Asymmetry and structural information in preferential attachment graphs

Graph symmetries intervene in diverse applications, from enumeration, to graph structure compression, to the discovery of graph dynamics (e.g., node arrival order inference). Whereas Erd\H{o}s-R\'enyi graphs are typically asymmetric, real networks are highly symmetric. So a natural question is whether preferential attachment graphs, where in each step a new node with $m$ edges is added, exhibit any symmetry. In recent work it was proved that preferential attachment graphs are symmetric for $m=1$, and there is some non-negligible probability of symmetry for $m=2$. It was conjectured that these graphs are asymmetric when $m \geq 3$. We settle this conjecture in the affirmative, then use it to estimate the structural entropy of the model. To do this, we also give bounds on the number of ways that the given graph structure could have arisen by preferential attachment. These results have further implications for information theoretic problems of interest on preferential attachment graphs.Comment: 24 pages; to appear in Random Structures & Algorithm

Complexity of Networks

Network or graph structures are ubiquitous in the study of complex systems. Often, we are interested in complexity trends of these system as it evolves under some dynamic. An example might be looking at the complexity of a food web as species enter an ecosystem via migration or speciation, and leave via extinction. In this paper, a complexity measure of networks is proposed based on the {\em complexity is information content} paradigm. To apply this paradigm to any object, one must fix two things: a representation language, in which strings of symbols from some alphabet describe, or stand for the objects being considered; and a means of determining when two such descriptions refer to the same object. With these two things set, the information content of an object can be computed in principle from the number of equivalent descriptions describing a particular object. I propose a simple representation language for undirected graphs that can be encoded as a bitstring, and equivalence is a topological equivalence. I also present an algorithm for computing the complexity of an arbitrary undirected network.Comment: Accepted for Australian Conference on Artificial Life (ACAL05). To appear in Advances in Natural Computation (World Scientific

A stochastic evolutionary model exhibiting power-law behaviour with an exponential cutoff

Recently several authors have proposed stochastic evolutionary models for the growth of complex networks that give rise to power-law distributions. These models are based on the notion of preferential attachment leading to the “rich get richer” phenomenon. Despite the generality of the proposed stochastic models, there are still some unexplained phenomena, which may arise due to the limited size of networks such as protein and e-mail networks. Such networks may in fact exhibit an exponential cutoff in the power-law scaling, although this cutoff may only be observable in the tail of the distribution for extremely large networks. We propose a modification of the basic stochastic evolutionary model, so that after a node is chosen preferentially, say according to the number of its inlinks, there is a small probability that this node will be discarded. We show that as a result of this modification, by viewing the stochastic process in terms of an urn transfer model, we obtain a power-law distribution with an exponential cutoff. Unlike many other models, the current model can capture instances where the exponent of the distribution is less than or equal to two. As a proof of concept, we demonstrate the consistency of our model by analysing a yeast protein interaction network, the distribution of which is known to follow a power law with an exponential cutoff

A model for collaboration networks giving rise to a power law distribution with exponential cutoff

Recently several authors have proposed stochastic evolutionary models for the growth of complex networks that give rise to power-law distributions. These models are based on the notion of preferential attachment leading to the ``rich get richer'' phenomenon. Despite the generality of the proposed stochastic models, there are still some unexplained phenomena, which may arise due to the limited size of networks such as protein, e-mail, actor and collaboration networks. Such networks may in fact exhibit an exponential cutoff in the power-law scaling, although this cutoff may only be observable in the tail of the distribution for extremely large networks. We propose a modification of the basic stochastic evolutionary model, so that after a node is chosen preferentially, say according to the number of its inlinks, there is a small probability that this node will become inactive. We show that as a result of this modification, by viewing the stochastic process in terms of an urn transfer model, we obtain a power-law distribution with an exponential cutoff. Unlike many other models, the current model can capture instances where the exponent of the distribution is less than or equal to two. As a proof of concept, we demonstrate the consistency of our model empirically by analysing the Mathematical Research collaboration network, the distribution of which is known to follow a power law with an exponential cutoff

Asymmetry and structural information in preferential attachment graphs

Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/151364/1/rsa20842.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/151364/2/rsa20842_am.pd