If there is a subfield in physics that studying network, the closest one must be the statistical mechanics, and the second should be the soft condensed matter physics.
In my opinion, there are three main branches of network study.
Part 1: Study of interaction in network
For this area of study, the focus is the equilibrium state and the statistical mechanics properties.
Strong correlation between elements can often give rise to emergent behaviors that do not appear in the underlying system.
If there are no interaction, these phenomena cannot occurs because the resulting system is only the sum of individual part.
Therefore, the network of interacting elements is important, and indeed, the precise structure of the network can determine whether the emergence phenomenon exist or not.
For a given problem in statistical physics, we can write down the Hamiltonian of the system,
say, Ising model
$\mathcal{H}=-\sum_{i,j} A_{ij} J_{ij}\sigma_{i}\sigma_{j}-h\sum_{i}\sigma_{i}$
where the $\sigma_{i}$ represents state (spin here) of a node and the $A_{ij}$ is the adjacency matrix of the network.
Note that $A_{ij}$ can have the topology of simple square lattice, or a complex network with non-local linkage.
Using this Hamiltonian, we can write down the corresponding partition function $Z = \sum_i {exp(-\beta \mathcal{E_i})}$,
which allows us to find the set of standard macroscopic properties.
In statistical mechanics, this model is interesting because it is the first model showing the phase transition and the critical phenomenon.
This model also show different behaviour when the topology is a complex network.
Within all properties, the most important one is the ground state and its energy
because their roles in physics and they are always defined for all system.
Furthermore, there is another reason for non-physics problem: the relationship with the optimization problem.
It is obvious that finding the ground state is an optimization problem because we need to find a state that has minimum energy.
Hence, it is related to, say, the minimizing of overall error in supervised learning or maximizing the some type of cost function in financial market.
An example of optimization problem is the community detection of network,
which classify each state as one community by minimizing the Hamiltonian.
In particular, most NP-complete problem also define an optimization problem.
And some of them can be mapped (polynomial time reduction) directly to problems in the statistical mechanics,
such as the problem of finding ground state of antiferromagnetic Ising model and MAXCUT problem in graph theory.
(see review Critical phenomena in complex networks)
Returning to the question of why the network topology can significantly change the problem, here is an example:
An antiferromagnetic Ising model on a square lattice (a bipartite network) has a simple unique ground state such that all nodes has spin opposite to its neighbor.
However, if we consider the model defined on a triangular lattice, the ground state is nondegenerate and the state can change easily because of the frustration of the neighboring spin. For this type of topology or other complex network, the finding of ground state a highly non-trivial task.
Idea in statistical mechanics can also be apply to other problems.
The idea of entropy (complexity) can possibly explain why a problem is hard to solve sometimes, but easy in other case when network topology is changed. The idea of temperature in statistical mechanics also benefit the study of randomized algorithm for optimization problems. The most obvious one is the Monte Carlo algorithm for finding ground state. In real world, if the truly optimized solution cannot be found, suboptimal solution is usually acceptable. The energy landscape of the problem can usually give an insight of the problem itself and an estimate of true ground state and the accessibility of the ground state.
Under construction...
