] and tc > 0 is a PA model with homophily and triadic closure.

get_expected_number_of_edges() → int¶

Computes and returns the expected number of edges based on minimum degree k and number of nodes n

Returns¶

int: Expected number of edges

get_metadata_as_dict() → dict¶: Returns metadata for a undirected.

get_target(source: int, available_nodes: list[int]) → int¶

Picks a random target node based on preferential attachment.

Parameters¶

source: int: Newly added node
available_nodes: List[int]: Potential (available) target nodes to connect to

Returns¶

int
Target node that an edge should be added to

info_computed()¶: Shows the computed properties of the graph.

info_params()¶: Shows the parameters of the model.

makecopy()¶: Makes a copy of the current object.

validate_parameters()¶: Validates the parameters of the undirected.

class netin.PA(n: int, k: int, f_m: float, seed: object | None = None)¶

Creates a new PA instance. An undirected graph with preferential attachment.

Parameters¶

n: int: number of nodes (minimum=2)
k: int: minimum degree of nodes (minimum=1)
f_m: float: fraction of minorities (minimum=1/n, maximum=(n-1)/n)
seed: object: seed for random number generator

Notes¶

The initialization is an undirected graph with n nodes and no edges. Then, everytime a node is selected as source, it gets connected to k target nodes. Target nodes are selected via preferential attachment (in-degree), see [BarabasiAlbert1999].

static fit(g, n=None, k=None, seed=None)¶

It fits the PA model to the given graph.

Parameters¶

g: netin.UnDiGraph: graph to fit the model to
n: int: number of nodes to override (e.g., to generate a smaller network)
k: int: minimum node degree to override (e.g., to generate a denser network k>1)
seed: object: seed for random number generator

Returns¶

netin.PA: fitted model

get_target_probabilities(source: int, available_nodes: list[int]) → tuple[array, list[int]]¶

Returns the probabilities of the target nodes to be selected given a source node. This probability is proportional to the degree of the target node.

Parameters¶

source: int: source node (id)
available_nodes: set: set of target nodes (ids)
special_targets: object: special available_nodes

Returns¶

probs: np.array: probabilities of the target nodes to be selected
available_nodes: set: set of target nodes (ids)

makecopy()¶: Makes a copy of the current object.

class netin.PATC(n: int, k: int, f_m: float, tc: float, seed: object | None = None)¶

Creates a new PATC instance. An undirected graph with preferential attachment and triadic closure.

Parameters¶

n: int: number of nodes (minimum=2)
k: int: minimum degree of nodes (minimum=1)
f_m: float: fraction of minorities (minimum=1/n, maximum=(n-1)/n)
tc: float: probability of a new edge to close a triad (minimum=0, maximum=1.)
seed: object: seed for random number generator

Notes¶

The initialization is an undirected graph with n nodes and no edges. Then, everytime a node is selected as source, it gets connected to k target nodes. Target nodes are selected via preferential attachment in-degree [BarabasiAlbert1999], or triadic closure tc [HolmeKim2002].

static fit(g, n=None, k=None, seed=None)¶

It fits the PATC model to the given graph.

Parameters¶

g: netin.UnDiGraph: graph to fit the model to
n: int: number of nodes to override (e.g., to generate a smaller network)
k: int: minimum node degree to override (e.g., to generate a denser network k>1)
seed: object: seed for random number generator

Returns¶

netin.PATC: fitted model

get_metadata_as_dict() → Dict[str, Any]¶

Returns the metadata information (input parameters of the model) of the graph as a dictionary.

Returns¶

dict: Dictionary with the graph’s metadata

get_target_probabilities_regular(source: int, target_list: List[int]) → Tuple[array, List[int]]¶

Returns the probabilities of selecting a target node from a set of nodes based on the preferential attachment.

Parameters¶

source: int: source node
target_list: set[int]: set of target nodes
special_targets: object: special available_nodes

Returns¶

Tuple[np.array, set[int]]: probabilities of selecting a target node from a set of nodes, and the set of target nodes

Parameters¶

n: int: number of nodes (minimum=2)
k: int: minimum degree of nodes (minimum=1)
f_m: float: fraction of minorities (minimum=1/n, maximum=(n-1)/n)
h_MM: float: homophily (similarity) between majority nodes (minimum=0, maximum=1.)
h_mm: float: homophily (similarity) between minority nodes (minimum=0, maximum=1.)
seed: object: seed for random number generator

Notes¶

The initialization is an undirected graph with n nodes, where f_m are the minority. Then, everytime a node is selected as source, it gets connected to k target nodes. Target nodes are selected via preferential attachment (in-degree) and homophily (h_**). This model is based on [Karimi2018] known as the “Barabasi model with homophily” or “BA Homophily”.

static fit(g, n=None, k=None, seed=None)¶

It fits the PAH model to the given graph.

Parameters¶

g: netin.UnDiGraph: graph to fit the model to
n: int: number of nodes to override (e.g., to generate a smaller network)
k: int: minimum node degree to override (e.g., to generate a denser network k>1)
seed: object: seed for random number generator

Returns¶

netin.PAH: fitted model

get_metadata_as_dict() → dict¶

Returns the metadata (parameters) of the model as a dictionary.

Returns¶

dict: metadata of the model

get_target_probabilities(source: int, available_nodes: list[int]) → tuple[array, list[int]]¶

Returns the probabilities of selecting a target node from a set of nodes based on the preferential attachment and homophily.

Parameters¶

source: int: source node
available_nodes: set[int]: set of target nodes
special_targets: object: special available_nodes

Returns¶

tuple[np.array, set[int]]: probabilities of selecting a target node from a set of nodes, and the set of target nodes

infer_homophily_values() → tuple[float, float]¶

Infers the level of homophily using the analytical solution of the model.

