h"+SrSSKrSSKrSSKJr SS/r\"S5\R"SS9S S j55r\"S5\R"SS9S S j55r g) zKatz centrality.N)not_implemented_forkatz_centralitykatz_centrality_numpy multigraphweight) edge_attrsc ^^[U5S:Xa0$UR5nUcUV s0sHoS_M sn mOUm[RU[ U55n [U5Hn Tm[RTS5mTH6n X H+n TU ==TU X U RUS5-- ss'M- M8 THn UTU -X-TU 'M [UU4SjT55nXU-:dMU(a&S[R"TR!56- nOSnTHn TU ==U-ss'M Ts $ [R$"U5es sn f![ [ [4a<n Un [U5[U5:wa[R"S5U eSn A GNMSn A ff=f!["a SnNf=f)uCompute the Katz centrality for the nodes of the graph G. Katz centrality computes the centrality for a node based on the centrality of its neighbors. It is a generalization of the eigenvector centrality. The Katz centrality for node $i$ is .. math:: x_i = \alpha \sum_{j} A_{ij} x_j + \beta, where $A$ is the adjacency matrix of graph G with eigenvalues $\lambda$. The parameter $\beta$ controls the initial centrality and .. math:: \alpha < \frac{1}{\lambda_{\max}}. Katz centrality computes the relative influence of a node within a network by measuring the number of the immediate neighbors (first degree nodes) and also all other nodes in the network that connect to the node under consideration through these immediate neighbors. Extra weight can be provided to immediate neighbors through the parameter $\beta$. Connections made with distant neighbors are, however, penalized by an attenuation factor $\alpha$ which should be strictly less than the inverse largest eigenvalue of the adjacency matrix in order for the Katz centrality to be computed correctly. More information is provided in [1]_. Parameters ---------- G : graph A NetworkX graph. alpha : float, optional (default=0.1) Attenuation factor beta : scalar or dictionary, optional (default=1.0) Weight attributed to the immediate neighborhood. If not a scalar, the dictionary must have a value for every node. max_iter : integer, optional (default=1000) Maximum number of iterations in power method. tol : float, optional (default=1.0e-6) Error tolerance used to check convergence in power method iteration. nstart : dictionary, optional Starting value of Katz iteration for each node. normalized : bool, optional (default=True) If True normalize the resulting values. weight : None or string, optional (default=None) If None, all edge weights are considered equal. Otherwise holds the name of the edge attribute used as weight. In this measure the weight is interpreted as the connection strength. Returns ------- nodes : dictionary Dictionary of nodes with Katz centrality as the value. Raises ------ NetworkXError If the parameter `beta` is not a scalar but lacks a value for at least one node PowerIterationFailedConvergence If the algorithm fails to converge to the specified tolerance within the specified number of iterations of the power iteration method. Examples -------- >>> import math >>> G = nx.path_graph(4) >>> phi = (1 + math.sqrt(5)) / 2.0 # largest eigenvalue of adj matrix >>> centrality = nx.katz_centrality(G, 1 / phi - 0.01) >>> for n, c in sorted(centrality.items()): ... print(f"{n} {c:.2f}") 0 0.37 1 0.60 2 0.60 3 0.37 See Also -------- katz_centrality_numpy eigenvector_centrality eigenvector_centrality_numpy :func:`~networkx.algorithms.link_analysis.pagerank_alg.pagerank` :func:`~networkx.algorithms.link_analysis.hits_alg.hits` Notes ----- Katz centrality was introduced by [2]_. This algorithm it uses the power method to find the eigenvector corresponding to the largest eigenvalue of the adjacency matrix of ``G``. The parameter ``alpha`` should be strictly less than the inverse of largest eigenvalue of the adjacency matrix for the algorithm to converge. You can use ``max(nx.adjacency_spectrum(G))`` to get $\lambda_{\max}$ the largest eigenvalue of the adjacency matrix. The iteration will stop after ``max_iter`` iterations or an error tolerance of ``number_of_nodes(G) * tol`` has been reached. For strongly connected graphs, as $\alpha \to 1/\lambda_{\max}$, and $\beta > 0$, Katz centrality approaches the results for eigenvector centrality. For directed graphs this finds "left" eigenvectors which corresponds to the in-edges in the graph. For out-edges Katz centrality, first reverse the graph with ``G.reverse()``. References ---------- .. [1] Mark E. J. Newman: Networks: An Introduction. Oxford University Press, USA, 2010, p. 720. .. [2] Leo Katz: A New Status Index Derived from Sociometric Index. Psychometrika 18(1):39–43, 1953 https://link.springer.com/content/pdf/10.1007/BF02289026.pdf rN0beta dictionary must have a value for every nodec3L># UHn[TUTU- 5v M g7f)N)abs).0nxxlasts U/var/www/html/env/lib/python3.13/site-packages/networkx/algorithms/centrality/katz.py "katz_centrality..s%4!QC!uQx((!s!