`nwspectral`

Graph Laplacian spectral analysis

Syntax

nwspectral
[netname]
[,
generate(newvarname)
bipartition
measure(string)
replace
silent]


`generate(newvarname)`	Required. Name of the Stata variable that stores each node’s own Fiedler-vector entry
`bipartition`	Also generate a two-way spectral partition (_fiedlersign, or generate()sign) from the sign of the Fiedler vector
`measure(binary\\|valued)`	Whether to use tie values (valued) or only presence/absence of ties (binary); default = valued for valued networks, binary otherwise
`replace`	Replace existing variable(s)
`silent`	Suppress display of results

Description

nwspectral computes the graph Laplacian L = D - W (D the diagonal weighted-degree matrix, W the adjacency/weight matrix) of a single network and its eigendecomposition - the standard starting point for spectral graph analysis. Always computed on the undirected, symmetrized network (no symmetrize option needed - a directed network is symmetrized automatically, the same convention nwcommunity/nwkcomponents already use, since the classical Laplacian spectrum results below assume a symmetric matrix).

Three classical results are reported directly:

**Connected components** — The MULTIPLICITY of eigenvalue 0 in the Laplacian spectrum exactly equals the number of connected components - r(components) counts eigenvalues within 1e-8 of 0, cross-checkable directly against nwcomponents.
**Algebraic connectivity** — The second-smallest eigenvalue (the “Fiedler value”, r(algebraic_connectivity)) - 0 for a disconnected network (matching the component-count result above), and otherwise a genuine measure of how well-connected the network is overall: larger values indicate a more robustly connected structure, harder to disconnect by removing few edges.
**Spectral bipartition** — The eigenvector belonging to the Fiedler value (the “Fiedler vector”) - stored per node via generate(newvarname) (required) - is a classical continuous relaxation of graph bisection: nodes with similar Fiedler-vector values tend to be well-connected to each other. bipartition additionally generates a discrete two-way split from its sign (_fiedlersign, or generate()sign when generate() is given).

For a network with more than one connected component, the Fiedler value is 0 and its own eigenvector is not uniquely defined (any vector constant on each component, summing to zero overall, is an equally valid choice) - nwspectral still reports whatever eigenvector the underlying decomposition happens to return in that case, but bipartition’s resulting split should not be interpreted as meaningful when r(algebraic_connectivity) is (near) 0; use nwcomponents directly instead for a disconnected network’s own true partition.

Examples

. nwwebuse florentine, nwclear
. nwspectral flomarriage, generate(_fiedler)
. nwspectral flomarriage, bipartition generate(_fiedler) replace

Supported network types

Binary: yes. Directed: yes, automatically symmetrized (the classical Laplacian spectrum results this command reports assume a symmetric matrix, the same reasoning nwcommunity/ nwkcomponents already apply). Weighted: yes, via measure(valued) (default for a valued network) - the Laplacian is built from tie weights directly rather than dichotomized presence/absence. Signed: not checked - a Laplacian built from mixed-sign weights is not guaranteed positive semi-definite, so results are not meaningful for a signed network. Two-mode: not checked - operates on the network’s own square adjacency matrix. Only a single network is accepted (unlike most other commands in this package, which accept a full netlist) - each network’s own Fiedler vector is a full per-node eigenvector, not a simple per-network scalar or per-node aggregate that stacks cleanly across multiple networks the way nwcomponents’s component id does.

Stored results

Scalars

r(algebraic_connectivity) second-smallest Laplacian eigenvalue (the Fiedler value)
r(components) number of Laplacian eigenvalues within 1e-8 of 0

Matrices

r(eigenvalues) all Laplacian eigenvalues, sorted ascending

References

Fiedler, M. (1973). Algebraic connectivity of graphs. Czechoslovak Mathematical Journal 23(2), 298-305.

von Luxburg, U. (2007). A tutorial on spectral clustering. Statistics and Computing 17(4), 395-416.

nwspectral