Potential energy of complex networks: a novel perspective
- URL: http://arxiv.org/abs/2002.04551v1
- Date: Tue, 11 Feb 2020 17:13:07 GMT
- Title: Potential energy of complex networks: a novel perspective
- Authors: Nicola Amoroso, Loredana Bellantuono, Saverio Pascazio, Angela
Lombardi, Alfonso Monaco, Sabina Tangaro, Roberto Bellotti
- Abstract summary: We present a novel characterization of complex networks, based on the potential of an associated Schr"odinger equation.
Crucial information is retained in the reconstructed potential, which provides a compact representation of the properties of the network structure.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: We present a novel characterization of complex networks, based on the
potential of an associated Schr\"odinger equation. The potential is designed so
that the energy spectrum of the Schr\"odinger equation coincides with the graph
spectrum of the normalized Laplacian. Crucial information is retained in the
reconstructed potential, which provides a compact representation of the
properties of the network structure. The median potential over several random
network realizations is fitted via a Landau-like function, and its length scale
is found to diverge as the critical connection probability is approached from
above. The ruggedness of the median potential profile is quantified using the
Higuchi fractal dimension, which displays a maximum at the critical connection
probability. This demonstrates that this technique can be successfully employed
in the study of random networks, as an alternative indicator of the percolation
phase transition. We apply the proposed approach to the investigation of
real-world networks describing infrastructures (US power grid). Curiously,
although no notion of phase transition can be given for such networks, the
fractality of the median potential displays signatures of criticality. We also
show that standard techniques (such as the scaling features of the largest
connected component) do not detect any signature or remnant of criticality.
Related papers
- Transmission through Cantor structured Dirac comb potential [0.0]
We introduce the Cantor-structured Dirac comb potential, referred to as the Cantor Dirac comb (CDC-$rho_N$) potential system.
This study is the first to investigate quantum tunneling through a fractal geometric Dirac comb potential.
arXiv Detail & Related papers (2024-10-11T09:32:31Z) - Chiral excitation flows of multinode network based on synthetic gauge
fields [0.0]
Chiral excitation flows have drawn a lot of attention for their unique unidirectionality.
Such flows have been studied in three-node networks with synthetic gauge fields (SGFs)
We propose a scheme to achieve chiral flows in $n$-node networks, where an auxiliary node is introduced to govern the system.
arXiv Detail & Related papers (2023-12-04T16:31:02Z) - On the Effective Number of Linear Regions in Shallow Univariate ReLU
Networks: Convergence Guarantees and Implicit Bias [50.84569563188485]
We show that gradient flow converges in direction when labels are determined by the sign of a target network with $r$ neurons.
Our result may already hold for mild over- parameterization, where the width is $tildemathcalO(r)$ and independent of the sample size.
arXiv Detail & Related papers (2022-05-18T16:57:10Z) - Geodesic statistics for random network families [0.0]
We derive measures of node and network connectivity that can contribute to explain such phenomena.
We provide specific results for widely used network families like block models, dot-product graphs, random graphs, and graphons.
Notably, the shortest path length distribution allows us to derive, for the network families above, important graph properties like the bond percolation threshold, size of the giant component, average shortest path length, and closeness and betweenness centralities.
arXiv Detail & Related papers (2021-11-03T16:25:39Z) - Towards Understanding Theoretical Advantages of Complex-Reaction
Networks [77.34726150561087]
We show that a class of functions can be approximated by a complex-reaction network using the number of parameters.
For empirical risk minimization, our theoretical result shows that the critical point set of complex-reaction networks is a proper subset of that of real-valued networks.
arXiv Detail & Related papers (2021-08-15T10:13:49Z) - Spectral Embedding of Graph Networks [76.27138343125985]
We introduce an unsupervised graph embedding that trades off local node similarity and connectivity, and global structure.
The embedding is based on a generalized graph Laplacian, whose eigenvectors compactly capture both network structure and neighborhood proximity in a single representation.
arXiv Detail & Related papers (2020-09-30T04:59:10Z) - Learning Connectivity of Neural Networks from a Topological Perspective [80.35103711638548]
We propose a topological perspective to represent a network into a complete graph for analysis.
By assigning learnable parameters to the edges which reflect the magnitude of connections, the learning process can be performed in a differentiable manner.
This learning process is compatible with existing networks and owns adaptability to larger search spaces and different tasks.
arXiv Detail & Related papers (2020-08-19T04:53:31Z) - Critical Phenomena in Complex Networks: from Scale-free to Random
Networks [77.34726150561087]
We study critical phenomena in a class of configuration network models with hidden variables controlling links between pairs of nodes.
We find analytical expressions for the average node degree, the expected number of edges, and the Landau and Helmholtz free energies, as a function of the temperature and number of nodes.
arXiv Detail & Related papers (2020-08-05T18:57:38Z) - Probabilistic bounds on neuron death in deep rectifier networks [6.167486561517023]
Neuron death is a complex phenomenon with implications for model trainability.
In this work, we derive both upper and lower bounds on the probability that a ReLU network is to a trainable point.
We show that it is possible to increase the depth of a network indefinitely, so long as the width increases as well.
arXiv Detail & Related papers (2020-07-13T05:15:04Z) - Complex networks with tuneable dimensions as a universality playground [0.0]
We discuss the role of a fundamental network parameter for universality, the spectral dimension.
By explicit computation we prove that the spectral dimension for this model can be tuned continuously from $1$ to infinity.
We propose our model as a tool to probe universal behaviour on inhomogeneous structures and comment on the possibility that the universal behaviour of correlated models on such networks mimics the one of continuous field theories in fractional Euclidean dimensions.
arXiv Detail & Related papers (2020-06-18T10:56:41Z) - On Infinite-Width Hypernetworks [101.03630454105621]
We show that hypernetworks do not guarantee to a global minima under descent.
We identify the functional priors of these architectures by deriving their corresponding GP and NTK kernels.
As part of this study, we make a mathematical contribution by deriving tight bounds on high order Taylor terms of standard fully connected ReLU networks.
arXiv Detail & Related papers (2020-03-27T00:50:29Z)
This list is automatically generated from the titles and abstracts of the papers in this site.
This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.