Package {locdiff}

Type: Package
Title: Local Diffusion Exponent
Version: 1.4
Maintainer: Chun-Xiao Nie <niechunxiao2009@163.com>
Description: The local diffusion exponent for undirected, unweighted networks characterizes the speed of local diffusion from a specified node and depends on the resistance distance.
License: GPL-3
Encoding: UTF-8
Depends: igraph, MASS
RoxygenNote: 7.3.2
NeedsCompilation: no
Packaged: 2026-06-20 10:31:45 UTC; niech
Author: Chun-Xiao Nie ORCID iD [aut, cre]
Repository: CRAN
Date/Publication: 2026-06-24 09:50:07 UTC

Compute local diffusion exponents with adaptive fitting range

Description

For each node, the mean squared resistance displacement is computed exactly via transition matrix powers. Then, for a fixed starting time t_min, we try all possible end times t_max from t_max_min to t_max_max, perform a log-log linear regression, and select the t_max that maximizes the coefficient of determination (R^2). The corresponding H_i is returned.

Usage

adaptivelocaldiffexp(adj, max_time, t_min, t_max_min, t_max_max)

Arguments

adj

Symmetric adjacency matrix (0/1, no self-loops) of a connected graph.

max_time

Maximum number of steps (t = 1, ..., max_time) for MSD calculation.

t_min

Fixed starting time for fitting (>= 1).

t_max_min

Minimum end time to try (must be >= t_min + 2 to have at least 3 points).

t_max_max

Maximum end time to try (must be <= max_time).

Value

A list with components:

H

Numeric vector of length n, estimated local diffusion exponents.

best_R2

Numeric vector of length n, the best R^2 achieved.

best_tmax

Integer vector of length n, the t_max that gave the best R^2.

msd

Matrix of size max_time x n, where rows are times 1:max_time and columns are nodes.

fit_objects

List of lm objects for the best fit (one per node).

References

Nie, Chun-Xiao. "Vertex‑level diffusion scaling from resistance distance." .Chaos, Under Review.

Examples

# Path graph with 4 nodes
adj <- matrix(c(0,1,0,0,
                1,0,1,0,
                0,1,0,1,
                0,0,1,0), nrow=4, byrow=TRUE)
res <- adaptivelocaldiffexp(adj, max_time = 50,
                                           t_min = 10, t_max_min = 20, t_max_max = 40)
print(res$H)

Compute local diffusion exponents for all nodes using exact analytical method

Description

For each node i, the mean squared resistance displacement is computed exactly via the transition matrix powers: MSD_i(t), for t = 1, 2, ..., max_time. Then a log-log linear regression is performed over the specified time window to estimate the local diffusion exponent H_i.

Usage

exactlocaldiffexp(adj, max_time, fit_range)

Arguments

adj

Symmetric adjacency matrix (0/1, no self-loops) of a connected graph.

max_time

Maximum number of steps (starting from t=1).

fit_range

Numeric vector of length 2, e.g. c(t_min, t_max), specifying the time window for fitting (1 <= t_min < t_max <= max_time).

Value

A list with components:

H

Numeric vector of length n, the estimated local diffusion exponents.

msd

Matrix of size max_time x n, where rows correspond to times 1:max_time and columns correspond to nodes. Entry [t, i] = MSD_i(t).

fit

List of lm objects (one per node), or NULL if fit failed.

References

Nie, Chun-Xiao. "Vertex‑level diffusion scaling from resistance distance." .Chaos, Under Review.

Examples

# Example: path graph with 4 nodes
adj <- matrix(c(0,1,0,0,
                1,0,1,0,
                0,1,0,1,
                0,0,1,0), nrow=4, byrow=TRUE)
res <- exactlocaldiffexp(adj, max_time = 50, fit_range = c(10, 40))
print(res$H)

Generalized Sierpiński graph

Description

Constructs the generalized Sierpiński graph S(n, G) as defined in "Generalized Sierpiński graphs" (Gravier et al.). The vertex set is \{1,\dots,k\}^n where k = |V(G)|, and edges are defined recursively based on the edges of G.

