How to construct a large graph (~1e5) that has max, min, and avg degree using the logical sparse adj matrix that computes faster. Also, how to ensure that the graph is not broken (there are no subsets of disconnected graphs in the main graph)?

I have the following code for generating a small (~1e3) graph that I can control the degree of (code 1). This has n (number of nodes), min, max and avg degree as the inputs.

clc;clear;
n=5000;
% dl=10; %dl=degree limit
mind=5; 
maxd=100;
avgd=10;
totd=avgd*n;
td=.5*n*(n-1);
a=zeros(n,n);
degct=zeros(1,n);
nd=2:n;
% generating adj
for i=1:n
    t=[];
    tn=mind-degct(i);
    if length(nd)<tn
        t=randperm(n-1,tn);
        
    a(i,t)=1;
    a(t,i)=1;

degct=sum(a);
break;
    end
        
    if tn>0
    t=randperm(length(nd),tn);
    a(i,nd(t))=1;
    a(nd(t),i)=1;

degct=sum(a);
    end
    ind=find(degct==mind);

nd=setdiff(nd,ind);
    if (isempty(nd))
        break;
    end
    
end

%%
% a=a+a';
a1=triu(a);

rd=totd-sum(degct);
at=-1*ones(n,n);

at=tril(at);

a1=a1+at;

indt=find(a1==0);

ind=randperm(length(indt),rd);

a1(indt(ind))=1;
a1=triu(a1);
a1=a1+a1';
a=a1;

%%
for i=1:n-1
    a(i,i+1)=1;
    a(i+1,i)=1;
end
for i=1:n
    a(i,i)=0;
end
%%
if issymmetric(a)
    disp('a is symmetric')
else
    disp('symmetricity error')
end

%%
filemat='file1.mat';
save(filemat,'a');
%%
sa=[];
sa=sum(a);
min(sa)
max(sa)

But then I have figured out a way to generate random graphs using a logical array that takes less time and memory to compute (code 2). This has only nodes and connectivity as the inputs.

function [a,ed] = Random_graph_genar_function_v4(nodes, connectivity)
   nPasses=10;
   a=sparse(false);
   
   for i=1:nPasses
    a=a | logical(sprandsym(nodes,connectivity/nPasses));
   end   
   a(1:nodes+1:end)=0; %zero the diagonal
    
    G=graph(a);
    
    ed=table2array(G.Edges).';
end

The final code need not have connectivity, but it must have n, min, max, and avg degree as the inputs.

I would like to integrate the two. I see that degree(graph) can return the degree and I can discard the nodes that do not have the desired degree. But, I am unsure about how to ensure that the graphs that I construct have the degree between (min, max). Thanks in advance.

Also, it need not be a function, if having a function induces function call lags. I am not sure if calling the function and retrieving the values can introduce the time lag or not.

[Matt J]

Does the result need to be random? Could you just generate a tree graph such that every node has the same degree, so that min_degree=max_degree=avg_degree?

[Jay Vaidya]

That is one of the many graphs I would need. But yes, that is one of my needed graphs.

The graphs need to be random because I would like to see how my algorithm works for different graphs.

[Jay Vaidya]

Please let me know if you have any suggestions. Thanks

[Walter Roberson]

http://complexnetworks.fr/wp-content/uploads/2011/01/random.pdf

[Jay Vaidya]

Thanks Walter. Would you be able to suggest some inputs to this please?https://www.mathworks.com/matlabcentral/answers/622008-simulate-and-sde-taking-really-long-time-how-to-execute-this-parallelly?s_tid=srchtitle