Kolmogorov Random Graphs and the Incompressibility Method

We investigate topological, combinatorial, statistical, and enumeration properties of finite graphs with high Kolmogorov complexity (almost all graphs) using the novel incompressibility method. Example results are: (i) the mean and variance of the number of (possibly overlapping) ordered labeled subgraphs of a labeled graph as a function of its randomness deficiency (how far it falls short of the maximum possible Kolmogorov complexity) and (ii) a new elementary proof for the number of unlabeled graphs.Comment: LaTeX 9 page

Space-Efficient Routing Tables for Almost All Networks and the Incompressibility Method

We use the incompressibility method based on Kolmogorov complexity to determine the total number of bits of routing information for almost all network topologies. In most models for routing, for almost all labeled graphs $\Theta (n^2)$ bits are necessary and sufficient for shortest path routing. By `almost all graphs' we mean the Kolmogorov random graphs which constitute a fraction of $1-1/n^c$ of all graphs on $n$ nodes, where $c > 0$ is an arbitrary fixed constant. There is a model for which the average case lower bound rises to $\Omega(n^2 \log n)$ and another model where the average case upper bound drops to $O(n \log^2 n)$. This clearly exposes the sensitivity of such bounds to the model under consideration. If paths have to be short, but need not be shortest (if the stretch factor may be larger than 1), then much less space is needed on average, even in the more demanding models. Full-information routing requires $\Theta (n^3)$ bits on average. For worst-case static networks we prove a $\Omega(n^2 \log n)$ lower bound for shortest path routing and all stretch factors $<2$ in some networks where free relabeling is not allowed.Comment: 19 pages, Latex, 1 table, 1 figure; SIAM J. Comput., To appea

On the existence of highly organized communities in networks of locally interacting agents

In this paper we investigate phenomena of spontaneous emergence or purposeful formation of highly organized structures in networks of related agents. We show that the formation of large organized structures requires exponentially large, in the size of the structures, networks. Our approach is based on Kolmogorov, or descriptional, complexity of networks viewed as finite size strings. We apply this approach to the study of the emergence or formation of simple organized, hierarchical, structures based on Sierpinski Graphs and we prove a Ramsey type theorem that bounds the number of vertices in Kolmogorov random graphs that contain Sierpinski Graphs as subgraphs. Moreover, we show that Sierpinski Graphs encompass close-knit relationships among their vertices that facilitate fast spread and learning of information when agents in their vertices are engaged in pairwise interactions modelled as two person games. Finally, we generalize our findings for any organized structure with succinct representations. Our work can be deployed, in particular, to study problems related to the security of networks by identifying conditions which enable or forbid the formation of sufficiently large insider subnetworks with malicious common goal to overtake the network or cause disruption of its operation

Algorithmic information and incompressibility of families of multidimensional networks

This article presents a theoretical investigation of string-based generalized representations of families of finite networks in a multidimensional space. First, we study the recursive labeling of networks with (finite) arbitrary node dimensions (or aspects), such as time instants or layers. In particular, we study these networks that are formalized in the form of multiaspect graphs. We show that, unlike classical graphs, the algorithmic information of a multidimensional network is not in general dominated by the algorithmic information of the binary sequence that determines the presence or absence of edges. This universal algorithmic approach sets limitations and conditions for irreducible information content analysis in comparing networks with a large number of dimensions, such as multilayer networks. Nevertheless, we show that there are particular cases of infinite nesting families of finite multidimensional networks with a unified recursive labeling such that each member of these families is incompressible. From these results, we study network topological properties and equivalences in irreducible information content of multidimensional networks in comparison to their isomorphic classical graph.Comment: Extended preprint version of the pape

Correlation of Automorphism Group Size and Topological Properties with Program-size Complexity Evaluations of Graphs and Complex Networks

We show that numerical approximations of Kolmogorov complexity (K) applied to graph adjacency matrices capture some group-theoretic and topological properties of graphs and empirical networks ranging from metabolic to social networks. That K and the size of the group of automorphisms of a graph are correlated opens up interesting connections to problems in computational geometry, and thus connects several measures and concepts from complexity science. We show that approximations of K characterise synthetic and natural networks by their generating mechanisms, assigning lower algorithmic randomness to complex network models (Watts-Strogatz and Barabasi-Albert networks) and high Kolmogorov complexity to (random) Erdos-Renyi graphs. We derive these results via two different Kolmogorov complexity approximation methods applied to the adjacency matrices of the graphs and networks. The methods used are the traditional lossless compression approach to Kolmogorov complexity, and a normalised version of a Block Decomposition Method (BDM) measure, based on algorithmic probability theory.Comment: 15 2-column pages, 20 figures. Forthcoming in Physica A: Statistical Mechanics and its Application

The Thermodynamics of Network Coding, and an Algorithmic Refinement of the Principle of Maximum Entropy

The principle of maximum entropy (Maxent) is often used to obtain prior probability distributions as a method to obtain a Gibbs measure under some restriction giving the probability that a system will be in a certain state compared to the rest of the elements in the distribution. Because classical entropy-based Maxent collapses cases confounding all distinct degrees of randomness and pseudo-randomness, here we take into consideration the generative mechanism of the systems considered in the ensemble to separate objects that may comply with the principle under some restriction and whose entropy is maximal but may be generated recursively from those that are actually algorithmically random offering a refinement to classical Maxent. We take advantage of a causal algorithmic calculus to derive a thermodynamic-like result based on how difficult it is to reprogram a computer code. Using the distinction between computable and algorithmic randomness we quantify the cost in information loss associated with reprogramming. To illustrate this we apply the algorithmic refinement to Maxent on graphs and introduce a Maximal Algorithmic Randomness Preferential Attachment (MARPA) Algorithm, a generalisation over previous approaches. We discuss practical implications of evaluation of network randomness. Our analysis provides insight in that the reprogrammability asymmetry appears to originate from a non-monotonic relationship to algorithmic probability. Our analysis motivates further analysis of the origin and consequences of the aforementioned asymmetries, reprogrammability, and computation.Comment: 30 page