Usage

generalized_sierpinski(G, n)

Arguments

G

An igraph object representing the base graph. Its vertices must be labelled 1, 2, ..., k (as characters or integers).

n

Integer, dimension (n \ge 1). n = 1 returns G.

Value

An igraph object representing S(n, G).

References

Gravier, Sylvain, Matjaž Kovše, and Aline Parreau. "Generalized Sierpiński graphs." Posters at EuroComb 11 (2011): 2016-0216.

Examples

library(igraph)

# Example 1: Base graph is a 4-cycle
G <- make_ring(4)
S2 <- generalized_sierpinski(G, 2)
plot(S2, vertex.size = 8, vertex.label = NA)

# Example 2: Base graph is a triangle (classical Sierpiński gasket)
G_tri <- make_graph(~ 1-2-3-1)
S3_tri <- generalized_sierpinski(G_tri, 3)
summary(S3_tri)  # vertices: 27, edges: 63

# Example 3: Base graph is a complete graph K4
G_K4 <- make_full_graph(4)
S2_K4 <- generalized_sierpinski(G_K4, 2)
summary(S2_K4)  # vertices: 16, edges: 96

Compute the local diffusion exponent for a given starting node

Description

Simulates random walks from a specified node on a graph and estimates the local diffusion exponent from the scaling of the mean squared resistance distance.

Usage

localdiffexp(
  R,
  start_node,
  adj,
  fit_range,
  num_paths = 100,
  max_time = 100,
  seed = NULL
)

Arguments

R

Resistance distance matrix (symmetric, zero diagonal).

start_node

Index or name of the starting node (must match row/column names of R).

adj

Adjacency matrix (symmetric, 0/1, zero diagonal).

fit_range

Numeric vector of length 2 specifying the time range for fitting, e.g. c(t_start, t_end) with 1 <= t_start < t_end <= max_time.

num_paths

Number of random walk paths to simulate. Default 100.

max_time

Maximum number of steps (from 0 to max_time). Default 100.

seed

Optional random seed for reproducibility.

Value

A list containing:

H

Estimated local diffusion exponent \(H_i\).

msd

Data frame with columns t (time) and msd (mean squared resistance distance).

fit

lm object from the log-log fit.

paths

Matrix of resistance distances for each path (rows = time steps, columns = paths).

References

Nie, Chun-Xiao. "Vertex‑level diffusion scaling from resistance distance." .Chaos, Under Review.

Examples

g<-make_lattice(length = 6, dim = 2)
adj <- as_adjacency_matrix(g,sparse = FALSE)
R <- resistance_distance_matrix(adj)
res <- localdiffexp(R, start_node = 1, adj = adj,fit_range = c(3, 30), max_time = 30)
print(res$H)

Compute Rényi's index for a wealth vector

Description

Compute Rényi's index for a wealth vector

Usage

renyiindex(w, alpha)

Arguments

w

Numeric vector of wealth values (non-negative, at least one positive).

alpha

Positive parameter controlling the sensitivity of the index.

Value

The Rényi index, a number between 0 and 1.

References

Eliazar, Iddo. "Randomness, evenness, and Rényi’s index." Physica A: Statistical Mechanics and its Applications 390.11 (2011): 1982-1990.

Examples

renyiindex(runif(100,0.1,1), 2)

Compute the resistance distance matrix of a connected undirected graph

Description

For a connected undirected graph, the resistance distance between two vertices is defined as the effective resistance when each edge is replaced by a unit resistor. It can be computed from the Moore–Penrose pseudoinverse of the graph Laplacian.

Usage

resistance_distance_matrix(adj_matrix)

Arguments

adj_matrix

A symmetric adjacency matrix (0/1, no self-loops) representing a connected undirected graph.

Value

A symmetric matrix R where R[i,j] is the resistance distance between vertices i and j. Diagonal entries are 0.

References

D.J.Klein and M.Randic, “Resistance distance,” Journal of Mathematical." Chemistry 12, 81–95 (1993).

Examples

# Example: a path graph with 3 vertices
adj <- matrix(c(0,1,0,
                1,0,1,
                0,1,0), nrow = 3, byrow = TRUE)
R <- resistance_distance_matrix(adj)
print(R)