matlab - Determine whether a network (graph) is connected based on its matrix -

how can determine given m x n matrix desribing two-mode network whether connected or not?

of course once can create data structure finding out – there mathematical solution?

i found out n x n matrices (one-mode network) easy: solution be: m * m , looking @ diagonal in resulting matrix. if there 0 on diagonal not connected. isn't true?

what suggestion both problems?

it not simple one-mode networks, you're close.

the network matrix m tells node connected other node in 1 step. m * m tells node connected other node in exactly two steps, m ^ 3 whether in exactly three steps, , on. "exactly" important, because if nodes not connected (zero diagonal elements), m * m not obtain two-step connectivity, loses one-step connectivity.

the products (powers) therefore more useful if nodes connected themselves:

a = (m + eye(size(m)) > 0) 

this step converts weighted matrix pure adjacency matrix. now

(a ^ > 0) 

gives information whether 2 nodes connected in i steps or less. in network n = size(m, 1) nodes, distance between 2 nodes can @ n - 1 steps. therefore

c = (a ^ (n - 1) > 0) 

gives information whether 2 nodes connected at all. whole network connected if there no pairs of unconnected nodes, i.e. if

connected = (sum(c(:)) == n ^ 2) 

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i'm not sure how 1 define connectedness in two-mode network. simple approach disregard difference between 2 types of nodes, , consider them part of one-mode network. if m original two-mode network matrix of size [m, n] = size(m),

m = [zeros(m, m) , m ; m' ; zeros(n, n)]; 

converts matrix describing corresponding one-mode network of size (m+n)x(m+n).

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these matrix powers not tell whether whole network connected, can used find connected sub-networks (network clusters) if not. since behavior of powers of matrix closely connected spectral decomposition (a.k.a. eigendecomposition), idea leads approach of spectral clustering. application weighted synchronization networks, see e.g. arxiv:0706.3375.