Evolutionary reconstruction of networks

Abstract

Can a graph specifying the pattern of connections of a dynamical network be reconstructed from statistical properties of a signal generated by such a system? In this model study, we present an evolutionary algorithm for reconstruction of graphs from their Laplacian spectra. Through a stochastic process of mutations and selection, evolving test networks converge to a reference graph. Applying the method to several examples of random graphs, clustered graphs, and small-world networks, we show that the proposed stochastic evolution allows exact reconstruction of relatively small networks and yields good approximations in the case of large sizes. [13]. Much effort is invested in the understanding how the structure of a network is mapped to its function and determines its operation. On the other hand, in applications ranging from bioengineering to neurosciences one also needs to design networks with a given function or reconstruct a network from its dynamics. Taking into account the great complexity of network dynamics, explicit solutions of such inverse problems of graph theory are difficult. But graph reconstruction may also be achieved without any knowledge of rules, by running an artificial evolution process through which a network learns to generate certain dynamics by adjusting its internal organization. Indeed, evolutionary algorithms are known to yield efficient solutions for complex optimization problems [14]. For the problem of graph reconstruction, such an approach has previously been proposed [15,16].In this Letter, we present an evolutionary algorithm and apply it to reconstruct graphs from their Laplacian spectra. Random graphs, small-world networks and networks with cluster organization are considered. We show that for relatively small graphs, exact reconstruction within a reasonable evolution time is possible. For larger graphs, the evolution leads to a network which provides a good approximation of the target graph. Both the spectral properties as well as other characteristic features of the reference network, such as the diameter, clustering coefficient, and the average degree, are well reproduced by the approximately reconstructed graph.Any graph G can be described by its adjacency matrix A such that A ij = 1 if the nodes i and j are connected, and A ij = 0 otherwise. A Laplacian spectrum of the graph G is defined [4] as the set of eigenvalues λ i of the matrix T with elements T ij = A ij − m i δ ij where m i = N j=1 A ij is the degree of node i and δ ij is the Kronecker symbol.

The Laplacian spectrum of neural networks

“…Given that similar spectra reflect similar network structures, it has been noted that the distribution of Laplacian eigenvalues can be used for classification of networks (Ipsen and Mikhailov, 2001; Vukadinović et al, 2002; Banerjee and Jost, 2008b; Cetinkaya et al, 2012). Spectral similarity was examined by visual inspection of the characteristics of the spectral plots and quantified by computing the average Euclidean distance between spectra.…”

“…The spectrum of the normalized Laplacian of undirected networks has the advantage that all eigenvalues are in the domain between 0 and 2, enabling the comparison of networks of different sizes (Banerjee, 2012). Furthermore, it has been noted that similarities between the spectra of networks can be used for the classification of networks (Ipsen and Mikhailov, 2001; Vukadinović et al, 2002; Banerjee and Jost, 2008b; Cetinkaya et al, 2012). Therefore, the spectra of the neural networks are examined in light of providing indications of general organizational characteristics of neural networks across species.…”

Stability Indicators in Network Reconstruction

“…The HIM distance [26] is a metric for network comparison combining an edit distance (Hamming [34], [35]) and a spectral one (Ipsen-Mikhailov [36]). As discussed in [37], edit distances are local, i.e.…”

“…Originally introduced in [36] as a tool for network reconstruction from its Laplacian spectrum, the definition of the Ipsen-Mikhailov metric follows the dynamical interpretation of a –nodes network as a –atoms molecule connected by identical elastic strings, where the pattern of connections is defined by the adjacency matrix of the corresponding network. In particular the connections between nodes in the network correspond to the bonds between atoms in the dynamical system and the adjacency matrix is its topological description.…”

Reconstruction of noise-driven nonlinear networks from node outputs by using high-order correlations

“…Second, in almost all realistic network systems various nonlinearities play crucial roles in generating diverse characteristic features and significant functions. So far most of works in treating network inference have made approximations either neglecting noise influences1516171819202122232425, or considering linear dynamics and interactions91011121314. These methods fail when both noise effects and nonlinearities of network structures are crucial for the data production.…”