On the Detection of Significant Pairwise Interactions in Complex Systems

Abstract

Many systems in nature, society and technology are complex systems, i.e., they are composed of numerous parts that interact in a non-linear way giving rise to positive and negative feedback. The dynamic organization of these systems often allows the emergence of intermediate structures that once formed profoundly influence the system and therefore play a key role in understanding its behavior. In the recent past our group has devised an effective method for identifying groups of interacting variables within a system, based on their observation. The result is a set of entities, each of which connects two or more nodes of the system: this result can therefore be represented by a hypergraph, which can be of considerable use for understanding the system under consideration. In particular, we use an index that allows us to evaluate the level of integration of a group of variables. In order for a group to be identified as significant, the value of this index must exceed a threshold that corresponds (under appropriate hypotheses) to a level of statistical significance decided by the user. In this work we propose a more elaborate approach to determining the significance threshold, which is (i) in itself theoretically interesting and (ii) of considerable practical utility. We use the new approach to determine collections of pairwise relationships in meaningful cases, such as relationships in gene regulatory networks.

Appendix A

The stochastic model presented in the paper can also be used to calculate the threshold in the case of three-level variables. To this aim, let us consider the ternary variables A and B and suppose that A takes value “0” with probability P_a0, “1” with probability P_a1 and “2” with probability P_a2, then B either copies A with probability P_c or takes value “0” with probability P_b0, “1” with probability P_b1 and “2” with probability P_b2.

In this case it is possible to directly calculate the entropy of the pair (A, B) as:

$$H\left(AB\right)=-\left({P}_{00}ln\left({P}_{00}\right)+{P}_{01}ln\left({P}_{01}\right)+{P}_{02}ln\left({P}_{02}\right)+{P}_{10}ln\left({P}_{10}\right)+{P}_{11}ln\left({P}_{11}\right)+{P}_{12}ln\left({P}_{12}\right)+{P}_{20}ln\left({P}_{20}\right)+{P}_{21}ln\left({P}_{21}\right)+{P}_{22}ln\left({P}_{22}\right)\right)$$

where:

$$ \left\{ {\begin{array}{*{20}l} {P_{00} = P_{a0} P_{c} + P_{a0} \left( {1 - P_{c} } \right)P_{b0} } \hfill \\ {\;\;\;\;\;P_{01} = P_{a0} \left( {1 - P_{c} } \right)P_{b1} } \hfill \\ {\;\;\;\;\;P_{02} = P_{a0} \left( {1 - P_{c} } \right)P_{b2} } \hfill \\ {\;\;\;\;\;P_{10} = P_{a1} \left( {1 - P_{c} } \right)P_{b0} } \hfill \\ {P_{11} = P_{a1} P_{c} + P_{a1} \left( {1 - P_{c} } \right)P_{b1} } \hfill \\ {\;\;\;\;\;P_{12} = P_{a1} \left( {1 - P_{c} } \right)P_{b2} } \hfill \\ {\;\;\;\;\;P_{20} = P_{a2} \left( {1 - P_{c} } \right)P_{b0} } \hfill \\ {\;\;\;\;\;P_{21} = P_{a2} \left( {1 - P_{c} } \right)P_{b1} } \hfill \\ {P_{22} = P_{a2} P_{c} + P_{a2} \left( {1 - P_{c} } \right)P_{b2} } \hfill \\ \end{array} } \right. $$

As before, we can calculate H(A) and H(B) from the marginal probabilities. It is then possible to directly calculate the integration I(S_AB(P_a0, P_a1, P_a2, P_b0, P_b1, P_b2, P_c)) of the pair (A, B) and therefore the value of the zI index in case of n observations.

This leads us to the new threshold:

$${\theta }_{zI}\left({P}_{a0},{P}_{a1},{P}_{a2},{P}_{b0},{P}_{b1},{P}_{b2}{,P}_{c}\right)=max\left(3.0,\frac{2nI\left({S}_{AB}\left({P}_{a0},{P}_{a1},{P}_{a2},{P}_{b0},{P}_{b1},{P}_{b2}{,P}_{c}\right)\right)-{d}_{k}}{\sqrt{2{d}_{k}}}\right)$$

where n is the number of observations, and the probabilities P_aj and P_bj(j = 0,1,2) can be estimated from data (thus making the threshold specific for the case under examination). Notice again that the threshold cannot be less than the normal value of 3.0, to avoid confusing noise with signal.