The Residual Connectedness Reliability (RCR) problem is as follows. We are given a graph G = (V, E) known as the base graph which models some complex system. To every vertex v we assign a binary random variable known as the state. For notational simplicity we assume that these states are independent and identically distributed. If the random variable takes value 1 then the part of the system represented by v is functioning correctly. If it takes value 0 then that part of the system has failed. We refer to these states as being up and down, respectively.

The system as a whole is considered to be functioning (or in the up state) if the set of up vertices induces a connected subgraph of G. The RCR problem is to compute the probability that the system as a whole is in the up state. This problem is #P complete, which means we need to turn to approximation methods such as Monte Carlo Methods.

This repository contains:

• Code for Monte Carlo methods for the RCR problem
• Visualisation code to help apply the methods
• Code related to exact computation for small graphs, and
• Utility code used to implement the previous three.

This repository contains implementations of the following methods under the monteCarloMethods/ folder:

• The crude Monte Carlo method (monteCarloMethods/crudeMC.cpp).
• Permutation Monte Carlo (monteCarloMethods/PMC.cpp). This computes RCR indirectly using the spectra.
• Stochastic Enumeration (monteCarloMethods/stochasticEnumeration*.cpp). This computes RCR indirectly using the spectra.
• The Recursive Variance Reduction technique (monteCarloMethods/recursiveVarianceReduction.cpp). For details see

Cancela H, Urquhart M (2002) Adapting RVR simulation techniques for residual connectedness
network reliability models. IEEE Transactions on Computers 51(4):439–443

• A conditional Monte Carlo method (monteCarloMethods/conditionalMC.cpp). For details see

Shah R, Hirsch C, Kroese DP, Schmidt V (2014) Rare event probability estimation for connectivity of large random graphs. In: Tolks A, Diallo SD, Ryzhov IO, Yilmaz L, Buckley S,
Miller JA (eds) Proceedings of the 2014 Winter Simulation Conference, Institute of Electrical and Electronics Engineers, Inc, Piscataway, New Jersey

• A basic splitting implementation (monteCarloMethods/splittingBasic.cpp).
• A splitting implementation that decomposes by biconnected components in the laste step (monteCarloMethods/usingBiconnectedSplitting.cpp).
• Sequential Monte Carlo methods using articulation vertices, (monteCarloMethods/articulationConditionangResampling.cpp and monteCarloMethods/articulationConditioningSplitting.cpp).

This repository contains code for:

• Determining the spectra exactly by brute-force, under folder exhaustiveSearch/.
• Using the spectra to compute the RCR exactly, under folder exhaustiveProbability/.
• Determining the expected number of vertices conditional on observing a connected graph, under folder expectedVertexCount/.
• Determining the minimum reliability using Mathematica and the exact spectra, under folder minimumReliability/.
• Computing the spectra of regular grid graphs exactly using the transfer matrix method, under folder transferMatrix/. For further details see

Klein DJ, Hite GE, Schmalz TG (1986) Transfer-matrix method for subgraph enumeration:
Applications to polypyrene fusenes. Journal of Computational Chemistry 7(4):443–456

• Folders gridCountSpecificSize/ and gridCountSpecificSize2/ contain code that count the number of connected subgraphs of a square grid-graph, which have a specific number of vertices. This code was used to check the transfer matrix method code.

• TODO

• Folder residualConnectivityCommon/ contains utility code used throughout.
• Folder transferMatrixMethodCommon/ contains utilitiy code used for the transfer matrix method computations.
• Folder RPackage contains an R package with bindings for some of the code listed, allowing it to be called from within R.
• Folder matlabPackage/ contains a matlab package.

The software is mostly organised as a collection of static libraries and executables. Static libraries include the utility code (residualConnectivityCommon/, transferMatrixCommon/) and implementations of the methods (monteCarloMethods/). These are then linked into command-line executables (E.g. executables/crudeMC, and the R and matlab packages).

You can build the software using CMake on Windows and Linux (Mac is untested, but should work). Required dependencies are:

• Cmake 3.*
• The Boost C++ libraries.
• The Eigen linear algebra libraries.
• The GNU MPFR and MPIR libraries.

Optional dependencies are:

• Mathematica for the code in getMinimumReliability/.
• Matlab for the code in matlabPackage.
• R for the code in RPackage/.
• Igraph and QT 5 for the visualisation code.

Building the Matlab package is controlled by the variable BUILD_MATLAB_PACKAGE (default: ON), building the R package is controlled by the variable BUILD_R_PACKAGE (default: ON), and building the visualisation code is controlled by the variable BUILD_VISUALISATION (default: ON). The minimal cmake command I have been using is:

cmake .. -DCMAKE_BUILD_TYPE=Release -DBOOST_ROOT=/home/uqrshah/Software/boost_1_58_0
-DEIGEN_INCLUDE_DIR=/home/uqrshah/Software/Eigen3.2.5 -DBUILD_VISUALISATION=OFF
-DRcpp_DIR=/home/uqrshah/Software/Rcpp/release
-DBOOST_LIBRARYDIR=/home/uqrshah/Software/boost_1_58_0/stage/lib-gcc5.2/

The only compiler that's likely to work on Windows is Visual Studio. To compile the R Package you will need the custom version of Rcpp available at rohan-shah/Rcpp.