## Contents

function [e1,e2,info] = bter(nd,cd,varargin)

%BTER Generate edges for BTER model.
%
%   [E1,E2] = BTER(ND,CD) generates a list of edges for the BTER model for
%   the given degree distribution and clustering coefficients per degree,
%   i.e., ND(d) = the number of nodes of degree d and CD(d) = mean
%   clustering coefficient for degree d. The edge list E = [E1;E2] is a
%   list of edges created by the BTER model from phases 1 and 2,
%   respectively. The edge list will have duplicates, which can be remedied
%   in post-processing.
%
%   [E1,E2,INFO] = BTER(...) returns extra information.
%
%   [...] = BTER(ND,CD,'param',value,...) takes a sequence of
%   parameter-value pairs to specify the operation of the method
%
%   - 'blowup'      - Multiplier for number of degree-1 nodes. Default: 1.
%   - 'rmloops'     - Remove self-links. Default: true.
%   - 'cleanup'     - Clean-up degree-1 nodes per blowup. Default: true.
%   - 'swap'        - Swap edge endpoints so least is 1st. Default: false.
%   - 'verbose'     - Print information. Default: true.
%   - 'rngseed'     - Seed for the random number generator.
%
%   References:
%   * C. Seshadhri, T. G. Kolda and A. Pinar. Community structure and
%     scale-free collections of Erdös-Rényi graphs, Physical Review E
%     85(5):056109, May 2012. (doi:10.1103/PhysRevE.85.056109)
%   * T. G. Kolda, A. Pinar, T. Plantenga and C. Seshadhri. A Scalable
%     Generative Graph Model with Community Structure,  arXiv:1302.6636,
%     March 2013. (http://arxiv.org/abs/1302.6636)
%
%
% Tamara G. Kolda, Ali Pinar, and others, FEASTPACK v1.1, Sandia National
% Laboratories, SAND2013-4136W, http://www.sandia.gov/~tgkolda/feastpack/,
% January 2014


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Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.

% ** Check inputs
if nargin < 1
error('At least one input is required');
end

if nargout < 1
error('At least one output is required');
end

% ** Process inputs
params = inputParser;
params.parse(varargin{:});

beta = params.Results.blowup;
if beta < 1
error('Must have ''blowup'' >= 1.');
end

swap = params.Results.swap;
rmloops = params.Results.rmloops;
cleanup = params.Results.cleanup;
verbose = params.Results.verbose;
rngseed = params.Results.rngseed;

% Set random number generator
if isempty(rngseed)
rngseed = rng;
end
rng(rngseed);

% Input checking
dmax = length(nd);
if length(cd) ~= dmax
error('Inputs ND and CD must be the same length');
end

if any(nd < 0)
error('Degree distribution cannot be negative');
end
if any(round(nd) ~= nd)
error('Degree distribution must be integral');
end
if any(cd < 0) || any(cd > 1)
error('Clustering coefficients must be between 0 and 1');
end

% ** Setup
tic;
[id,wd,ndfill,rdfill,ig,wg,bg,ng] = bter_setup(nd, cd, beta);
time_setup = toc;

if (verbose)
fprintf('--- BTER HPC Set-up ---\n');
fprintf('Desired number of nodes: %d\n', sum(nd));
fprintf('Desired number of edges: %.0f\n', sum(nd.*(1:dmax)')/2);
fprintf('Multiplier to degree-1 nodes: %g\n', beta);
fprintf('Maximum degree: %d\n', dmax);
fprintf('Number of groups: %d\n', length(ig));
fprintf('Number of blocks: %d\n', sum(bg));
fprintf('Phase 1 total weight: %.0f\n', sum(wg));
fprintf('Phase 2 total weight: %.0f\n', sum(wd));
fprintf('Time for setup (sec): %.2f\n', time_setup);
end

% ** Determine number of samples per phase
tic;

w1 = sum(wg);
w2 = sum(wd);
w = w1+w2;
nsmp = round(w);
r = rand(nsmp,1);
s1 = sum(r < w1/w);
s2 = nsmp - s1;

time_split = toc;

if (verbose)
fprintf('Determined phase for %d edges in %f seconds\n', nsmp, time_split);
end

% ** Phase 1 Samples
tic;

grp_smp = random_sample(wg, s1);
blk_r = rand(s1,1);
blk_b = bg(grp_smp);
blk_i = ig(grp_smp);
blk_n = ng(grp_smp);
e1_shift = blk_i + floor(blk_r .* blk_b) .* blk_n;
e1_r = rand(s1,2);
e1(:,1) = floor(e1_r(:,1) .* blk_n) + e1_shift;
e1(:,2) = floor(e1_r(:,2) .* (blk_n - 1)) + e1_shift;
e1(:,2) = e1(:,2) + (e1(:,2) >= e1(:,1));

time_p1 = toc;

if (verbose)
fprintf('Sampled %d edges for phase 1 in %f seconds\n', s1, time_p1);
end

% ** Phase 2 Samples
tic;

% Setup
idfill = id;
idbulk = id + ndfill;
ndbulk = nd - ndfill;
ndbulk(1) = 0;
% Sample
d_smp = random_sample(wd, 2*s2);
d_smp = reshape(d_smp, s2, 2);
r = rand(s2,2);
tf_fill = r < rdfill(d_smp);
e2_shift_fill = idfill(d_smp);
e2_sz_fill = ndfill(d_smp);
e2_shift_bulk = idbulk(d_smp);
e2_sz_bulk = ndbulk(d_smp);

r = rand(s2,2);
e2_fill = e2_shift_fill + floor(r .* e2_sz_fill);
e2_bulk = e2_shift_bulk + floor(r .* e2_sz_bulk);
e2 = tf_fill .* (e2_fill) + ~tf_fill .* (e2_bulk);

time_p2 = toc;

if (verbose)
fprintf('Sampled %d edges for phase 2 in %f seconds\n', s2, time_p2);
end