Returns¶

tuple[float, float]: homophily between majority nodes, and homophily between minority nodes

Notes¶

See derivations in [Karimi2018].

info_computed()¶: Shows the computed properties of the graph.

info_params()¶: Shows the parameters of the model.

initialize(class_attribute: str = 'm', class_values: list | None = None, class_labels: list | None = None)¶

Initializes the model.

Parameters¶

class_attribute: str: name of the attribute that represents the class
class_values: list: values of the class attribute
class_labels: list: labels of the class attribute mapping the class_values.

makecopy()¶: Makes a copy of the current object.

validate_parameters()¶: Validates the parameters of the undirected.

class netin.PATCH(n: int, k: int, f_m: float, h_mm: float, h_MM: float, tc: float, tc_uniform: bool = True, seed: object | None = None)¶

Creates a new PATCH instance. An undirected graph with preferential attachment, homophily, and triadic closure.

Parameters¶

n: int: number of nodes (minimum=2)
k: int: minimum degree of nodes (minimum=1)
f_m: float: fraction of minorities (minimum=1/n, maximum=(n-1)/n)
h_MM: float: homophily (similarity) between majority nodes (minimum=0, maximum=1.)
h_mm: float: homophily (similarity) between minority nodes (minimum=0, maximum=1.)
tc: float: probability of a new edge to close a triad (minimum=0, maximum=1.)
tc_uniform: bool: specifies whether the triadic closure target is chosen uniform at random or if it follows the regular link formation mechanisms (e.g., homophily) (default=True)

Notes¶

The initialization is an undirected graph with n nodes and no edges. Then, everytime a node is selected as source, it gets connected to k target nodes. Target nodes are selected via preferential attachment (in-degree) [BarabasiAlbert1999] and homophily (h_**; see netin.Homophily) [Karimi2018] with probability 1-p_{TC}, and with probability p_{TC} via triadic closure (see netin.GraphTC) [HolmeKim2002].

Note that this model is still work in progress and not fully implemented yet.

get_metadata_as_dict() → Dict[str, Any]¶

Returns a dictionary with the metadata of the PATCH graph.

Returns¶

dict: the graph metadata as a dictionary

get_target_probabilities(source: int, available_nodes: List[int]) → Tuple[Any, List[int]]¶

Returns the probabilities of nodes to be selected as target nodes.

Parameters¶

source: int: source node id
available_nodes: List[int]: list of available nodes to connect to

Returns¶

Tuple[np.ndarray, List[int]]: probabilities of nodes to be selected as target nodes, and list of target of nodes

get_target_probabilities_regular(source: int, target_list: List[int]) → Tuple[ndarray, List[int]]¶

Returns the probability of nodes to be selected as target nodes using the preferential attachment with homophily mechanism.

Parameters¶

source: int: source node id
available_nodes: set: set of target node ids
special_targets: dict: dictionary of special target node ids to be considered

Returns¶

tuple: probabilities of nodes to be selected as target nodes, and set of target of nodes

infer_homophily_values() → tuple[float, float]¶

Infers analytically the homophily values of the graph.

Returns¶

tuple: homophily values of the graph (majority, minority)

infer_triadic_closure() → float¶

Infers analytically the triadic closure value of the graph.

Returns¶

float: triadic closure probability of the graph

info_computed()¶: Shows the (computed) properties of the graph.

info_params()¶: Shows the (input) parameters of the graph.

makecopy()¶: Makes a copy of the current object.

validate_parameters()¶: Validates the parameters of the undirected.

class netin.TCH(n: int, k: int, f_m: float, h_mm: float, h_MM: float, tc: float, tc_uniform: bool = True, seed: object | None = None)¶

Creates a new TCH graph. An undirected graph with homophily and triadic closure as link formation mechanisms.

Parameters¶

n: int: number of nodes (minimum=2)
k: int: minimum degree of nodes (minimum=1)
f_m: float: fraction of minorities (minimum=1/n, maximum=(n-1)/n)
h_MM: float: homophily (similarity) between majority nodes (minimum=0, maximum=1.)
h_mm: float: homophily (similarity) between minority nodes (minimum=0, maximum=1.)
tc: float: probability of a new edge to close a triad (minimum=0, maximum=1.)
tc_uniform: bool: specifies whether the triadic closure target is chosen uniform at random or if it follows the regular link formation mechanisms (e.g., homophily) (default=True)

Notes¶

The initialization is an undirected graph with n nodes and no edges. Then, everytime a node is selected as source, it gets connected to k target nodes. Target nodes are selected via homophily (h_**; see netin.Homophily) [Karimi2018] with probability 1-p_{TC}, and with probability p_{TC} via triadic closure (see netin.TriadicClosure) [HolmeKim2002].

Note that this model is still work in progress and not fully implemented yet.

get_target_probabilities_regular(source: int, target_list: list[int]) → tuple[ndarray, list[int]]¶

Returns the probability of nodes to be selected as target nodes using the homophily mechanism.

Parameters¶

source: int: source node id
target_list: set: set of target node ids
special_targets: dict: dictionary of special target node ids to be considered

Returns¶

tuple: probabilities of nodes to be selected as target nodes, and set of target of nodes

infer_triadic_closure() → float¶: Infers analytically the triadic closure value of the graph. @TODO: This still needs to be implemented. Returns ——- float

triadic closure probability of the graph

info_computed()¶: Shows the (computed) properties of the graph.

info_params()¶: Shows the (input) parameters of the graph.

makecopy()¶: Makes a copy of the current object.

Undirected Graphs¶

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