$?)lennumber_of_nodesdictfromkeysfloat TypeError ValueErrorAttributeErrorsetnx NetworkXErrorrangegetsummathhypotvaluesZeroDivisionErrorPowerIterationFailedConvergence)Galphabetamax_itertolnstart normalizedrnnodesrberr_nbrerrorsrrs @@rrr sT 1v{    F ~ 1aT1   MM!U4[ )8_ MM% #At#%(QT#Y]]61%===A1Q4 2  t9A ""B  2)As. EE$F2F/31F**F/2 GGcSSKn[U5S:Xa0$UR5n[U5[U5:wa[R "S5eUR [UR55[S9n[R"XUS9R5R n U R"Sn UR$R'UR)X5X-- U5R+5n U(a6UR-[/U 55UR$R1U 5-OSn [3[5XkU - R7555$![ac [U5nUR[U5S45U-nGN![[[4an[R "S5UeSnAff=ff=f)u% Compute the Katz centrality for the graph G. Katz centrality computes the centrality for a node based on the centrality of its neighbors. It is a generalization of the eigenvector centrality. The Katz centrality for node $i$ is .. math:: x_i = \alpha \sum_{j} A_{ij} x_j + \beta, where $A$ is the adjacency matrix of graph G with eigenvalues $\lambda$. The parameter $\beta$ controls the initial centrality and .. math:: \alpha < \frac{1}{\lambda_{\max}}. Katz centrality computes the relative influence of a node within a network by measuring the number of the immediate neighbors (first degree nodes) and also all other nodes in the network that connect to the node under consideration through these immediate neighbors. Extra weight can be provided to immediate neighbors through the parameter $\beta$. Connections made with distant neighbors are, however, penalized by an attenuation factor $\alpha$ which should be strictly less than the inverse largest eigenvalue of the adjacency matrix in order for the Katz centrality to be computed correctly. More information is provided in [1]_. Parameters ---------- G : graph A NetworkX graph alpha : float Attenuation factor beta : scalar or dictionary, optional (default=1.0) Weight attributed to the immediate neighborhood. If not a scalar the dictionary must have an value for every node. normalized : bool If True normalize the resulting values. weight : None or string, optional If None, all edge weights are considered equal. Otherwise holds the name of the edge attribute used as weight. In this measure the weight is interpreted as the connection strength. Returns ------- nodes : dictionary Dictionary of nodes with Katz centrality as the value. Raises ------ NetworkXError If the parameter `beta` is not a scalar but lacks a value for at least one node Examples -------- >>> import math >>> G = nx.path_graph(4) >>> phi = (1 + math.sqrt(5)) / 2.0 # largest eigenvalue of adj matrix >>> centrality = nx.katz_centrality_numpy(G, 1 / phi) >>> for n, c in sorted(centrality.items()): ... print(f"{n} {c:.2f}") 0 0.37 1 0.60 2 0.60 3 0.37 See Also -------- katz_centrality eigenvector_centrality_numpy eigenvector_centrality :func:`~networkx.algorithms.link_analysis.pagerank_alg.pagerank` :func:`~networkx.algorithms.link_analysis.hits_alg.hits` Notes ----- Katz centrality was introduced by [2]_. This algorithm uses a direct linear solver to solve the above equation. The parameter ``alpha`` should be strictly less than the inverse of largest eigenvalue of the adjacency matrix for there to be a solution. You can use ``max(nx.adjacency_spectrum(G))`` to get $\lambda_{\max}$ the largest eigenvalue of the adjacency matrix. For strongly connected graphs, as $\alpha \to 1/\lambda_{\max}$, and $\beta > 0$, Katz centrality approaches the results for eigenvector centrality. For directed graphs this finds "left" eigenvectors which corresponds to the in-edges in the graph. For out-edges Katz centrality, first reverse the graph with ``G.reverse()``. References ---------- .. [1] Mark E. J. Newman: Networks: An Introduction. Oxford University Press, USA, 2010, p. 173. .. [2] Leo Katz: A New Status Index Derived from Sociometric Index. Psychometrika 18(1):39–43, 1953 https://link.springer.com/content/pdf/10.1007/BF02289026.pdf rNr )dtyper zbeta must be a number)nodelistr)numpyrkeysrrr arraylistr&rronesrradjacency_matrixtodenseTshapelinalgsolveeyesqueezesignr#normrziptolist) r)r*r+r/rnpr9r1r2Ar centralityrHs rrrsz` 1v{  E99; x=CF """#UV V HHT$++-(H 6 A@HHJLLA  A!;Q?GGIJEO2773z? #biinnZ&@ @TUD HD088:; << E7 EX*+d2A:~6 E""#:; D E Es*A)EG 4FG+GGG )皙?rigư>NTN)rNrTN) __doc__r$networkxrnetworkx.utilsr__all__ _dispatchablerrrrVs . 5 6\"X&    u7'#u7p\"X&D='#D=rU