| Summary: | This dissertation concerns modeling two aspects of dynamics of complex networks: (1)
response dynamics and (2) growth and formation.
A particularly challenging class of networks are ones in which both nodes and links are
evolving over time – the most prominent example is a financial network. In the first part
of the dissertation we present a model for the response dynamics in networks near a metastable
point. We start with a Landau-Ginzburg approach and show that the most general
lowest order Lagrangians for dynamical weighted networks can be used to derive conditions
for stability under external shocks. Using a closely related model, which is easier to solve
numerically, we propose a powerful and intuitive set of equations for response dynamics
of financial networks. We find the stability conditions of the model and find two phases:
“calm” phase , in which changes are sub-exponential and where the system moves to a new,
close-by equilibrium; “frantic” phase, where changes are exponential, with negative blows
resulting in crashes and positive ones leading to formation of "bubbles". We empirically
verify these claims by analyzing data from Eurozone crisis of 2009-2012 and stock markets.
We show that the model correctly identifies the time-line of the Eurozone crisis, and in the stock market data it correctly reproduces the auto-correlations and phases observed in the
data.
The second half of the dissertation addresses the following question: Do networks that
form due to local interactions (local in real space, or in an abstract parameter space) have
characteristics different from networks formed of random or non-local interactions? Using
interacting fields obeying Fokker-Planck equations we show that many network characteristics
such as degree distribution, degree-degree correlation and clustering can either be
derived analytically or there are analytical bounds on their behaviour. In particular, we
derive recursive equations for all powers of the ensemble average of the adjacency matrix.
We analyze a few real world networks and show that some networks that seem to form from
local interactions indeed have characteristics almost identical to simulations based on our
model, in contrast with many other networks.
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