Hidden Connectivity in Networks with Vulnerable Classes of Nodes

Abstract

In many complex systems representable as networks, nodes can be separated into different classes. Often these classes can be linked to a mutually shared vulnerability. Shared vulnerabilities may be due to a shared eavesdropper or correlated failures. In this paper, we show the impact of shared vulnerabilities on robust connectivity and how the heterogeneity of node classes can be exploited to maintain functionality by utilizing multiple paths. Percolation is the field of statistical physics that is generally used to analyze connectivity in complex networks, but in its existing forms, it cannot treat the heterogeneity of multiple vulnerable classes. To analyze the connectivity under these constraints, we describe each class as a color and develop a "color-avoiding" percolation. We present an analytic theory for random networks and a numerical algorithm for all networks, with which we can determine which nodes are color-avoiding connected and whether the maximal set percolates in the system. We find that the interaction of topology and color distribution implies a rich critical behavior, with critical values and critical exponents depending both on the topology and on the color distribution. Applying our physics-based theory to the Internet, we show how color-avoiding percolation can be used as the basis for new topologically aware secure communication protocols. Beyond applications to cybersecurity, our framework reveals a new layer of hidden structure in a wide range of natural and technological systems.

Critical field-exponents for secure message-passing in modular networks

“…where the probabilities u i, j satisfy the system of self-consistent equations . For k E =0 we recover the equations obtained by Krause et al in [3,5].…”

“…We note that any node in the GSC can securely communicate with any other node in the GSC. To find the GSC we generalize the framework of 'color-avoiding percolation' (CAP) [3,5] to study a more realistic case of secure message-passing in a communication network with a given community structure and different classes of adversaries (vulnerabilities).…”

“…Here we consider CAP on networks with given community structure, a realistic case for many networks [6][7][8][9][10][11][12][13][14]. Continuing the above example of internet routers, in each country most of the routers presumably belong to that country with a smaller number of routers belonging to other countries [3,15,16]. To study the community structure we use the stochastic block model [17,18], where each community is recognized as a 'block' in an adjacency matrix, and assign a certain color to dominate each module.…”

“…This same framework can be used to describe networks where the links are correlated by color (see SM). To identify the GSC, we find the standard giant component under the removal of nodes of a single color, and then add back nodes of the removed color which have a direct link to the largest component (reflecting the assumption that the endpoints of every path are secure) [3]. This is done for each color and then the intersection of all these components is the GSC.…”

“…β=1 [3]. To extract information about the universality class of the model, we will now measure the scaling exponents δ and γ which relate to the properties of the field.…”