Information Technology Reference
In-Depth Information
eigenvalue unity for equilibrium. That means that each of these eigenvalues of C
reflect the decreasing exponential rates of decrease of the associated eigenvector as
the system approaches equilibrium as
approaches infinity in M= e
λ
C
. This insight
allows us to see that all of the Renyi entropy values are increasing as the system
approaches equilibrium, which is normally the state of all nodes having the same
value of this hypothetical probability. The use here of this 'artificial flow of
probability under M' provides us with more than just a method of encapsulating the
topology with generalized entropy values, it also gives an intuitive model for the
eigenvectors and eigenvalues for C and sheds light on the graph isomerism problem
(different topologies having the same eigenvalue spectra).
λ
6 Conclusion. Potential Applications to Large Internet Networks
Based upon the arguments above, we suggest that for real networks such as the
internet, that the appropriate connection matrix be formed, from source and
destination information transfers, where both asymmetry and levels of connection are
to be maintained in the C(t) matrix values during that window of time about that time
instant. Specifically, this means that if a connection is made multiple times in that
time interval, then that C element should reflect the appropriate weight of
connectivity as this adds substantial value to the entropy functions. We then suggest
that at each instant, the column and row entropy spectra be computed along with the
total row and column entropy and entropy of entropies and that this be done for lower
order Renyi entropies as well as lower order values in the expansion of the Markov
parameter
that includes higher order connectivity of the topology. We are currently
performing tests to see how effective these entropy metrics are in detecting abnormal
changes in topologies that could be associated with attacks, intrusions, malicious
processes, and system failures. We are performing these experiments on both
mathematical simulations of networks with changing topologies in know ways, and
also on real network data both in raw forms and in forms simulated from raw data.
The objective is to see if these metrics can be useful in the practical sense of
monitoring sections of the internet and other computer networks. In addition to the
two values of total entropy and entropy of entropy that summarize the column (or
row) spectral distribution, we are looking at other natural expansions of this function
in terms of functions or orthogonal polynomials that summarize the general behavior
in useful ways thus providing other summary metric variables for the entropy spectra.
λ
Acknowledgements
The author benefited from extensive collaborations and conversations with Dr.
Vladimir Gudkov.
References
1.
Renyi, A.: Probability Theory, North-Holland Series in Applied Mathematics and
Mechanics, North-Holland Pub. Co (1970) 670 pages.
2.
Renyi, A.: Selected Papers of Alfred Renyi, Akademia Kiado, Buadapest, Vol. 2 of 3
volumes (1976)
