Optimal network modification for spectral radius dependent phase transitions

Abstract

The dynamics of contact processes on networks is often determined by the spectral radius of the networks adjacency matrices. A decrease of the spectral radius can prevent the outbreak of an epidemic, or impact the synchronization among systems of coupled oscillators. The spectral radius is thus tightly linked to network dynamics and function. As such, finding the minimal change in network structure necessary to reach the intended spectral radius is important theoretically and practically. Given contemporary big data resources such as large scale communication or social networks, this problem should be solved with a low runtime complexity. We introduce a novel method for the minimal decrease in weights of edges required to reach a given spectral radius. The problem is formulated as a convex optimization problem, where a global optimum is guaranteed. The method can be easily adjusted to an efficient discrete removal of edges. We introduce a variant of the method which finds optimal decrease with a focus on weights of vertices. The proposed algorithm is exceptionally scalable, solving the problem for real networks of tens of millions of edges in a short time.Motivated by the interplay between the spectral radius and dynamics, strategies for affecting phase transitions were first proposed in percolation problems. An early solution was a ranking of vertices by their connectivity [18,19]. This ranking was improved by a centrality measure specifically designed for ranking edges or vertices via spectral radius reduction-the dynamical importance method [20]. Taking a perturbative approach, the dynamical importance method is defined as the first order approximation of the relative decrease in the radius when an edge or vertex is removed from the network. This work was further expanded to include structural perturbations of removing or adding a small subgraph and second order terms [21]. A different line of research is focused on a closely related problem, where minimization of the spectral radius is constrained by the number of edges allowed to be removed. Interestingly, the proposed algorithms for this problem [22,23], offer similar ranking to that of the dynamical importance algorithm [20].The aforementioned algorithms share a common strategy-they perform a full removal of edges, which results in discrete solutions. In the case of weighted networks, where it is possible to regulate the weights of edges continuously, this all-or-none discrete edge removal is a drastic measure which provides a suboptimal solution. This strategy of discrete removal has two major shortcomings.First, in many real world applications, continuous reduction of weights of edges is favorable. One such example is found in epidemiological models on weighted networks. The aforementioned criticality of the spectral radius and the infection rate also applies to these models [5,24]. Worldwide epidemic spread is mediated through transportation networks, e.g., the spread of Influenza, SARS or Ebola through the global flight network [25][26][27][28]. In the...