Many natural phenomenon such as human activities, wild species evolution and epidemic spreading, man-made complex systems like the World Wide Web, the power grids and the transportation networks
can all be modeled as a networked system
and characterized as a network,
in which the interaction between nodes are governed by a highly coupling and nonlinear dynamics.
The network representation facilitates the understandings of the inner structure and interactions,
hence advances to exercise control over the systems for desired behaviors.
However, the incomplete systems are the norm in practice.
Some components of the systems may be absent due to data inaccessibility,
or corrupted by stochastic noise.
To reconstruct the entire system from incomplete information,
it often involves several related and challenging sub-tasks,
such as recovering the full network (e.g., link prediction),
identifying the explicit dynamics, predicting the steady-states
(e.g., infection rates in epidemic systems, biomass in biological systems),
estimating the topological statistics (e.g., individual degrees, average degree or network motif)
of the complete network, or revealing some crucial behaviors (e.g., traffic breakdown, abrupt blackout)
exhibited during evolution. This dissertation develops a collection of mean-field based approaches, and concentrates on addressing some of these related sub-tasks.
With incomplete information,
we approach reliable inferences of
(i) the equilibrium states and (ii) the individual nodal degrees for general networks,
(iii) the complete structure of the nuclear reaction network and
(iv) the edge dynamics during training for artificial neural networks.
This research studies various complex networks and nonlinear governing dynamics
based on the fundamental mean-field theory.
From an initial state, the system can iteratively evolve into a steady-state.
The steady-state may describe the abundances of different species in a biological system,
or infection rates of different communities in a disease spreading system
when the entire system arrives into an equilibrium point.
The predictability of the steady-state is highly demanding in determining possible human intervention,
especially when the state is undesired or even disastrous to the system.
The research maps the dynamics of the unseen part of the network to a single node,
and it allows us to recover accurate estimates of steady-state on as few as five observed vertices
in domains ranging from ecology to social networks to gene regulation. Another critical part in understanding the dynamical behavior of a complex system is to obtain the full characteristics (e.g., the network size, the degree distribution, the average degree)
of the network from observed data.
Prior studies usually refer to the structure-based estimation,
little effort attempts to estimate the specific degree of each vertex from a sampled induced graph,
which prevents us from measuring the lethality of nodes in protein networks and influencers in social networks.
The current approaches dramatically fail for a tiny sampled induced graph and require a specific sampling method and a large sample size. These approaches neglect information of the vertex state, representing the dynamical behavior of the networked system, such as the biomass of species or expression of a gene, which is useful for degree estimation.
This research fills this gap by integrating the mean-field theory with combinatorial optimization,
and infers individual vertex degrees with both information of the sampled topology and nodes' states.
Experimental results on a variety of real systems demonstrate
that the framework can produce reliable degree estimates and dramatically improve existing link prediction methods by replacing the sampled degrees with the proposed estimates of the degrees. The third task is primarily on the nuclear reaction network. It assembles from a set of existing nuclear reactions a reaction network, whose
degree distribution is found to be bimodal.
That significantly deviates from the common power-law distribution of scale-free networks and
Poisson distribution of random networks.
The research develops a parametric spatial degree model to capture the bimodality,
and proposes a network growth mechanism with three rules to model the structural evolution.
Under the framework, the full reaction network can be reconstructed
by filling the missing links with possible new reactions not yet discovered. Finally, this research brings out a comprehensive analysis of artificial neural networks.The ultimate goal is an efficient model ranking,
identifying a robust neural network model from a set of candidates.
It is a fundamental yet challenging task in deep learning.
Current practice often requires expensive computational costs in training for performance prediction.
This research builds a linear graph representation of a neural network,
then reformulates the stochastic gradient descend based training algorithm to an edge dynamics,
modeling the interactions between synaptic connections.
The analysis is built on the fact that back-propagation during neural network training
is equivalent to the dynamical evolution over the synaptic connections.
Therefore, a converged neural network is associated
with a steady-state of a networked system composed of those edges.
Furthermore, a neural capacitance metric is derived from the edge dynamics as a predictive measure,
universally capturing the performance of the neural network
by observing only a short segment of the early learning curve.
Extensive experiments on a set of popular pre-trained ImageNet models and five benchmark datasets
show that the proposed approach is effective and outperforms the state-of-the-art with as few as five observation points on the learning curves.;
May 2022; School of Science
Dept. of Computer Science;
Rensselaer Polytechnic Institute, Troy, NY
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