TERM PAPER: ADAPTIVE RANDOM RESISTOR NETWORKS
MATTHEW REICHL
While studying for my Q-exam I stumbled upon a problem, infamous among
physics students, asking for the equivalent resistance between two points on a large
grid of resistors [1]. The problem immediately led me to wonder: how does the
equivalent resistance change if I randomly rewire the grid? Or, restated, how does
the equivalent resistance between arbitrary points on a random network of resistors
depend on the global structure of the network? Unfortunately (for the purposes of
writing this term paper), I soon discovered that much progress has already been
made in answering this question. Resistor networks have been studied since the 70’s
(see Ref.[2] for the groundbreaking paper), and more recent work [3, 4] has studied
the interplay between network structure and equivalent resistance in random resistor
networks.
To my knowledge, however, no one has studied models of random resistor networks
that rewire themselves to optimize the total equivalent resistance (i.e. adaptive
resistor networks). The simplest such model I can think of implements a Metropolis
Monte Carlo algorithm to rewire the network:
• Create an initial random network with a large number of nodes N. Perhaps
the simplest initial network would be an Erdõs-Renyı́ network where each
node is connected randomly with some probability p independent of other
nodes.
P
• Compute the total equivalent resistance Ro = 12 ij Rij , where Rij is the
equivalent resistance between nodes i and j computed using using Kirchhoff’s
laws (see Ref.[4] for details). For simplicity, assume all resistors , represented
by links in the network, have equal resistance.
• Choose a node i at random, and choose a random node j directly connected
to i and a random node k not connected to i.
• Consider a rewiring of the network:
– Temporarily disconnect i from j and
P connect i to k. Compute the new
total equivalent resistance Rf = 21 ij Rij .
– If Rf < Ro , accept the rewiring. Otherwise, reject the rewiring.
• Repeat from the second step. N iterations of this algorithm defines one Monte
Carlo (MC) step.
There are several interesting questions to pursue with this model. First, does the
total equivalent resistance reach a steady state value for a large number of MC steps?
Second, and perhaps more interestingly, what is the average degree distribution of
the networks that result from this simple rewiring procedure (where the average is
taken over an ensemble of initial networks)? The degree distribution, defined as the
average number of nodes with a given number of connections (the node’s “degree”),
provides a simple but useful quantitative measure of the structure of a network.
Finally, a third question to explore is how the resultant degree distribution depends
on the structure of the initial networks. For instance, one might ask: how does the
resultant degree distribution depend on the probability p used to build the ErdõsRenyı́ networks that initialize the model?
There are many reasons, besides novelty, why this model would be interesting
and useful to explore. Broadly speaking, networks are ubiquitous in nature, and
understanding their dynamical and structural properties is useful for a wide group
of subjects including sociology, genetics, physics, and mathematics [5]. Equivalent
resistance provides an easily computed measure of the connectedness of a network [6],
and is also useful for understanding transport properties and diffusion on the network
[3]. The adaptive network model proposed here may give insight into how naturally
occurring networks reach self-organized steady states that optimize transport and
the flow of information (or “current”). Another reason to study this model is that it
is very simple. One can easily imagine many different variations of this model that
might be more directly relevant to physical systems; studying this simplified model
may provide an intuition for more complicated, realistic models.
References
[1] “Infinite Grid of Resistors”, http://www.mathpages.com/home/kmath668/kmath668.htm
[2] S. Kirkpatrick, “Classical transport in disordered- media: scaling and effective-medium theories”, Phys. Rev. Lett., 27, 1722 (1971).
[3] E. Lòpez, “Anomalous transport in scale-free networks” Phys. Rev. Lett., 94, 248701 (2005).
[4] G. Korniss et al, “Scaling in small-world resistor networks”, Phys. Lett. A, 350 ,324 (2006).
[5] R. Albert and A. Barabàsi, “Statistical mechanics of complex networks”, Rev. Mod. Phys. 74,
47 (2002).
[6] A. Ghosh, S. Boyd, and A. Saberi, “Minimizing effective resistance of a graph”, Proceedings of
the 17th International Symposium on Mathematical Theory of Networks and Systems, (2006).