% ** Remove loops (Phase 2 only)
if rmloops
tic;
tf = (e2(:,1) == e2(:,2));
e2 = e2(~tf,:);
time_rmloops = toc;
if (verbose)
fprintf('Removed %d loops in %f seconds\n', sum(tf), time_rmloops);
end
end

% ** Reorder edges so lower index is first
if swap
tic;
e1 = bter_swap(e1);
e2 = bter_swap(e2);
time_swap = toc;
if (verbose)
fprintf('Swapped low degree first in %f seconds\n', time_swap);
end
end

% ** Reindexing "degree-1" vertices to only keep those that have a link
if (beta > 1) && (cleanup)
tic;
idx = id(1); % Index of first degree-1 node
dtrue = accumarray(e2(:),1);
tf = ones(size(dtrue)) > 0;
tf(idx:end) = dtrue(idx:end) > 0;
n_old = length(dtrue);
n_new = sum(tf);
old2newidx = zeros(length(dtrue),1);
old2newidx(tf) = (1:n_new)';
e2 = old2newidx(e2);
time_cleanup = toc;
if (verbose)
fprintf('Removed %d spurious degree-1 nodes in %f seconds\n', ...
n_old - n_new, time_cleanup);
end
end

% ** Finishing up
if (verbose)
fprintf('--- BTER HPC Complete ---\n');
end

% Fill in INFO
if nargout >= 3
S = whos;
info = struct;
for i = 1:length(S);
var = S(i).name;
if (var(1) == 'e') || strcmp(var(1:min(3,end)),'blk') || ...
strcmp(var, 'old2newidx') || strcmp(var, 'd_smp') || ...
strcmp(var, 'grp_smp') || strcmp(var, 'tf') || ...
strcmp(var, 'dtrue') || strcmp(var, 'r') || ...
strcmp(var,'tf_fill')
continue;
end
eval(['info.' var ' =  ' var ';']);
end
end

function edges = bter_swap(edges)
%BTER_SWAP Given a list of edges, make lower index first.

tf = edges(:,1) > edges(:,2);
tmp = edges(tf,2);
edges(tf,2) = edges(tf,1);
edges(tf,1) = tmp;

function [id,wd,ndfill,rdfill,ig,wg,bg,ng] = bter_setup(nd, cd, beta)
%BTER_SETUP Creates data for generating BTER graphs.

% Compute maximum degree
dmax = length(nd);

% Set up arrays (max # groups = dmax)
id = zeros(dmax,1); % i_d
wd = zeros(dmax,1); % w_d
rdfill = zeros(dmax,1); % r^{\rm fill}_d
ndfill = zeros(dmax,1); % n^{\rm fill}_d
wg = zeros(dmax,1); % w_g
ig = zeros(dmax,1); % i_g
bg = zeros(dmax,1); % b_g
ng = zeros(dmax,1); % n_g

% Index of first node for each degree.
% Degree 1 vertices are numbered last.
tmp = cumsum(nd(2:end));
id(2) = 1;
id(3:end) = tmp(1:end-1)+1;
id(1) = tmp(end)+1;

% Compute number of nodes with degree greater than d
tmp = flipud(cumsum(flipud(nd)));
ndprime = zeros(dmax, 1);
ndprime(2:end-1) = tmp(3:end);

% Handle degree-1 nodes
ndfill(1) = nd(1) * beta;
wd(1) = 0.5 * nd(1);
rdfill(1) = 1;

% Main loop
g = 0;
nfillblk = 0;
intdeg = 0;
for d = 2:dmax

if nfillblk > 0
ndfill(d) = min( nfillblk, nd(d) );
nfillblk = nfillblk - ndfill(d);
wdfilltmp = 0.5 * ndfill(d) * (d - intdeg);
else
ndfill(d) = 0;
wdfilltmp = 0;
end

ndbulktmp = nd(d) - ndfill(d);

if ndbulktmp > 0
g = g + 1;
ig(g) = id(d) + ndfill(d);
bg(g) = ceil(ndbulktmp / (d+1));
ng(g) = d+1;
if (bg(g) * (d+1)) > (ndprime(d) + ndbulktmp)
if bg(g) ~= 1
error('Last group has more than 1 block');
end
ng(g) = ndprime(d) + ndbulktmp;
end
rho = nthroot(cd(d), 3);
intdeg = (ng(g) - 1) * rho;
wdbulktmp = 0.5 * ndbulktmp * (d - intdeg);
wg(g) = bg(g) * 0.5 * ng(g) * (ng(g) - 1) * log (1/(1-rho));
nfillblk = bg(g) * ng(g) - ndbulktmp;
else
wdbulktmp = 0;
end

wd(d) = wdbulktmp + wdfilltmp;
if (wd(d) > 0)
rdfill(d) = wdfilltmp / wd(d);
else
rdfill(d) = 0;
end
end

% Shorten the group arrays
ig = ig(1:g);
wg = wg(1:g);
bg = bg(1:g);
ng = ng(1:g);